Provided by: liblapack-doc_3.8.0-2_all

**NAME**

doubleOTHERauxiliary

**SYNOPSIS**

Functionssubroutinedlabrd(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)DLABRDreduces the first nb rows and columns of a general matrix to a bidiagonal form. subroutinedlacn2(N, V, X, ISGN, EST, KASE, ISAVE)DLACN2estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutinedlacon(N, V, X, ISGN, EST, KASE)DLACONestimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutinedladiv(A, B, C, D, P, Q)DLADIVperforms complex division in real arithmetic, avoiding unnecessary overflow. subroutinedladiv1(A, B, C, D, P, Q) double precision functiondladiv2(A, B, C, D, R, T) subroutinedlaein(RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO)DLAEINcomputes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. subroutinedlaexc(WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, INFO)DLAEXCswaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation. subroutinedlag2(A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)DLAG2computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. subroutinedlag2s(M, N, A, LDA, SA, LDSA, INFO)DLAG2Sconverts a double precision matrix to a single precision matrix. subroutinedlags2(UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)DLAGS2computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel. subroutinedlagtm(TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)DLAGTMperforms a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1. subroutinedlagv2(A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR, SNR)DLAGV2computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. subroutinedlahqr(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)DLAHQRcomputes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. subroutinedlahr2(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)DLAHR2reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. subroutinedlaic1(JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)DLAIC1applies one step of incremental condition estimation. subroutinedlaln2(LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO)DLALN2solves a 1-by-1 or 2-by-2 linear system of equations of the specified form. double precision functiondlangt(NORM, N, DL, D, DU)DLANGTreturns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. double precision functiondlanhs(NORM, N, A, LDA, WORK)DLANHSreturns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix. double precision functiondlansb(NORM, UPLO, N, K, AB, LDAB, WORK)DLANSBreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix. double precision functiondlansp(NORM, UPLO, N, AP, WORK)DLANSPreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form. double precision functiondlantb(NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)DLANTBreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. double precision functiondlantp(NORM, UPLO, DIAG, N, AP, WORK)DLANTPreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form. double precision functiondlantr(NORM, UPLO, DIAG, M, N, A, LDA, WORK)DLANTRreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix. subroutinedlanv2(A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN)DLANV2computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form. subroutinedlapll(N, X, INCX, Y, INCY, SSMIN)DLAPLLmeasures the linear dependence of two vectors. subroutinedlapmr(FORWRD, M, N, X, LDX, K)DLAPMRrearranges rows of a matrix as specified by a permutation vector. subroutinedlapmt(FORWRD, M, N, X, LDX, K)DLAPMTperforms a forward or backward permutation of the columns of a matrix. subroutinedlaqp2(M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)DLAQP2computes a QR factorization with column pivoting of the matrix block. subroutinedlaqps(M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)DLAQPScomputes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. subroutinedlaqr0(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)DLAQR0computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutinedlaqr1(N, H, LDH, SR1, SI1, SR2, SI2, V)DLAQR1sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts. subroutinedlaqr2(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)DLAQR2performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutinedlaqr3(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)DLAQR3performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutinedlaqr4(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)DLAQR4computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutinedlaqr5(WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)DLAQR5performs a single small-bulge multi-shift QR sweep. subroutinedlaqsb(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)DLAQSBscales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ. subroutinedlaqsp(UPLO, N, AP, S, SCOND, AMAX, EQUED)DLAQSPscales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ. subroutinedlaqtr(LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, INFO)DLAQTRsolves a real quasi-triangular system of equations, or a complex quasi- triangular system of special form, in real arithmetic. subroutinedlar1v(N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)DLAR1Vcomputes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. subroutinedlar2v(N, X, Y, Z, INCX, C, S, INCC)DLAR2Vapplies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices. subroutinedlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)DLARFapplies an elementary reflector to a general rectangular matrix. subroutinedlarfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)DLARFBapplies a block reflector or its transpose to a general rectangular matrix. subroutinedlarfg(N, ALPHA, X, INCX, TAU)DLARFGgenerates an elementary reflector (Householder matrix). subroutinedlarfgp(N, ALPHA, X, INCX, TAU)DLARFGPgenerates an elementary reflector (Householder matrix) with non-negative beta. subroutinedlarft(DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)DLARFTforms the triangular factor T of a block reflector H = I - vtvH subroutinedlarfx(SIDE, M, N, V, TAU, C, LDC, WORK)DLARFXapplies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10. subroutinedlargv(N, X, INCX, Y, INCY, C, INCC)DLARGVgenerates a vector of plane rotations with real cosines and real sines. subroutinedlarrv(N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)DLARRVcomputes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. subroutinedlartv(N, X, INCX, Y, INCY, C, S, INCC)DLARTVapplies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors. subroutinedlaswp(N, A, LDA, K1, K2, IPIV, INCX)DLASWPperforms a series of row interchanges on a general rectangular matrix. subroutinedlat2s(UPLO, N, A, LDA, SA, LDSA, INFO)DLAT2Sconverts a double-precision triangular matrix to a single-precision triangular matrix. subroutinedlatbs(UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)DLATBSsolves a triangular banded system of equations. subroutinedlatdf(IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)DLATDFuses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutinedlatps(UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)DLATPSsolves a triangular system of equations with the matrix held in packed storage. subroutinedlatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)DLATRDreduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. subroutinedlatrs(UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)DLATRSsolves a triangular system of equations with the scale factor set to prevent overflow. subroutinedlauu2(UPLO, N, A, LDA, INFO)DLAUU2computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm). subroutinedlauum(UPLO, N, A, LDA, INFO)DLAUUMcomputes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm). subroutinedrscl(N, SA, SX, INCX)DRSCLmultiplies a vector by the reciprocal of a real scalar. subroutinedtprfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)DTPRFBapplies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks. subroutineslatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)SLATRDreduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

**Detailed** **Description**

This is the group of double other auxiliary routines

**Function** **Documentation**

subroutinedlabrd(integerM,integerN,integerNB,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)TAUQ,doubleprecision,dimension(*)TAUP,doubleprecision,dimension(ldx,*)X,integerLDX,doubleprecision,dimension(ldy,*)Y,integerLDY)DLABRDreduces the first nb rows and columns of a general matrix to a bidiagonal form.Purpose:DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q**T * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form. This is an auxiliary routine called by DGEBRDParameters:MM is INTEGER The number of rows in the matrix A.NN is INTEGER The number of columns in the matrix A.NBNB is INTEGER The number of leading rows and columns of A to be reduced.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).DD is DOUBLE PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).EE is DOUBLE PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.TAUQTAUQ is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.TAUPTAUP is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.XX is DOUBLE PRECISION array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).YY is DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.LDYLDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017FurtherDetails:The matrices Q and P are represented as products of elementary reflectors: Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T where tauq and taup are real scalars, and v and u are real vectors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**T which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y**T - X*U**T. The contents of A on exit are illustrated by the following examples with nb = 2: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).subroutinedlacn2(integerN,doubleprecision,dimension(*)V,doubleprecision,dimension(*)X,integer,dimension(*)ISGN,doubleprecisionEST,integerKASE,integer,dimension(3)ISAVE)DLACN2estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.Purpose:DLACN2 estimates the 1-norm of a square, real matrix A. Reverse communication is used for evaluating matrix-vector products.Parameters:NN is INTEGER The order of the matrix. N >= 1.VV is DOUBLE PRECISION array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned).XX is DOUBLE PRECISION array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**T * X, if KASE=2, and DLACN2 must be re-called with all the other parameters unchanged.ISGNISGN is INTEGER array, dimension (N)ESTEST is DOUBLE PRECISION On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be unchanged from the previous call to DLACN2. On exit, EST is an estimate (a lower bound) for norm(A).KASEKASE is INTEGER On the initial call to DLACN2, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A**T * X. On the final return from DLACN2, KASE will again be 0.ISAVEISAVE is INTEGER array, dimension (3) ISAVE is used to save variables between calls to DLACN2Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:Originally named SONEST, dated March 16, 1988. This is a thread safe version of DLACON, which uses the array ISAVE in place of a SAVE statement, as follows: DLACON DLACN2 JUMP ISAVE(1) J ISAVE(2) ITER ISAVE(3)Contributors:Nick Higham, University of ManchesterReferences:N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.subroutinedlacon(integerN,doubleprecision,dimension(*)V,doubleprecision,dimension(*)X,integer,dimension(*)ISGN,doubleprecisionEST,integerKASE)DLACONestimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.Purpose:DLACON estimates the 1-norm of a square, real matrix A. Reverse communication is used for evaluating matrix-vector products.Parameters:NN is INTEGER The order of the matrix. N >= 1.VV is DOUBLE PRECISION array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned).XX is DOUBLE PRECISION array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**T * X, if KASE=2, and DLACON must be re-called with all the other parameters unchanged.ISGNISGN is INTEGER array, dimension (N)ESTEST is DOUBLE PRECISION On entry with KASE = 1 or 2 and JUMP = 3, EST should be unchanged from the previous call to DLACON. On exit, EST is an estimate (a lower bound) for norm(A).KASEKASE is INTEGER On the initial call to DLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A**T * X. On the final return from DLACON, KASE will again be 0.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:Nick Higham, University of Manchester. Originally named SONEST, dated March 16, 1988.References:N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.subroutinedladiv(doubleprecisionA,doubleprecisionB,doubleprecisionC,doubleprecisionD,doubleprecisionP,doubleprecisionQ)DLADIVperforms complex division in real arithmetic, avoiding unnecessary overflow.Purpose:DLADIV performs complex division in real arithmetic a + i*b p + i*q = --------- c + i*d The algorithm is due to Michael Baudin and Robert L. Smith and can be found in the paper "A Robust Complex Division in Scilab"Parameters:AA is DOUBLE PRECISIONBB is DOUBLE PRECISIONCC is DOUBLE PRECISIONDD is DOUBLE PRECISION The scalars a, b, c, and d in the above expression.PP is DOUBLE PRECISIONQQ is DOUBLE PRECISION The scalars p and q in the above expression.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:January 2013subroutinedlaein(logicalRIGHTV,logicalNOINIT,integerN,doubleprecision,dimension(ldh,*)H,integerLDH,doubleprecisionWR,doubleprecisionWI,doubleprecision,dimension(*)VR,doubleprecision,dimension(*)VI,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(*)WORK,doubleprecisionEPS3,doubleprecisionSMLNUM,doubleprecisionBIGNUM,integerINFO)DLAEINcomputes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.Purpose:DLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg matrix H.Parameters:RIGHTVRIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector.NOINITNOINIT is LOGICAL = .TRUE. : no initial vector supplied in (VR,VI). = .FALSE.: initial vector supplied in (VR,VI).NN is INTEGER The order of the matrix H. N >= 0.HH is DOUBLE PRECISION array, dimension (LDH,N) The upper Hessenberg matrix H.LDHLDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).WRWR is DOUBLE PRECISIONWIWI is DOUBLE PRECISION The real and imaginary parts of the eigenvalue of H whose corresponding right or left eigenvector is to be computed.VRVR is DOUBLE PRECISION array, dimension (N)VIVI is DOUBLE PRECISION array, dimension (N) On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain a real starting vector for inverse iteration using the real eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI must contain the real and imaginary parts of a complex starting vector for inverse iteration using the complex eigenvalue (WR,WI); otherwise VR and VI need not be set. On exit, if WI = 0.0 (real eigenvalue), VR contains the computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), VR and VI contain the real and imaginary parts of the computed complex eigenvector. The eigenvector is normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|. VI is not referenced if WI = 0.0.BB is DOUBLE PRECISION array, dimension (LDB,N)LDBLDB is INTEGER The leading dimension of the array B. LDB >= N+1.WORKWORK is DOUBLE PRECISION array, dimension (N)EPS3EPS3 is DOUBLE PRECISION A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots.SMLNUMSMLNUM is DOUBLE PRECISION A machine-dependent value close to the underflow threshold.BIGNUMBIGNUM is DOUBLE PRECISION A machine-dependent value close to the overflow threshold.INFOINFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; VR is set to the last iterate, and so is VI if WI.ne.0.0.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaexc(logicalWANTQ,integerN,doubleprecision,dimension(ldt,*)T,integerLDT,doubleprecision,dimension(ldq,*)Q,integerLDQ,integerJ1,integerN1,integerN2,doubleprecision,dimension(*)WORK,integerINFO)DLAEXCswaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.Purpose:DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in an upper quasi-triangular matrix T by an orthogonal similarity transformation. T must be in Schur canonical form, that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elemnts equal and its off-diagonal elements of opposite sign.Parameters:WANTQWANTQ is LOGICAL = .TRUE. : accumulate the transformation in the matrix Q; = .FALSE.: do not accumulate the transformation.NN is INTEGER The order of the matrix T. N >= 0.TT is DOUBLE PRECISION array, dimension (LDT,N) On entry, the upper quasi-triangular matrix T, in Schur canonical form. On exit, the updated matrix T, again in Schur canonical form.LDTLDT is INTEGER The leading dimension of the array T. LDT >= max(1,N).QQ is DOUBLE PRECISION array, dimension (LDQ,N) On entry, if WANTQ is .TRUE., the orthogonal matrix Q. On exit, if WANTQ is .TRUE., the updated matrix Q. If WANTQ is .FALSE., Q is not referenced.LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.J1J1 is INTEGER The index of the first row of the first block T11.N1N1 is INTEGER The order of the first block T11. N1 = 0, 1 or 2.N2N2 is INTEGER The order of the second block T22. N2 = 0, 1 or 2.WORKWORK is DOUBLE PRECISION array, dimension (N)INFOINFO is INTEGER = 0: successful exit = 1: the transformed matrix T would be too far from Schur form; the blocks are not swapped and T and Q are unchanged.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlag2(doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecisionSAFMIN,doubleprecisionSCALE1,doubleprecisionSCALE2,doubleprecisionWR1,doubleprecisionWR2,doubleprecisionWI)DLAG2computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.Purpose:DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflow. The scaling factor "s" results in a modified eigenvalue equation s A - w B where s is a non-negative scaling factor chosen so that w, w B, and s A do not overflow and, if possible, do not underflow, either.Parameters:AA is DOUBLE PRECISION array, dimension (LDA, 2) On entry, the 2 x 2 matrix A. It is assumed that its 1-norm is less than 1/SAFMIN. Entries less than sqrt(SAFMIN)*norm(A) are subject to being treated as zero.LDALDA is INTEGER The leading dimension of the array A. LDA >= 2.BB is DOUBLE PRECISION array, dimension (LDB, 2) On entry, the 2 x 2 upper triangular matrix B. It is assumed that the one-norm of B is less than 1/SAFMIN. The diagonals should be at least sqrt(SAFMIN) times the largest element of B (in absolute value); if a diagonal is smaller than that, then +/- sqrt(SAFMIN) will be used instead of that diagonal.LDBLDB is INTEGER The leading dimension of the array B. LDB >= 2.SAFMINSAFMIN is DOUBLE PRECISION The smallest positive number s.t. 1/SAFMIN does not overflow. (This should always be DLAMCH('S') -- it is an argument in order to avoid having to call DLAMCH frequently.)SCALE1SCALE1 is DOUBLE PRECISION A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the first eigenvalue. If the eigenvalues are complex, then the eigenvalues are ( WR1 +/- WI i ) / SCALE1 (which may lie outside the exponent range of the machine), SCALE1=SCALE2, and SCALE1 will always be positive. If the eigenvalues are real, then the first (real) eigenvalue is WR1 / SCALE1 , but this may overflow or underflow, and in fact, SCALE1 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large.SCALE2SCALE2 is DOUBLE PRECISION A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the second eigenvalue. If the eigenvalues are complex, then SCALE2=SCALE1. If the eigenvalues are real, then the second (real) eigenvalue is WR2 / SCALE2 , but this may overflow or underflow, and in fact, SCALE2 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large.WR1WR1 is DOUBLE PRECISION If the eigenvalue is real, then WR1 is SCALE1 times the eigenvalue closest to the (2,2) element of A B**(-1). If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues.WR2WR2 is DOUBLE PRECISION If the eigenvalue is real, then WR2 is SCALE2 times the other eigenvalue. If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues.WIWI is DOUBLE PRECISION If the eigenvalue is real, then WI is zero. If the eigenvalue is complex, then WI is SCALE1 times the imaginary part of the eigenvalues. WI will always be non-negative.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinedlag2s(integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,real,dimension(ldsa,*)SA,integerLDSA,integerINFO)DLAG2Sconverts a double precision matrix to a single precision matrix.Purpose:DLAG2S converts a DOUBLE PRECISION matrix, SA, to a SINGLE PRECISION matrix, A. RMAX is the overflow for the SINGLE PRECISION arithmetic DLAG2S checks that all the entries of A are between -RMAX and RMAX. If not the conversion is aborted and a flag is raised. This is an auxiliary routine so there is no argument checking.Parameters:MM 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.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N coefficient matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).SASA is REAL array, dimension (LDSA,N) On exit, if INFO=0, the M-by-N coefficient matrix SA; if INFO>0, the content of SA is unspecified.LDSALDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M).INFOINFO is INTEGER = 0: successful exit. = 1: an entry of the matrix A is greater than the SINGLE PRECISION overflow threshold, in this case, the content of SA in exit is unspecified.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlags2(logicalUPPER,doubleprecisionA1,doubleprecisionA2,doubleprecisionA3,doubleprecisionB1,doubleprecisionB2,doubleprecisionB3,doubleprecisionCSU,doubleprecisionSNU,doubleprecisionCSV,doubleprecisionSNV,doubleprecisionCSQ,doubleprecisionSNQ)DLAGS2computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.Purpose:DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such that if ( UPPER ) then U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 ) ( 0 A3 ) ( x x ) and V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 ) ( 0 B3 ) ( x x ) or if ( .NOT.UPPER ) then U**T *A*Q = U**T *( A1 0 )*Q = ( x x ) ( A2 A3 ) ( 0 x ) and V**T*B*Q = V**T*( B1 0 )*Q = ( x x ) ( B2 B3 ) ( 0 x ) The rows of the transformed A and B are parallel, where U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) Z**T denotes the transpose of Z.Parameters:UPPERUPPER is LOGICAL = .TRUE.: the input matrices A and B are upper triangular. = .FALSE.: the input matrices A and B are lower triangular.A1A1 is DOUBLE PRECISIONA2A2 is DOUBLE PRECISIONA3A3 is DOUBLE PRECISION On entry, A1, A2 and A3 are elements of the input 2-by-2 upper (lower) triangular matrix A.B1B1 is DOUBLE PRECISIONB2B2 is DOUBLE PRECISIONB3B3 is DOUBLE PRECISION On entry, B1, B2 and B3 are elements of the input 2-by-2 upper (lower) triangular matrix B.CSUCSU is DOUBLE PRECISIONSNUSNU is DOUBLE PRECISION The desired orthogonal matrix U.CSVCSV is DOUBLE PRECISIONSNVSNV is DOUBLE PRECISION The desired orthogonal matrix V.CSQCSQ is DOUBLE PRECISIONSNQSNQ is DOUBLE PRECISION The desired orthogonal matrix Q.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlagtm(characterTRANS,integerN,integerNRHS,doubleprecisionALPHA,doubleprecision,dimension(*)DL,doubleprecision,dimension(*)D,doubleprecision,dimension(*)DU,doubleprecision,dimension(ldx,*)X,integerLDX,doubleprecisionBETA,doubleprecision,dimension(ldb,*)B,integerLDB)DLAGTMperforms a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.Purpose:DLAGTM performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1.Parameters:TRANSTRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': No transpose, B := alpha * A * X + beta * B = 'T': Transpose, B := alpha * A'* X + beta * B = 'C': Conjugate transpose = TransposeNN is INTEGER The order of the matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices X and B.ALPHAALPHA is DOUBLE PRECISION The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise, it is assumed to be 0.DLDL is DOUBLE PRECISION array, dimension (N-1) The (n-1) sub-diagonal elements of T.DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of T.DUDU is DOUBLE PRECISION array, dimension (N-1) The (n-1) super-diagonal elements of T.XX is DOUBLE PRECISION array, dimension (LDX,NRHS) The N by NRHS matrix X.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(N,1).BETABETA is DOUBLE PRECISION The scalar beta. BETA must be 0., 1., or -1.; otherwise, it is assumed to be 1.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N by NRHS matrix B. On exit, B is overwritten by the matrix expression B := alpha * A * X + beta * B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(N,1).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlagv2(doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(2)ALPHAR,doubleprecision,dimension(2)ALPHAI,doubleprecision,dimension(2)BETA,doubleprecisionCSL,doubleprecisionSNL,doubleprecisionCSR,doubleprecisionSNR)DLAGV2computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.Purpose:DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. This routine computes orthogonal (rotation) matrices given by CSL, SNL and CSR, SNR such that 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0 types), then [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ], 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues, then [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ] where b11 >= b22 > 0.Parameters:AA is DOUBLE PRECISION array, dimension (LDA, 2) On entry, the 2 x 2 matrix A. On exit, A is overwritten by the ``A-part'' of the generalized Schur form.LDALDA is INTEGER THe leading dimension of the array A. LDA >= 2.BB is DOUBLE PRECISION array, dimension (LDB, 2) On entry, the upper triangular 2 x 2 matrix B. On exit, B is overwritten by the ``B-part'' of the generalized Schur form.LDBLDB is INTEGER THe leading dimension of the array B. LDB >= 2.ALPHARALPHAR is DOUBLE PRECISION array, dimension (2)ALPHAIALPHAI is DOUBLE PRECISION array, dimension (2)BETABETA is DOUBLE PRECISION array, dimension (2) (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may be zero.CSLCSL is DOUBLE PRECISION The cosine of the left rotation matrix.SNLSNL is DOUBLE PRECISION The sine of the left rotation matrix.CSRCSR is DOUBLE PRECISION The cosine of the right rotation matrix.SNRSNR is DOUBLE PRECISION The sine of the right rotation matrix.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:Mark Fahey, Department of Mathematics, Univ. of Kentucky, USAsubroutinedlahqr(logicalWANTT,logicalWANTZ,integerN,integerILO,integerIHI,doubleprecision,dimension(ldh,*)H,integerLDH,doubleprecision,dimension(*)WR,doubleprecision,dimension(*)WI,integerILOZ,integerIHIZ,doubleprecision,dimension(ldz,*)Z,integerLDZ,integerINFO)DLAHQRcomputes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.Purpose:DLAHQR is an auxiliary routine called by DHSEQR to update the eigenvalues and Schur decomposition already computed by DHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI.Parameters:WANTTWANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.WANTZWANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.NN is INTEGER The order of the matrix H. N >= 0.ILOILO is INTEGERIHIIHI is INTEGER It is assumed that H is already upper quasi-triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). DLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N.HH is DOUBLE PRECISION array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO is zero and if WANTT is .TRUE., H is upper quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in standard form. If INFO is zero and WANTT is .FALSE., the contents of H are unspecified on exit. The output state of H if INFO is nonzero is given below under the description of INFO.LDHLDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).WRWR is DOUBLE PRECISION array, dimension (N)WIWI is DOUBLE PRECISION array, dimension (N) The real and imaginary parts, respectively, of the computed eigenvalues ILO to IHI are stored in the corresponding elements of WR and WI. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i), and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).ILOZILOZ is INTEGERIHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.ZZ is DOUBLE PRECISION array, dimension (LDZ,N) If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by DHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N).INFOINFO is INTEGER = 0: successful exit .GT. 0: If INFO = i, DLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed. If INFO .GT. 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO thorugh INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthognal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.)Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:02-96 Based on modifications by David Day, Sandia National Laboratory, USA 12-04 Further modifications by Ralph Byers, University of Kansas, USA This is a modified version of DLAHQR from LAPACK version 3.0. It is (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997).subroutinedlahr2(integerN,integerK,integerNB,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(nb)TAU,doubleprecision,dimension(ldt,nb)T,integerLDT,doubleprecision,dimension(ldy,nb)Y,integerLDY)DLAHR2reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.Purpose:DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T. This is an auxiliary routine called by DGEHRD.Parameters:NN is INTEGER The order of the matrix A.KK is INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. K < N.NBNB is INTEGER The number of columns to be reduced.AA is DOUBLE PRECISION array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).TAUTAU is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.TT is DOUBLE PRECISION array, dimension (LDT,NB) The upper triangular matrix T.LDTLDT is INTEGER The leading dimension of the array T. LDT >= NB.YY is DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y.LDYLDY is INTEGER The leading dimension of the array Y. LDY >= N.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i). The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**T) * (A - Y*V**T). The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). This subroutine is a slight modification of LAPACK-3.0's DLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3.0's DLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)References:Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.subroutinedlaic1(integerJOB,integerJ,doubleprecision,dimension(j)X,doubleprecisionSEST,doubleprecision,dimension(j)W,doubleprecisionGAMMA,doubleprecisionSESTPR,doubleprecisionS,doubleprecisionC)DLAIC1applies one step of incremental condition estimation.Purpose:DLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that twonorm(L*x) = sest Then DLAIC1 computes sestpr, s, c such that the vector [ s*x ] xhat = [ c ] is an approximate singular vector of [ L 0 ] Lhat = [ w**T gamma ] in the sense that twonorm(Lhat*xhat) = sestpr. Depending on JOB, an estimate for the largest or smallest singular value is computed. Note that [s c]**T and sestpr**2 is an eigenpair of the system diag(sest*sest, 0) + [alpha gamma] * [ alpha ] [ gamma ] where alpha = x**T*w.Parameters:JOBJOB is INTEGER = 1: an estimate for the largest singular value is computed. = 2: an estimate for the smallest singular value is computed.JJ is INTEGER Length of X and WXX is DOUBLE PRECISION array, dimension (J) The j-vector x.SESTSEST is DOUBLE PRECISION Estimated singular value of j by j matrix LWW is DOUBLE PRECISION array, dimension (J) The j-vector w.GAMMAGAMMA is DOUBLE PRECISION The diagonal element gamma.SESTPRSESTPR is DOUBLE PRECISION Estimated singular value of (j+1) by (j+1) matrix Lhat.SS is DOUBLE PRECISION Sine needed in forming xhat.CC is DOUBLE PRECISION Cosine needed in forming xhat.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaln2(logicalLTRANS,integerNA,integerNW,doubleprecisionSMIN,doubleprecisionCA,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecisionD1,doubleprecisionD2,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecisionWR,doubleprecisionWI,doubleprecision,dimension(ldx,*)X,integerLDX,doubleprecisionSCALE,doubleprecisionXNORM,integerINFO)DLALN2solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.Purpose:DLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A**T - w D) X = s B with possible scaling ("s") and perturbation of A. (A**T means A-transpose.) A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or complex value, and X and B are NA x 1 matrices -- real if w is real, complex if w is complex. NA may be 1 or 2. If w is complex, X and B are represented as NA x 2 matrices, the first column of each being the real part and the second being the imaginary part. "s" is a scaling factor (.LE. 1), computed by DLALN2, which is so chosen that X can be computed without overflow. X is further scaled if necessary to assure that norm(ca A - w D)*norm(X) is less than overflow. If both singular values of (ca A - w D) are less than SMIN, SMIN*identity will be used instead of (ca A - w D). If only one singular value is less than SMIN, one element of (ca A - w D) will be perturbed enough to make the smallest singular value roughly SMIN. If both singular values are at least SMIN, (ca A - w D) will not be perturbed. In any case, the perturbation will be at most some small multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values are computed by infinity-norm approximations, and thus will only be correct to a factor of 2 or so. Note: all input quantities are assumed to be smaller than overflow by a reasonable factor. (See BIGNUM.)Parameters:LTRANSLTRANS is LOGICAL =.TRUE.: A-transpose will be used. =.FALSE.: A will be used (not transposed.)NANA is INTEGER The size of the matrix A. It may (only) be 1 or 2.NWNW is INTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2.SMINSMIN is DOUBLE PRECISION The desired lower bound on the singular values of A. This should be a safe distance away from underflow or overflow, say, between (underflow/machine precision) and (machine precision * overflow ). (See BIGNUM and ULP.)CACA is DOUBLE PRECISION The coefficient c, which A is multiplied by.AA is DOUBLE PRECISION array, dimension (LDA,NA) The NA x NA matrix A.LDALDA is INTEGER The leading dimension of A. It must be at least NA.D1D1 is DOUBLE PRECISION The 1,1 element in the diagonal matrix D.D2D2 is DOUBLE PRECISION The 2,2 element in the diagonal matrix D. Not used if NA=1.BB is DOUBLE PRECISION array, dimension (LDB,NW) The NA x NW matrix B (right-hand side). If NW=2 ("w" is complex), column 1 contains the real part of B and column 2 contains the imaginary part.LDBLDB is INTEGER The leading dimension of B. It must be at least NA.WRWR is DOUBLE PRECISION The real part of the scalar "w".WIWI is DOUBLE PRECISION The imaginary part of the scalar "w". Not used if NW=1.XX is DOUBLE PRECISION array, dimension (LDX,NW) The NA x NW matrix X (unknowns), as computed by DLALN2. If NW=2 ("w" is complex), on exit, column 1 will contain the real part of X and column 2 will contain the imaginary part.LDXLDX is INTEGER The leading dimension of X. It must be at least NA.SCALESCALE is DOUBLE PRECISION The scale factor that B must be multiplied by to insure that overflow does not occur when computing X. Thus, (ca A - w D) X will be SCALE*B, not B (ignoring perturbations of A.) It will be at most 1.XNORMXNORM is DOUBLE PRECISION The infinity-norm of X, when X is regarded as an NA x NW real matrix.INFOINFO is INTEGER An error flag. It will be set to zero if no error occurs, a negative number if an argument is in error, or a positive number if ca A - w D had to be perturbed. The possible values are: = 0: No error occurred, and (ca A - w D) did not have to be perturbed. = 1: (ca A - w D) had to be perturbed to make its smallest (or only) singular value greater than SMIN. NOTE: In the interests of speed, this routine does not check the inputs for errors.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlangt(characterNORM,integerN,doubleprecision,dimension(*)DL,doubleprecision,dimension(*)D,doubleprecision,dimension(*)DU)DLANGTreturns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.Purpose:DLANGT 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 tridiagonal matrix A.Returns:DLANGT DLANGT = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANGT as described above.NN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANGT is set to zero.DLDL is DOUBLE PRECISION array, dimension (N-1) The (n-1) sub-diagonal elements of A.DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of A.DUDU is DOUBLE PRECISION array, dimension (N-1) The (n-1) super-diagonal elements of A.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlanhs(characterNORM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)WORK)DLANHSreturns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.Purpose:DLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A.Returns:DLANHS DLANHS = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANHS as described above.NN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANHS is set to zero.AA is DOUBLE PRECISION array, dimension (LDA,N) The n by n upper Hessenberg matrix A; the part of A below the first sub-diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(N,1).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlansb(characterNORM,characterUPLO,integerN,integerK,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)WORK)DLANSBreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.Purpose:DLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals.Returns:DLANSB DLANSB = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANSB as described above.UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the band matrix A is supplied. = 'U': Upper triangular part is supplied = 'L': Lower triangular part is suppliedNN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANSB is set to zero.KK is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangle of the symmetric band matrix A, stored in the first K+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlansp(characterNORM,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)WORK)DLANSPreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.Purpose:DLANSP 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 matrix A, supplied in packed form.Returns:DLANSP DLANSP = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANSP as described above.UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is supplied. = 'U': Upper triangular part of A is supplied = 'L': Lower triangular part of A is suppliedNN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANSP is set to zero.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlantb(characterNORM,characterUPLO,characterDIAG,integerN,integerK,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)WORK)DLANTBreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.Purpose:DLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals.Returns:DLANTB DLANTB = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANTB as described above.UPLOUPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangularDIAGDIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangularNN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANTB is set to zero.KK is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals of the matrix A if UPLO = 'L'. K >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first k+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). Note that when DIAG = 'U', the elements of the array AB corresponding to the diagonal elements of the matrix A are not referenced, but are assumed to be one.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlantp(characterNORM,characterUPLO,characterDIAG,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)WORK)DLANTPreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.Purpose:DLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form.Returns:DLANTP DLANTP = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANTP as described above.UPLOUPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangularDIAGDIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangularNN is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANTP is set to zero.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. Note that when DIAG = 'U', the elements of the array AP corresponding to the diagonal elements of the matrix A are not referenced, but are assumed to be one.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016doubleprecisionfunctiondlantr(characterNORM,characterUPLO,characterDIAG,integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)WORK)DLANTRreturns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.Purpose:DLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A.Returns:DLANTR DLANTR = ( 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.Parameters:NORMNORM is CHARACTER*1 Specifies the value to be returned in DLANTR as described above.UPLOUPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower trapezoidal. = 'U': Upper trapezoidal = 'L': Lower trapezoidal Note that A is triangular instead of trapezoidal if M = N.DIAGDIAG is CHARACTER*1 Specifies whether or not the matrix A has unit diagonal. = 'N': Non-unit diagonal = 'U': Unit diagonalMM is INTEGER The number of rows of the matrix A. M >= 0, and if UPLO = 'U', M <= N. When M = 0, DLANTR is set to zero.NN is INTEGER The number of columns of the matrix A. N >= 0, and if UPLO = 'L', N <= M. When N = 0, DLANTR is set to zero.AA is DOUBLE PRECISION array, dimension (LDA,N) The trapezoidal matrix A (A is triangular if M = N). If UPLO = 'U', the leading m by n upper trapezoidal part of the array A contains the upper trapezoidal matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading m by n lower trapezoidal part of the array A contains the lower trapezoidal matrix, and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be one.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlanv2(doubleprecisionA,doubleprecisionB,doubleprecisionC,doubleprecisionD,doubleprecisionRT1R,doubleprecisionRT1I,doubleprecisionRT2R,doubleprecisionRT2I,doubleprecisionCS,doubleprecisionSN)DLANV2computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.Purpose:DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form: [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] [ C D ] [ SN CS ] [ CC DD ] [-SN CS ] where either 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex conjugate eigenvalues.Parameters:AA is DOUBLE PRECISIONBB is DOUBLE PRECISIONCC is DOUBLE PRECISIONDD is DOUBLE PRECISION On entry, the elements of the input matrix. On exit, they are overwritten by the elements of the standardised Schur form.RT1RRT1R is DOUBLE PRECISIONRT1IRT1I is DOUBLE PRECISIONRT2RRT2R is DOUBLE PRECISIONRT2IRT2I is DOUBLE PRECISION The real and imaginary parts of the eigenvalues. If the eigenvalues are a complex conjugate pair, RT1I > 0.CSCS is DOUBLE PRECISIONSNSN is DOUBLE PRECISION Parameters of the rotation matrix.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:Modified by V. Sima, Research Institute for Informatics, Bucharest, Romania, to reduce the risk of cancellation errors, when computing real eigenvalues, and to ensure, if possible, that abs(RT1R) >= abs(RT2R).subroutinedlapll(integerN,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)Y,integerINCY,doubleprecisionSSMIN)DLAPLLmeasures the linear dependence of two vectors.Purpose:Given two column vectors X and Y, let A = ( X Y ). The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y.Parameters:NN is INTEGER The length of the vectors X and Y.XX is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) On entry, X contains the N-vector X. On exit, X is overwritten.INCXINCX is INTEGER The increment between successive elements of X. INCX > 0.YY is DOUBLE PRECISION array, dimension (1+(N-1)*INCY) On entry, Y contains the N-vector Y. On exit, Y is overwritten.INCYINCY is INTEGER The increment between successive elements of Y. INCY > 0.SSMINSSMIN is DOUBLE PRECISION The smallest singular value of the N-by-2 matrix A = ( X Y ).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlapmr(logicalFORWRD,integerM,integerN,doubleprecision,dimension(ldx,*)X,integerLDX,integer,dimension(*)K)DLAPMRrearranges rows of a matrix as specified by a permutation vector.Purpose:DLAPMR rearranges the rows of the M by N matrix X as specified by the permutation K(1),K(2),...,K(M) of the integers 1,...,M. If FORWRD = .TRUE., forward permutation: X(K(I),*) is moved X(I,*) for I = 1,2,...,M. If FORWRD = .FALSE., backward permutation: X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.Parameters:FORWRDFORWRD is LOGICAL = .TRUE., forward permutation = .FALSE., backward permutationMM is INTEGER The number of rows of the matrix X. M >= 0.NN is INTEGER The number of columns of the matrix X. N >= 0.XX is DOUBLE PRECISION array, dimension (LDX,N) On entry, the M by N matrix X. On exit, X contains the permuted matrix X.LDXLDX is INTEGER The leading dimension of the array X, LDX >= MAX(1,M).KK is INTEGER array, dimension (M) On entry, K contains the permutation vector. K is used as internal workspace, but reset to its original value on output.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlapmt(logicalFORWRD,integerM,integerN,doubleprecision,dimension(ldx,*)X,integerLDX,integer,dimension(*)K)DLAPMTperforms a forward or backward permutation of the columns of a matrix.Purpose:DLAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: X(*,K(J)) is moved X(*,J) for J = 1,2,...,N. If FORWRD = .FALSE., backward permutation: X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.Parameters:FORWRDFORWRD is LOGICAL = .TRUE., forward permutation = .FALSE., backward permutationMM is INTEGER The number of rows of the matrix X. M >= 0.NN is INTEGER The number of columns of the matrix X. N >= 0.XX is DOUBLE PRECISION array, dimension (LDX,N) On entry, the M by N matrix X. On exit, X contains the permuted matrix X.LDXLDX is INTEGER The leading dimension of the array X, LDX >= MAX(1,M).KK is INTEGER array, dimension (N) On entry, K contains the permutation vector. K is used as internal workspace, but reset to its original value on output.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaqp2(integerM,integerN,integerOFFSET,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)JPVT,doubleprecision,dimension(*)TAU,doubleprecision,dimension(*)VN1,doubleprecision,dimension(*)VN2,doubleprecision,dimension(*)WORK)DLAQP2computes a QR factorization with column pivoting of the matrix block.Purpose:DLAQP2 computes a QR factorization with column pivoting of the block A(OFFSET+1:M,1:N). The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.Parameters: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.OFFSETOFFSET is INTEGER The number of rows of the matrix A that must be pivoted but no factorized. OFFSET >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the upper triangle of block A(OFFSET+1:M,1:N) is the triangular factor obtained; the elements in block A(OFFSET+1:M,1:N) below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. Block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).JPVTJPVT is INTEGER array, dimension (N) On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted to the front of A*P (a leading column); if JPVT(i) = 0, the i-th column of A is a free column. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A.TAUTAU is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors.VN1VN1 is DOUBLE PRECISION array, dimension (N) The vector with the partial column norms.VN2VN2 is DOUBLE PRECISION array, dimension (N) The vector with the exact column norms.WORKWORK is DOUBLE PRECISION array, dimension (N)Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.References:LAPACK Working Note 176subroutinedlaqps(integerM,integerN,integerOFFSET,integerNB,integerKB,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)JPVT,doubleprecision,dimension(*)TAU,doubleprecision,dimension(*)VN1,doubleprecision,dimension(*)VN2,doubleprecision,dimension(*)AUXV,doubleprecision,dimension(ldf,*)F,integerLDF)DLAQPScomputes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.Purpose:DLAQPS computes a step of QR factorization with column pivoting of a real M-by-N matrix A by using Blas-3. It tries to factorize NB columns from A starting from the row OFFSET+1, and updates all of the matrix with Blas-3 xGEMM. In some cases, due to catastrophic cancellations, it cannot factorize NB columns. Hence, the actual number of factorized columns is returned in KB. Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.Parameters: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 >= 0OFFSETOFFSET is INTEGER The number of rows of A that have been factorized in previous steps.NBNB is INTEGER The number of columns to factorize.KBKB is INTEGER The number of columns actually factorized.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, block A(OFFSET+1:M,1:KB) is the triangular factor obtained and block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized. The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has been updated.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).JPVTJPVT is INTEGER array, dimension (N) JPVT(I) = K <==> Column K of the full matrix A has been permuted into position I in AP.TAUTAU is DOUBLE PRECISION array, dimension (KB) The scalar factors of the elementary reflectors.VN1VN1 is DOUBLE PRECISION array, dimension (N) The vector with the partial column norms.VN2VN2 is DOUBLE PRECISION array, dimension (N) The vector with the exact column norms.AUXVAUXV is DOUBLE PRECISION array, dimension (NB) Auxiliar vector.FF is DOUBLE PRECISION array, dimension (LDF,NB) Matrix F**T = L*Y**T*A.LDFLDF is INTEGER The leading dimension of the array F. LDF >= max(1,N).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.References:LAPACK Working Note 176subroutinedlaqr0(logicalWANTT,logicalWANTZ,integerN,integerILO,integerIHI,doubleprecision,dimension(ldh,*)H,integerLDH,doubleprecision,dimension(*)WR,doubleprecision,dimension(*)WI,integerILOZ,integerIHIZ,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DLAQR0computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.Purpose:DLAQR0 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.Parameters:WANTTWANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.WANTZWANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.NN is INTEGER The order of the matrix H. N .GE. 0.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 and, if ILO.GT.1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to DGEBAL, and then passed to DGEHRD when the matrix output by DGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.HH is DOUBLE PRECISION array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.LDHLDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N).WRWR is DOUBLE PRECISION array, dimension (IHI)WIWI is DOUBLE PRECISION array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).ILOZILOZ is INTEGERIHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.ZZ is DOUBLE PRECISION array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.)LDZLDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.WORKWORK is DOUBLE PRECISION array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then DLAQR0 does a workspace query. In this case, DLAQR0 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.INFOINFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, DLAQR0 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthogonal matrix. The final value of H is upper Hessenberg and quasi-triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*) (regard- less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed.Contributors:Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USAReferences:K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaqr1(integerN,doubleprecision,dimension(ldh,*)H,integerLDH,doubleprecisionSR1,doubleprecisionSI1,doubleprecisionSR2,doubleprecisionSI2,doubleprecision,dimension(*)V)DLAQR1sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.Purpose:Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a scalar multiple of the first column of the product (*) K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I) scaling to avoid overflows and most underflows. It is assumed that either 1) sr1 = sr2 and si1 = -si2 or 2) si1 = si2 = 0. This is useful for starting double implicit shift bulges in the QR algorithm.Parameters:NN is INTEGER Order of the matrix H. N must be either 2 or 3.HH is DOUBLE PRECISION array, dimension (LDH,N) The 2-by-2 or 3-by-3 matrix H in (*).LDHLDH is INTEGER The leading dimension of H as declared in the calling procedure. LDH.GE.NSR1SR1 is DOUBLE PRECISIONSI1SI1 is DOUBLE PRECISIONSR2SR2 is DOUBLE PRECISIONSI2SI2 is DOUBLE PRECISION The shifts in (*).VV is DOUBLE PRECISION array, dimension (N) A scalar multiple of the first column of the matrix K in (*).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USAsubroutinedlaqr2(logicalWANTT,logicalWANTZ,integerN,integerKTOP,integerKBOT,integerNW,doubleprecision,dimension(ldh,*)H,integerLDH,integerILOZ,integerIHIZ,doubleprecision,dimension(ldz,*)Z,integerLDZ,integerNS,integerND,doubleprecision,dimension(*)SR,doubleprecision,dimension(*)SI,doubleprecision,dimension(ldv,*)V,integerLDV,integerNH,doubleprecision,dimension(ldt,*)T,integerLDT,integerNV,doubleprecision,dimension(ldwv,*)WV,integerLDWV,doubleprecision,dimension(*)WORK,integerLWORK)DLAQR2performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).Purpose:DLAQR2 is identical to DLAQR3 except that it avoids recursion by calling DLAHQR instead of DLAQR4. Aggressive early deflation: This subroutine accepts as input an upper Hessenberg matrix H and performs an orthogonal similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an orthogonal similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.Parameters:WANTTWANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues.WANTZWANTZ is LOGICAL If .TRUE., then the orthogonal matrix Z is updated so so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.NN is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z.KTOPKTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.KBOTKBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.NWNW is INTEGER Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).HH is DOUBLE PRECISION array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by an orthogonal similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.LDHLDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N .LE. LDHILOZILOZ is INTEGERIHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.ZZ is DOUBLE PRECISION array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced.LDZLDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 .LE. LDZ.NSNS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine.NDND is INTEGER The number of converged eigenvalues uncovered by this subroutine.SRSR is DOUBLE PRECISION array, dimension (KBOT)SISI is DOUBLE PRECISION array, dimension (KBOT) On output, the real and imaginary parts of approximate eigenvalues that may be used for shifts are stored in SR(KBOT-ND-NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively. The real and imaginary parts of converged eigenvalues are stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1) through SI(KBOT), respectively.VV is DOUBLE PRECISION array, dimension (LDV,NW) An NW-by-NW work array.LDVLDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW .LE. LDVNHNH is INTEGER The number of columns of T. NH.GE.NW.TT is DOUBLE PRECISION array, dimension (LDT,NW)LDTLDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW .LE. LDTNVNV is INTEGER The number of rows of work array WV available for workspace. NV.GE.NW.WVWV is DOUBLE PRECISION array, dimension (LDWV,NW)LDWVLDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW .LE. LDVWORKWORK is DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT.LWORKLWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. If LWORK = -1, then a workspace query is assumed; DLAQR2 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USAsubroutinedlaqr3(logicalWANTT,logicalWANTZ,integerN,integerKTOP,integerKBOT,integerNW,doubleprecision,dimension(ldh,*)H,integerLDH,integerILOZ,integerIHIZ,doubleprecision,dimension(ldz,*)Z,integerLDZ,integerNS,integerND,doubleprecision,dimension(*)SR,doubleprecision,dimension(*)SI,doubleprecision,dimension(ldv,*)V,integerLDV,integerNH,doubleprecision,dimension(ldt,*)T,integerLDT,integerNV,doubleprecision,dimension(ldwv,*)WV,integerLDWV,doubleprecision,dimension(*)WORK,integerLWORK)DLAQR3performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).Purpose:Aggressive early deflation: DLAQR3 accepts as input an upper Hessenberg matrix H and performs an orthogonal similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an orthogonal similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.Parameters:WANTTWANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues.WANTZWANTZ is LOGICAL If .TRUE., then the orthogonal matrix Z is updated so so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.NN is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z.KTOPKTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.KBOTKBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.NWNW is INTEGER Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).HH is DOUBLE PRECISION array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by an orthogonal similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.LDHLDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N .LE. LDHILOZILOZ is INTEGERIHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.ZZ is DOUBLE PRECISION array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced.LDZLDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 .LE. LDZ.NSNS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine.NDND is INTEGER The number of converged eigenvalues uncovered by this subroutine.SRSR is DOUBLE PRECISION array, dimension (KBOT)SISI is DOUBLE PRECISION array, dimension (KBOT) On output, the real and imaginary parts of approximate eigenvalues that may be used for shifts are stored in SR(KBOT-ND-NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively. The real and imaginary parts of converged eigenvalues are stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1) through SI(KBOT), respectively.VV is DOUBLE PRECISION array, dimension (LDV,NW) An NW-by-NW work array.LDVLDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW .LE. LDVNHNH is INTEGER The number of columns of T. NH.GE.NW.TT is DOUBLE PRECISION array, dimension (LDT,NW)LDTLDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW .LE. LDTNVNV is INTEGER The number of rows of work array WV available for workspace. NV.GE.NW.WVWV is DOUBLE PRECISION array, dimension (LDWV,NW)LDWVLDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW .LE. LDVWORKWORK is DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT.LWORKLWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. If LWORK = -1, then a workspace query is assumed; DLAQR3 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016Contributors:Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USAsubroutinedlaqr4(logicalWANTT,logicalWANTZ,integerN,integerILO,integerIHI,doubleprecision,dimension(ldh,*)H,integerLDH,doubleprecision,dimension(*)WR,doubleprecision,dimension(*)WI,integerILOZ,integerIHIZ,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DLAQR4computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.Purpose:DLAQR4 implements one level of recursion for DLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by DLAQR0 and, for large enough deflation window size, it may be called by DLAQR3. This subroutine is identical to DLAQR0 except that it calls DLAQR2 instead of DLAQR3. DLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.Parameters:WANTTWANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.WANTZWANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.NN is INTEGER The order of the matrix H. N .GE. 0.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 and, if ILO.GT.1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to DGEBAL, and then passed to DGEHRD when the matrix output by DGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.HH is DOUBLE PRECISION array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.LDHLDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N).WRWR is DOUBLE PRECISION array, dimension (IHI)WIWI is DOUBLE PRECISION array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).ILOZILOZ is INTEGERIHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.ZZ is DOUBLE PRECISION array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.)LDZLDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.WORKWORK is DOUBLE PRECISION array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then DLAQR4 does a workspace query. In this case, DLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.INFOINFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, DLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is a orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*) (regard- less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USAReferences:K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.subroutinedlaqr5(logicalWANTT,logicalWANTZ,integerKACC22,integerN,integerKTOP,integerKBOT,integerNSHFTS,doubleprecision,dimension(*)SR,doubleprecision,dimension(*)SI,doubleprecision,dimension(ldh,*)H,integerLDH,integerILOZ,integerIHIZ,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(ldv,*)V,integerLDV,doubleprecision,dimension(ldu,*)U,integerLDU,integerNV,doubleprecision,dimension(ldwv,*)WV,integerLDWV,integerNH,doubleprecision,dimension(ldwh,*)WH,integerLDWH)DLAQR5performs a single small-bulge multi-shift QR sweep.Purpose:DLAQR5, called by DLAQR0, performs a single small-bulge multi-shift QR sweep.Parameters:WANTTWANTT is LOGICAL WANTT = .true. if the quasi-triangular Schur factor is being computed. WANTT is set to .false. otherwise.WANTZWANTZ is LOGICAL WANTZ = .true. if the orthogonal Schur factor is being computed. WANTZ is set to .false. otherwise.KACC22KACC22 is INTEGER with value 0, 1, or 2. Specifies the computation mode of far-from-diagonal orthogonal updates. = 0: DLAQR5 does not accumulate reflections and does not use matrix-matrix multiply to update far-from-diagonal matrix entries. = 1: DLAQR5 accumulates reflections and uses matrix-matrix multiply to update the far-from-diagonal matrix entries. = 2: DLAQR5 accumulates reflections, uses matrix-matrix multiply to update the far-from-diagonal matrix entries, and takes advantage of 2-by-2 block structure during matrix multiplies.NN is INTEGER N is the order of the Hessenberg matrix H upon which this subroutine operates.KTOPKTOP is INTEGERKBOTKBOT is INTEGER These are the first and last rows and columns of an isolated diagonal block upon which the QR sweep is to be applied. It is assumed without a check that either KTOP = 1 or H(KTOP,KTOP-1) = 0 and either KBOT = N or H(KBOT+1,KBOT) = 0.NSHFTSNSHFTS is INTEGER NSHFTS gives the number of simultaneous shifts. NSHFTS must be positive and even.SRSR is DOUBLE PRECISION array, dimension (NSHFTS)SISI is DOUBLE PRECISION array, dimension (NSHFTS) SR contains the real parts and SI contains the imaginary parts of the NSHFTS shifts of origin that define the multi-shift QR sweep. On output SR and SI may be reordered.HH is DOUBLE PRECISION array, dimension (LDH,N) On input H contains a Hessenberg matrix. On output a multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied to the isolated diagonal block in rows and columns KTOP through KBOT.LDHLDH is INTEGER LDH is the leading dimension of H just as declared in the calling procedure. LDH.GE.MAX(1,N).ILOZILOZ is INTEGERIHIZIHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. NZZ is DOUBLE PRECISION array, dimension (LDZ,IHIZ) If WANTZ = .TRUE., then the QR Sweep orthogonal similarity transformation is accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ = .FALSE., then Z is unreferenced.LDZLDZ is INTEGER LDA is the leading dimension of Z just as declared in the calling procedure. LDZ.GE.N.VV is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)LDVLDV is INTEGER LDV is the leading dimension of V as declared in the calling procedure. LDV.GE.3.UU is DOUBLE PRECISION array, dimension (LDU,3*NSHFTS-3)LDULDU is INTEGER LDU is the leading dimension of U just as declared in the in the calling subroutine. LDU.GE.3*NSHFTS-3.NHNH is INTEGER NH is the number of columns in array WH available for workspace. NH.GE.1.WHWH is DOUBLE PRECISION array, dimension (LDWH,NH)LDWHLDWH is INTEGER Leading dimension of WH just as declared in the calling procedure. LDWH.GE.3*NSHFTS-3.NVNV is INTEGER NV is the number of rows in WV agailable for workspace. NV.GE.1.WVWV is DOUBLE PRECISION array, dimension (LDWV,3*NSHFTS-3)LDWVLDWV is INTEGER LDWV is the leading dimension of WV as declared in the in the calling subroutine. LDWV.GE.NV.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016Contributors:Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USAReferences:K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.subroutinedlaqsb(characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)S,doubleprecisionSCOND,doubleprecisionAMAX,characterEQUED)DLAQSBscales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.Purpose:DLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.SS is DOUBLE PRECISION array, dimension (N) The scale factors for A.SCONDSCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).AMAXAMAX is DOUBLE PRECISION Absolute value of largest matrix entry.EQUEDEQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).InternalParameters:THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaqsp(characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)S,doubleprecisionSCOND,doubleprecisionAMAX,characterEQUED)DLAQSPscales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.Purpose:DLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrix A. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the equilibrated matrix: diag(S) * A * diag(S), in the same storage format as A.SS is DOUBLE PRECISION array, dimension (N) The scale factors for A.SCONDSCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).AMAXAMAX is DOUBLE PRECISION Absolute value of largest matrix entry.EQUEDEQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).InternalParameters:THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaqtr(logicalLTRAN,logicalLREAL,integerN,doubleprecision,dimension(ldt,*)T,integerLDT,doubleprecision,dimension(*)B,doubleprecisionW,doubleprecisionSCALE,doubleprecision,dimension(*)X,doubleprecision,dimension(*)WORK,integerINFO)DLAQTRsolves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.Purpose:DLAQTR solves the real quasi-triangular system op(T)*p = scale*c, if LREAL = .TRUE. or the complex quasi-triangular systems op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE. in real arithmetic, where T is upper quasi-triangular. If LREAL = .FALSE., then the first diagonal block of T must be 1 by 1, B is the specially structured matrix B = [ b(1) b(2) ... b(n) ] [ w ] [ w ] [ . ] [ w ] op(A) = A or A**T, A**T denotes the transpose of matrix A. On input, X = [ c ]. On output, X = [ p ]. [ d ] [ q ] This subroutine is designed for the condition number estimation in routine DTRSNA.Parameters:LTRANLTRAN is LOGICAL On entry, LTRAN specifies the option of conjugate transpose: = .FALSE., op(T+i*B) = T+i*B, = .TRUE., op(T+i*B) = (T+i*B)**T.LREALLREAL is LOGICAL On entry, LREAL specifies the input matrix structure: = .FALSE., the input is complex = .TRUE., the input is realNN is INTEGER On entry, N specifies the order of T+i*B. N >= 0.TT is DOUBLE PRECISION array, dimension (LDT,N) On entry, T contains a matrix in Schur canonical form. If LREAL = .FALSE., then the first diagonal block of T mu be 1 by 1.LDTLDT is INTEGER The leading dimension of the matrix T. LDT >= max(1,N).BB is DOUBLE PRECISION array, dimension (N) On entry, B contains the elements to form the matrix B as described above. If LREAL = .TRUE., B is not referenced.WW is DOUBLE PRECISION On entry, W is the diagonal element of the matrix B. If LREAL = .TRUE., W is not referenced.SCALESCALE is DOUBLE PRECISION On exit, SCALE is the scale factor.XX is DOUBLE PRECISION array, dimension (2*N) On entry, X contains the right hand side of the system. On exit, X is overwritten by the solution.WORKWORK is DOUBLE PRECISION array, dimension (N)INFOINFO is INTEGER On exit, INFO is set to 0: successful exit. 1: the some diagonal 1 by 1 block has been perturbed by a small number SMIN to keep nonsingularity. 2: the some diagonal 2 by 2 block has been perturbed by a small number in DLALN2 to keep nonsingularity. NOTE: In the interests of speed, this routine does not check the inputs for errors.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlar1v(integerN,integerB1,integerBN,doubleprecisionLAMBDA,doubleprecision,dimension(*)D,doubleprecision,dimension(*)L,doubleprecision,dimension(*)LD,doubleprecision,dimension(*)LLD,doubleprecisionPIVMIN,doubleprecisionGAPTOL,doubleprecision,dimension(*)Z,logicalWANTNC,integerNEGCNT,doubleprecisionZTZ,doubleprecisionMINGMA,integerR,integer,dimension(*)ISUPPZ,doubleprecisionNRMINV,doubleprecisionRESID,doubleprecisionRQCORR,doubleprecision,dimension(*)WORK)DLAR1Vcomputes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.Purpose:DLAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D L**T - sigma I. When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation : (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T, (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T, (c) Computation of the diagonal elements of the inverse of L D L**T - sigma I by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude. (d) Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the the stationary and the bottom part of the progressive transform.Parameters:NN is INTEGER The order of the matrix L D L**T.B1B1 is INTEGER First index of the submatrix of L D L**T.BNBN is INTEGER Last index of the submatrix of L D L**T.LAMBDALAMBDA is DOUBLE PRECISION The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T.LL is DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.DD is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the diagonal matrix D.LDLD is DOUBLE PRECISION array, dimension (N-1) The n-1 elements L(i)*D(i).LLDLLD is DOUBLE PRECISION array, dimension (N-1) The n-1 elements L(i)*L(i)*D(i).PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence.GAPTOLGAPTOL is DOUBLE PRECISION Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.ZZ is DOUBLE PRECISION array, dimension (N) On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1.WANTNCWANTNC is LOGICAL Specifies whether NEGCNT has to be computed.NEGCNTNEGCNT is INTEGER If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.ZTZZTZ is DOUBLE PRECISION The square of the 2-norm of Z.MINGMAMINGMA is DOUBLE PRECISION The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I.RR is INTEGER The twist index for the twisted factorization used to compute Z. On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L**T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector.ISUPPZISUPPZ is INTEGER array, dimension (2) The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).NRMINVNRMINV is DOUBLE PRECISION NRMINV = 1/SQRT( ZTZ )RESIDRESID is DOUBLE PRECISION The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )RQCORRRQCORR is DOUBLE PRECISION The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMPWORKWORK is DOUBLE PRECISION array, dimension (4*N)Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 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, USAsubroutinedlar2v(integerN,doubleprecision,dimension(*)X,doubleprecision,dimension(*)Y,doubleprecision,dimension(*)Z,integerINCX,doubleprecision,dimension(*)C,doubleprecision,dimension(*)S,integerINCC)DLAR2Vapplies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.Purpose:DLAR2V applies a vector of real plane rotations from both sides to a sequence of 2-by-2 real symmetric matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) ) ( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) )Parameters:NN is INTEGER The number of plane rotations to be applied.XX is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) The vector x.YY is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) The vector y.ZZ is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) The vector z.INCXINCX is INTEGER The increment between elements of X, Y and Z. INCX > 0.CC is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The cosines of the plane rotations.SS is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The sines of the plane rotations.INCCINCC is INTEGER The increment between elements of C and S. INCC > 0.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlarf(characterSIDE,integerM,integerN,doubleprecision,dimension(*)V,integerINCV,doubleprecisionTAU,doubleprecision,dimension(ldc,*)C,integerLDC,doubleprecision,dimension(*)WORK)DLARFapplies an elementary reflector to a general rectangular matrix.Purpose:DLARF applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form H = I - tau * v * v**T where tau is a real scalar and v is a real vector. If tau = 0, then H is taken to be the unit matrix.Parameters:SIDESIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * HMM is INTEGER The number of rows of the matrix C.NN is INTEGER The number of columns of the matrix C.VV is DOUBLE PRECISION array, dimension (1 + (M-1)*abs(INCV)) if SIDE = 'L' or (1 + (N-1)*abs(INCV)) if SIDE = 'R' The vector v in the representation of H. V is not used if TAU = 0.INCVINCV is INTEGER The increment between elements of v. INCV <> 0.TAUTAU is DOUBLE PRECISION The value tau in the representation of H.CC is DOUBLE PRECISION array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'.LDCLDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).WORKWORK is DOUBLE PRECISION array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R'Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlarfb(characterSIDE,characterTRANS,characterDIRECT,characterSTOREV,integerM,integerN,integerK,doubleprecision,dimension(ldv,*)V,integerLDV,doubleprecision,dimension(ldt,*)T,integerLDT,doubleprecision,dimension(ldc,*)C,integerLDC,doubleprecision,dimension(ldwork,*)WORK,integerLDWORK)DLARFBapplies a block reflector or its transpose to a general rectangular matrix.Purpose:DLARFB applies a real block reflector H or its transpose H**T to a real m by n matrix C, from either the left or the right.Parameters:SIDESIDE is CHARACTER*1 = 'L': apply H or H**T from the Left = 'R': apply H or H**T from the RightTRANSTRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'T': apply H**T (Transpose)DIRECTDIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward)STOREVSTOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columnwise = 'R': RowwiseMM is INTEGER The number of rows of the matrix C.NN is INTEGER The number of columns of the matrix C.KK is INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector).VV is DOUBLE PRECISION array, dimension (LDV,K) if STOREV = 'C' (LDV,M) if STOREV = 'R' and SIDE = 'L' (LDV,N) if STOREV = 'R' and SIDE = 'R' The matrix V. See Further Details.LDVLDV is INTEGER The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K.TT is DOUBLE PRECISION array, dimension (LDT,K) The triangular k by k matrix T in the representation of the block reflector.LDTLDT is INTEGER The leading dimension of the array T. LDT >= K.CC is DOUBLE PRECISION array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.LDCLDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).WORKWORK is DOUBLE PRECISION array, dimension (LDWORK,K)LDWORKLDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= max(1,N); if SIDE = 'R', LDWORK >= max(1,M).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2013FurtherDetails:The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ( v1 v2 v3 ) ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) ( 1 )subroutinedlarfg(integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionTAU)DLARFGgenerates an elementary reflector (Householder matrix).Purpose:DLARFG generates a real elementary reflector H of order n, such that H * ( alpha ) = ( beta ), H**T * H = I. ( x ) ( 0 ) where alpha and beta are scalars, and x is an (n-1)-element real vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**T ) , ( v ) where tau is a real scalar and v is a real (n-1)-element vector. If the elements of x are all zero, then tau = 0 and H is taken to be the unit matrix. Otherwise 1 <= tau <= 2.Parameters:NN is INTEGER The order of the elementary reflector.ALPHAALPHA is DOUBLE PRECISION On entry, the value alpha. On exit, it is overwritten with the value beta.XX is DOUBLE PRECISION array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v.INCXINCX is INTEGER The increment between elements of X. INCX > 0.TAUTAU is DOUBLE PRECISION The value tau.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:November 2017subroutinedlarfgp(integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionTAU)DLARFGPgenerates an elementary reflector (Householder matrix) with non-negative beta.Purpose:DLARFGP generates a real elementary reflector H of order n, such that H * ( alpha ) = ( beta ), H**T * H = I. ( x ) ( 0 ) where alpha and beta are scalars, beta is non-negative, and x is an (n-1)-element real vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**T ) , ( v ) where tau is a real scalar and v is a real (n-1)-element vector. If the elements of x are all zero, then tau = 0 and H is taken to be the unit matrix.Parameters:NN is INTEGER The order of the elementary reflector.ALPHAALPHA is DOUBLE PRECISION On entry, the value alpha. On exit, it is overwritten with the value beta.XX is DOUBLE PRECISION array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v.INCXINCX is INTEGER The increment between elements of X. INCX > 0.TAUTAU is DOUBLE PRECISION The value tau.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:November 2017subroutinedlarft(characterDIRECT,characterSTOREV,integerN,integerK,doubleprecision,dimension(ldv,*)V,integerLDV,doubleprecision,dimension(*)TAU,doubleprecision,dimension(ldt,*)T,integerLDT)DLARFTforms the triangular factor T of a block reflector H = I - vtvHPurpose:DLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and H = I - V * T * V**T If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and H = I - V**T * T * VParameters:DIRECTDIRECT is CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward)STOREVSTOREV is CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): = 'C': columnwise = 'R': rowwiseNN is INTEGER The order of the block reflector H. N >= 0.KK is INTEGER The order of the triangular factor T (= the number of elementary reflectors). K >= 1.VV is DOUBLE PRECISION array, dimension (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details.LDVLDV is INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.TAUTAU is DOUBLE PRECISION array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i).TT is DOUBLE PRECISION array, dimension (LDT,K) The k by k triangular factor T of the block reflector. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. The rest of the array is not used.LDTLDT is INTEGER The leading dimension of the array T. LDT >= K.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ( v1 v2 v3 ) ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) ( 1 )subroutinedlarfx(characterSIDE,integerM,integerN,doubleprecision,dimension(*)V,doubleprecisionTAU,doubleprecision,dimension(ldc,*)C,integerLDC,doubleprecision,dimension(*)WORK)DLARFXapplies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.Purpose:DLARFX applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form H = I - tau * v * v**T where tau is a real scalar and v is a real vector. If tau = 0, then H is taken to be the unit matrix This version uses inline code if H has order < 11.Parameters:SIDESIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * HMM is INTEGER The number of rows of the matrix C.NN is INTEGER The number of columns of the matrix C.VV is DOUBLE PRECISION array, dimension (M) if SIDE = 'L' or (N) if SIDE = 'R' The vector v in the representation of H.TAUTAU is DOUBLE PRECISION The value tau in the representation of H.CC is DOUBLE PRECISION array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'.LDCLDC is INTEGER The leading dimension of the array C. LDA >= (1,M).WORKWORK is DOUBLE PRECISION array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R' WORK is not referenced if H has order < 11.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlargv(integerN,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)Y,integerINCY,doubleprecision,dimension(*)C,integerINCC)DLARGVgenerates a vector of plane rotations with real cosines and real sines.Purpose:DLARGV generates a vector of real plane rotations, determined by elements of the real vectors x and y. For i = 1,2,...,n ( c(i) s(i) ) ( x(i) ) = ( a(i) ) ( -s(i) c(i) ) ( y(i) ) = ( 0 )Parameters:NN is INTEGER The number of plane rotations to be generated.XX is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) On entry, the vector x. On exit, x(i) is overwritten by a(i), for i = 1,...,n.INCXINCX is INTEGER The increment between elements of X. INCX > 0.YY is DOUBLE PRECISION array, dimension (1+(N-1)*INCY) On entry, the vector y. On exit, the sines of the plane rotations.INCYINCY is INTEGER The increment between elements of Y. INCY > 0.CC is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The cosines of the plane rotations.INCCINCC is INTEGER The increment between elements of C. INCC > 0.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlarrv(integerN,doubleprecisionVL,doubleprecisionVU,doubleprecision,dimension(*)D,doubleprecision,dimension(*)L,doubleprecisionPIVMIN,integer,dimension(*)ISPLIT,integerM,integerDOL,integerDOU,doubleprecisionMINRGP,doubleprecisionRTOL1,doubleprecisionRTOL2,doubleprecision,dimension(*)W,doubleprecision,dimension(*)WERR,doubleprecision,dimension(*)WGAP,integer,dimension(*)IBLOCK,integer,dimension(*)INDEXW,doubleprecision,dimension(*)GERS,doubleprecision,dimension(ldz,*)Z,integerLDZ,integer,dimension(*)ISUPPZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DLARRVcomputes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.Purpose:DLARRV computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by DLARRE.Parameters:NN is INTEGER The order of the matrix. N >= 0.VLVL is DOUBLE PRECISION Lower bound of the interval that contains the desired eigenvalues. VL < VU. Needed to compute gaps on the left or right end of the extremal eigenvalues in the desired RANGE.VUVU is DOUBLE PRECISION Upper bound of the interval that contains the desired eigenvalues. VL < VU. Note: VU is currently not used by this implementation of DLARRV, VU is passed to DLARRV because it could be used compute gaps on the right end of the extremal eigenvalues. However, with not much initial accuracy in LAMBDA and VU, the formula can lead to an overestimation of the right gap and thus to inadequately early RQI 'convergence'. This is currently prevented this by forcing a small right gap. And so it turns out that VU is currently not used by this implementation of DLARRV.DD is DOUBLE PRECISION array, dimension (N) On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten.LL is DOUBLE PRECISION array, dimension (N) On entry, the (N-1) subdiagonal elements of the unit bidiagonal matrix L are in elements 1 to N-1 of L (if the matrix is not split.) At the end of each block is stored the corresponding shift as given by DLARRE. On exit, L is overwritten.PIVMINPIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence.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.MM is INTEGER The total number of input eigenvalues. 0 <= M <= N.DOLDOL is INTEGERDOUDOU is INTEGER If the user wants to compute only selected eigenvectors from all the eigenvalues supplied, he can specify an index range DOL:DOU. Or else the setting DOL=1, DOU=M should be applied. Note that DOL and DOU refer to the order in which the eigenvalues are stored in W. If the user wants to compute only selected eigenpairs, then the columns DOL-1 to DOU+1 of the eigenvector space Z contain the computed eigenvectors. All other columns of Z are set to zero.MINRGPMINRGP is DOUBLE PRECISIONRTOL1RTOL1 is DOUBLE PRECISIONRTOL2RTOL2 is DOUBLE PRECISION Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )WW is DOUBLE PRECISION array, dimension (N) The first M elements of W contain the APPROXIMATE eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block ( The output array W from DLARRE is expected here ). Furthermore, they are with respect to the shift of the corresponding root representation for their block. On exit, W holds the eigenvalues of the UNshifted matrix.WERRWERR is DOUBLE PRECISION array, dimension (N) The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in WWGAPWGAP is DOUBLE PRECISION array, dimension (N) The separation from the right neighbor eigenvalue in W.IBLOCKIBLOCK 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 the second block.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)). The Gerschgorin intervals should be computed from the original UNshifted matrix.ZZ is DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) If INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the input eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). Note: the user must ensure that at least max(1,M) columns are supplied in the array Z.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).ISUPPZISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The I-th eigenvector is nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).WORKWORK is DOUBLE PRECISION array, dimension (12*N)IWORKIWORK is INTEGER array, dimension (7*N)INFOINFO is INTEGER = 0: successful exit > 0: A problem occurred in DLARRV. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =-1: Problem in DLARRB when refining a child's eigenvalues. =-2: Problem in DLARRF when computing the RRR of a child. When a child is inside a tight cluster, it can be difficult to find an RRR. A partial remedy from the user's point of view is to make the parameter MINRGP smaller and recompile. However, as the orthogonality of the computed vectors is proportional to 1/MINRGP, the user should be aware that he might be trading in precision when he decreases MINRGP. =-3: Problem in DLARRB when refining a single eigenvalue after the Rayleigh correction was rejected. = 5: The Rayleigh Quotient Iteration failed to converge to full accuracy in MAXITR steps.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 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, USAsubroutinedlartv(integerN,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)Y,integerINCY,doubleprecision,dimension(*)C,doubleprecision,dimension(*)S,integerINCC)DLARTVapplies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors.Purpose:DLARTV applies a vector of real plane rotations to elements of the real vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) ( y(i) ) ( -s(i) c(i) ) ( y(i) )Parameters:NN is INTEGER The number of plane rotations to be applied.XX is DOUBLE PRECISION array, dimension (1+(N-1)*INCX) The vector x.INCXINCX is INTEGER The increment between elements of X. INCX > 0.YY is DOUBLE PRECISION array, dimension (1+(N-1)*INCY) The vector y.INCYINCY is INTEGER The increment between elements of Y. INCY > 0.CC is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The cosines of the plane rotations.SS is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The sines of the plane rotations.INCCINCC is INTEGER The increment between elements of C and S. INCC > 0.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlaswp(integerN,doubleprecision,dimension(lda,*)A,integerLDA,integerK1,integerK2,integer,dimension(*)IPIV,integerINCX)DLASWPperforms a series of row interchanges on a general rectangular matrix.Purpose:DLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A.Parameters:NN is INTEGER The number of columns of the matrix A.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the matrix of column dimension N to which the row interchanges will be applied. On exit, the permuted matrix.LDALDA is INTEGER The leading dimension of the array A.K1K1 is INTEGER The first element of IPIV for which a row interchange will be done.K2K2 is INTEGER (K2-K1+1) is the number of elements of IPIV for which a row interchange will be done.IPIVIPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX)) The vector of pivot indices. Only the elements in positions K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed. IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be interchanged.INCXINCX is INTEGER The increment between successive values of IPIV. If INCX is negative, the pivots are applied in reverse order.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017FurtherDetails:Modified by R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USAsubroutinedlat2s(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,real,dimension(ldsa,*)SA,integerLDSA,integerINFO)DLAT2Sconverts a double-precision triangular matrix to a single-precision triangular matrix.Purpose:DLAT2S converts a DOUBLE PRECISION triangular matrix, SA, to a SINGLE PRECISION triangular matrix, A. RMAX is the overflow for the SINGLE PRECISION arithmetic DLAS2S checks that all the entries of A are between -RMAX and RMAX. If not the conversion is aborted and a flag is raised. This is an auxiliary routine so there is no argument checking.Parameters:UPLOUPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular.NN is INTEGER The number of rows and columns of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N triangular coefficient matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).SASA is REAL array, dimension (LDSA,N) Only the UPLO part of SA is referenced. On exit, if INFO=0, the N-by-N coefficient matrix SA; if INFO>0, the content of the UPLO part of SA is unspecified.LDSALDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M).INFOINFO is INTEGER = 0: successful exit. = 1: an entry of the matrix A is greater than the SINGLE PRECISION overflow threshold, in this case, the content of the UPLO part of SA in exit is unspecified.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlatbs(characterUPLO,characterTRANS,characterDIAG,characterNORMIN,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)X,doubleprecisionSCALE,doubleprecision,dimension(*)CNORM,integerINFO)DLATBSsolves a triangular banded system of equations.Purpose:DLATBS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangularTRANSTRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)DIAGDIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangularNORMINNORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.XX is DOUBLE PRECISION array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.SCALESCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.CNORMCNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:A rough bound on x is computed; if that is less than overflow, DTBSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T*x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).subroutinedlatdf(integerIJOB,integerN,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)RHS,doubleprecisionRDSUM,doubleprecisionRDSCAL,integer,dimension(*)IPIV,integer,dimension(*)JPIV)DLATDFuses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.Purpose:DLATDF uses the LU factorization of the n-by-n matrix Z computed by DGETC2 and computes a contribution to the reciprocal Dif-estimate by solving Z * x = b for x, and choosing the r.h.s. b such that the norm of x is as large as possible. On entry RHS = b holds the contribution from earlier solved sub-systems, and on return RHS = x. The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, where P and Q are permutation matrices. L is lower triangular with unit diagonal elements and U is upper triangular.Parameters:IJOBIJOB is INTEGER IJOB = 2: First compute an approximative null-vector e of Z using DGECON, e is normalized and solve for Zx = +-e - f with the sign giving the greater value of 2-norm(x). About 5 times as expensive as Default. IJOB .ne. 2: Local look ahead strategy where all entries of the r.h.s. b is chosen as either +1 or -1 (Default).NN is INTEGER The number of columns of the matrix Z.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by DGETC2: Z = P * L * U * QLDZLDZ is INTEGER The leading dimension of the array Z. LDA >= max(1, N).RHSRHS is DOUBLE PRECISION array, dimension (N) On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries acoording to the value of IJOB (see above).RDSUMRDSUM is DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Dif-estimate under computation by DTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.RDSCALRDSCAL is DOUBLE PRECISION On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when DTGSY2 is called by DTGSYL.IPIVIPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).JPIVJPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016FurtherDetails:This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.Contributors:Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.References:[1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. [2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report IMINF-95.05, Departement of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.subroutinedlatps(characterUPLO,characterTRANS,characterDIAG,characterNORMIN,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)X,doubleprecisionSCALE,doubleprecision,dimension(*)CNORM,integerINFO)DLATPSsolves a triangular system of equations with the matrix held in packed storage.Purpose:DLATPS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangularTRANSTRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)DIAGDIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangularNORMINNORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.NN is INTEGER The order of the matrix A. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.XX is DOUBLE PRECISION array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.SCALESCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.CNORMCNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:A rough bound on x is computed; if that is less than overflow, DTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T*x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).subroutinedlatrd(characterUPLO,integerN,integerNB,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,doubleprecision,dimension(*)TAU,doubleprecision,dimension(ldw,*)W,integerLDW)DLATRDreduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.Purpose:DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q**T * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = 'U', DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. This is an auxiliary routine called by DSYTRD.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrix A.NBNB is INTEGER The number of rows and columns to be reduced.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.LDALDA is INTEGER The leading dimension of the array A. LDA >= (1,N).EE is DOUBLE PRECISION array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.TAUTAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details.WW is DOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.LDWLDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) . . . H(n-nb+1). Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(nb). Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i). The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W**T - W*V**T. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).subroutinedlatrs(characterUPLO,characterTRANS,characterDIAG,characterNORMIN,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,doubleprecisionSCALE,doubleprecision,dimension(*)CNORM,integerINFO)DLATRSsolves a triangular system of equations with the scale factor set to prevent overflow.Purpose:DLATRS solves one of the triangular systems A *x = s*b or A**T *x = s*b with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangularTRANSTRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)DIAGDIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangularNORMINNORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1.LDALDA is INTEGER The leading dimension of the array A. LDA >= max (1,N).XX is DOUBLE PRECISION array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.SCALESCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.CNORMCNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:A rough bound on x is computed; if that is less than overflow, DTRSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T*x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).subroutinedlauu2(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integerINFO)DLAUU2computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).Purpose:DLAUU2 computes the product U * U**T or L**T * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the unblocked form of the algorithm, calling Level 2 BLAS.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the triangular factor stored in the array A is upper or lower triangular: = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the triangular factor U or L. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the triangular factor U or L. On exit, if UPLO = 'U', the upper triangle of A is overwritten with the upper triangle of the product U * U**T; if UPLO = 'L', the lower triangle of A is overwritten with the lower triangle of the product L**T * L.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlauum(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integerINFO)DLAUUMcomputes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).Purpose:DLAUUM computes the product U * U**T or L**T * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the blocked form of the algorithm, calling Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the triangular factor stored in the array A is upper or lower triangular: = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the triangular factor U or L. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the triangular factor U or L. On exit, if UPLO = 'U', the upper triangle of A is overwritten with the upper triangle of the product U * U**T; if UPLO = 'L', the lower triangle of A is overwritten with the lower triangle of the product L**T * L.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedrscl(integerN,doubleprecisionSA,doubleprecision,dimension(*)SX,integerINCX)DRSCLmultiplies a vector by the reciprocal of a real scalar.Purpose:DRSCL multiplies an n-element real vector x by the real scalar 1/a. This is done without overflow or underflow as long as the final result x/a does not overflow or underflow.Parameters:NN is INTEGER The number of components of the vector x.SASA is DOUBLE PRECISION The scalar a which is used to divide each component of x. SA must be >= 0, or the subroutine will divide by zero.SXSX is DOUBLE PRECISION array, dimension (1+(N-1)*abs(INCX)) The n-element vector x.INCXINCX is INTEGER The increment between successive values of the vector SX. > 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= nAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:November 2017subroutinedtprfb(characterSIDE,characterTRANS,characterDIRECT,characterSTOREV,integerM,integerN,integerK,integerL,doubleprecision,dimension(ldv,*)V,integerLDV,doubleprecision,dimension(ldt,*)T,integerLDT,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(ldwork,*)WORK,integerLDWORK)DTPRFBapplies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.Purpose:DTPRFB applies a real "triangular-pentagonal" block reflector H or its transpose H**T to a real matrix C, which is composed of two blocks A and B, either from the left or right.Parameters:SIDESIDE is CHARACTER*1 = 'L': apply H or H**T from the Left = 'R': apply H or H**T from the RightTRANSTRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'T': apply H**T (Transpose)DIRECTDIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward)STOREVSTOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columns = 'R': RowsMM is INTEGER The number of rows of the matrix B. M >= 0.NN is INTEGER The number of columns of the matrix B. N >= 0.KK is INTEGER The order of the matrix T, i.e. the number of elementary reflectors whose product defines the block reflector. K >= 0.LL is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.VV is DOUBLE PRECISION array, dimension (LDV,K) if STOREV = 'C' (LDV,M) if STOREV = 'R' and SIDE = 'L' (LDV,N) if STOREV = 'R' and SIDE = 'R' The pentagonal matrix V, which contains the elementary reflectors H(1), H(2), ..., H(K). See Further Details.LDVLDV is INTEGER The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K.TT is DOUBLE PRECISION array, dimension (LDT,K) The triangular K-by-K matrix T in the representation of the block reflector.LDTLDT is INTEGER The leading dimension of the array T. LDT >= K.AA is DOUBLE PRECISION array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of H*C or H**T*C or C*H or C*H**T. See Further Details.LDALDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDC >= max(1,K); If SIDE = 'R', LDC >= max(1,M).BB is DOUBLE PRECISION array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of H*C or H**T*C or C*H or C*H**T. See Further Details.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).WORKWORK is DOUBLE PRECISION array, dimension (LDWORK,N) if SIDE = 'L', (LDWORK,K) if SIDE = 'R'.LDWORKLDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= K; if SIDE = 'R', LDWORK >= M.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:The matrix C is a composite matrix formed from blocks A and B. The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K, and if SIDE = 'L', A is of size K-by-N. If SIDE = 'R' and DIRECT = 'F', C = [A B]. If SIDE = 'L' and DIRECT = 'F', C = [A] [B]. If SIDE = 'R' and DIRECT = 'B', C = [B A]. If SIDE = 'L' and DIRECT = 'B', C = [B] [A]. The pentagonal matrix V is composed of a rectangular block V1 and a trapezoidal block V2. The size of the trapezoidal block is determined by the parameter L, where 0<=L<=K. If L=K, the V2 block of V is triangular; if L=0, there is no trapezoidal block, thus V = V1 is rectangular. If DIRECT = 'F' and STOREV = 'C': V = [V1] [V2] - V2 is upper trapezoidal (first L rows of K-by-K upper triangular) If DIRECT = 'F' and STOREV = 'R': V = [V1 V2] - V2 is lower trapezoidal (first L columns of K-by-K lower triangular) If DIRECT = 'B' and STOREV = 'C': V = [V2] [V1] - V2 is lower trapezoidal (last L rows of K-by-K lower triangular) If DIRECT = 'B' and STOREV = 'R': V = [V2 V1] - V2 is upper trapezoidal (last L columns of K-by-K upper triangular) If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K. If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K. If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L. If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.subroutineslatrd(characterUPLO,integerN,integerNB,real,dimension(lda,*)A,integerLDA,real,dimension(*)E,real,dimension(*)TAU,real,dimension(ldw,*)W,integerLDW)SLATRDreduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.Purpose:SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q**T * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = 'U', SLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', SLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. This is an auxiliary routine called by SSYTRD.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrix A.NBNB is INTEGER The number of rows and columns to be reduced.AA is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.LDALDA is INTEGER The leading dimension of the array A. LDA >= (1,N).EE is REAL array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.TAUTAU is REAL array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details.WW is REAL array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.LDWLDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) . . . H(n-nb+1). Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(nb). Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i). The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W**T - W*V**T. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

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