Provided by: liblapack-doc_3.9.0-1build1_all

**NAME**

OTHERauxiliary

**SYNOPSIS**

Modulesdoublerealcomplexcomplex16Functionssubroutinedcombssq(V1, V2)DCOMBSSQadds two scaled sum of squares quantities. logical functiondisnan(DIN)DISNANtests input for NaN. subroutinedlabad(SMALL, LARGE)DLABADsubroutinedlacpy(UPLO, M, N, A, LDA, B, LDB)DLACPYcopies all or part of one two-dimensional array to another. subroutinedlae2(A, B, C, RT1, RT2)DLAE2computes the eigenvalues of a 2-by-2 symmetric matrix. subroutinedlaebz(IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO)DLAEBZcomputes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz. subroutinedlaev2(A, B, C, RT1, RT2, CS1, SN1)DLAEV2computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix. subroutinedlagts(JOB, N, A, B, C, D, IN, Y, TOL, INFO)DLAGTSsolves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf. logical functiondlaisnan(DIN1, DIN2)DLAISNANtests input for NaN by comparing two arguments for inequality. integer functiondlaneg(N, D, LLD, SIGMA, PIVMIN, R)DLANEGcomputes the Sturm count. double precision functiondlanst(NORM, N, D, E)DLANSTreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix. double precision functiondlapy2(X, Y)DLAPY2returns sqrt(x2+y2). double precision functiondlapy3(X, Y, Z)DLAPY3returns sqrt(x2+y2+z2). subroutinedlarnv(IDIST, ISEED, N, X)DLARNVreturns a vector of random numbers from a uniform or normal distribution. subroutinedlarra(N, D, E, E2, SPLTOL, TNRM, NSPLIT, ISPLIT, INFO)DLARRAcomputes the splitting points with the specified threshold. subroutinedlarrb(N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO)DLARRBprovides limited bisection to locate eigenvalues for more accuracy. subroutinedlarrc(JOBT, N, VL, VU, D, E, PIVMIN, EIGCNT, LCNT, RCNT, INFO)DLARRCcomputes the number of eigenvalues of the symmetric tridiagonal matrix. subroutinedlarrd(RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO)DLARRDcomputes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy. subroutinedlarre(RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO)DLARREgiven the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. subroutinedlarrf(N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK, INFO)DLARRFfinds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated. subroutinedlarrj(N, D, E2, IFIRST, ILAST, RTOL, OFFSET, W, WERR, WORK, IWORK, PIVMIN, SPDIAM, INFO)DLARRJperforms refinement of the initial estimates of the eigenvalues of the matrix T. subroutinedlarrk(N, IW, GL, GU, D, E2, PIVMIN, RELTOL, W, WERR, INFO)DLARRKcomputes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. subroutinedlarrr(N, D, E, INFO)DLARRRperforms tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues. subroutinedlartg(F, G, CS, SN, R)DLARTGgenerates a plane rotation with real cosine and real sine. subroutinedlartgp(F, G, CS, SN, R)DLARTGPgenerates a plane rotation so that the diagonal is nonnegative. subroutinedlaruv(ISEED, N, X)DLARUVreturns a vector of n random real numbers from a uniform distribution. subroutinedlas2(F, G, H, SSMIN, SSMAX)DLAS2computes singular values of a 2-by-2 triangular matrix. subroutinedlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)DLASCLmultiplies a general rectangular matrix by a real scalar defined as cto/cfrom. subroutinedlasd0(N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK, WORK, INFO)DLASD0computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc. subroutinedlasd1(NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO)DLASD1computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc. subroutinedlasd2(NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO)DLASD2merges the two sets of singular values together into a single sorted set. Used by sbdsdc. subroutinedlasd3(NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO)DLASD3finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc. subroutinedlasd4(N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO)DLASD4computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc. subroutinedlasd5(I, D, Z, DELTA, RHO, DSIGMA, WORK)DLASD5computes the square root of the i-th eigenvalue of a positive symmetric rank- one modification of a 2-by-2 diagonal matrix. Used by sbdsdc. subroutinedlasd6(ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO)DLASD6computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. subroutinedlasd7(ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO)DLASD7merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc. subroutinedlasd8(ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, DSIGMA, WORK, INFO)DLASD8finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc. subroutinedlasda(ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)DLASDAcomputes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc. subroutinedlasdq(UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)DLASDQcomputes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc. subroutinedlasdt(N, LVL, ND, INODE, NDIML, NDIMR, MSUB)DLASDTcreates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc. subroutinedlaset(UPLO, M, N, ALPHA, BETA, A, LDA)DLASETinitializes the off-diagonal elements and the diagonal elements of a matrix to given values. subroutinedlasr(SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)DLASRapplies a sequence of plane rotations to a general rectangular matrix. subroutinedlassq(N, X, INCX, SCALE, SUMSQ)DLASSQupdates a sum of squares represented in scaled form. subroutinedlasv2(F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL)DLASV2computes the singular value decomposition of a 2-by-2 triangular matrix. integer functionieeeck(ISPEC, ZERO, ONE)IEEECKinteger functioniladlc(M, N, A, LDA)ILADLCscans a matrix for its last non-zero column. integer functioniladlr(M, N, A, LDA)ILADLRscans a matrix for its last non-zero row. integer functionilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)ILAENVinteger functionilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)ILAENV2STAGEinteger functioniparmq(ISPEC, NAME, OPTS, N, ILO, IHI, LWORK)IPARMQlogical functionlsamen(N, CA, CB)LSAMENsubroutinescombssq(V1, V2)SCOMBSSQadds two scaled sum of squares quantities logical functionsisnan(SIN)SISNANtests input for NaN. subroutineslabad(SMALL, LARGE)SLABADsubroutineslacpy(UPLO, M, N, A, LDA, B, LDB)SLACPYcopies all or part of one two-dimensional array to another. subroutineslae2(A, B, C, RT1, RT2)SLAE2computes the eigenvalues of a 2-by-2 symmetric matrix. subroutineslaebz(IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO)SLAEBZcomputes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz. subroutineslaev2(A, B, C, RT1, RT2, CS1, SN1)SLAEV2computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix. subroutineslag2d(M, N, SA, LDSA, A, LDA, INFO)SLAG2Dconverts a single precision matrix to a double precision matrix. subroutineslagts(JOB, N, A, B, C, D, IN, Y, TOL, INFO)SLAGTSsolves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf. logical functionslaisnan(SIN1, SIN2)SLAISNANtests input for NaN by comparing two arguments for inequality. integer functionslaneg(N, D, LLD, SIGMA, PIVMIN, R)SLANEGcomputes the Sturm count. real functionslanst(NORM, N, D, E)SLANSTreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix. real functionslapy2(X, Y)SLAPY2returns sqrt(x2+y2). real functionslapy3(X, Y, Z)SLAPY3returns sqrt(x2+y2+z2). subroutineslarnv(IDIST, ISEED, N, X)SLARNVreturns a vector of random numbers from a uniform or normal distribution. subroutineslarra(N, D, E, E2, SPLTOL, TNRM, NSPLIT, ISPLIT, INFO)SLARRAcomputes the splitting points with the specified threshold. subroutineslarrb(N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO)SLARRBprovides limited bisection to locate eigenvalues for more accuracy. subroutineslarrc(JOBT, N, VL, VU, D, E, PIVMIN, EIGCNT, LCNT, RCNT, INFO)SLARRCcomputes the number of eigenvalues of the symmetric tridiagonal matrix. subroutineslarrd(RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO)SLARRDcomputes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy. subroutineslarre(RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO)SLARREgiven the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. subroutineslarrf(N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK, INFO)SLARRFfinds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated. subroutineslarrj(N, D, E2, IFIRST, ILAST, RTOL, OFFSET, W, WERR, WORK, IWORK, PIVMIN, SPDIAM, INFO)SLARRJperforms refinement of the initial estimates of the eigenvalues of the matrix T. subroutineslarrk(N, IW, GL, GU, D, E2, PIVMIN, RELTOL, W, WERR, INFO)SLARRKcomputes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. subroutineslarrr(N, D, E, INFO)SLARRRperforms tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues. subroutineslartg(F, G, CS, SN, R)SLARTGgenerates a plane rotation with real cosine and real sine. subroutineslartgp(F, G, CS, SN, R)SLARTGPgenerates a plane rotation so that the diagonal is nonnegative. subroutineslaruv(ISEED, N, X)SLARUVreturns a vector of n random real numbers from a uniform distribution. subroutineslas2(F, G, H, SSMIN, SSMAX)SLAS2computes singular values of a 2-by-2 triangular matrix. subroutineslascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)SLASCLmultiplies a general rectangular matrix by a real scalar defined as cto/cfrom. subroutineslasd0(N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK, WORK, INFO)SLASD0computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc. subroutineslasd1(NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO)SLASD1computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc. subroutineslasd2(NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO)SLASD2merges the two sets of singular values together into a single sorted set. Used by sbdsdc. subroutineslasd3(NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO)SLASD3finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc. subroutineslasd4(N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO)SLASD4computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc. subroutineslasd5(I, D, Z, DELTA, RHO, DSIGMA, WORK)SLASD5computes the square root of the i-th eigenvalue of a positive symmetric rank- one modification of a 2-by-2 diagonal matrix. Used by sbdsdc. subroutineslasd6(ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO)SLASD6computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. subroutineslasd7(ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO)SLASD7merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc. subroutineslasd8(ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, DSIGMA, WORK, INFO)SLASD8finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc. subroutineslasda(ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)SLASDAcomputes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc. subroutineslasdq(UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)SLASDQcomputes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc. subroutineslasdt(N, LVL, ND, INODE, NDIML, NDIMR, MSUB)SLASDTcreates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc. subroutineslaset(UPLO, M, N, ALPHA, BETA, A, LDA)SLASETinitializes the off-diagonal elements and the diagonal elements of a matrix to given values. subroutineslasr(SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)SLASRapplies a sequence of plane rotations to a general rectangular matrix. subroutineslassq(N, X, INCX, SCALE, SUMSQ)SLASSQupdates a sum of squares represented in scaled form. subroutineslasv2(F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL)SLASV2computes the singular value decomposition of a 2-by-2 triangular matrix. subroutinexerbla(SRNAME, INFO)XERBLAsubroutinexerbla_array(SRNAME_ARRAY, SRNAME_LEN, INFO)XERBLA_ARRAY

**Detailed** **Description**

This is the group of Other Auxiliary routines

**Function** **Documentation**

subroutinedcombssq(doubleprecision,dimension(2)V1,doubleprecision,dimension(2)V2)DCOMBSSQadds two scaled sum of squares quantities.Purpose:DCOMBSSQ adds two scaled sum of squares quantities, V1 := V1 + V2. That is, V1_scale**2 * V1_sumsq := V1_scale**2 * V1_sumsq + V2_scale**2 * V2_sumsqParametersV1V1 is DOUBLE PRECISION array, dimension (2). The first scaled sum. V1(1) = V1_scale, V1(2) = V1_sumsq.V2V2 is DOUBLE PRECISION array, dimension (2). The second scaled sum. V2(1) = V2_scale, V2(2) = V2_sumsq.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateNovember 2018logicalfunctiondisnan(doubleprecision,intent(in)DIN)DISNANtests input for NaN.Purpose:DISNAN returns .TRUE. if its argument is NaN, and .FALSE. otherwise. To be replaced by the Fortran 2003 intrinsic in the future.ParametersDINDIN is DOUBLE PRECISION Input to test for NaN.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017subroutinedlabad(doubleprecisionSMALL,doubleprecisionLARGE)DLABADPurpose:DLABAD takes as input the values computed by DLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by DLAMCH. This subroutine is needed because DLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray.ParametersSMALLSMALL is DOUBLE PRECISION On entry, the underflow threshold as computed by DLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of SMALL, otherwise unchanged.LARGELARGE is DOUBLE PRECISION On entry, the overflow threshold as computed by DLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of LARGE, otherwise unchanged.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlacpy(characterUPLO,integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB)DLACPYcopies all or part of one two-dimensional array to another.Purpose:DLACPY copies all or part of a two-dimensional matrix A to another matrix B.ParametersUPLOUPLO is CHARACTER*1 Specifies the part of the matrix A to be copied to B. = 'U': Upper triangular part = 'L': Lower triangular part Otherwise: All of the matrix AMM is INTEGER The number of rows of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. If UPLO = 'U', only the upper triangle or trapezoid is accessed; if UPLO = 'L', only the lower triangle or trapezoid is accessed.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).BB is DOUBLE PRECISION array, dimension (LDB,N) On exit, B = A in the locations specified by UPLO.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlae2(doubleprecisionA,doubleprecisionB,doubleprecisionC,doubleprecisionRT1,doubleprecisionRT2)DLAE2computes the eigenvalues of a 2-by-2 symmetric matrix.Purpose:DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, and RT2 is the eigenvalue of smaller absolute value.ParametersAA is DOUBLE PRECISION The (1,1) element of the 2-by-2 matrix.BB is DOUBLE PRECISION The (1,2) and (2,1) elements of the 2-by-2 matrix.CC is DOUBLE PRECISION The (2,2) element of the 2-by-2 matrix.RT1RT1 is DOUBLE PRECISION The eigenvalue of larger absolute value.RT2RT2 is DOUBLE PRECISION The eigenvalue of smaller absolute value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*C-B*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.subroutinedlaebz(integerIJOB,integerNITMAX,integerN,integerMMAX,integerMINP,integerNBMIN,doubleprecisionABSTOL,doubleprecisionRELTOL,doubleprecisionPIVMIN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)E2,integer,dimension(*)NVAL,doubleprecision,dimension(mmax,*)AB,doubleprecision,dimension(*)C,integerMOUT,integer,dimension(mmax,*)NAB,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DLAEBZcomputes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.Purpose:DLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases half-open intervals, i.e., of the form (a,b] , which includes b but not a . To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966 Note: the arguments are, in general, *not* checked for unreasonable values.ParametersIJOBIJOB is INTEGER Specifies what is to be done: = 1: Compute NAB for the initial intervals. = 2: Perform bisection iteration to find eigenvalues of T. = 3: Perform bisection iteration to invert N(w), i.e., to find a point which has a specified number of eigenvalues of T to its left. Other values will cause DLAEBZ to return with INFO=-1.NITMAXNITMAX is INTEGER The maximum number of "levels" of bisection to be performed, i.e., an interval of width W will not be made smaller than 2^(-NITMAX) * W. If not all intervals have converged after NITMAX iterations, then INFO is set to the number of non-converged intervals.NN is INTEGER The dimension n of the tridiagonal matrix T. It must be at least 1.MMAXMMAX is INTEGER The maximum number of intervals. If more than MMAX intervals are generated, then DLAEBZ will quit with INFO=MMAX+1.MINPMINP is INTEGER The initial number of intervals. It may not be greater than MMAX.NBMINNBMIN is INTEGER The smallest number of intervals that should be processed using a vector loop. If zero, then only the scalar loop will be used.ABSTOLABSTOL is DOUBLE PRECISION The minimum (absolute) width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. This must be at least zero.RELTOLRELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.PIVMINPIVMIN is DOUBLE PRECISION The minimum absolute value of a "pivot" in the Sturm sequence loop. This must be at least max |e(j)**2|*safe_min and at least safe_min, where safe_min is at least the smallest number that can divide one without overflow.DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T.EE is DOUBLE PRECISION array, dimension (N) The offdiagonal elements of the tridiagonal matrix T in positions 1 through N-1. E(N) is arbitrary.E2E2 is DOUBLE PRECISION array, dimension (N) The squares of the offdiagonal elements of the tridiagonal matrix T. E2(N) is ignored.NVALNVAL is INTEGER array, dimension (MINP) If IJOB=1 or 2, not referenced. If IJOB=3, the desired values of N(w). The elements of NVAL will be reordered to correspond with the intervals in AB. Thus, NVAL(j) on output will not, in general be the same as NVAL(j) on input, but it will correspond with the interval (AB(j,1),AB(j,2)] on output.ABAB is DOUBLE PRECISION array, dimension (MMAX,2) The endpoints of the intervals. AB(j,1) is a(j), the left endpoint of the j-th interval, and AB(j,2) is b(j), the right endpoint of the j-th interval. The input intervals will, in general, be modified, split, and reordered by the calculation.CC is DOUBLE PRECISION array, dimension (MMAX) If IJOB=1, ignored. If IJOB=2, workspace. If IJOB=3, then on input C(j) should be initialized to the first search point in the binary search.MOUTMOUT is INTEGER If IJOB=1, the number of eigenvalues in the intervals. If IJOB=2 or 3, the number of intervals output. If IJOB=3, MOUT will equal MINP.NABNAB is INTEGER array, dimension (MMAX,2) If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). If IJOB=2, then on input, NAB(i,j) should be set. It must satisfy the condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with IJOB=1. On output, NAB(i,j) will contain max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input interval that the output interval (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2). If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless N(w) > NVAL(i) for all search points w , in which case NAB(i,1) will not be modified, i.e., the output value will be the same as the input value (modulo reorderings -- see NVAL and AB), or unless N(w) < NVAL(i) for all search points w , in which case NAB(i,2) will not be modified. Normally, NAB should be set to some distinctive value(s) before DLAEBZ is called.WORKWORK is DOUBLE PRECISION array, dimension (MMAX) Workspace.IWORKIWORK is INTEGER array, dimension (MMAX) Workspace.INFOINFO is INTEGER = 0: All intervals converged. = 1--MMAX: The last INFO intervals did not converge. = MMAX+1: More than MMAX intervals were generated.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:This routine is intended to be called only by other LAPACK routines, thus the interface is less user-friendly. It is intended for two purposes: (a) finding eigenvalues. In this case, DLAEBZ should have one or more initial intervals set up in AB, and DLAEBZ should be called with IJOB=1. This sets up NAB, and also counts the eigenvalues. Intervals with no eigenvalues would usually be thrown out at this point. Also, if not all the eigenvalues in an interval i are desired, NAB(i,1) can be increased or NAB(i,2) decreased. For example, set NAB(i,1)=NAB(i,2)-1 to get the largest eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX no smaller than the value of MOUT returned by the call with IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1 through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the tolerance specified by ABSTOL and RELTOL. (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l). In this case, start with a Gershgorin interval (a,b). Set up AB to contain 2 search intervals, both initially (a,b). One NVAL element should contain f-1 and the other should contain l , while C should contain a and b, resp. NAB(i,1) should be -1 and NAB(i,2) should be N+1, to flag an error if the desired interval does not lie in (a,b). DLAEBZ is then called with IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals -- j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and w(l-r)=...=w(l+k) are handled similarly.subroutinedlaev2(doubleprecisionA,doubleprecisionB,doubleprecisionC,doubleprecisionRT1,doubleprecisionRT2,doubleprecisionCS1,doubleprecisionSN1)DLAEV2computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.Purpose:DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ] [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].ParametersAA is DOUBLE PRECISION The (1,1) element of the 2-by-2 matrix.BB is DOUBLE PRECISION The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.CC is DOUBLE PRECISION The (2,2) element of the 2-by-2 matrix.RT1RT1 is DOUBLE PRECISION The eigenvalue of larger absolute value.RT2RT2 is DOUBLE PRECISION The eigenvalue of smaller absolute value.CS1CS1 is DOUBLE PRECISIONSN1SN1 is DOUBLE PRECISION The vector (CS1, SN1) is a unit right eigenvector for RT1.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*C-B*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. CS1 and SN1 are accurate to a few ulps barring over/underflow. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.subroutinedlagts(integerJOB,integerN,doubleprecision,dimension(*)A,doubleprecision,dimension(*)B,doubleprecision,dimension(*)C,doubleprecision,dimension(*)D,integer,dimension(*)IN,doubleprecision,dimension(*)Y,doubleprecisionTOL,integerINFO)DLAGTSsolves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.Purpose:DLAGTS may be used to solve one of the systems of equations (T - lambda*I)*x = y or (T - lambda*I)**T*x = y, where T is an n by n tridiagonal matrix, for x, following the factorization of (T - lambda*I) as (T - lambda*I) = P*L*U , by routine DLAGTF. The choice of equation to be solved is controlled by the argument JOB, and in each case there is an option to perturb zero or very small diagonal elements of U, this option being intended for use in applications such as inverse iteration.ParametersJOBJOB is INTEGER Specifies the job to be performed by DLAGTS as follows: = 1: The equations (T - lambda*I)x = y are to be solved, but diagonal elements of U are not to be perturbed. = -1: The equations (T - lambda*I)x = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. = 2: The equations (T - lambda*I)**Tx = y are to be solved, but diagonal elements of U are not to be perturbed. = -2: The equations (T - lambda*I)**Tx = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below.NN is INTEGER The order of the matrix T.AA is DOUBLE PRECISION array, dimension (N) On entry, A must contain the diagonal elements of U as returned from DLAGTF.BB is DOUBLE PRECISION array, dimension (N-1) On entry, B must contain the first super-diagonal elements of U as returned from DLAGTF.CC is DOUBLE PRECISION array, dimension (N-1) On entry, C must contain the sub-diagonal elements of L as returned from DLAGTF.DD is DOUBLE PRECISION array, dimension (N-2) On entry, D must contain the second super-diagonal elements of U as returned from DLAGTF.ININ is INTEGER array, dimension (N) On entry, IN must contain details of the matrix P as returned from DLAGTF.YY is DOUBLE PRECISION array, dimension (N) On entry, the right hand side vector y. On exit, Y is overwritten by the solution vector x.TOLTOL is DOUBLE PRECISION On entry, with JOB < 0, TOL should be the minimum perturbation to be made to very small diagonal elements of U. TOL should normally be chosen as about eps*norm(U), where eps is the relative machine precision, but if TOL is supplied as non-positive, then it is reset to eps*max( abs( u(i,j) ) ). If JOB > 0 then TOL is not referenced. On exit, TOL is changed as described above, only if TOL is non-positive on entry. Otherwise TOL is unchanged.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: overflow would occur when computing the INFO(th) element of the solution vector x. This can only occur when JOB is supplied as positive and either means that a diagonal element of U is very small, or that the elements of the right-hand side vector y are very large.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016logicalfunctiondlaisnan(doubleprecision,intent(in)DIN1,doubleprecision,intent(in)DIN2)DLAISNANtests input for NaN by comparing two arguments for inequality.Purpose:This routine is not for general use. It exists solely to avoid over-optimization in DISNAN. DLAISNAN checks for NaNs by comparing its two arguments for inequality. NaN is the only floating-point value where NaN != NaN returns .TRUE. To check for NaNs, pass the same variable as both arguments. A compiler must assume that the two arguments are not the same variable, and the test will not be optimized away. Interprocedural or whole-program optimization may delete this test. The ISNAN functions will be replaced by the correct Fortran 03 intrinsic once the intrinsic is widely available.ParametersDIN1DIN1 is DOUBLE PRECISIONDIN2DIN2 is DOUBLE PRECISION Two numbers to compare for inequality.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017integerfunctiondlaneg(integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)LLD,doubleprecisionSIGMA,doubleprecisionPIVMIN,integerR)DLANEGcomputes the Sturm count.Purpose:DLANEG computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T - sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from DLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE-754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is non-zero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)ParametersNN is INTEGER The order of the matrix.DD is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D.LLDLLD is DOUBLE PRECISION array, dimension (N-1) The (N-1) elements L(i)*L(i)*D(i).SIGMASIGMA is DOUBLE PRECISION Shift amount in T - sigma I = L D L^T.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence. May be used when zero pivots are encountered on non-IEEE-754 architectures.RR is INTEGER The twist index for the twisted factorization that is used for the negcount.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA Jason Riedy, University of California, Berkeley, USAdoubleprecisionfunctiondlanst(characterNORM,integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E)DLANSTreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.Purpose:DLANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A.ReturnsDLANST DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.ParametersNORMNORM is CHARACTER*1 Specifies the value to be returned in DLANST as described above.NN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANST is set to zero.DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of A.EE is DOUBLE PRECISION array, dimension (N-1) The (n-1) sub-diagonal or super-diagonal elements of A.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016doubleprecisionfunctiondlapy2(doubleprecisionX,doubleprecisionY)DLAPY2returns sqrt(x2+y2).Purpose:DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary overflow.ParametersXX is DOUBLE PRECISIONYY is DOUBLE PRECISION X and Y specify the values x and y.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017doubleprecisionfunctiondlapy3(doubleprecisionX,doubleprecisionY,doubleprecisionZ)DLAPY3returns sqrt(x2+y2+z2).Purpose:DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause unnecessary overflow.ParametersXX is DOUBLE PRECISIONYY is DOUBLE PRECISIONZZ is DOUBLE PRECISION X, Y and Z specify the values x, y and z.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlarnv(integerIDIST,integer,dimension(4)ISEED,integerN,doubleprecision,dimension(*)X)DLARNVreturns a vector of random numbers from a uniform or normal distribution.Purpose:DLARNV returns a vector of n random real numbers from a uniform or normal distribution.ParametersIDISTIDIST is INTEGER Specifies the distribution of the random numbers: = 1: uniform (0,1) = 2: uniform (-1,1) = 3: normal (0,1)ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated.NN is INTEGER The number of random numbers to be generated.XX is DOUBLE PRECISION array, dimension (N) The generated random numbers.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:This routine calls the auxiliary routine DLARUV to generate random real numbers from a uniform (0,1) distribution, in batches of up to 128 using vectorisable code. The Box-Muller method is used to transform numbers from a uniform to a normal distribution.subroutinedlarra(integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)E2,doubleprecisionSPLTOL,doubleprecisionTNRM,integerNSPLIT,integer,dimension(*)ISPLIT,integerINFO)DLARRAcomputes the splitting points with the specified threshold.Purpose:Compute the splitting points with threshold SPLTOL. DLARRA sets any "small" off-diagonal elements to zero.ParametersNN is INTEGER The order of the matrix. N > 0.DD is DOUBLE PRECISION array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T.EE is DOUBLE PRECISION array, dimension (N) On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, are set to zero, the other entries of E are untouched.E2E2 is DOUBLE PRECISION array, dimension (N) On entry, the first (N-1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zeroSPLTOLSPLTOL is DOUBLE PRECISION The threshold for splitting. Two criteria can be used: SPLTOL<0 : criterion based on absolute off-diagonal value SPLTOL>0 : criterion that preserves relative accuracyTNRMTNRM is DOUBLE PRECISION The norm of the matrix.NSPLITNSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N.ISPLITISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.INFOINFO is INTEGER = 0: successful exitAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlarrb(integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)LLD,integerIFIRST,integerILAST,doubleprecisionRTOL1,doubleprecisionRTOL2,integerOFFSET,doubleprecision,dimension(*)W,doubleprecision,dimension(*)WGAP,doubleprecision,dimension(*)WERR,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,doubleprecisionPIVMIN,doubleprecisionSPDIAM,integerTWIST,integerINFO)DLARRBprovides limited bisection to locate eigenvalues for more accuracy.Purpose:Given the relatively robust representation(RRR) L D L^T, DLARRB does "limited" bisection to refine the eigenvalues of L D L^T, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses and their gaps are input in WERR and WGAP, respectively. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.ParametersNN is INTEGER The order of the matrix.DD is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D.LLDLLD is DOUBLE PRECISION array, dimension (N-1) The (N-1) elements L(i)*L(i)*D(i).IFIRSTIFIRST is INTEGER The index of the first eigenvalue to be computed.ILASTILAST is INTEGER The index of the last eigenvalue to be computed.RTOL1RTOL1 is DOUBLE PRECISIONRTOL2RTOL2 is DOUBLE PRECISION Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the (estimated) distance to the nearest eigenvalue.OFFSETOFFSET is INTEGER Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.WW is DOUBLE PRECISION array, dimension (N) On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.WGAPWGAP is DOUBLE PRECISION array, dimension (N-1) On input, the (estimated) gaps between consecutive eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues I and I+1. Note that if IFIRST = ILAST then WGAP(IFIRST-OFFSET) must be set to ZERO. On output, these gaps are refined.WERRWERR is DOUBLE PRECISION array, dimension (N) On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.WORKWORK is DOUBLE PRECISION array, dimension (2*N) Workspace.IWORKIWORK is INTEGER array, dimension (2*N) Workspace.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence.SPDIAMSPDIAM is DOUBLE PRECISION The spectral diameter of the matrix.TWISTTWIST is INTEGER The twist index for the twisted factorization that is used for the negcount. TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)INFOINFO is INTEGER Error flag.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlarrc(characterJOBT,integerN,doubleprecisionVL,doubleprecisionVU,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecisionPIVMIN,integerEIGCNT,integerLCNT,integerRCNT,integerINFO)DLARRCcomputes the number of eigenvalues of the symmetric tridiagonal matrix.Purpose:Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.ParametersJOBTJOBT is CHARACTER*1 = 'T': Compute Sturm count for matrix T. = 'L': Compute Sturm count for matrix L D L^T.NN is INTEGER The order of the matrix. N > 0.VLVL is DOUBLE PRECISION The lower bound for the eigenvalues.VUVU is DOUBLE PRECISION The upper bound for the eigenvalues.DD is DOUBLE PRECISION array, dimension (N) JOBT = 'T': The N diagonal elements of the tridiagonal matrix T. JOBT = 'L': The N diagonal elements of the diagonal matrix D.EE is DOUBLE PRECISION array, dimension (N) JOBT = 'T': The N-1 offdiagonal elements of the matrix T. JOBT = 'L': The N-1 offdiagonal elements of the matrix L.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence for T.EIGCNTEIGCNT is INTEGER The number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU]LCNTLCNT is INTEGERRCNTRCNT is INTEGER The left and right negcounts of the interval.INFOINFO is INTEGERAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlarrd(characterRANGE,characterORDER,integerN,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecision,dimension(*)GERS,doubleprecisionRELTOL,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)E2,doubleprecisionPIVMIN,integerNSPLIT,integer,dimension(*)ISPLIT,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(*)WERR,doubleprecisionWL,doubleprecisionWU,integer,dimension(*)IBLOCK,integer,dimension(*)INDEXW,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DLARRDcomputes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.Purpose:DLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.ParametersRANGERANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.ORDERORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest.NN is INTEGER The order of the tridiagonal matrix T. N >= 0.VLVL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.VUVU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.ILIL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IUIU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.GERSGERS is DOUBLE PRECISION array, dimension (2*N) The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)).RELTOLRELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.DD is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T.EE is DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T.E2E2 is DOUBLE PRECISION array, dimension (N-1) The (n-1) squared off-diagonal elements of the tridiagonal matrix T.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence for T.NSPLITNSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.ISPLITISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.)MM is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)WW is DOUBLE PRECISION array, dimension (N) On exit, the first M elements of W will contain the eigenvalue approximations. DLARRD computes an interval I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue approximation is given as the interval midpoint W(j)= ( a_j + b_j)/2. The corresponding error is bounded by WERR(j) = abs( a_j - b_j)/2WERRWERR is DOUBLE PRECISION array, dimension (N) The error bound on the corresponding eigenvalue approximation in W.WLWL is DOUBLE PRECISIONWUWU is DOUBLE PRECISION The interval (WL, WU] contains all the wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin bounds on the spectrum. If RANGE='I', then WL and WU are computed by DLAEBZ from the index range specified.IBLOCKIBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (DLARRD may use the remaining N-M elements as workspace.)INDEXWINDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th eigenvalue W(i) is the j-th eigenvalue in block k.WORKWORK is DOUBLE PRECISION array, dimension (4*N)IWORKIWORK is INTEGER array, dimension (3*N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1-IL Cause: non-monotonic arithmetic, causing the Sturm sequence to be non-monotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floating-point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again.InternalParameters:FUDGE DOUBLE PRECISION, default = 2 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution.Contributors:W. Kahan, University of California, Berkeley, USA Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016subroutinedlarre(characterRANGE,integerN,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)E2,doubleprecisionRTOL1,doubleprecisionRTOL2,doubleprecisionSPLTOL,integerNSPLIT,integer,dimension(*)ISPLIT,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(*)WERR,doubleprecision,dimension(*)WGAP,integer,dimension(*)IBLOCK,integer,dimension(*)INDEXW,doubleprecision,dimension(*)GERS,doubleprecisionPIVMIN,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DLARREgiven the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.Purpose:To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, DLARRE sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by DSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to conpute all and then discard any unwanted one. As an added benefit, DLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.ParametersRANGERANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.NN is INTEGER The order of the matrix. N > 0.VLVL is DOUBLE PRECISION If RANGE='V', the lower bound for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', DLARRE computes bounds on the desired part of the spectrum.VUVU is DOUBLE PRECISION If RANGE='V', the upper bound for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', DLARRE computes bounds on the desired part of the spectrum.ILIL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N.IUIU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N.DD is DOUBLE PRECISION array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T. On exit, the N diagonal elements of the diagonal matrices D_i.EE is DOUBLE PRECISION array, dimension (N) On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, E contains the subdiagonal elements of the unit bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, contain the base points sigma_i on output.E2E2 is DOUBLE PRECISION array, dimension (N) On entry, the first (N-1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zeroRTOL1RTOL1 is DOUBLE PRECISIONRTOL2RTOL2 is DOUBLE PRECISION Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )SPLTOLSPLTOL is DOUBLE PRECISION The threshold for splitting.NSPLITNSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N.ISPLITISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.MM is INTEGER The total number of eigenvalues (of all L_i D_i L_i^T) found.WW is DOUBLE PRECISION array, dimension (N) The first M elements contain the eigenvalues. The eigenvalues of each of the blocks, L_i D_i L_i^T, are sorted in ascending order ( DLARRE may use the remaining N-M elements as workspace).WERRWERR is DOUBLE PRECISION array, dimension (N) The error bound on the corresponding eigenvalue in W.WGAPWGAP is DOUBLE PRECISION array, dimension (N) The separation from the right neighbor eigenvalue in W. The gap is only with respect to the eigenvalues of the same block as each block has its own representation tree. Exception: at the right end of a block we store the left gapIBLOCKIBLOCK is INTEGER array, dimension (N) The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc.INDEXWINDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th eigenvalue W(i) is the 10-th eigenvalue in block 2GERSGERS is DOUBLE PRECISION array, dimension (2*N) The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)).PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence for T.WORKWORK is DOUBLE PRECISION array, dimension (6*N) Workspace.IWORKIWORK is INTEGER array, dimension (5*N) Workspace.INFOINFO is INTEGER = 0: successful exit > 0: A problem occurred in DLARRE. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =-1: Problem in DLARRD. = 2: No base representation could be found in MAXTRY iterations. Increasing MAXTRY and recompilation might be a remedy. =-3: Problem in DLARRB when computing the refined root representation for DLASQ2. =-4: Problem in DLARRB when preforming bisection on the desired part of the spectrum. =-5: Problem in DLASQ2. =-6: Problem in DLASQ2.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016FurtherDetails:The base representations are required to suffer very little element growth and consequently define all their eigenvalues to high relative accuracy.Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlarrf(integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)L,doubleprecision,dimension(*)LD,integerCLSTRT,integerCLEND,doubleprecision,dimension(*)W,doubleprecision,dimension(*)WGAP,doubleprecision,dimension(*)WERR,doubleprecisionSPDIAM,doubleprecisionCLGAPL,doubleprecisionCLGAPR,doubleprecisionPIVMIN,doubleprecisionSIGMA,doubleprecision,dimension(*)DPLUS,doubleprecision,dimension(*)LPLUS,doubleprecision,dimension(*)WORK,integerINFO)DLARRFfinds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.Purpose:Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), DLARRF finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.ParametersNN is INTEGER The order of the matrix (subblock, if the matrix split).DD is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D.LL is DOUBLE PRECISION array, dimension (N-1) The (N-1) subdiagonal elements of the unit bidiagonal matrix L.LDLD is DOUBLE PRECISION array, dimension (N-1) The (N-1) elements L(i)*D(i).CLSTRTCLSTRT is INTEGER The index of the first eigenvalue in the cluster.CLENDCLEND is INTEGER The index of the last eigenvalue in the cluster.WW is DOUBLE PRECISION array, dimension dimension is >= (CLEND-CLSTRT+1) The eigenvalue APPROXIMATIONS of L D L^T in ascending order. W( CLSTRT ) through W( CLEND ) form the cluster of relatively close eigenalues.WGAPWGAP is DOUBLE PRECISION array, dimension dimension is >= (CLEND-CLSTRT+1) The separation from the right neighbor eigenvalue in W.WERRWERR is DOUBLE PRECISION array, dimension dimension is >= (CLEND-CLSTRT+1) WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in WSPDIAMSPDIAM is DOUBLE PRECISION estimate of the spectral diameter obtained from the Gerschgorin intervalsCLGAPLCLGAPL is DOUBLE PRECISIONCLGAPRCLGAPR is DOUBLE PRECISION absolute gap on each end of the cluster. Set by the calling routine to protect against shifts too close to eigenvalues outside the cluster.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence.SIGMASIGMA is DOUBLE PRECISION The shift used to form L(+) D(+) L(+)^T.DPLUSDPLUS is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D(+).LPLUSLPLUS is DOUBLE PRECISION array, dimension (N-1) The first (N-1) elements of LPLUS contain the subdiagonal elements of the unit bidiagonal matrix L(+).WORKWORK is DOUBLE PRECISION array, dimension (2*N) Workspace.INFOINFO is INTEGER Signals processing OK (=0) or failure (=1)AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlarrj(integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E2,integerIFIRST,integerILAST,doubleprecisionRTOL,integerOFFSET,doubleprecision,dimension(*)W,doubleprecision,dimension(*)WERR,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,doubleprecisionPIVMIN,doubleprecisionSPDIAM,integerINFO)DLARRJperforms refinement of the initial estimates of the eigenvalues of the matrix T.Purpose:Given the initial eigenvalue approximations of T, DLARRJ does bisection to refine the eigenvalues of T, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses in WERR. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.ParametersNN is INTEGER The order of the matrix.DD is DOUBLE PRECISION array, dimension (N) The N diagonal elements of T.E2E2 is DOUBLE PRECISION array, dimension (N-1) The Squares of the (N-1) subdiagonal elements of T.IFIRSTIFIRST is INTEGER The index of the first eigenvalue to be computed.ILASTILAST is INTEGER The index of the last eigenvalue to be computed.RTOLRTOL is DOUBLE PRECISION Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).OFFSETOFFSET is INTEGER Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.WW is DOUBLE PRECISION array, dimension (N) On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.WERRWERR is DOUBLE PRECISION array, dimension (N) On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.WORKWORK is DOUBLE PRECISION array, dimension (2*N) Workspace.IWORKIWORK is INTEGER array, dimension (2*N) Workspace.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence for T.SPDIAMSPDIAM is DOUBLE PRECISION The spectral diameter of T.INFOINFO is INTEGER Error flag.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlarrk(integerN,integerIW,doubleprecisionGL,doubleprecisionGU,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E2,doubleprecisionPIVMIN,doubleprecisionRELTOL,doubleprecisionW,doubleprecisionWERR,integerINFO)DLARRKcomputes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.Purpose:DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.ParametersNN is INTEGER The order of the tridiagonal matrix T. N >= 0.IWIW is INTEGER The index of the eigenvalues to be returned.GLGL is DOUBLE PRECISIONGUGU is DOUBLE PRECISION An upper and a lower bound on the eigenvalue.DD is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T.E2E2 is DOUBLE PRECISION array, dimension (N-1) The (n-1) squared off-diagonal elements of the tridiagonal matrix T.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence for T.RELTOLRELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.WW is DOUBLE PRECISIONWERRWERR is DOUBLE PRECISION The error bound on the corresponding eigenvalue approximation in W.INFOINFO is INTEGER = 0: Eigenvalue converged = -1: Eigenvalue did NOT convergeInternalParameters:FUDGE DOUBLE PRECISION, default = 2 A "fudge factor" to widen the Gershgorin intervals.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017subroutinedlarrr(integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,integerINFO)DLARRRperforms tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.Purpose:Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.ParametersNN is INTEGER The order of the matrix. N > 0.DD is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the tridiagonal matrix T.EE is DOUBLE PRECISION array, dimension (N) On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) is set to ZERO.INFOINFO is INTEGER INFO = 0(default) : the matrix warrants computations preserving relative accuracy. INFO = 1 : the matrix warrants computations guaranteeing only absolute accuracy.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutinedlartg(doubleprecisionF,doubleprecisionG,doubleprecisionCS,doubleprecisionSN,doubleprecisionR)DLARTGgenerates a plane rotation with real cosine and real sine.Purpose:DLARTG generate a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ -SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the BLAS1 routine DROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any floating point operations (saves work in DBDSQR when there are zeros on the diagonal). If F exceeds G in magnitude, CS will be positive.ParametersFF is DOUBLE PRECISION The first component of vector to be rotated.GG is DOUBLE PRECISION The second component of vector to be rotated.CSCS is DOUBLE PRECISION The cosine of the rotation.SNSN is DOUBLE PRECISION The sine of the rotation.RR is DOUBLE PRECISION The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlartgp(doubleprecisionF,doubleprecisionG,doubleprecisionCS,doubleprecisionSN,doubleprecisionR)DLARTGPgenerates a plane rotation so that the diagonal is nonnegative.Purpose:DLARTGP generates a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ -SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the Level 1 BLAS routine DROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=(+/-)1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=(+/-)1. The sign is chosen so that R >= 0.ParametersFF is DOUBLE PRECISION The first component of vector to be rotated.GG is DOUBLE PRECISION The second component of vector to be rotated.CSCS is DOUBLE PRECISION The cosine of the rotation.SNSN is DOUBLE PRECISION The sine of the rotation.RR is DOUBLE PRECISION The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlaruv(integer,dimension(4)ISEED,integerN,doubleprecision,dimension(n)X)DLARUVreturns a vector of n random real numbers from a uniform distribution.Purpose:DLARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). This is an auxiliary routine called by DLARNV and ZLARNV.ParametersISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated.NN is INTEGER The number of random numbers to be generated. N <= 128.XX is DOUBLE PRECISION array, dimension (N) The generated random numbers.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:This routine uses a multiplicative congruential method with modulus 2**48 and multiplier 33952834046453 (see G.S.Fishman, 'Multiplicative congruential random number generators with modulus 2**b: an exhaustive analysis for b = 32 and a partial analysis for b = 48', Math. Comp. 189, pp 331-344, 1990). 48-bit integers are stored in 4 integer array elements with 12 bits per element. Hence the routine is portable across machines with integers of 32 bits or more.subroutinedlas2(doubleprecisionF,doubleprecisionG,doubleprecisionH,doubleprecisionSSMIN,doubleprecisionSSMAX)DLAS2computes singular values of a 2-by-2 triangular matrix.Purpose:DLAS2 computes the singular values of the 2-by-2 matrix [ F G ] [ 0 H ]. On return, SSMIN is the smaller singular value and SSMAX is the larger singular value.ParametersFF is DOUBLE PRECISION The (1,1) element of the 2-by-2 matrix.GG is DOUBLE PRECISION The (1,2) element of the 2-by-2 matrix.HH is DOUBLE PRECISION The (2,2) element of the 2-by-2 matrix.SSMINSSMIN is DOUBLE PRECISION The smaller singular value.SSMAXSSMAX is DOUBLE PRECISION The larger singular value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Barring over/underflow, all output quantities are correct to within a few units in the last place (ulps), even in the absence of a guard digit in addition/subtraction. In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows, or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.subroutinedlascl(characterTYPE,integerKL,integerKU,doubleprecisionCFROM,doubleprecisionCTO,integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integerINFO)DLASCLmultiplies a general rectangular matrix by a real scalar defined as cto/cfrom.Purpose:DLASCL multiplies the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.ParametersTYPETYPE is CHARACTER*1 TYPE indices the storage type of the input matrix. = 'G': A is a full matrix. = 'L': A is a lower triangular matrix. = 'U': A is an upper triangular matrix. = 'H': A is an upper Hessenberg matrix. = 'B': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the lower half stored. = 'Q': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the upper half stored. = 'Z': A is a band matrix with lower bandwidth KL and upper bandwidth KU. See DGBTRF for storage details.KLKL is INTEGER The lower bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'.KUKU is INTEGER The upper bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'.CFROMCFROM is DOUBLE PRECISIONCTOCTO is DOUBLE PRECISION The matrix A is multiplied by CTO/CFROM. A(I,J) is computed without over/underflow if the final result CTO*A(I,J)/CFROM can be represented without over/underflow. CFROM must be nonzero.MM is INTEGER The number of rows of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The matrix to be multiplied by CTO/CFROM. See TYPE for the storage type.LDALDA is INTEGER The leading dimension of the array A. If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M); TYPE = 'B', LDA >= KL+1; TYPE = 'Q', LDA >= KU+1; TYPE = 'Z', LDA >= 2*KL+KU+1.INFOINFO is INTEGER 0 - successful exit <0 - if INFO = -i, the i-th argument had an illegal value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016subroutinedlasd0(integerN,integerSQRE,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(ldu,*)U,integerLDU,doubleprecision,dimension(ldvt,*)VT,integerLDVT,integerSMLSIZ,integer,dimension(*)IWORK,doubleprecision,dimension(*)WORK,integerINFO)DLASD0computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc.Purpose:Using a divide and conquer approach, DLASD0 computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes orthogonal matrices U and VT such that B = U * S * VT. The singular values S are overwritten on D. A related subroutine, DLASDA, computes only the singular values, and optionally, the singular vectors in compact form.ParametersNN is INTEGER On entry, the row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D.SQRESQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N+1;DD is DOUBLE PRECISION array, dimension (N) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values.EE is DOUBLE PRECISION array, dimension (M-1) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed.UU is DOUBLE PRECISION array, dimension (LDU, N) On exit, U contains the left singular vectors.LDULDU is INTEGER On entry, leading dimension of U.VTVT is DOUBLE PRECISION array, dimension (LDVT, M) On exit, VT**T contains the right singular vectors.LDVTLDVT is INTEGER On entry, leading dimension of VT.SMLSIZSMLSIZ is INTEGER On entry, maximum size of the subproblems at the bottom of the computation tree.IWORKIWORK is INTEGER array, dimension (8*N)WORKWORK is DOUBLE PRECISION array, dimension (3*M**2+2*M)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd1(integerNL,integerNR,integerSQRE,doubleprecision,dimension(*)D,doubleprecisionALPHA,doubleprecisionBETA,doubleprecision,dimension(ldu,*)U,integerLDU,doubleprecision,dimension(ldvt,*)VT,integerLDVT,integer,dimension(*)IDXQ,integer,dimension(*)IWORK,doubleprecision,dimension(*)WORK,integerINFO)DLASD1computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.Purpose:DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0. A related subroutine DLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired. DLASD1 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLASD2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine DLASD4 (as called by DLASD3). This routine also calculates the singular vectors of the current problem. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.ParametersNLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.DD is DOUBLE PRECISION array, dimension (N = NL+NR+1). On entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.ALPHAALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row.BETABETA is DOUBLE PRECISION Contains the off-diagonal element associated with the added row.UU is DOUBLE PRECISION array, dimension(LDU,N) On entry U(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. On exit U contains the left singular vectors of the bidiagonal matrix.LDULDU is INTEGER The leading dimension of the array U. LDU >= max( 1, N ).VTVT is DOUBLE PRECISION array, dimension(LDVT,M) where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. On exit VT**T contains the right singular vectors of the bidiagonal matrix.LDVTLDVT is INTEGER The leading dimension of the array VT. LDVT >= max( 1, M ).IDXQIDXQ is INTEGER array, dimension(N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order.IWORKIWORK is INTEGER array, dimension( 4 * N )WORKWORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd2(integerNL,integerNR,integerSQRE,integerK,doubleprecision,dimension(*)D,doubleprecision,dimension(*)Z,doubleprecisionALPHA,doubleprecisionBETA,doubleprecision,dimension(ldu,*)U,integerLDU,doubleprecision,dimension(ldvt,*)VT,integerLDVT,doubleprecision,dimension(*)DSIGMA,doubleprecision,dimension(ldu2,*)U2,integerLDU2,doubleprecision,dimension(ldvt2,*)VT2,integerLDVT2,integer,dimension(*)IDXP,integer,dimension(*)IDX,integer,dimension(*)IDXC,integer,dimension(*)IDXQ,integer,dimension(*)COLTYP,integerINFO)DLASD2merges the two sets of singular values together into a single sorted set. Used by sbdsdc.Purpose:DLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. DLASD2 is called from DLASD1.ParametersNLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.KK is INTEGER Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.DD is DOUBLE PRECISION array, dimension(N) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order.ZZ is DOUBLE PRECISION array, dimension(N) On exit Z contains the updating row vector in the secular equation.ALPHAALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row.BETABETA is DOUBLE PRECISION Contains the off-diagonal element associated with the added row.UU is DOUBLE PRECISION array, dimension(LDU,N) On entry U contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (N-K) updated left singular vectors (those which were deflated) in its last N-K columns.LDULDU is INTEGER The leading dimension of the array U. LDU >= N.VTVT is DOUBLE PRECISION array, dimension(LDVT,M) On entry VT**T contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT**T contains the trailing (N-K) updated right singular vectors (those which were deflated) in its last N-K columns. In case SQRE =1, the last row of VT spans the right null space.LDVTLDVT is INTEGER The leading dimension of the array VT. LDVT >= M.DSIGMADSIGMA is DOUBLE PRECISION array, dimension (N) Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation.U2U2 is DOUBLE PRECISION array, dimension(LDU2,N) Contains a copy of the first K-1 left singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains non-zero entries only at and above NL; the third contains non-zero entries only below NL+1; and the fourth is dense.LDU2LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N.VT2VT2 is DOUBLE PRECISION array, dimension(LDVT2,N) VT2**T contains a copy of the first K right singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains non-zeros only at and before NL +1; the third block contains non-zeros only at and after NL +2.LDVT2LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= M.IDXPIDXP is INTEGER array, dimension(N) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values.IDXIDX is INTEGER array, dimension(N) This will contain the permutation used to sort the contents of D into ascending order.IDXCIDXC is INTEGER array, dimension(N) This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains non-zero entries only at and above NL, the second contains non-zero entries only below NL+2, and the third is dense.IDXQIDXQ is INTEGER array, dimension(N) This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values.COLTYPCOLTYP is INTEGER array, dimension(N) As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is: 1 : non-zero in the upper half only 2 : non-zero in the lower half only 3 : dense 4 : deflated On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the I-th type columns.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd3(integerNL,integerNR,integerSQRE,integerK,doubleprecision,dimension(*)D,doubleprecision,dimension(ldq,*)Q,integerLDQ,doubleprecision,dimension(*)DSIGMA,doubleprecision,dimension(ldu,*)U,integerLDU,doubleprecision,dimension(ldu2,*)U2,integerLDU2,doubleprecision,dimension(ldvt,*)VT,integerLDVT,doubleprecision,dimension(ldvt2,*)VT2,integerLDVT2,integer,dimension(*)IDXC,integer,dimension(*)CTOT,doubleprecision,dimension(*)Z,integerINFO)DLASD3finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.Purpose:DLASD3 finds all the square roots of the roots of the secular equation, as defined by the values in D and Z. It makes the appropriate calls to DLASD4 and then updates the singular vectors by matrix multiplication. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. DLASD3 is called from DLASD1.ParametersNLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.KK is INTEGER The size of the secular equation, 1 =< K = < N.DD is DOUBLE PRECISION array, dimension(K) On exit the square roots of the roots of the secular equation, in ascending order.QQ is DOUBLE PRECISION array, dimension (LDQ,K)LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= K.DSIGMADSIGMA is DOUBLE PRECISION array, dimension(K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.UU is DOUBLE PRECISION array, dimension (LDU, N) The last N - K columns of this matrix contain the deflated left singular vectors.LDULDU is INTEGER The leading dimension of the array U. LDU >= N.U2U2 is DOUBLE PRECISION array, dimension (LDU2, N) The first K columns of this matrix contain the non-deflated left singular vectors for the split problem.LDU2LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N.VTVT is DOUBLE PRECISION array, dimension (LDVT, M) The last M - K columns of VT**T contain the deflated right singular vectors.LDVTLDVT is INTEGER The leading dimension of the array VT. LDVT >= N.VT2VT2 is DOUBLE PRECISION array, dimension (LDVT2, N) The first K columns of VT2**T contain the non-deflated right singular vectors for the split problem.LDVT2LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= N.IDXCIDXC is INTEGER array, dimension ( N ) The permutation used to arrange the columns of U (and rows of VT) into three groups: the first group contains non-zero entries only at and above (or before) NL +1; the second contains non-zero entries only at and below (or after) NL+2; and the third is dense. The first column of U and the row of VT are treated separately, however. The rows of the singular vectors found by DLASD4 must be likewise permuted before the matrix multiplies can take place.CTOTCTOT is INTEGER array, dimension ( 4 ) A count of the total number of the various types of columns in U (or rows in VT), as described in IDXC. The fourth column type is any column which has been deflated.ZZ is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating row vector.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd4(integerN,integerI,doubleprecision,dimension(*)D,doubleprecision,dimension(*)Z,doubleprecision,dimension(*)DELTA,doubleprecisionRHO,doubleprecisionSIGMA,doubleprecision,dimension(*)WORK,integerINFO)DLASD4computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.Purpose:This subroutine computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.ParametersNN is INTEGER The length of all arrays.II is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.DD is DOUBLE PRECISION array, dimension ( N ) The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J.ZZ is DOUBLE PRECISION array, dimension ( N ) The components of the updating vector.DELTADELTA is DOUBLE PRECISION array, dimension ( N ) If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th component. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvectors.RHORHO is DOUBLE PRECISION The scalar in the symmetric updating formula.SIGMASIGMA is DOUBLE PRECISION The computed sigma_I, the I-th updated eigenvalue.WORKWORK is DOUBLE PRECISION array, dimension ( N ) If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th component. If N = 1, then WORK( 1 ) = 1.INFOINFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.InternalParameters:Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ren-Cang Li, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd5(integerI,doubleprecision,dimension(2)D,doubleprecision,dimension(2)Z,doubleprecision,dimension(2)DELTA,doubleprecisionRHO,doubleprecisionDSIGMA,doubleprecision,dimension(2)WORK)DLASD5computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.Purpose:This subroutine computes the square root of the I-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) * diag( D ) + RHO * Z * transpose(Z) . The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.ParametersII is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2.DD is DOUBLE PRECISION array, dimension ( 2 ) The original eigenvalues. We assume 0 <= D(1) < D(2).ZZ is DOUBLE PRECISION array, dimension ( 2 ) The components of the updating vector.DELTADELTA is DOUBLE PRECISION array, dimension ( 2 ) Contains (D(j) - sigma_I) in its j-th component. The vector DELTA contains the information necessary to construct the eigenvectors.RHORHO is DOUBLE PRECISION The scalar in the symmetric updating formula.DSIGMADSIGMA is DOUBLE PRECISION The computed sigma_I, the I-th updated eigenvalue.WORKWORK is DOUBLE PRECISION array, dimension ( 2 ) WORK contains (D(j) + sigma_I) in its j-th component.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ren-Cang Li, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd6(integerICOMPQ,integerNL,integerNR,integerSQRE,doubleprecision,dimension(*)D,doubleprecision,dimension(*)VF,doubleprecision,dimension(*)VL,doubleprecisionALPHA,doubleprecisionBETA,integer,dimension(*)IDXQ,integer,dimension(*)PERM,integerGIVPTR,integer,dimension(ldgcol,*)GIVCOL,integerLDGCOL,doubleprecision,dimension(ldgnum,*)GIVNUM,integerLDGNUM,doubleprecision,dimension(ldgnum,*)POLES,doubleprecision,dimension(*)DIFL,doubleprecision,dimension(*)DIFR,doubleprecision,dimension(*)Z,integerK,doubleprecisionC,doubleprecisionS,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DLASD6computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.Purpose:DLASD6 computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row. This routine is used only for the problem which requires all singular values and optionally singular vector matrices in factored form. B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine, DLASD1, handles the case in which all singular values and singular vectors of the bidiagonal matrix are desired. DLASD6 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The singular values of B can be computed using D1, D2, the first components of all the right singular vectors of the lower block, and the last components of all the right singular vectors of the upper block. These components are stored and updated in VF and VL, respectively, in DLASD6. Hence U and VT are not explicitly referenced. The singular values are stored in D. The algorithm consists of two stages: The first stage consists of deflating the size of the problem when there are multiple singular values or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLASD7. The second stage consists of calculating the updated singular values. This is done by finding the roots of the secular equation via the routine DLASD4 (as called by DLASD8). This routine also updates VF and VL and computes the distances between the updated singular values and the old singular values. DLASD6 is called from DLASDA.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well.NLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.DD is DOUBLE PRECISION array, dimension ( NL+NR+1 ). On entry D(1:NL,1:NL) contains the singular values of the upper block, and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.VFVF is DOUBLE PRECISION array, dimension ( M ) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix.VLVL is DOUBLE PRECISION array, dimension ( M ) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix.ALPHAALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row.BETABETA is DOUBLE PRECISION Contains the off-diagonal element associated with the added row.IDXQIDXQ is INTEGER array, dimension ( N ) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order.PERMPERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0.GIVPTRGIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.GIVCOLGIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0.LDGCOLLDGCOL is INTEGER leading dimension of GIVCOL, must be at least N.GIVNUMGIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0.LDGNUMLDGNUM is INTEGER The leading dimension of GIVNUM and POLES, must be at least N.POLESPOLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) On exit, POLES(1,*) is an array containing the new singular values obtained from solving the secular equation, and POLES(2,*) is an array containing the poles in the secular equation. Not referenced if ICOMPQ = 0.DIFLDIFL is DOUBLE PRECISION array, dimension ( N ) On exit, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.DIFRDIFR is DOUBLE PRECISION array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. See DLASD8 for details on DIFL and DIFR.ZZ is DOUBLE PRECISION array, dimension ( M ) The first elements of this array contain the components of the deflation-adjusted updating row vector.KK is INTEGER Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.CC is DOUBLE PRECISION C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.SS is DOUBLE PRECISION S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.WORKWORK is DOUBLE PRECISION array, dimension ( 4 * M )IWORKIWORK is INTEGER array, dimension ( 3 * N )INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd7(integerICOMPQ,integerNL,integerNR,integerSQRE,integerK,doubleprecision,dimension(*)D,doubleprecision,dimension(*)Z,doubleprecision,dimension(*)ZW,doubleprecision,dimension(*)VF,doubleprecision,dimension(*)VFW,doubleprecision,dimension(*)VL,doubleprecision,dimension(*)VLW,doubleprecisionALPHA,doubleprecisionBETA,doubleprecision,dimension(*)DSIGMA,integer,dimension(*)IDX,integer,dimension(*)IDXP,integer,dimension(*)IDXQ,integer,dimension(*)PERM,integerGIVPTR,integer,dimension(ldgcol,*)GIVCOL,integerLDGCOL,doubleprecision,dimension(ldgnum,*)GIVNUM,integerLDGNUM,doubleprecisionC,doubleprecisionS,integerINFO)DLASD7merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.Purpose:DLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. DLASD7 is called from DLASD6.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows: = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form.NLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.KK is INTEGER Contains the dimension of the non-deflated matrix, this is the order of the related secular equation. 1 <= K <=N.DD is DOUBLE PRECISION array, dimension ( N ) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order.ZZ is DOUBLE PRECISION array, dimension ( M ) On exit Z contains the updating row vector in the secular equation.ZWZW is DOUBLE PRECISION array, dimension ( M ) Workspace for Z.VFVF is DOUBLE PRECISION array, dimension ( M ) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix.VFWVFW is DOUBLE PRECISION array, dimension ( M ) Workspace for VF.VLVL is DOUBLE PRECISION array, dimension ( M ) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix.VLWVLW is DOUBLE PRECISION array, dimension ( M ) Workspace for VL.ALPHAALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row.BETABETA is DOUBLE PRECISION Contains the off-diagonal element associated with the added row.DSIGMADSIGMA is DOUBLE PRECISION array, dimension ( N ) Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation.IDXIDX is INTEGER array, dimension ( N ) This will contain the permutation used to sort the contents of D into ascending order.IDXPIDXP is INTEGER array, dimension ( N ) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values.IDXQIDXQ is INTEGER array, dimension ( N ) This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first half of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values.PERMPERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each singular block. Not referenced if ICOMPQ = 0.GIVPTRGIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.GIVCOLGIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0.LDGCOLLDGCOL is INTEGER The leading dimension of GIVCOL, must be at least N.GIVNUMGIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0.LDGNUMLDGNUM is INTEGER The leading dimension of GIVNUM, must be at least N.CC is DOUBLE PRECISION C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.SS is DOUBLE PRECISION S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasd8(integerICOMPQ,integerK,doubleprecision,dimension(*)D,doubleprecision,dimension(*)Z,doubleprecision,dimension(*)VF,doubleprecision,dimension(*)VL,doubleprecision,dimension(*)DIFL,doubleprecision,dimension(lddifr,*)DIFR,integerLDDIFR,doubleprecision,dimension(*)DSIGMA,doubleprecision,dimension(*)WORK,integerINFO)DLASD8finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.Purpose:DLASD8 finds the square roots of the roots of the secular equation, as defined by the values in DSIGMA and Z. It makes the appropriate calls to DLASD4, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA). It also updates the arrays VF and VL, the first and last components of all the right singular vectors of the original bidiagonal matrix. DLASD8 is called from DLASD6.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form in the calling routine: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well.KK is INTEGER The number of terms in the rational function to be solved by DLASD4. K >= 1.DD is DOUBLE PRECISION array, dimension ( K ) On output, D contains the updated singular values.ZZ is DOUBLE PRECISION array, dimension ( K ) On entry, the first K elements of this array contain the components of the deflation-adjusted updating row vector. On exit, Z is updated.VFVF is DOUBLE PRECISION array, dimension ( K ) On entry, VF contains information passed through DBEDE8. On exit, VF contains the first K components of the first components of all right singular vectors of the bidiagonal matrix.VLVL is DOUBLE PRECISION array, dimension ( K ) On entry, VL contains information passed through DBEDE8. On exit, VL contains the first K components of the last components of all right singular vectors of the bidiagonal matrix.DIFLDIFL is DOUBLE PRECISION array, dimension ( K ) On exit, DIFL(I) = D(I) - DSIGMA(I).DIFRDIFR is DOUBLE PRECISION array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix.LDDIFRLDDIFR is INTEGER The leading dimension of DIFR, must be at least K.DSIGMADSIGMA is DOUBLE PRECISION array, dimension ( K ) On entry, the first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. On exit, the elements of DSIGMA may be very slightly altered in value.WORKWORK is DOUBLE PRECISION array, dimension (3*K)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasda(integerICOMPQ,integerSMLSIZ,integerN,integerSQRE,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(ldu,*)U,integerLDU,doubleprecision,dimension(ldu,*)VT,integer,dimension(*)K,doubleprecision,dimension(ldu,*)DIFL,doubleprecision,dimension(ldu,*)DIFR,doubleprecision,dimension(ldu,*)Z,doubleprecision,dimension(ldu,*)POLES,integer,dimension(*)GIVPTR,integer,dimension(ldgcol,*)GIVCOL,integerLDGCOL,integer,dimension(ldgcol,*)PERM,doubleprecision,dimension(ldu,*)GIVNUM,doubleprecision,dimension(*)C,doubleprecision,dimension(*)S,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DLASDAcomputes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.Purpose:Using a divide and conquer approach, DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes the singular values in the SVD B = U * S * VT. The orthogonal matrices U and VT are optionally computed in compact form. A related subroutine, DLASD0, computes the singular values and the singular vectors in explicit form.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form.SMLSIZSMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree.NN is INTEGER The row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D.SQRESQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N + 1.DD is DOUBLE PRECISION array, dimension ( N ) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values.EE is DOUBLE PRECISION array, dimension ( M-1 ) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed.UU is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left singular vector matrices of all subproblems at the bottom level.LDULDU is INTEGER, LDU = > N. The leading dimension of arrays U, VT, DIFL, DIFR, POLES, GIVNUM, and Z.VTVT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right singular vector matrices of all subproblems at the bottom level.KK is INTEGER array, dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th secular equation on the computation tree.DIFLDIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ), where NLVL = floor(log_2 (N/SMLSIZ))).DIFRDIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) record distances between singular values on the I-th level and singular values on the (I -1)-th level, and DIFR(1:N, 2 * I ) contains the normalizing factors for the right singular vector matrix. See DLASD8 for details.ZZ is DOUBLE PRECISION array, dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. The first K elements of Z(1, I) contain the components of the deflation-adjusted updating row vector for subproblems on the I-th level.POLESPOLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and POLES(1, 2*I) contain the new and old singular values involved in the secular equations on the I-th level.GIVPTRGIVPTR is INTEGER array, dimension ( N ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records the number of Givens rotations performed on the I-th problem on the computation tree.GIVCOLGIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations of Givens rotations performed on the I-th level on the computation tree.LDGCOLLDGCOL is INTEGER, LDGCOL = > N. The leading dimension of arrays GIVCOL and PERM.PERMPERM is INTEGER array, dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records permutations done on the I-th level of the computation tree.GIVNUMGIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- values of Givens rotations performed on the I-th level on the computation tree.CC is DOUBLE PRECISION array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the I-th subproblem is not square, on exit, C( I ) contains the C-value of a Givens rotation related to the right null space of the I-th subproblem.SS is DOUBLE PRECISION array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the I-th subproblem is not square, on exit, S( I ) contains the S-value of a Givens rotation related to the right null space of the I-th subproblem.WORKWORK is DOUBLE PRECISION array, dimension (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).IWORKIWORK is INTEGER array, dimension (7*N)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasdq(characterUPLO,integerSQRE,integerN,integerNCVT,integerNRU,integerNCC,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(ldvt,*)VT,integerLDVT,doubleprecision,dimension(ldu,*)U,integerLDU,doubleprecision,dimension(ldc,*)C,integerLDC,doubleprecision,dimension(*)WORK,integerINFO)DLASDQcomputes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.Purpose:DLASDQ computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired. Letting B denote the input bidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose of P). The singular values S are overwritten on D. The input matrix U is changed to U * Q if desired. The input matrix VT is changed to P**T * VT if desired. The input matrix C is changed to Q**T * C if desired. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm.ParametersUPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the input bidiagonal matrix is upper or lower bidiagonal, and whether it is square are not. UPLO = 'U' or 'u' B is upper bidiagonal. UPLO = 'L' or 'l' B is lower bidiagonal.SQRESQRE is INTEGER = 0: then the input matrix is N-by-N. = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and (N+1)-by-N if UPLU = 'L'. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.NN is INTEGER On entry, N specifies the number of rows and columns in the matrix. N must be at least 0.NCVTNCVT is INTEGER On entry, NCVT specifies the number of columns of the matrix VT. NCVT must be at least 0.NRUNRU is INTEGER On entry, NRU specifies the number of rows of the matrix U. NRU must be at least 0.NCCNCC is INTEGER On entry, NCC specifies the number of columns of the matrix C. NCC must be at least 0.DD is DOUBLE PRECISION array, dimension (N) On entry, D contains the diagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in ascending order.EE is DOUBLE PRECISION array. dimension is (N-1) if SQRE = 0 and N if SQRE = 1. On entry, the entries of E contain the offdiagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, E will contain 0. If the algorithm does not converge, D and E will contain the diagonal and superdiagonal entries of a bidiagonal matrix orthogonally equivalent to the one given as input.VTVT is DOUBLE PRECISION array, dimension (LDVT, NCVT) On entry, contains a matrix which on exit has been premultiplied by P**T, dimension N-by-NCVT if SQRE = 0 and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).LDVTLDVT is INTEGER On entry, LDVT specifies the leading dimension of VT as declared in the calling (sub) program. LDVT must be at least 1. If NCVT is nonzero LDVT must also be at least N.UU is DOUBLE PRECISION array, dimension (LDU, N) On entry, contains a matrix which on exit has been postmultiplied by Q, dimension NRU-by-N if SQRE = 0 and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).LDULDU is INTEGER On entry, LDU specifies the leading dimension of U as declared in the calling (sub) program. LDU must be at least max( 1, NRU ) .CC is DOUBLE PRECISION array, dimension (LDC, NCC) On entry, contains an N-by-NCC matrix which on exit has been premultiplied by Q**T dimension N-by-NCC if SQRE = 0 and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).LDCLDC is INTEGER On entry, LDC specifies the leading dimension of C as declared in the calling (sub) program. LDC must be at least 1. If NCC is nonzero, LDC must also be at least N.WORKWORK is DOUBLE PRECISION array, dimension (4*N) Workspace. Only referenced if one of NCVT, NRU, or NCC is nonzero, and if N is at least 2.INFOINFO is INTEGER On exit, a value of 0 indicates a successful exit. If INFO < 0, argument number -INFO is illegal. If INFO > 0, the algorithm did not converge, and INFO specifies how many superdiagonals did not converge.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlasdt(integerN,integerLVL,integerND,integer,dimension(*)INODE,integer,dimension(*)NDIML,integer,dimension(*)NDIMR,integerMSUB)DLASDTcreates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.Purpose:DLASDT creates a tree of subproblems for bidiagonal divide and conquer.ParametersNN is INTEGER On entry, the number of diagonal elements of the bidiagonal matrix.LVLLVL is INTEGER On exit, the number of levels on the computation tree.NDND is INTEGER On exit, the number of nodes on the tree.INODEINODE is INTEGER array, dimension ( N ) On exit, centers of subproblems.NDIMLNDIML is INTEGER array, dimension ( N ) On exit, row dimensions of left children.NDIMRNDIMR is INTEGER array, dimension ( N ) On exit, row dimensions of right children.MSUBMSUB is INTEGER On entry, the maximum row dimension each subproblem at the bottom of the tree can be of.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutinedlaset(characterUPLO,integerM,integerN,doubleprecisionALPHA,doubleprecisionBETA,doubleprecision,dimension(lda,*)A,integerLDA)DLASETinitializes the off-diagonal elements and the diagonal elements of a matrix to given values.Purpose:DLASET initializes an m-by-n matrix A to BETA on the diagonal and ALPHA on the offdiagonals.ParametersUPLOUPLO is CHARACTER*1 Specifies the part of the matrix A to be set. = 'U': Upper triangular part is set; the strictly lower triangular part of A is not changed. = 'L': Lower triangular part is set; the strictly upper triangular part of A is not changed. Otherwise: All of the matrix A is set.MM is INTEGER The number of rows of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.ALPHAALPHA is DOUBLE PRECISION The constant to which the offdiagonal elements are to be set.BETABETA is DOUBLE PRECISION The constant to which the diagonal elements are to be set.AA is DOUBLE PRECISION array, dimension (LDA,N) On exit, the leading m-by-n submatrix of A is set as follows: if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n, if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n, otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j, and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlasr(characterSIDE,characterPIVOT,characterDIRECT,integerM,integerN,doubleprecision,dimension(*)C,doubleprecision,dimension(*)S,doubleprecision,dimension(lda,*)A,integerLDA)DLASRapplies a sequence of plane rotations to a general rectangular matrix.Purpose:DLASR applies a sequence of plane rotations to a real matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form A := P*A and when SIDE = 'R', the transformation takes the form A := A*P**T where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and P**T is the transpose of P. When DIRECT = 'F' (Forward sequence), then P = P(z-1) * ... * P(2) * P(1) and when DIRECT = 'B' (Backward sequence), then P = P(1) * P(2) * ... * P(z-1) where P(k) is a plane rotation matrix defined by the 2-by-2 rotation R(k) = ( c(k) s(k) ) = ( -s(k) c(k) ). When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears as a rank-2 modification to the identity matrix in rows and columns k and k+1. When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears in rows and columns 1 and k+1. Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.ParametersSIDESIDE is CHARACTER*1 Specifies whether the plane rotation matrix P is applied to A on the left or the right. = 'L': Left, compute A := P*A = 'R': Right, compute A:= A*P**TPIVOTPIVOT is CHARACTER*1 Specifies the plane for which P(k) is a plane rotation matrix. = 'V': Variable pivot, the plane (k,k+1) = 'T': Top pivot, the plane (1,k+1) = 'B': Bottom pivot, the plane (k,z)DIRECTDIRECT is CHARACTER*1 Specifies whether P is a forward or backward sequence of plane rotations. = 'F': Forward, P = P(z-1)*...*P(2)*P(1) = 'B': Backward, P = P(1)*P(2)*...*P(z-1)MM is INTEGER The number of rows of the matrix A. If m <= 1, an immediate return is effected.NN is INTEGER The number of columns of the matrix A. If n <= 1, an immediate return is effected.CC is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations.SS is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The sines s(k) of the plane rotations. The 2-by-2 plane rotation part of the matrix P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( -s(k) c(k) ).AA is DOUBLE PRECISION array, dimension (LDA,N) The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE = 'L' or by A*P**T if SIDE = 'R'.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlassq(integerN,doubleprecision,dimension(*)X,integerINCX,doubleprecisionSCALE,doubleprecisionSUMSQ)DLASSQupdates a sum of squares represented in scaled form.Purpose:DLASSQ returns the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is assumed to be non-negative and scl returns the value scl = max( scale, abs( x( i ) ) ). scale and sumsq must be supplied in SCALE and SUMSQ and scl and smsq are overwritten on SCALE and SUMSQ respectively. The routine makes only one pass through the vector x.ParametersNN is INTEGER The number of elements to be used from the vector X.XX is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) The vector for which a scaled sum of squares is computed. x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.INCXINCX is INTEGER The increment between successive values of the vector X. INCX > 0.SCALESCALE is DOUBLE PRECISION On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares.SUMSQSUMSQ is DOUBLE PRECISION On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinedlasv2(doubleprecisionF,doubleprecisionG,doubleprecisionH,doubleprecisionSSMIN,doubleprecisionSSMAX,doubleprecisionSNR,doubleprecisionCSR,doubleprecisionSNL,doubleprecisionCSL)DLASV2computes the singular value decomposition of a 2-by-2 triangular matrix.Purpose:DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix [ F G ] [ 0 H ]. On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and right singular vectors for abs(SSMAX), giving the decomposition [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ] [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].ParametersFF is DOUBLE PRECISION The (1,1) element of the 2-by-2 matrix.GG is DOUBLE PRECISION The (1,2) element of the 2-by-2 matrix.HH is DOUBLE PRECISION The (2,2) element of the 2-by-2 matrix.SSMINSSMIN is DOUBLE PRECISION abs(SSMIN) is the smaller singular value.SSMAXSSMAX is DOUBLE PRECISION abs(SSMAX) is the larger singular value.SNLSNL is DOUBLE PRECISIONCSLCSL is DOUBLE PRECISION The vector (CSL, SNL) is a unit left singular vector for the singular value abs(SSMAX).SNRSNR is DOUBLE PRECISIONCSRCSR is DOUBLE PRECISION The vector (CSR, SNR) is a unit right singular vector for the singular value abs(SSMAX).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Any input parameter may be aliased with any output parameter. Barring over/underflow and assuming a guard digit in subtraction, all output quantities are correct to within a few units in the last place (ulps). In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.integerfunctionieeeck(integerISPEC,realZERO,realONE)IEEECKPurpose:IEEECK is called from the ILAENV to verify that Infinity and possibly NaN arithmetic is safe (i.e. will not trap).ParametersISPECISPEC is INTEGER Specifies whether to test just for inifinity arithmetic or whether to test for infinity and NaN arithmetic. = 0: Verify infinity arithmetic only. = 1: Verify infinity and NaN arithmetic.ZEROZERO is REAL Must contain the value 0.0 This is passed to prevent the compiler from optimizing away this code.ONEONE is REAL Must contain the value 1.0 This is passed to prevent the compiler from optimizing away this code. RETURN VALUE: INTEGER = 0: Arithmetic failed to produce the correct answers = 1: Arithmetic produced the correct answersAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016integerfunctioniladlc(integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA)ILADLCscans a matrix for its last non-zero column.Purpose:ILADLC scans A for its last non-zero column.ParametersMM is INTEGER The number of rows of the matrix A.NN is INTEGER The number of columns of the matrix A.AA is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016integerfunctioniladlr(integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA)ILADLRscans a matrix for its last non-zero row.Purpose:ILADLR scans A for its last non-zero row.ParametersMM is INTEGER The number of rows of the matrix A.NN is INTEGER The number of columns of the matrix A.AA is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016integerfunctionilaenv(integerISPEC,character*(*)NAME,character*(*)OPTS,integerN1,integerN2,integerN3,integerN4)ILAENVPurpose:ILAENV is called from the LAPACK routines to choose problem-dependent parameters for the local environment. See ISPEC for a description of the parameters. ILAENV returns an INTEGER if ILAENV >= 0: ILAENV returns the value of the parameter specified by ISPEC if ILAENV < 0: if ILAENV = -k, the k-th argument had an illegal value. This version provides a set of parameters which should give good, but not optimal, performance on many of the currently available computers. Users are encouraged to modify this subroutine to set the tuning parameters for their particular machine using the option and problem size information in the arguments. This routine will not function correctly if it is converted to all lower case. Converting it to all upper case is allowed.ParametersISPECISPEC is INTEGER Specifies the parameter to be returned as the value of ILAENV. = 1: the optimal blocksize; if this value is 1, an unblocked algorithm will give the best performance. = 2: the minimum block size for which the block routine should be used; if the usable block size is less than this value, an unblocked routine should be used. = 3: the crossover point (in a block routine, for N less than this value, an unblocked routine should be used) = 4: the number of shifts, used in the nonsymmetric eigenvalue routines (DEPRECATED) = 5: the minimum column dimension for blocking to be used; rectangular blocks must have dimension at least k by m, where k is given by ILAENV(2,...) and m by ILAENV(5,...) = 6: the crossover point for the SVD (when reducing an m by n matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds this value, a QR factorization is used first to reduce the matrix to a triangular form.) = 7: the number of processors = 8: the crossover point for the multishift QR method for nonsymmetric eigenvalue problems (DEPRECATED) = 9: maximum size of the subproblems at the bottom of the computation tree in the divide-and-conquer algorithm (used by xGELSD and xGESDD) =10: ieee NaN arithmetic can be trusted not to trap =11: infinity arithmetic can be trusted not to trap 12 <= ISPEC <= 16: xHSEQR or related subroutines, see IPARMQ for detailed explanationNAMENAME is CHARACTER*(*) The name of the calling subroutine, in either upper case or lower case.OPTSOPTS is CHARACTER*(*) The character options to the subroutine NAME, concatenated into a single character string. For example, UPLO = 'U', TRANS = 'T', and DIAG = 'N' for a triangular routine would be specified as OPTS = 'UTN'.N1N1 is INTEGERN2N2 is INTEGERN3N3 is INTEGERN4N4 is INTEGER Problem dimensions for the subroutine NAME; these may not all be required.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateNovember 2019FurtherDetails:The following conventions have been used when calling ILAENV from the LAPACK routines: 1) OPTS is a concatenation of all of the character options to subroutine NAME, in the same order that they appear in the argument list for NAME, even if they are not used in determining the value of the parameter specified by ISPEC. 2) The problem dimensions N1, N2, N3, N4 are specified in the order that they appear in the argument list for NAME. N1 is used first, N2 second, and so on, and unused problem dimensions are passed a value of -1. 3) The parameter value returned by ILAENV is checked for validity in the calling subroutine. For example, ILAENV is used to retrieve the optimal blocksize for STRTRI as follows: NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 ) IF( NB.LE.1 ) NB = MAX( 1, N )integerfunctionilaenv2stage(integerISPEC,character*(*)NAME,character*(*)OPTS,integerN1,integerN2,integerN3,integerN4)ILAENV2STAGEPurpose:ILAENV2STAGE is called from the LAPACK routines to choose problem-dependent parameters for the local environment. See ISPEC for a description of the parameters. It sets problem and machine dependent parameters useful for *_2STAGE and related subroutines. ILAENV2STAGE returns an INTEGER if ILAENV2STAGE >= 0: ILAENV2STAGE returns the value of the parameter specified by ISPEC if ILAENV2STAGE < 0: if ILAENV2STAGE = -k, the k-th argument had an illegal value. This version provides a set of parameters which should give good, but not optimal, performance on many of the currently available computers for the 2-stage solvers. Users are encouraged to modify this subroutine to set the tuning parameters for their particular machine using the option and problem size information in the arguments. This routine will not function correctly if it is converted to all lower case. Converting it to all upper case is allowed.ParametersISPECISPEC is INTEGER Specifies the parameter to be returned as the value of ILAENV2STAGE. = 1: the optimal blocksize nb for the reduction to BAND = 2: the optimal blocksize ib for the eigenvectors singular vectors update routine = 3: The length of the array that store the Housholder representation for the second stage Band to Tridiagonal or Bidiagonal = 4: The workspace needed for the routine in input. = 5: For future release.NAMENAME is CHARACTER*(*) The name of the calling subroutine, in either upper case or lower case.OPTSOPTS is CHARACTER*(*) The character options to the subroutine NAME, concatenated into a single character string. For example, UPLO = 'U', TRANS = 'T', and DIAG = 'N' for a triangular routine would be specified as OPTS = 'UTN'.N1N1 is INTEGERN2N2 is INTEGERN3N3 is INTEGERN4N4 is INTEGER Problem dimensions for the subroutine NAME; these may not all be required.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Nick R. PapiorDateJuly 2017FurtherDetails:The following conventions have been used when calling ILAENV2STAGE from the LAPACK routines: 1) OPTS is a concatenation of all of the character options to subroutine NAME, in the same order that they appear in the argument list for NAME, even if they are not used in determining the value of the parameter specified by ISPEC. 2) The problem dimensions N1, N2, N3, N4 are specified in the order that they appear in the argument list for NAME. N1 is used first, N2 second, and so on, and unused problem dimensions are passed a value of -1. 3) The parameter value returned by ILAENV2STAGE is checked for validity in the calling subroutine.integerfunctioniparmq(integerISPEC,character,dimension(*)NAME,character,dimension(*)OPTS,integerN,integerILO,integerIHI,integerLWORK)IPARMQPurpose:This program sets problem and machine dependent parameters useful for xHSEQR and related subroutines for eigenvalue problems. It is called whenever IPARMQ is called with 12 <= ISPEC <= 16ParametersISPECISPEC is INTEGER ISPEC specifies which tunable parameter IPARMQ should return. ISPEC=12: (INMIN) Matrices of order nmin or less are sent directly to xLAHQR, the implicit double shift QR algorithm. NMIN must be at least 11. ISPEC=13: (INWIN) Size of the deflation window. This is best set greater than or equal to the number of simultaneous shifts NS. Larger matrices benefit from larger deflation windows. ISPEC=14: (INIBL) Determines when to stop nibbling and invest in an (expensive) multi-shift QR sweep. If the aggressive early deflation subroutine finds LD converged eigenvalues from an order NW deflation window and LD > (NW*NIBBLE)/100, then the next QR sweep is skipped and early deflation is applied immediately to the remaining active diagonal block. Setting IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a multi-shift QR sweep whenever early deflation finds a converged eigenvalue. Setting IPARMQ(ISPEC=14) greater than or equal to 100 prevents TTQRE from skipping a multi-shift QR sweep. ISPEC=15: (NSHFTS) The number of simultaneous shifts in a multi-shift QR iteration. ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the following meanings. 0: During the multi-shift QR/QZ sweep, blocked eigenvalue reordering, blocked Hessenberg-triangular reduction, reflections and/or rotations are not accumulated when updating the far-from-diagonal matrix entries. 1: During the multi-shift QR/QZ sweep, blocked eigenvalue reordering, blocked Hessenberg-triangular reduction, reflections and/or rotations are accumulated, and matrix-matrix multiplication is used to update the far-from-diagonal matrix entries. 2: During the multi-shift QR/QZ sweep, blocked eigenvalue reordering, blocked Hessenberg-triangular reduction, reflections and/or rotations are accumulated, and 2-by-2 block structure is exploited during matrix-matrix multiplies. (If xTRMM is slower than xGEMM, then IPARMQ(ISPEC=16)=1 may be more efficient than IPARMQ(ISPEC=16)=2 despite the greater level of arithmetic work implied by the latter choice.)NAMENAME is CHARACTER string Name of the calling subroutineOPTSOPTS is CHARACTER string This is a concatenation of the string arguments to TTQRE.NN is INTEGER N is the order of the Hessenberg matrix H.ILOILO is INTEGERIHIIHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N.LWORKLWORK is INTEGER The amount of workspace available.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017FurtherDetails:Little is known about how best to choose these parameters. It is possible to use different values of the parameters for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR. It is probably best to choose different parameters for different matrices and different parameters at different times during the iteration, but this has not been implemented --- yet. The best choices of most of the parameters depend in an ill-understood way on the relative execution rate of xLAQR3 and xLAQR5 and on the nature of each particular eigenvalue problem. Experiment may be the only practical way to determine which choices are most effective. Following is a list of default values supplied by IPARMQ. These defaults may be adjusted in order to attain better performance in any particular computational environment. IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point. Default: 75. (Must be at least 11.) IPARMQ(ISPEC=13) Recommended deflation window size. This depends on ILO, IHI and NS, the number of simultaneous shifts returned by IPARMQ(ISPEC=15). The default for (IHI-ILO+1) <= 500 is NS. The default for (IHI-ILO+1) > 500 is 3*NS/2. IPARMQ(ISPEC=14) Nibble crossover point. Default: 14. IPARMQ(ISPEC=15) Number of simultaneous shifts, NS. a multi-shift QR iteration. If IHI-ILO+1 is ... greater than ...but less ... the or equal to ... than default is 0 30 NS = 2+ 30 60 NS = 4+ 60 150 NS = 10 150 590 NS = ** 590 3000 NS = 64 3000 6000 NS = 128 6000 infinity NS = 256 (+) By default matrices of this order are passed to the implicit double shift routine xLAHQR. See IPARMQ(ISPEC=12) above. These values of NS are used only in case of a rare xLAHQR failure. (**) The asterisks (**) indicate an ad-hoc function increasing from 10 to 64. IPARMQ(ISPEC=16) Select structured matrix multiply. (See ISPEC=16 above for details.) Default: 3.logicalfunctionlsamen(integerN,character*(*)CA,character*(*)CB)LSAMENPurpose:LSAMEN tests if the first N letters of CA are the same as the first N letters of CB, regardless of case. LSAMEN returns .TRUE. if CA and CB are equivalent except for case and .FALSE. otherwise. LSAMEN also returns .FALSE. if LEN( CA ) or LEN( CB ) is less than N.ParametersNN is INTEGER The number of characters in CA and CB to be compared.CACA is CHARACTER*(*)CBCB is CHARACTER*(*) CA and CB specify two character strings of length at least N. Only the first N characters of each string will be accessed.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinescombssq(real,dimension(2)V1,real,dimension(2)V2)SCOMBSSQadds two scaled sum of squares quantitiesPurpose:SCOMBSSQ adds two scaled sum of squares quantities, V1 := V1 + V2. That is, V1_scale**2 * V1_sumsq := V1_scale**2 * V1_sumsq + V2_scale**2 * V2_sumsqParametersV1V1 is REAL array, dimension (2). The first scaled sum. V1(1) = V1_scale, V1(2) = V1_sumsq.V2V2 is REAL array, dimension (2). The second scaled sum. V2(1) = V2_scale, V2(2) = V2_sumsq.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateNovember 2018logicalfunctionsisnan(real,intent(in)SIN)SISNANtests input for NaN.Purpose:SISNAN returns .TRUE. if its argument is NaN, and .FALSE. otherwise. To be replaced by the Fortran 2003 intrinsic in the future.ParametersSINSIN is REAL Input to test for NaN.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017subroutineslabad(realSMALL,realLARGE)SLABADPurpose:SLABAD takes as input the values computed by SLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by SLAMCH. This subroutine is needed because SLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray.ParametersSMALLSMALL is REAL On entry, the underflow threshold as computed by SLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of SMALL, otherwise unchanged.LARGELARGE is REAL On entry, the overflow threshold as computed by SLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of LARGE, otherwise unchanged.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslacpy(characterUPLO,integerM,integerN,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB)SLACPYcopies all or part of one two-dimensional array to another.Purpose:SLACPY copies all or part of a two-dimensional matrix A to another matrix B.ParametersUPLOUPLO is CHARACTER*1 Specifies the part of the matrix A to be copied to B. = 'U': Upper triangular part = 'L': Lower triangular part Otherwise: All of the matrix AMM is INTEGER The number of rows of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.AA is REAL array, dimension (LDA,N) The m by n matrix A. If UPLO = 'U', only the upper triangle or trapezoid is accessed; if UPLO = 'L', only the lower triangle or trapezoid is accessed.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).BB is REAL array, dimension (LDB,N) On exit, B = A in the locations specified by UPLO.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslae2(realA,realB,realC,realRT1,realRT2)SLAE2computes the eigenvalues of a 2-by-2 symmetric matrix.Purpose:SLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, and RT2 is the eigenvalue of smaller absolute value.ParametersAA is REAL The (1,1) element of the 2-by-2 matrix.BB is REAL The (1,2) and (2,1) elements of the 2-by-2 matrix.CC is REAL The (2,2) element of the 2-by-2 matrix.RT1RT1 is REAL The eigenvalue of larger absolute value.RT2RT2 is REAL The eigenvalue of smaller absolute value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*C-B*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.subroutineslaebz(integerIJOB,integerNITMAX,integerN,integerMMAX,integerMINP,integerNBMIN,realABSTOL,realRELTOL,realPIVMIN,real,dimension(*)D,real,dimension(*)E,real,dimension(*)E2,integer,dimension(*)NVAL,real,dimension(mmax,*)AB,real,dimension(*)C,integerMOUT,integer,dimension(mmax,*)NAB,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SLAEBZcomputes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.Purpose:SLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases half-open intervals, i.e., of the form (a,b] , which includes b but not a . To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966 Note: the arguments are, in general, *not* checked for unreasonable values.ParametersIJOBIJOB is INTEGER Specifies what is to be done: = 1: Compute NAB for the initial intervals. = 2: Perform bisection iteration to find eigenvalues of T. = 3: Perform bisection iteration to invert N(w), i.e., to find a point which has a specified number of eigenvalues of T to its left. Other values will cause SLAEBZ to return with INFO=-1.NITMAXNITMAX is INTEGER The maximum number of "levels" of bisection to be performed, i.e., an interval of width W will not be made smaller than 2^(-NITMAX) * W. If not all intervals have converged after NITMAX iterations, then INFO is set to the number of non-converged intervals.NN is INTEGER The dimension n of the tridiagonal matrix T. It must be at least 1.MMAXMMAX is INTEGER The maximum number of intervals. If more than MMAX intervals are generated, then SLAEBZ will quit with INFO=MMAX+1.MINPMINP is INTEGER The initial number of intervals. It may not be greater than MMAX.NBMINNBMIN is INTEGER The smallest number of intervals that should be processed using a vector loop. If zero, then only the scalar loop will be used.ABSTOLABSTOL is REAL The minimum (absolute) width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. This must be at least zero.RELTOLRELTOL is REAL The minimum relative width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.PIVMINPIVMIN is REAL The minimum absolute value of a "pivot" in the Sturm sequence loop. This must be at least max |e(j)**2|*safe_min and at least safe_min, where safe_min is at least the smallest number that can divide one without overflow.DD is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T.EE is REAL array, dimension (N) The offdiagonal elements of the tridiagonal matrix T in positions 1 through N-1. E(N) is arbitrary.E2E2 is REAL array, dimension (N) The squares of the offdiagonal elements of the tridiagonal matrix T. E2(N) is ignored.NVALNVAL is INTEGER array, dimension (MINP) If IJOB=1 or 2, not referenced. If IJOB=3, the desired values of N(w). The elements of NVAL will be reordered to correspond with the intervals in AB. Thus, NVAL(j) on output will not, in general be the same as NVAL(j) on input, but it will correspond with the interval (AB(j,1),AB(j,2)] on output.ABAB is REAL array, dimension (MMAX,2) The endpoints of the intervals. AB(j,1) is a(j), the left endpoint of the j-th interval, and AB(j,2) is b(j), the right endpoint of the j-th interval. The input intervals will, in general, be modified, split, and reordered by the calculation.CC is REAL array, dimension (MMAX) If IJOB=1, ignored. If IJOB=2, workspace. If IJOB=3, then on input C(j) should be initialized to the first search point in the binary search.MOUTMOUT is INTEGER If IJOB=1, the number of eigenvalues in the intervals. If IJOB=2 or 3, the number of intervals output. If IJOB=3, MOUT will equal MINP.NABNAB is INTEGER array, dimension (MMAX,2) If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). If IJOB=2, then on input, NAB(i,j) should be set. It must satisfy the condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, NAB(i,j)=N(AB(i,j)), from a previous call to SLAEBZ with IJOB=1. On output, NAB(i,j) will contain max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input interval that the output interval (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2). If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless N(w) > NVAL(i) for all search points w , in which case NAB(i,1) will not be modified, i.e., the output value will be the same as the input value (modulo reorderings -- see NVAL and AB), or unless N(w) < NVAL(i) for all search points w , in which case NAB(i,2) will not be modified. Normally, NAB should be set to some distinctive value(s) before SLAEBZ is called.WORKWORK is REAL array, dimension (MMAX) Workspace.IWORKIWORK is INTEGER array, dimension (MMAX) Workspace.INFOINFO is INTEGER = 0: All intervals converged. = 1--MMAX: The last INFO intervals did not converge. = MMAX+1: More than MMAX intervals were generated.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:This routine is intended to be called only by other LAPACK routines, thus the interface is less user-friendly. It is intended for two purposes: (a) finding eigenvalues. In this case, SLAEBZ should have one or more initial intervals set up in AB, and SLAEBZ should be called with IJOB=1. This sets up NAB, and also counts the eigenvalues. Intervals with no eigenvalues would usually be thrown out at this point. Also, if not all the eigenvalues in an interval i are desired, NAB(i,1) can be increased or NAB(i,2) decreased. For example, set NAB(i,1)=NAB(i,2)-1 to get the largest eigenvalue. SLAEBZ is then called with IJOB=2 and MMAX no smaller than the value of MOUT returned by the call with IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1 through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the tolerance specified by ABSTOL and RELTOL. (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l). In this case, start with a Gershgorin interval (a,b). Set up AB to contain 2 search intervals, both initially (a,b). One NVAL element should contain f-1 and the other should contain l , while C should contain a and b, resp. NAB(i,1) should be -1 and NAB(i,2) should be N+1, to flag an error if the desired interval does not lie in (a,b). SLAEBZ is then called with IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals -- j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and w(l-r)=...=w(l+k) are handled similarly.subroutineslaev2(realA,realB,realC,realRT1,realRT2,realCS1,realSN1)SLAEV2computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.Purpose:SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ] [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].ParametersAA is REAL The (1,1) element of the 2-by-2 matrix.BB is REAL The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.CC is REAL The (2,2) element of the 2-by-2 matrix.RT1RT1 is REAL The eigenvalue of larger absolute value.RT2RT2 is REAL The eigenvalue of smaller absolute value.CS1CS1 is REALSN1SN1 is REAL The vector (CS1, SN1) is a unit right eigenvector for RT1.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*C-B*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. CS1 and SN1 are accurate to a few ulps barring over/underflow. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.subroutineslag2d(integerM,integerN,real,dimension(ldsa,*)SA,integerLDSA,doubleprecision,dimension(lda,*)A,integerLDA,integerINFO)SLAG2Dconverts a single precision matrix to a double precision matrix.Purpose:SLAG2D converts a SINGLE PRECISION matrix, SA, to a DOUBLE PRECISION matrix, A. Note that while it is possible to overflow while converting from double to single, it is not possible to overflow when converting from single to double. This is an auxiliary routine so there is no argument checking.ParametersMM is INTEGER The number of lines of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.SASA is REAL array, dimension (LDSA,N) On entry, the M-by-N coefficient matrix SA.LDSALDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M).AA is DOUBLE PRECISION array, dimension (LDA,N) On exit, the M-by-N coefficient matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).INFOINFO is INTEGER = 0: successful exitAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslagts(integerJOB,integerN,real,dimension(*)A,real,dimension(*)B,real,dimension(*)C,real,dimension(*)D,integer,dimension(*)IN,real,dimension(*)Y,realTOL,integerINFO)SLAGTSsolves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.Purpose:SLAGTS may be used to solve one of the systems of equations (T - lambda*I)*x = y or (T - lambda*I)**T*x = y, where T is an n by n tridiagonal matrix, for x, following the factorization of (T - lambda*I) as (T - lambda*I) = P*L*U , by routine SLAGTF. The choice of equation to be solved is controlled by the argument JOB, and in each case there is an option to perturb zero or very small diagonal elements of U, this option being intended for use in applications such as inverse iteration.ParametersJOBJOB is INTEGER Specifies the job to be performed by SLAGTS as follows: = 1: The equations (T - lambda*I)x = y are to be solved, but diagonal elements of U are not to be perturbed. = -1: The equations (T - lambda*I)x = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. = 2: The equations (T - lambda*I)**Tx = y are to be solved, but diagonal elements of U are not to be perturbed. = -2: The equations (T - lambda*I)**Tx = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below.NN is INTEGER The order of the matrix T.AA is REAL array, dimension (N) On entry, A must contain the diagonal elements of U as returned from SLAGTF.BB is REAL array, dimension (N-1) On entry, B must contain the first super-diagonal elements of U as returned from SLAGTF.CC is REAL array, dimension (N-1) On entry, C must contain the sub-diagonal elements of L as returned from SLAGTF.DD is REAL array, dimension (N-2) On entry, D must contain the second super-diagonal elements of U as returned from SLAGTF.ININ is INTEGER array, dimension (N) On entry, IN must contain details of the matrix P as returned from SLAGTF.YY is REAL array, dimension (N) On entry, the right hand side vector y. On exit, Y is overwritten by the solution vector x.TOLTOL is REAL On entry, with JOB < 0, TOL should be the minimum perturbation to be made to very small diagonal elements of U. TOL should normally be chosen as about eps*norm(U), where eps is the relative machine precision, but if TOL is supplied as non-positive, then it is reset to eps*max( abs( u(i,j) ) ). If JOB > 0 then TOL is not referenced. On exit, TOL is changed as described above, only if TOL is non-positive on entry. Otherwise TOL is unchanged.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: overflow would occur when computing the INFO(th) element of the solution vector x. This can only occur when JOB is supplied as positive and either means that a diagonal element of U is very small, or that the elements of the right-hand side vector y are very large.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016logicalfunctionslaisnan(real,intent(in)SIN1,real,intent(in)SIN2)SLAISNANtests input for NaN by comparing two arguments for inequality.Purpose:This routine is not for general use. It exists solely to avoid over-optimization in SISNAN. SLAISNAN checks for NaNs by comparing its two arguments for inequality. NaN is the only floating-point value where NaN != NaN returns .TRUE. To check for NaNs, pass the same variable as both arguments. A compiler must assume that the two arguments are not the same variable, and the test will not be optimized away. Interprocedural or whole-program optimization may delete this test. The ISNAN functions will be replaced by the correct Fortran 03 intrinsic once the intrinsic is widely available.ParametersSIN1SIN1 is REALSIN2SIN2 is REAL Two numbers to compare for inequality.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017integerfunctionslaneg(integerN,real,dimension(*)D,real,dimension(*)LLD,realSIGMA,realPIVMIN,integerR)SLANEGcomputes the Sturm count.Purpose:SLANEG computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T - sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from SLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE-754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is non-zero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)ParametersNN is INTEGER The order of the matrix.DD is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D.LLDLLD is REAL array, dimension (N-1) The (N-1) elements L(i)*L(i)*D(i).SIGMASIGMA is REAL Shift amount in T - sigma I = L D L^T.PIVMINPIVMIN is REAL The minimum pivot in the Sturm sequence. May be used when zero pivots are encountered on non-IEEE-754 architectures.RR is INTEGER The twist index for the twisted factorization that is used for the negcount.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA Jason Riedy, University of California, Berkeley, USArealfunctionslanst(characterNORM,integerN,real,dimension(*)D,real,dimension(*)E)SLANSTreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.Purpose:SLANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A.ReturnsSLANST SLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.ParametersNORMNORM is CHARACTER*1 Specifies the value to be returned in SLANST as described above.NN is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANST is set to zero.DD is REAL array, dimension (N) The diagonal elements of A.EE is REAL array, dimension (N-1) The (n-1) sub-diagonal or super-diagonal elements of A.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016realfunctionslapy2(realX,realY)SLAPY2returns sqrt(x2+y2).Purpose:SLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary overflow.ParametersXX is REALYY is REAL X and Y specify the values x and y.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017realfunctionslapy3(realX,realY,realZ)SLAPY3returns sqrt(x2+y2+z2).Purpose:SLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause unnecessary overflow.ParametersXX is REALYY is REALZZ is REAL X, Y and Z specify the values x, y and z.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslarnv(integerIDIST,integer,dimension(4)ISEED,integerN,real,dimension(*)X)SLARNVreturns a vector of random numbers from a uniform or normal distribution.Purpose:SLARNV returns a vector of n random real numbers from a uniform or normal distribution.ParametersIDISTIDIST is INTEGER Specifies the distribution of the random numbers: = 1: uniform (0,1) = 2: uniform (-1,1) = 3: normal (0,1)ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated.NN is INTEGER The number of random numbers to be generated.XX is REAL array, dimension (N) The generated random numbers.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:This routine calls the auxiliary routine SLARUV to generate random real numbers from a uniform (0,1) distribution, in batches of up to 128 using vectorisable code. The Box-Muller method is used to transform numbers from a uniform to a normal distribution.subroutineslarra(integerN,real,dimension(*)D,real,dimension(*)E,real,dimension(*)E2,realSPLTOL,realTNRM,integerNSPLIT,integer,dimension(*)ISPLIT,integerINFO)SLARRAcomputes the splitting points with the specified threshold.Purpose:Compute the splitting points with threshold SPLTOL. SLARRA sets any "small" off-diagonal elements to zero.ParametersNN is INTEGER The order of the matrix. N > 0.DD is REAL array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T.EE is REAL array, dimension (N) On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, are set to zero, the other entries of E are untouched.E2E2 is REAL array, dimension (N) On entry, the first (N-1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zeroSPLTOLSPLTOL is REAL The threshold for splitting. Two criteria can be used: SPLTOL<0 : criterion based on absolute off-diagonal value SPLTOL>0 : criterion that preserves relative accuracyTNRMTNRM is REAL The norm of the matrix.NSPLITNSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N.ISPLITISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.INFOINFO is INTEGER = 0: successful exitAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslarrb(integerN,real,dimension(*)D,real,dimension(*)LLD,integerIFIRST,integerILAST,realRTOL1,realRTOL2,integerOFFSET,real,dimension(*)W,real,dimension(*)WGAP,real,dimension(*)WERR,real,dimension(*)WORK,integer,dimension(*)IWORK,realPIVMIN,realSPDIAM,integerTWIST,integerINFO)SLARRBprovides limited bisection to locate eigenvalues for more accuracy.Purpose:Given the relatively robust representation(RRR) L D L^T, SLARRB does "limited" bisection to refine the eigenvalues of L D L^T, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses and their gaps are input in WERR and WGAP, respectively. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.ParametersNN is INTEGER The order of the matrix.DD is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D.LLDLLD is REAL array, dimension (N-1) The (N-1) elements L(i)*L(i)*D(i).IFIRSTIFIRST is INTEGER The index of the first eigenvalue to be computed.ILASTILAST is INTEGER The index of the last eigenvalue to be computed.RTOL1RTOL1 is REALRTOL2RTOL2 is REAL Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the (estimated) distance to the nearest eigenvalue.OFFSETOFFSET is INTEGER Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.WW is REAL array, dimension (N) On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.WGAPWGAP is REAL array, dimension (N-1) On input, the (estimated) gaps between consecutive eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues I and I+1. Note that if IFIRST = ILAST then WGAP(IFIRST-OFFSET) must be set to ZERO. On output, these gaps are refined.WERRWERR is REAL array, dimension (N) On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.WORKWORK is REAL array, dimension (2*N) Workspace.IWORKIWORK is INTEGER array, dimension (2*N) Workspace.PIVMINPIVMIN is REAL The minimum pivot in the Sturm sequence.SPDIAMSPDIAM is REAL The spectral diameter of the matrix.TWISTTWIST is INTEGER The twist index for the twisted factorization that is used for the negcount. TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)INFOINFO is INTEGER Error flag.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslarrc(characterJOBT,integerN,realVL,realVU,real,dimension(*)D,real,dimension(*)E,realPIVMIN,integerEIGCNT,integerLCNT,integerRCNT,integerINFO)SLARRCcomputes the number of eigenvalues of the symmetric tridiagonal matrix.Purpose:Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.ParametersJOBTJOBT is CHARACTER*1 = 'T': Compute Sturm count for matrix T. = 'L': Compute Sturm count for matrix L D L^T.NN is INTEGER The order of the matrix. N > 0.VLVL is REAL The lower bound for the eigenvalues.VUVU is REAL The upper bound for the eigenvalues.DD is REAL array, dimension (N) JOBT = 'T': The N diagonal elements of the tridiagonal matrix T. JOBT = 'L': The N diagonal elements of the diagonal matrix D.EE is REAL array, dimension (N) JOBT = 'T': The N-1 offdiagonal elements of the matrix T. JOBT = 'L': The N-1 offdiagonal elements of the matrix L.PIVMINPIVMIN is REAL The minimum pivot in the Sturm sequence for T.EIGCNTEIGCNT is INTEGER The number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU]LCNTLCNT is INTEGERRCNTRCNT is INTEGER The left and right negcounts of the interval.INFOINFO is INTEGERAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslarrd(characterRANGE,characterORDER,integerN,realVL,realVU,integerIL,integerIU,real,dimension(*)GERS,realRELTOL,real,dimension(*)D,real,dimension(*)E,real,dimension(*)E2,realPIVMIN,integerNSPLIT,integer,dimension(*)ISPLIT,integerM,real,dimension(*)W,real,dimension(*)WERR,realWL,realWU,integer,dimension(*)IBLOCK,integer,dimension(*)INDEXW,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SLARRDcomputes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.Purpose:SLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.ParametersRANGERANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.ORDERORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest.NN is INTEGER The order of the tridiagonal matrix T. N >= 0.VLVL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.VUVU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.ILIL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IUIU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.GERSGERS is REAL array, dimension (2*N) The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)).RELTOLRELTOL is REAL The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.DD is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T.EE is REAL array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T.E2E2 is REAL array, dimension (N-1) The (n-1) squared off-diagonal elements of the tridiagonal matrix T.PIVMINPIVMIN is REAL The minimum pivot allowed in the Sturm sequence for T.NSPLITNSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.ISPLITISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.)MM is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)WW is REAL array, dimension (N) On exit, the first M elements of W will contain the eigenvalue approximations. SLARRD computes an interval I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue approximation is given as the interval midpoint W(j)= ( a_j + b_j)/2. The corresponding error is bounded by WERR(j) = abs( a_j - b_j)/2WERRWERR is REAL array, dimension (N) The error bound on the corresponding eigenvalue approximation in W.WLWL is REALWUWU is REAL The interval (WL, WU] contains all the wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin bounds on the spectrum. If RANGE='I', then WL and WU are computed by SLAEBZ from the index range specified.IBLOCKIBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (SLARRD may use the remaining N-M elements as workspace.)INDEXWINDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th eigenvalue W(i) is the j-th eigenvalue in block k.WORKWORK is REAL array, dimension (4*N)IWORKIWORK is INTEGER array, dimension (3*N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1-IL Cause: non-monotonic arithmetic, causing the Sturm sequence to be non-monotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floating-point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again.InternalParameters:FUDGE REAL, default = 2 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution.Contributors:W. Kahan, University of California, Berkeley, USA Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016subroutineslarre(characterRANGE,integerN,realVL,realVU,integerIL,integerIU,real,dimension(*)D,real,dimension(*)E,real,dimension(*)E2,realRTOL1,realRTOL2,realSPLTOL,integerNSPLIT,integer,dimension(*)ISPLIT,integerM,real,dimension(*)W,real,dimension(*)WERR,real,dimension(*)WGAP,integer,dimension(*)IBLOCK,integer,dimension(*)INDEXW,real,dimension(*)GERS,realPIVMIN,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SLARREgiven the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.Purpose:To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, SLARRE sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by SSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to conpute all and then discard any unwanted one. As an added benefit, SLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.ParametersRANGERANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.NN is INTEGER The order of the matrix. N > 0.VLVL is REAL If RANGE='V', the lower bound for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', SLARRE computes bounds on the desired part of the spectrum.VUVU is REAL If RANGE='V', the upper bound for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', SLARRE computes bounds on the desired part of the spectrum.ILIL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N.IUIU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N.DD is REAL array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T. On exit, the N diagonal elements of the diagonal matrices D_i.EE is REAL array, dimension (N) On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, E contains the subdiagonal elements of the unit bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, contain the base points sigma_i on output.E2E2 is REAL array, dimension (N) On entry, the first (N-1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zeroRTOL1RTOL1 is REALRTOL2RTOL2 is REAL Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )SPLTOLSPLTOL is REAL The threshold for splitting.NSPLITNSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N.ISPLITISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.MM is INTEGER The total number of eigenvalues (of all L_i D_i L_i^T) found.WW is REAL array, dimension (N) The first M elements contain the eigenvalues. The eigenvalues of each of the blocks, L_i D_i L_i^T, are sorted in ascending order ( SLARRE may use the remaining N-M elements as workspace).WERRWERR is REAL array, dimension (N) The error bound on the corresponding eigenvalue in W.WGAPWGAP is REAL array, dimension (N) The separation from the right neighbor eigenvalue in W. The gap is only with respect to the eigenvalues of the same block as each block has its own representation tree. Exception: at the right end of a block we store the left gapIBLOCKIBLOCK is INTEGER array, dimension (N) The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc.INDEXWINDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th eigenvalue W(i) is the 10-th eigenvalue in block 2GERSGERS is REAL array, dimension (2*N) The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)).PIVMINPIVMIN is REAL The minimum pivot in the Sturm sequence for T.WORKWORK is REAL array, dimension (6*N) Workspace.IWORKIWORK is INTEGER array, dimension (5*N) Workspace.INFOINFO is INTEGER = 0: successful exit > 0: A problem occurred in SLARRE. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =-1: Problem in SLARRD. = 2: No base representation could be found in MAXTRY iterations. Increasing MAXTRY and recompilation might be a remedy. =-3: Problem in SLARRB when computing the refined root representation for SLASQ2. =-4: Problem in SLARRB when preforming bisection on the desired part of the spectrum. =-5: Problem in SLASQ2. =-6: Problem in SLASQ2.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016FurtherDetails:The base representations are required to suffer very little element growth and consequently define all their eigenvalues to high relative accuracy.Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslarrf(integerN,real,dimension(*)D,real,dimension(*)L,real,dimension(*)LD,integerCLSTRT,integerCLEND,real,dimension(*)W,real,dimension(*)WGAP,real,dimension(*)WERR,realSPDIAM,realCLGAPL,realCLGAPR,realPIVMIN,realSIGMA,real,dimension(*)DPLUS,real,dimension(*)LPLUS,real,dimension(*)WORK,integerINFO)SLARRFfinds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.Purpose:Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), SLARRF finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.ParametersNN is INTEGER The order of the matrix (subblock, if the matrix split).DD is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D.LL is REAL array, dimension (N-1) The (N-1) subdiagonal elements of the unit bidiagonal matrix L.LDLD is REAL array, dimension (N-1) The (N-1) elements L(i)*D(i).CLSTRTCLSTRT is INTEGER The index of the first eigenvalue in the cluster.CLENDCLEND is INTEGER The index of the last eigenvalue in the cluster.WW is REAL array, dimension dimension is >= (CLEND-CLSTRT+1) The eigenvalue APPROXIMATIONS of L D L^T in ascending order. W( CLSTRT ) through W( CLEND ) form the cluster of relatively close eigenalues.WGAPWGAP is REAL array, dimension dimension is >= (CLEND-CLSTRT+1) The separation from the right neighbor eigenvalue in W.WERRWERR is REAL array, dimension dimension is >= (CLEND-CLSTRT+1) WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in WSPDIAMSPDIAM is REAL estimate of the spectral diameter obtained from the Gerschgorin intervalsCLGAPLCLGAPL is REALCLGAPRCLGAPR is REAL absolute gap on each end of the cluster. Set by the calling routine to protect against shifts too close to eigenvalues outside the cluster.PIVMINPIVMIN is REAL The minimum pivot allowed in the Sturm sequence.SIGMASIGMA is REAL The shift used to form L(+) D(+) L(+)^T.DPLUSDPLUS is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D(+).LPLUSLPLUS is REAL array, dimension (N-1) The first (N-1) elements of LPLUS contain the subdiagonal elements of the unit bidiagonal matrix L(+).WORKWORK is REAL array, dimension (2*N) Workspace.INFOINFO is INTEGER Signals processing OK (=0) or failure (=1)AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslarrj(integerN,real,dimension(*)D,real,dimension(*)E2,integerIFIRST,integerILAST,realRTOL,integerOFFSET,real,dimension(*)W,real,dimension(*)WERR,real,dimension(*)WORK,integer,dimension(*)IWORK,realPIVMIN,realSPDIAM,integerINFO)SLARRJperforms refinement of the initial estimates of the eigenvalues of the matrix T.Purpose:Given the initial eigenvalue approximations of T, SLARRJ does bisection to refine the eigenvalues of T, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses in WERR. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.ParametersNN is INTEGER The order of the matrix.DD is REAL array, dimension (N) The N diagonal elements of T.E2E2 is REAL array, dimension (N-1) The Squares of the (N-1) subdiagonal elements of T.IFIRSTIFIRST is INTEGER The index of the first eigenvalue to be computed.ILASTILAST is INTEGER The index of the last eigenvalue to be computed.RTOLRTOL is REAL Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).OFFSETOFFSET is INTEGER Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.WW is REAL array, dimension (N) On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.WERRWERR is REAL array, dimension (N) On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.WORKWORK is REAL array, dimension (2*N) Workspace.IWORKIWORK is INTEGER array, dimension (2*N) Workspace.PIVMINPIVMIN is REAL The minimum pivot in the Sturm sequence for T.SPDIAMSPDIAM is REAL The spectral diameter of T.INFOINFO is INTEGER Error flag.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslarrk(integerN,integerIW,realGL,realGU,real,dimension(*)D,real,dimension(*)E2,realPIVMIN,realRELTOL,realW,realWERR,integerINFO)SLARRKcomputes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.Purpose:SLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.ParametersNN is INTEGER The order of the tridiagonal matrix T. N >= 0.IWIW is INTEGER The index of the eigenvalues to be returned.GLGL is REALGUGU is REAL An upper and a lower bound on the eigenvalue.DD is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T.E2E2 is REAL array, dimension (N-1) The (n-1) squared off-diagonal elements of the tridiagonal matrix T.PIVMINPIVMIN is REAL The minimum pivot allowed in the Sturm sequence for T.RELTOLRELTOL is REAL The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.WW is REALWERRWERR is REAL The error bound on the corresponding eigenvalue approximation in W.INFOINFO is INTEGER = 0: Eigenvalue converged = -1: Eigenvalue did NOT convergeInternalParameters:FUDGE REAL , default = 2 A "fudge factor" to widen the Gershgorin intervals.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017subroutineslarrr(integerN,real,dimension(*)D,real,dimension(*)E,integerINFO)SLARRRperforms tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.Purpose:Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.ParametersNN is INTEGER The order of the matrix. N > 0.DD is REAL array, dimension (N) The N diagonal elements of the tridiagonal matrix T.EE is REAL array, dimension (N) On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) is set to ZERO.INFOINFO is INTEGER INFO = 0(default) : the matrix warrants computations preserving relative accuracy. INFO = 1 : the matrix warrants computations guaranteeing only absolute accuracy.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USAsubroutineslartg(realF,realG,realCS,realSN,realR)SLARTGgenerates a plane rotation with real cosine and real sine.Purpose:SLARTG generate a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ -SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the BLAS1 routine SROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any floating point operations (saves work in SBDSQR when there are zeros on the diagonal). If F exceeds G in magnitude, CS will be positive.ParametersFF is REAL The first component of vector to be rotated.GG is REAL The second component of vector to be rotated.CSCS is REAL The cosine of the rotation.SNSN is REAL The sine of the rotation.RR is REAL The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslartgp(realF,realG,realCS,realSN,realR)SLARTGPgenerates a plane rotation so that the diagonal is nonnegative.Purpose:SLARTGP generates a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ -SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the Level 1 BLAS routine SROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=(+/-)1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=(+/-)1. The sign is chosen so that R >= 0.ParametersFF is REAL The first component of vector to be rotated.GG is REAL The second component of vector to be rotated.CSCS is REAL The cosine of the rotation.SNSN is REAL The sine of the rotation.RR is REAL The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslaruv(integer,dimension(4)ISEED,integerN,real,dimension(n)X)SLARUVreturns a vector of n random real numbers from a uniform distribution.Purpose:SLARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). This is an auxiliary routine called by SLARNV and CLARNV.ParametersISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated.NN is INTEGER The number of random numbers to be generated. N <= 128.XX is REAL array, dimension (N) The generated random numbers.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:This routine uses a multiplicative congruential method with modulus 2**48 and multiplier 33952834046453 (see G.S.Fishman, 'Multiplicative congruential random number generators with modulus 2**b: an exhaustive analysis for b = 32 and a partial analysis for b = 48', Math. Comp. 189, pp 331-344, 1990). 48-bit integers are stored in 4 integer array elements with 12 bits per element. Hence the routine is portable across machines with integers of 32 bits or more.subroutineslas2(realF,realG,realH,realSSMIN,realSSMAX)SLAS2computes singular values of a 2-by-2 triangular matrix.Purpose:SLAS2 computes the singular values of the 2-by-2 matrix [ F G ] [ 0 H ]. On return, SSMIN is the smaller singular value and SSMAX is the larger singular value.ParametersFF is REAL The (1,1) element of the 2-by-2 matrix.GG is REAL The (1,2) element of the 2-by-2 matrix.HH is REAL The (2,2) element of the 2-by-2 matrix.SSMINSSMIN is REAL The smaller singular value.SSMAXSSMAX is REAL The larger singular value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Barring over/underflow, all output quantities are correct to within a few units in the last place (ulps), even in the absence of a guard digit in addition/subtraction. In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows, or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.subroutineslascl(characterTYPE,integerKL,integerKU,realCFROM,realCTO,integerM,integerN,real,dimension(lda,*)A,integerLDA,integerINFO)SLASCLmultiplies a general rectangular matrix by a real scalar defined as cto/cfrom.Purpose:SLASCL multiplies the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.ParametersTYPETYPE is CHARACTER*1 TYPE indices the storage type of the input matrix. = 'G': A is a full matrix. = 'L': A is a lower triangular matrix. = 'U': A is an upper triangular matrix. = 'H': A is an upper Hessenberg matrix. = 'B': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the lower half stored. = 'Q': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the upper half stored. = 'Z': A is a band matrix with lower bandwidth KL and upper bandwidth KU. See SGBTRF for storage details.KLKL is INTEGER The lower bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'.KUKU is INTEGER The upper bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'.CFROMCFROM is REALCTOCTO is REAL The matrix A is multiplied by CTO/CFROM. A(I,J) is computed without over/underflow if the final result CTO*A(I,J)/CFROM can be represented without over/underflow. CFROM must be nonzero.MM is INTEGER The number of rows of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.AA is REAL array, dimension (LDA,N) The matrix to be multiplied by CTO/CFROM. See TYPE for the storage type.LDALDA is INTEGER The leading dimension of the array A. If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M); TYPE = 'B', LDA >= KL+1; TYPE = 'Q', LDA >= KU+1; TYPE = 'Z', LDA >= 2*KL+KU+1.INFOINFO is INTEGER 0 - successful exit <0 - if INFO = -i, the i-th argument had an illegal value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016subroutineslasd0(integerN,integerSQRE,real,dimension(*)D,real,dimension(*)E,real,dimension(ldu,*)U,integerLDU,real,dimension(ldvt,*)VT,integerLDVT,integerSMLSIZ,integer,dimension(*)IWORK,real,dimension(*)WORK,integerINFO)SLASD0computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc.Purpose:Using a divide and conquer approach, SLASD0 computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes orthogonal matrices U and VT such that B = U * S * VT. The singular values S are overwritten on D. A related subroutine, SLASDA, computes only the singular values, and optionally, the singular vectors in compact form.ParametersNN is INTEGER On entry, the row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D.SQRESQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N+1;DD is REAL array, dimension (N) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values.EE is REAL array, dimension (M-1) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed.UU is REAL array, dimension (LDU, N) On exit, U contains the left singular vectors.LDULDU is INTEGER On entry, leading dimension of U.VTVT is REAL array, dimension (LDVT, M) On exit, VT**T contains the right singular vectors.LDVTLDVT is INTEGER On entry, leading dimension of VT.SMLSIZSMLSIZ is INTEGER On entry, maximum size of the subproblems at the bottom of the computation tree.IWORKIWORK is INTEGER array, dimension (8*N)WORKWORK is REAL array, dimension (3*M**2+2*M)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasd1(integerNL,integerNR,integerSQRE,real,dimension(*)D,realALPHA,realBETA,real,dimension(ldu,*)U,integerLDU,real,dimension(ldvt,*)VT,integerLDVT,integer,dimension(*)IDXQ,integer,dimension(*)IWORK,real,dimension(*)WORK,integerINFO)SLASD1computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.Purpose:SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0. A related subroutine SLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired. SLASD1 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLASD2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine SLASD4 (as called by SLASD3). This routine also calculates the singular vectors of the current problem. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.ParametersNLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.DD is REAL array, dimension (NL+NR+1). N = NL+NR+1 On entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.ALPHAALPHA is REAL Contains the diagonal element associated with the added row.BETABETA is REAL Contains the off-diagonal element associated with the added row.UU is REAL array, dimension (LDU,N) On entry U(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. On exit U contains the left singular vectors of the bidiagonal matrix.LDULDU is INTEGER The leading dimension of the array U. LDU >= max( 1, N ).VTVT is REAL array, dimension (LDVT,M) where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. On exit VT**T contains the right singular vectors of the bidiagonal matrix.LDVTLDVT is INTEGER The leading dimension of the array VT. LDVT >= max( 1, M ).IDXQIDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order.IWORKIWORK is INTEGER array, dimension (4*N)WORKWORK is REAL array, dimension (3*M**2+2*M)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasd2(integerNL,integerNR,integerSQRE,integerK,real,dimension(*)D,real,dimension(*)Z,realALPHA,realBETA,real,dimension(ldu,*)U,integerLDU,real,dimension(ldvt,*)VT,integerLDVT,real,dimension(*)DSIGMA,real,dimension(ldu2,*)U2,integerLDU2,real,dimension(ldvt2,*)VT2,integerLDVT2,integer,dimension(*)IDXP,integer,dimension(*)IDX,integer,dimension(*)IDXC,integer,dimension(*)IDXQ,integer,dimension(*)COLTYP,integerINFO)SLASD2merges the two sets of singular values together into a single sorted set. Used by sbdsdc.Purpose:SLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. SLASD2 is called from SLASD1.ParametersNLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.KK is INTEGER Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.DD is REAL array, dimension (N) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order.ZZ is REAL array, dimension (N) On exit Z contains the updating row vector in the secular equation.ALPHAALPHA is REAL Contains the diagonal element associated with the added row.BETABETA is REAL Contains the off-diagonal element associated with the added row.UU is REAL array, dimension (LDU,N) On entry U contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (N-K) updated left singular vectors (those which were deflated) in its last N-K columns.LDULDU is INTEGER The leading dimension of the array U. LDU >= N.VTVT is REAL array, dimension (LDVT,M) On entry VT**T contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT**T contains the trailing (N-K) updated right singular vectors (those which were deflated) in its last N-K columns. In case SQRE =1, the last row of VT spans the right null space.LDVTLDVT is INTEGER The leading dimension of the array VT. LDVT >= M.DSIGMADSIGMA is REAL array, dimension (N) Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation.U2U2 is REAL array, dimension (LDU2,N) Contains a copy of the first K-1 left singular vectors which will be used by SLASD3 in a matrix multiply (SGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains non-zero entries only at and above NL; the third contains non-zero entries only below NL+1; and the fourth is dense.LDU2LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N.VT2VT2 is REAL array, dimension (LDVT2,N) VT2**T contains a copy of the first K right singular vectors which will be used by SLASD3 in a matrix multiply (SGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains non-zeros only at and before NL +1; the third block contains non-zeros only at and after NL +2.LDVT2LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= M.IDXPIDXP is INTEGER array, dimension (N) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values.IDXIDX is INTEGER array, dimension (N) This will contain the permutation used to sort the contents of D into ascending order.IDXCIDXC is INTEGER array, dimension (N) This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains non-zero entries only at and above NL, the second contains non-zero entries only below NL+2, and the third is dense.IDXQIDXQ is INTEGER array, dimension (N) This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values.COLTYPCOLTYP is INTEGER array, dimension (N) As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is: 1 : non-zero in the upper half only 2 : non-zero in the lower half only 3 : dense 4 : deflated On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the I-th type columns.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasd3(integerNL,integerNR,integerSQRE,integerK,real,dimension(*)D,real,dimension(ldq,*)Q,integerLDQ,real,dimension(*)DSIGMA,real,dimension(ldu,*)U,integerLDU,real,dimension(ldu2,*)U2,integerLDU2,real,dimension(ldvt,*)VT,integerLDVT,real,dimension(ldvt2,*)VT2,integerLDVT2,integer,dimension(*)IDXC,integer,dimension(*)CTOT,real,dimension(*)Z,integerINFO)SLASD3finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.Purpose:SLASD3 finds all the square roots of the roots of the secular equation, as defined by the values in D and Z. It makes the appropriate calls to SLASD4 and then updates the singular vectors by matrix multiplication. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. SLASD3 is called from SLASD1.ParametersNLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.KK is INTEGER The size of the secular equation, 1 =< K = < N.DD is REAL array, dimension(K) On exit the square roots of the roots of the secular equation, in ascending order.QQ is REAL array, dimension (LDQ,K)LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= K.DSIGMADSIGMA is REAL array, dimension(K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.UU is REAL array, dimension (LDU, N) The last N - K columns of this matrix contain the deflated left singular vectors.LDULDU is INTEGER The leading dimension of the array U. LDU >= N.U2U2 is REAL array, dimension (LDU2, N) The first K columns of this matrix contain the non-deflated left singular vectors for the split problem.LDU2LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N.VTVT is REAL array, dimension (LDVT, M) The last M - K columns of VT**T contain the deflated right singular vectors.LDVTLDVT is INTEGER The leading dimension of the array VT. LDVT >= N.VT2VT2 is REAL array, dimension (LDVT2, N) The first K columns of VT2**T contain the non-deflated right singular vectors for the split problem.LDVT2LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= N.IDXCIDXC is INTEGER array, dimension (N) The permutation used to arrange the columns of U (and rows of VT) into three groups: the first group contains non-zero entries only at and above (or before) NL +1; the second contains non-zero entries only at and below (or after) NL+2; and the third is dense. The first column of U and the row of VT are treated separately, however. The rows of the singular vectors found by SLASD4 must be likewise permuted before the matrix multiplies can take place.CTOTCTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in U (or rows in VT), as described in IDXC. The fourth column type is any column which has been deflated.ZZ is REAL array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating row vector.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasd4(integerN,integerI,real,dimension(*)D,real,dimension(*)Z,real,dimension(*)DELTA,realRHO,realSIGMA,real,dimension(*)WORK,integerINFO)SLASD4computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.Purpose:This subroutine computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.ParametersNN is INTEGER The length of all arrays.II is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.DD is REAL array, dimension ( N ) The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J.ZZ is REAL array, dimension ( N ) The components of the updating vector.DELTADELTA is REAL array, dimension ( N ) If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th component. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvectors.RHORHO is REAL The scalar in the symmetric updating formula.SIGMASIGMA is REAL The computed sigma_I, the I-th updated eigenvalue.WORKWORK is REAL array, dimension ( N ) If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th component. If N = 1, then WORK( 1 ) = 1.INFOINFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.InternalParameters:Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ren-Cang Li, Computer Science Division, University of California at Berkeley, USAsubroutineslasd5(integerI,real,dimension(2)D,real,dimension(2)Z,real,dimension(2)DELTA,realRHO,realDSIGMA,real,dimension(2)WORK)SLASD5computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.Purpose:This subroutine computes the square root of the I-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) * diag( D ) + RHO * Z * transpose(Z) . The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.ParametersII is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2.DD is REAL array, dimension (2) The original eigenvalues. We assume 0 <= D(1) < D(2).ZZ is REAL array, dimension (2) The components of the updating vector.DELTADELTA is REAL array, dimension (2) Contains (D(j) - sigma_I) in its j-th component. The vector DELTA contains the information necessary to construct the eigenvectors.RHORHO is REAL The scalar in the symmetric updating formula.DSIGMADSIGMA is REAL The computed sigma_I, the I-th updated eigenvalue.WORKWORK is REAL array, dimension (2) WORK contains (D(j) + sigma_I) in its j-th component.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ren-Cang Li, Computer Science Division, University of California at Berkeley, USAsubroutineslasd6(integerICOMPQ,integerNL,integerNR,integerSQRE,real,dimension(*)D,real,dimension(*)VF,real,dimension(*)VL,realALPHA,realBETA,integer,dimension(*)IDXQ,integer,dimension(*)PERM,integerGIVPTR,integer,dimension(ldgcol,*)GIVCOL,integerLDGCOL,real,dimension(ldgnum,*)GIVNUM,integerLDGNUM,real,dimension(ldgnum,*)POLES,real,dimension(*)DIFL,real,dimension(*)DIFR,real,dimension(*)Z,integerK,realC,realS,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SLASD6computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.Purpose:SLASD6 computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row. This routine is used only for the problem which requires all singular values and optionally singular vector matrices in factored form. B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine, SLASD1, handles the case in which all singular values and singular vectors of the bidiagonal matrix are desired. SLASD6 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The singular values of B can be computed using D1, D2, the first components of all the right singular vectors of the lower block, and the last components of all the right singular vectors of the upper block. These components are stored and updated in VF and VL, respectively, in SLASD6. Hence U and VT are not explicitly referenced. The singular values are stored in D. The algorithm consists of two stages: The first stage consists of deflating the size of the problem when there are multiple singular values or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLASD7. The second stage consists of calculating the updated singular values. This is done by finding the roots of the secular equation via the routine SLASD4 (as called by SLASD8). This routine also updates VF and VL and computes the distances between the updated singular values and the old singular values. SLASD6 is called from SLASDA.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well.NLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.DD is REAL array, dimension (NL+NR+1). On entry D(1:NL,1:NL) contains the singular values of the upper block, and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.VFVF is REAL array, dimension (M) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix.VLVL is REAL array, dimension (M) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix.ALPHAALPHA is REAL Contains the diagonal element associated with the added row.BETABETA is REAL Contains the off-diagonal element associated with the added row.IDXQIDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order.PERMPERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0.GIVPTRGIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.GIVCOLGIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0.LDGCOLLDGCOL is INTEGER leading dimension of GIVCOL, must be at least N.GIVNUMGIVNUM is REAL array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0.LDGNUMLDGNUM is INTEGER The leading dimension of GIVNUM and POLES, must be at least N.POLESPOLES is REAL array, dimension ( LDGNUM, 2 ) On exit, POLES(1,*) is an array containing the new singular values obtained from solving the secular equation, and POLES(2,*) is an array containing the poles in the secular equation. Not referenced if ICOMPQ = 0.DIFLDIFL is REAL array, dimension ( N ) On exit, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.DIFRDIFR is REAL array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. See SLASD8 for details on DIFL and DIFR.ZZ is REAL array, dimension ( M ) The first elements of this array contain the components of the deflation-adjusted updating row vector.KK is INTEGER Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.CC is REAL C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.SS is REAL S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.WORKWORK is REAL array, dimension ( 4 * M )IWORKIWORK is INTEGER array, dimension ( 3 * N )INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasd7(integerICOMPQ,integerNL,integerNR,integerSQRE,integerK,real,dimension(*)D,real,dimension(*)Z,real,dimension(*)ZW,real,dimension(*)VF,real,dimension(*)VFW,real,dimension(*)VL,real,dimension(*)VLW,realALPHA,realBETA,real,dimension(*)DSIGMA,integer,dimension(*)IDX,integer,dimension(*)IDXP,integer,dimension(*)IDXQ,integer,dimension(*)PERM,integerGIVPTR,integer,dimension(ldgcol,*)GIVCOL,integerLDGCOL,real,dimension(ldgnum,*)GIVNUM,integerLDGNUM,realC,realS,integerINFO)SLASD7merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.Purpose:SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. SLASD7 is called from SLASD6.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows: = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form.NLNL is INTEGER The row dimension of the upper block. NL >= 1.NRNR is INTEGER The row dimension of the lower block. NR >= 1.SQRESQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.KK is INTEGER Contains the dimension of the non-deflated matrix, this is the order of the related secular equation. 1 <= K <=N.DD is REAL array, dimension ( N ) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order.ZZ is REAL array, dimension ( M ) On exit Z contains the updating row vector in the secular equation.ZWZW is REAL array, dimension ( M ) Workspace for Z.VFVF is REAL array, dimension ( M ) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix.VFWVFW is REAL array, dimension ( M ) Workspace for VF.VLVL is REAL array, dimension ( M ) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix.VLWVLW is REAL array, dimension ( M ) Workspace for VL.ALPHAALPHA is REAL Contains the diagonal element associated with the added row.BETABETA is REAL Contains the off-diagonal element associated with the added row.DSIGMADSIGMA is REAL array, dimension ( N ) Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation.IDXIDX is INTEGER array, dimension ( N ) This will contain the permutation used to sort the contents of D into ascending order.IDXPIDXP is INTEGER array, dimension ( N ) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values.IDXQIDXQ is INTEGER array, dimension ( N ) This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first half of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values.PERMPERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each singular block. Not referenced if ICOMPQ = 0.GIVPTRGIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.GIVCOLGIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0.LDGCOLLDGCOL is INTEGER The leading dimension of GIVCOL, must be at least N.GIVNUMGIVNUM is REAL array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0.LDGNUMLDGNUM is INTEGER The leading dimension of GIVNUM, must be at least N.CC is REAL C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.SS is REAL S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasd8(integerICOMPQ,integerK,real,dimension(*)D,real,dimension(*)Z,real,dimension(*)VF,real,dimension(*)VL,real,dimension(*)DIFL,real,dimension(lddifr,*)DIFR,integerLDDIFR,real,dimension(*)DSIGMA,real,dimension(*)WORK,integerINFO)SLASD8finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.Purpose:SLASD8 finds the square roots of the roots of the secular equation, as defined by the values in DSIGMA and Z. It makes the appropriate calls to SLASD4, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA). It also updates the arrays VF and VL, the first and last components of all the right singular vectors of the original bidiagonal matrix. SLASD8 is called from SLASD6.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form in the calling routine: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well.KK is INTEGER The number of terms in the rational function to be solved by SLASD4. K >= 1.DD is REAL array, dimension ( K ) On output, D contains the updated singular values.ZZ is REAL array, dimension ( K ) On entry, the first K elements of this array contain the components of the deflation-adjusted updating row vector. On exit, Z is updated.VFVF is REAL array, dimension ( K ) On entry, VF contains information passed through DBEDE8. On exit, VF contains the first K components of the first components of all right singular vectors of the bidiagonal matrix.VLVL is REAL array, dimension ( K ) On entry, VL contains information passed through DBEDE8. On exit, VL contains the first K components of the last components of all right singular vectors of the bidiagonal matrix.DIFLDIFL is REAL array, dimension ( K ) On exit, DIFL(I) = D(I) - DSIGMA(I).DIFRDIFR is REAL array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix.LDDIFRLDDIFR is INTEGER The leading dimension of DIFR, must be at least K.DSIGMADSIGMA is REAL array, dimension ( K ) On entry, the first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. On exit, the elements of DSIGMA may be very slightly altered in value.WORKWORK is REAL array, dimension (3*K)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2017Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasda(integerICOMPQ,integerSMLSIZ,integerN,integerSQRE,real,dimension(*)D,real,dimension(*)E,real,dimension(ldu,*)U,integerLDU,real,dimension(ldu,*)VT,integer,dimension(*)K,real,dimension(ldu,*)DIFL,real,dimension(ldu,*)DIFR,real,dimension(ldu,*)Z,real,dimension(ldu,*)POLES,integer,dimension(*)GIVPTR,integer,dimension(ldgcol,*)GIVCOL,integerLDGCOL,integer,dimension(ldgcol,*)PERM,real,dimension(ldu,*)GIVNUM,real,dimension(*)C,real,dimension(*)S,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SLASDAcomputes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.Purpose:Using a divide and conquer approach, SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes the singular values in the SVD B = U * S * VT. The orthogonal matrices U and VT are optionally computed in compact form. A related subroutine, SLASD0, computes the singular values and the singular vectors in explicit form.ParametersICOMPQICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form.SMLSIZSMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree.NN is INTEGER The row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D.SQRESQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N + 1.DD is REAL array, dimension ( N ) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values.EE is REAL array, dimension ( M-1 ) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed.UU is REAL array, dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left singular vector matrices of all subproblems at the bottom level.LDULDU is INTEGER, LDU = > N. The leading dimension of arrays U, VT, DIFL, DIFR, POLES, GIVNUM, and Z.VTVT is REAL array, dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right singular vector matrices of all subproblems at the bottom level.KK is INTEGER array, dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th secular equation on the computation tree.DIFLDIFL is REAL array, dimension ( LDU, NLVL ), where NLVL = floor(log_2 (N/SMLSIZ))).DIFRDIFR is REAL array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) record distances between singular values on the I-th level and singular values on the (I -1)-th level, and DIFR(1:N, 2 * I ) contains the normalizing factors for the right singular vector matrix. See SLASD8 for details.ZZ is REAL array, dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. The first K elements of Z(1, I) contain the components of the deflation-adjusted updating row vector for subproblems on the I-th level.POLESPOLES is REAL array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and POLES(1, 2*I) contain the new and old singular values involved in the secular equations on the I-th level.GIVPTRGIVPTR is INTEGER array, dimension ( N ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records the number of Givens rotations performed on the I-th problem on the computation tree.GIVCOLGIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations of Givens rotations performed on the I-th level on the computation tree.LDGCOLLDGCOL is INTEGER, LDGCOL = > N. The leading dimension of arrays GIVCOL and PERM.PERMPERM is INTEGER array, dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records permutations done on the I-th level of the computation tree.GIVNUMGIVNUM is REAL array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- values of Givens rotations performed on the I-th level on the computation tree.CC is REAL array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the I-th subproblem is not square, on exit, C( I ) contains the C-value of a Givens rotation related to the right null space of the I-th subproblem.SS is REAL array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the I-th subproblem is not square, on exit, S( I ) contains the S-value of a Givens rotation related to the right null space of the I-th subproblem.WORKWORK is REAL array, dimension (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).IWORKIWORK is INTEGER array, dimension (7*N).INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not convergeAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasdq(characterUPLO,integerSQRE,integerN,integerNCVT,integerNRU,integerNCC,real,dimension(*)D,real,dimension(*)E,real,dimension(ldvt,*)VT,integerLDVT,real,dimension(ldu,*)U,integerLDU,real,dimension(ldc,*)C,integerLDC,real,dimension(*)WORK,integerINFO)SLASDQcomputes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.Purpose:SLASDQ computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired. Letting B denote the input bidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose of P). The singular values S are overwritten on D. The input matrix U is changed to U * Q if desired. The input matrix VT is changed to P**T * VT if desired. The input matrix C is changed to Q**T * C if desired. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm.ParametersUPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the input bidiagonal matrix is upper or lower bidiagonal, and whether it is square are not. UPLO = 'U' or 'u' B is upper bidiagonal. UPLO = 'L' or 'l' B is lower bidiagonal.SQRESQRE is INTEGER = 0: then the input matrix is N-by-N. = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and (N+1)-by-N if UPLU = 'L'. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.NN is INTEGER On entry, N specifies the number of rows and columns in the matrix. N must be at least 0.NCVTNCVT is INTEGER On entry, NCVT specifies the number of columns of the matrix VT. NCVT must be at least 0.NRUNRU is INTEGER On entry, NRU specifies the number of rows of the matrix U. NRU must be at least 0.NCCNCC is INTEGER On entry, NCC specifies the number of columns of the matrix C. NCC must be at least 0.DD is REAL array, dimension (N) On entry, D contains the diagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in ascending order.EE is REAL array. dimension is (N-1) if SQRE = 0 and N if SQRE = 1. On entry, the entries of E contain the offdiagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, E will contain 0. If the algorithm does not converge, D and E will contain the diagonal and superdiagonal entries of a bidiagonal matrix orthogonally equivalent to the one given as input.VTVT is REAL array, dimension (LDVT, NCVT) On entry, contains a matrix which on exit has been premultiplied by P**T, dimension N-by-NCVT if SQRE = 0 and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).LDVTLDVT is INTEGER On entry, LDVT specifies the leading dimension of VT as declared in the calling (sub) program. LDVT must be at least 1. If NCVT is nonzero LDVT must also be at least N.UU is REAL array, dimension (LDU, N) On entry, contains a matrix which on exit has been postmultiplied by Q, dimension NRU-by-N if SQRE = 0 and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).LDULDU is INTEGER On entry, LDU specifies the leading dimension of U as declared in the calling (sub) program. LDU must be at least max( 1, NRU ) .CC is REAL array, dimension (LDC, NCC) On entry, contains an N-by-NCC matrix which on exit has been premultiplied by Q**T dimension N-by-NCC if SQRE = 0 and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).LDCLDC is INTEGER On entry, LDC specifies the leading dimension of C as declared in the calling (sub) program. LDC must be at least 1. If NCC is nonzero, LDC must also be at least N.WORKWORK is REAL array, dimension (4*N) Workspace. Only referenced if one of NCVT, NRU, or NCC is nonzero, and if N is at least 2.INFOINFO is INTEGER On exit, a value of 0 indicates a successful exit. If INFO < 0, argument number -INFO is illegal. If INFO > 0, the algorithm did not converge, and INFO specifies how many superdiagonals did not converge.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateJune 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslasdt(integerN,integerLVL,integerND,integer,dimension(*)INODE,integer,dimension(*)NDIML,integer,dimension(*)NDIMR,integerMSUB)SLASDTcreates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.Purpose:SLASDT creates a tree of subproblems for bidiagonal divide and conquer.ParametersNN is INTEGER On entry, the number of diagonal elements of the bidiagonal matrix.LVLLVL is INTEGER On exit, the number of levels on the computation tree.NDND is INTEGER On exit, the number of nodes on the tree.INODEINODE is INTEGER array, dimension ( N ) On exit, centers of subproblems.NDIMLNDIML is INTEGER array, dimension ( N ) On exit, row dimensions of left children.NDIMRNDIMR is INTEGER array, dimension ( N ) On exit, row dimensions of right children.MSUBMSUB is INTEGER On entry, the maximum row dimension each subproblem at the bottom of the tree can be of.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Contributors:Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USAsubroutineslaset(characterUPLO,integerM,integerN,realALPHA,realBETA,real,dimension(lda,*)A,integerLDA)SLASETinitializes the off-diagonal elements and the diagonal elements of a matrix to given values.Purpose:SLASET initializes an m-by-n matrix A to BETA on the diagonal and ALPHA on the offdiagonals.ParametersUPLOUPLO is CHARACTER*1 Specifies the part of the matrix A to be set. = 'U': Upper triangular part is set; the strictly lower triangular part of A is not changed. = 'L': Lower triangular part is set; the strictly upper triangular part of A is not changed. Otherwise: All of the matrix A is set.MM is INTEGER The number of rows of the matrix A. M >= 0.NN is INTEGER The number of columns of the matrix A. N >= 0.ALPHAALPHA is REAL The constant to which the offdiagonal elements are to be set.BETABETA is REAL The constant to which the diagonal elements are to be set.AA is REAL array, dimension (LDA,N) On exit, the leading m-by-n submatrix of A is set as follows: if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n, if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n, otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j, and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslasr(characterSIDE,characterPIVOT,characterDIRECT,integerM,integerN,real,dimension(*)C,real,dimension(*)S,real,dimension(lda,*)A,integerLDA)SLASRapplies a sequence of plane rotations to a general rectangular matrix.Purpose:SLASR applies a sequence of plane rotations to a real matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form A := P*A and when SIDE = 'R', the transformation takes the form A := A*P**T where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and P**T is the transpose of P. When DIRECT = 'F' (Forward sequence), then P = P(z-1) * ... * P(2) * P(1) and when DIRECT = 'B' (Backward sequence), then P = P(1) * P(2) * ... * P(z-1) where P(k) is a plane rotation matrix defined by the 2-by-2 rotation R(k) = ( c(k) s(k) ) = ( -s(k) c(k) ). When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears as a rank-2 modification to the identity matrix in rows and columns k and k+1. When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears in rows and columns 1 and k+1. Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.ParametersSIDESIDE is CHARACTER*1 Specifies whether the plane rotation matrix P is applied to A on the left or the right. = 'L': Left, compute A := P*A = 'R': Right, compute A:= A*P**TPIVOTPIVOT is CHARACTER*1 Specifies the plane for which P(k) is a plane rotation matrix. = 'V': Variable pivot, the plane (k,k+1) = 'T': Top pivot, the plane (1,k+1) = 'B': Bottom pivot, the plane (k,z)DIRECTDIRECT is CHARACTER*1 Specifies whether P is a forward or backward sequence of plane rotations. = 'F': Forward, P = P(z-1)*...*P(2)*P(1) = 'B': Backward, P = P(1)*P(2)*...*P(z-1)MM is INTEGER The number of rows of the matrix A. If m <= 1, an immediate return is effected.NN is INTEGER The number of columns of the matrix A. If n <= 1, an immediate return is effected.CC is REAL array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations.SS is REAL array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The sines s(k) of the plane rotations. The 2-by-2 plane rotation part of the matrix P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( -s(k) c(k) ).AA is REAL array, dimension (LDA,N) The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE = 'R' or by A*P**T if SIDE = 'L'.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslassq(integerN,real,dimension(*)X,integerINCX,realSCALE,realSUMSQ)SLASSQupdates a sum of squares represented in scaled form.Purpose:SLASSQ returns the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is assumed to be non-negative and scl returns the value scl = max( scale, abs( x( i ) ) ). scale and sumsq must be supplied in SCALE and SUMSQ and scl and smsq are overwritten on SCALE and SUMSQ respectively. The routine makes only one pass through the vector x.ParametersNN is INTEGER The number of elements to be used from the vector X.XX is REAL array, dimension (1+(N-1)*INCX) The vector for which a scaled sum of squares is computed. x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.INCXINCX is INTEGER The increment between successive values of the vector X. INCX > 0.SCALESCALE is REAL On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares.SUMSQSUMSQ is REAL On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutineslasv2(realF,realG,realH,realSSMIN,realSSMAX,realSNR,realCSR,realSNL,realCSL)SLASV2computes the singular value decomposition of a 2-by-2 triangular matrix.Purpose:SLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix [ F G ] [ 0 H ]. On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and right singular vectors for abs(SSMAX), giving the decomposition [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ] [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].ParametersFF is REAL The (1,1) element of the 2-by-2 matrix.GG is REAL The (1,2) element of the 2-by-2 matrix.HH is REAL The (2,2) element of the 2-by-2 matrix.SSMINSSMIN is REAL abs(SSMIN) is the smaller singular value.SSMAXSSMAX is REAL abs(SSMAX) is the larger singular value.SNLSNL is REALCSLCSL is REAL The vector (CSL, SNL) is a unit left singular vector for the singular value abs(SSMAX).SNRSNR is REALCSRCSR is REAL The vector (CSR, SNR) is a unit right singular vector for the singular value abs(SSMAX).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Any input parameter may be aliased with any output parameter. Barring over/underflow and assuming a guard digit in subtraction, all output quantities are correct to within a few units in the last place (ulps). In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.subroutinexerbla(character*(*)SRNAME,integerINFO)XERBLAPurpose:XERBLA is an error handler for the LAPACK routines. It is called by an LAPACK routine if an input parameter has an invalid value. A message is printed and execution stops. Installers may consider modifying the STOP statement in order to call system-specific exception-handling facilities.ParametersSRNAMESRNAME is CHARACTER*(*) The name of the routine which called XERBLA.INFOINFO is INTEGER The position of the invalid parameter in the parameter list of the calling routine.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinexerbla_array(character(1),dimension(srname_len)SRNAME_ARRAY,integerSRNAME_LEN,integerINFO)XERBLA_ARRAYPurpose:XERBLA_ARRAY assists other languages in calling XERBLA, the LAPACK and BLAS error handler. Rather than taking a Fortran string argument as the function's name, XERBLA_ARRAY takes an array of single characters along with the array's length. XERBLA_ARRAY then copies up to 32 characters of that array into a Fortran string and passes that to XERBLA. If called with a non-positive SRNAME_LEN, XERBLA_ARRAY will call XERBLA with a string of all blank characters. Say some macro or other device makes XERBLA_ARRAY available to C99 by a name lapack_xerbla and with a common Fortran calling convention. Then a C99 program could invoke XERBLA via: { int flen = strlen(__func__); lapack_xerbla(__func__, &flen, &info); } Providing XERBLA_ARRAY is not necessary for intercepting LAPACK errors. XERBLA_ARRAY calls XERBLA.ParametersSRNAME_ARRAYSRNAME_ARRAY is CHARACTER(1) array, dimension (SRNAME_LEN) The name of the routine which called XERBLA_ARRAY.SRNAME_LENSRNAME_LEN is INTEGER The length of the name in SRNAME_ARRAY.INFOINFO is INTEGER The position of the invalid parameter in the parameter list of the calling routine.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016

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