Provided by: liblapack-doc_3.7.1-4ubuntu1_all

**NAME**

doubleSYcomputational

**SYNOPSIS**

Functionssubroutinedla_syamv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)DLA_SYAMVcomputes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. double precision functiondla_syrcond(UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)DLA_SYRCONDestimates the Skeel condition number for a symmetric indefinite matrix. subroutinedla_syrfsx_extended(PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)DLA_SYRFSX_EXTENDEDimproves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. double precision functiondla_syrpvgrw(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)DLA_SYRPVGRWcomputes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. subroutinedlasyf(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)DLASYFcomputes a partial factorization of a real symmetric matrix using the Bunch- Kaufman diagonal pivoting method. subroutinedlasyf_aa(UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK)DLASYF_AAsubroutinedlasyf_rk(UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)DLASYF_RKcomputes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. subroutinedlasyf_rook(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)DLASYF_ROOK*> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. subroutinedsycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)DSYCONsubroutinedsycon_3(UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, IWORK, INFO)DSYCON_3subroutinedsycon_rook(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)DSYCON_ROOKsubroutinedsyconv(UPLO, WAY, N, A, LDA, IPIV, E, INFO)DSYCONVsubroutinedsyconvf(UPLO, WAY, N, A, LDA, E, IPIV, INFO)DSYCONVFsubroutinedsyconvf_rook(UPLO, WAY, N, A, LDA, E, IPIV, INFO)DSYCONVF_ROOKsubroutinedsyequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)DSYEQUBsubroutinedsygs2(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)DSYGS2reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm). subroutinedsygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)DSYGSTsubroutinedsyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)DSYRFSsubroutinedsyrfsx(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)DSYRFSXsubroutinedsytd2(UPLO, N, A, LDA, D, E, TAU, INFO)DSYTD2reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). subroutinedsytf2(UPLO, N, A, LDA, IPIV, INFO)DSYTF2computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). subroutinedsytf2_rk(UPLO, N, A, LDA, E, IPIV, INFO)DSYTF2_RKcomputes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). subroutinedsytf2_rook(UPLO, N, A, LDA, IPIV, INFO)DSYTF2_ROOKcomputes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). subroutinedsytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)DSYTRDsubroutinedsytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)DSYTRD_2STAGEsubroutinedsytrd_sy2sb(UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)DSYTRD_SY2SBsubroutinedsytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)DSYTRFsubroutinedsytrf_aa(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)DSYTRF_AAsubroutinedsytrf_rk(UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)DSYTRF_RKcomputes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). subroutinedsytrf_rook(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)DSYTRF_ROOKsubroutinedsytri(UPLO, N, A, LDA, IPIV, WORK, INFO)DSYTRIsubroutinedsytri2(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)DSYTRI2subroutinedsytri2x(UPLO, N, A, LDA, IPIV, WORK, NB, INFO)DSYTRI2Xsubroutinedsytri_3(UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)DSYTRI_3subroutinedsytri_3x(UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO)DSYTRI_3Xsubroutinedsytri_rook(UPLO, N, A, LDA, IPIV, WORK, INFO)DSYTRI_ROOKsubroutinedsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)DSYTRSsubroutinedsytrs2(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)DSYTRS2subroutinedsytrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)DSYTRS_3subroutinedsytrs_aa(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)DSYTRS_AAsubroutinedsytrs_rook(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)DSYTRS_ROOKsubroutinedtgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)DTGSYLsubroutinedtrsyl(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)DTRSYL

**Detailed** **Description**

This is the group of double computational functions for SY matrices

**Function** **Documentation**

subroutinedla_syamv(integerUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionBETA,doubleprecision,dimension(*)Y,integerINCY)DLA_SYAMVcomputes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.Purpose:DLA_SYAMV performs the matrix-vector operation y := alpha*abs(A)*abs(x) + beta*abs(y), where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix. This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, "symbolically" zero components are not perturbed. A zero entry is considered "symbolic" if all multiplications involved in computing that entry have at least one zero multiplicand.Parameters:UPLOUPLO is INTEGER On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = BLAS_UPPER Only the upper triangular part of A is to be referenced. UPLO = BLAS_LOWER Only the lower triangular part of A is to be referenced. Unchanged on exit.NN is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.ALPHAALPHA is DOUBLE PRECISION . On entry, ALPHA specifies the scalar alpha. Unchanged on exit.AA is DOUBLE PRECISION array, dimension ( LDA, n ). Before entry, the leading m by n part of the array A must contain the matrix of coefficients. Unchanged on exit.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Unchanged on exit.XX is DOUBLE PRECISION array, dimension ( 1 + ( n - 1 )*abs( INCX ) ) Before entry, the incremented array X must contain the vector x. Unchanged on exit.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.BETABETA is DOUBLE PRECISION . On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit.YY is DOUBLE PRECISION array, dimension ( 1 + ( n - 1 )*abs( INCY ) ) Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017FurtherDetails:Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. -- Modified for the absolute-value product, April 2006 Jason Riedy, UC Berkeleydoubleprecisionfunctiondla_syrcond(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldaf,*)AF,integerLDAF,integer,dimension(*)IPIV,integerCMODE,doubleprecision,dimension(*)C,integerINFO,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK)DLA_SYRCONDestimates the Skeel condition number for a symmetric indefinite matrix.Purpose:DLA_SYRCOND estimates the Skeel condition number of op(A) * op2(C) where op2 is determined by CMODE as follows CMODE = 1 op2(C) = C CMODE = 0 op2(C) = I CMODE = -1 op2(C) = inv(C) The Skeel condition number cond(A) = norminf( |inv(A)||A| ) is computed by computing scaling factors R such that diag(R)*A*op2(C) is row equilibrated and computing the standard infinity-norm condition number.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF is DOUBLE PRECISION array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.CMODECMODE is INTEGER Determines op2(C) in the formula op(A) * op2(C) as follows: CMODE = 1 op2(C) = C CMODE = 0 op2(C) = I CMODE = -1 op2(C) = inv(C)CC is DOUBLE PRECISION array, dimension (N) The vector C in the formula op(A) * op2(C).INFOINFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid.WORKWORK is DOUBLE PRECISION array, dimension (3*N). Workspace.IWORKIWORK is INTEGER array, dimension (N). Workspace.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedla_syrfsx_extended(integerPREC_TYPE,characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldaf,*)AF,integerLDAF,integer,dimension(*)IPIV,logicalCOLEQU,doubleprecision,dimension(*)C,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(ldy,*)Y,integerLDY,doubleprecision,dimension(*)BERR_OUT,integerN_NORMS,doubleprecision,dimension(nrhs,*)ERR_BNDS_NORM,doubleprecision,dimension(nrhs,*)ERR_BNDS_COMP,doubleprecision,dimension(*)RES,doubleprecision,dimension(*)AYB,doubleprecision,dimension(*)DY,doubleprecision,dimension(*)Y_TAIL,doubleprecisionRCOND,integerITHRESH,doubleprecisionRTHRESH,doubleprecisionDZ_UB,logicalIGNORE_CWISE,integerINFO)DLA_SYRFSX_EXTENDEDimproves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.Purpose:DLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. This subroutine is called by DSYRFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. Note that this subroutine is only resonsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP.Parameters:PREC_TYPEPREC_TYPE is INTEGER Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P = 'S': Single = 'D': Double = 'I': Indigenous = 'X', 'E': ExtraUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The number of linear equations, i.e., 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 matrix B.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF is DOUBLE PRECISION array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.COLEQUCOLEQU is LOGICAL If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.CC is DOUBLE PRECISION array, dimension (N) The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) The right-hand-side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).YY is DOUBLE PRECISION array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by DSYTRS. On exit, the improved solution matrix Y.LDYLDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).BERR_OUTBERR_OUT is DOUBLE PRECISION array, dimension (NRHS) On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by DLA_LIN_BERR.N_NORMSN_NORMS is INTEGER Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.ERR_BNDS_NORMERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i)) The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.ERR_BNDS_COMPERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i)) The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.RESRES is DOUBLE PRECISION array, dimension (N) Workspace to hold the intermediate residual.AYBAYB is DOUBLE PRECISION array, dimension (N) Workspace. This can be the same workspace passed for Y_TAIL.DYDY is DOUBLE PRECISION array, dimension (N) Workspace to hold the intermediate solution.Y_TAILY_TAIL is DOUBLE PRECISION array, dimension (N) Workspace to hold the trailing bits of the intermediate solution.RCONDRCOND is DOUBLE PRECISION Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned.ITHRESHITHRESH is INTEGER The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.RTHRESHRTHRESH is DOUBLE PRECISION Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.DZ_UBDZ_UB is DOUBLE PRECISION Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we definte as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.IGNORE_CWISEIGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE..INFOINFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to DLA_SYRFSX_EXTENDED had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017doubleprecisionfunctiondla_syrpvgrw(character*1UPLO,integerN,integerINFO,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldaf,*)AF,integerLDAF,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK)DLA_SYRPVGRWcomputes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.Purpose:DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.INFOINFO is INTEGER The value of INFO returned from DSYTRF, .i.e., the pivot in column INFO is exactly 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF is DOUBLE PRECISION array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.WORKWORK is DOUBLE PRECISION array, dimension (2*N)Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedlasyf(characterUPLO,integerN,integerNB,integerKB,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldw,*)W,integerLDW,integerINFO)DLASYFcomputes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.Purpose:DLASYF computes a partial factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12**T U22**T ) A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').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.NBNB is INTEGER The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks.KBKB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.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, A contains details of the partial factorization.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': Only the last KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': Only the first KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.WW is DOUBLE PRECISION array, dimension (LDW,NB)LDWLDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).INFOINFO is INTEGER = 0: successful exit > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:November 2013Contributors:November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeleysubroutinedlasyf_aa(characterUPLO,integerJ1,integerM,integerNB,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldh,*)H,integerLDH,doubleprecision,dimension(*)WORK)DLASYF_AAPurpose:DLATRF_AA factorizes a panel of a real symmetric matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L. In order to factorize the panel, the Aasen's algorithm requires the last row, or column, of the previous panel. The first row, or column, of A is set to be the first row, or column, of an identity matrix, which is used to factorize the first panel. The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.J1J1 is INTEGER The location of the first row, or column, of the panel within the submatrix of A, passed to this routine, e.g., when called by DSYTRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2.MM is INTEGER The dimension of the submatrix. M >= 0.NBNB is INTEGER The dimension of the panel to be facotorized.AA is DOUBLE PRECISION array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels. On entry, A contains the last row, or column, of the previous panel, and the trailing submatrix of A to be factorized, except for the first panel, only the panel is passed. On exit, the leading panel is factorized.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).IPIVIPIV is INTEGER array, dimension (M) Details of the row and column interchanges, the row and column k were interchanged with the row and column IPIV(k).HH is DOUBLE PRECISION workspace, dimension (LDH,NB).LDHLDH is INTEGER The leading dimension of the workspace H. LDH >= max(1,M).WORKWORK is DOUBLE PRECISION workspace, dimension (M).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017subroutinedlasyf_rk(characterUPLO,integerN,integerNB,integerKB,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,doubleprecision,dimension(ldw,*)W,integerLDW,integerINFO)DLASYF_RKcomputes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.Purpose:DLASYF_RK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12**T U22**T ) A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L', ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').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.NBNB is INTEGER The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks.KBKB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.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, contains: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output.WW is DOUBLE PRECISION array, dimension (LDW,NB)LDWLDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedlasyf_rook(characterUPLO,integerN,integerNB,integerKB,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldw,*)W,integerLDW,integerINFO)DLASYF_ROOK*> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.Purpose:DLASYF_ROOK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12**T U22**T ) A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. DLASYF_ROOK is an auxiliary routine called by DSYTRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').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.NBNB is INTEGER The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks.KBKB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.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, A contains details of the partial factorization.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': Only the last KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': Only the first KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.WW is DOUBLE PRECISION array, dimension (LDW,NB)LDWLDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).INFOINFO is INTEGER = 0: successful exit > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:November 2013Contributors:November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsycon(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecisionANORM,doubleprecisionRCOND,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DSYCONPurpose:DSYCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.ANORMANORM is DOUBLE PRECISION The 1-norm of the original matrix A.RCONDRCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORKWORK is DOUBLE PRECISION array, dimension (2*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsycon_3(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,doubleprecisionANORM,doubleprecisionRCOND,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DSYCON_3Purpose:DSYCON_3 estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK: A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). This routine uses BLAS3 solver DSYTRS_3.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.ANORMANORM is DOUBLE PRECISION The 1-norm of the original matrix A.RCONDRCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORKWORK is DOUBLE PRECISION array, dimension (2*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsycon_rook(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecisionANORM,doubleprecisionRCOND,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DSYCON_ROOKPurpose:DSYCON_ROOK estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_ROOK.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_ROOK.ANORMANORM is DOUBLE PRECISION The 1-norm of the original matrix A.RCONDRCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORKWORK is DOUBLE PRECISION array, dimension (2*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:April 2012Contributors:April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsyconv(characterUPLO,characterWAY,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)E,integerINFO)DSYCONVPurpose:DSYCONV convert A given by TRF into L and D and vice-versa. Get Non-diag elements of D (returned in workspace) and apply or reverse permutation done in TRF.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.WAYWAY is CHARACTER*1 = 'C': Convert = 'R': RevertNN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.EE is DOUBLE PRECISION array, dimension (N) E stores the supdiagonal/subdiagonal of the symmetric 1-by-1 or 2-by-2 block diagonal matrix D in LDLT.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsyconvf(characterUPLO,characterWAY,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,integerINFO)DSYCONVFPurpose:If parameter WAY = 'C': DSYCONVF converts the factorization output format used in DSYTRF provided on entry in parameter A into the factorization output format used in DSYTRF_RK (or DSYTRF_BK) that is stored on exit in parameters A and E. It also coverts in place details of the intechanges stored in IPIV from the format used in DSYTRF into the format used in DSYTRF_RK (or DSYTRF_BK). If parameter WAY = 'R': DSYCONVF performs the conversion in reverse direction, i.e. converts the factorization output format used in DSYTRF_RK (or DSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in DSYTRF that is stored on exit in parameter A. It also coverts in place details of the intechanges stored in IPIV from the format used in DSYTRF_RK (or DSYTRF_BK) into the format used in DSYTRF.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix A. = 'U': Upper triangular = 'L': Lower triangularWAYWAY is CHARACTER*1 = 'C': Convert = 'R': RevertNN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) 1) If WAY ='C': On entry, contains factorization details in format used in DSYTRF: a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A. On exit, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A. 2) If WAY = 'R': On entry, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A. On exit, contains factorization details in format used in DSYTRF: a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) 1) If WAY ='C': On entry, just a workspace. On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. 2) If WAY = 'R': On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. On exit, is not changedIPIVIPIV is INTEGER array, dimension (N) 1) If WAY ='C': On entry, details of the interchanges and the block structure of D in the format used in DSYTRF. On exit, details of the interchanges and the block structure of D in the format used in DSYTRF_RK ( or DSYTRF_BK). 1) If WAY ='R': On entry, details of the interchanges and the block structure of D in the format used in DSYTRF_RK ( or DSYTRF_BK). On exit, details of the interchanges and the block structure of D in the format used in DSYTRF.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeleysubroutinedsyconvf_rook(characterUPLO,characterWAY,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,integerINFO)DSYCONVF_ROOKPurpose:If parameter WAY = 'C': DSYCONVF_ROOK converts the factorization output format used in DSYTRF_ROOK provided on entry in parameter A into the factorization output format used in DSYTRF_RK (or DSYTRF_BK) that is stored on exit in parameters A and E. IPIV format for DSYTRF_ROOK and DSYTRF_RK (or DSYTRF_BK) is the same and is not converted. If parameter WAY = 'R': DSYCONVF_ROOK performs the conversion in reverse direction, i.e. converts the factorization output format used in DSYTRF_RK (or DSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in DSYTRF_ROOK that is stored on exit in parameter A. IPIV format for DSYTRF_ROOK and DSYTRF_RK (or DSYTRF_BK) is the same and is not converted.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix A. = 'U': Upper triangular = 'L': Lower triangularWAYWAY is CHARACTER*1 = 'C': Convert = 'R': RevertNN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) 1) If WAY ='C': On entry, contains factorization details in format used in DSYTRF_ROOK: a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A. On exit, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A. 2) If WAY = 'R': On entry, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A. On exit, contains factorization details in format used in DSYTRF_ROOK: a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) 1) If WAY ='C': On entry, just a workspace. On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. 2) If WAY = 'R': On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. On exit, is not changedIPIVIPIV is INTEGER array, dimension (N) On entry, details of the interchanges and the block structure of D as determined: 1) by DSYTRF_ROOK, if WAY ='C'; 2) by DSYTRF_RK (or DSYTRF_BK), if WAY ='R'. The IPIV format is the same for all these routines. On exit, is not changed.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeleysubroutinedsyequb(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)S,doubleprecisionSCOND,doubleprecisionAMAX,doubleprecision,dimension(*)WORK,integerINFO)DSYEQUBPurpose:DSYEQUB computes row and column scalings intended to equilibrate a symmetric matrix A (with respect to the Euclidean norm) and reduce its condition number. The scale factors S are computed by the BIN algorithm (see references) so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The N-by-N symmetric matrix whose scaling factors are to be computed.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).SS is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A.SCONDSCOND is DOUBLE PRECISION If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.AMAXAMAX is DOUBLE PRECISION Largest absolute value of any matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.WORKWORK is DOUBLE PRECISION array, dimension (2*N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016References:Livne, O.E. and Golub, G.H., 'Scaling by Binormalization', Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. DOI 10.1023/B:NUMA.0000016606.32820.69 Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679subroutinedsygs2(integerITYPE,characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,integerINFO)DSYGS2reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).Purpose:DSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T) If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L. B must have been previously factorized as U**T *U or L*L**T by DPOTRF.Parameters:ITYPEITYPE is INTEGER = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); = 2 or 3: compute U*A*U**T or L**T *A*L.UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored, and how B has been factorized. = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrices A and B. N >= 0.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 INFO = 0, the transformed matrix, stored in the same format as A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).BB is DOUBLE PRECISION array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by DPOTRF.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsygst(integerITYPE,characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,integerINFO)DSYGSTPurpose:DSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T) If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L. B must have been previously factorized as U**T*U or L*L**T by DPOTRF.Parameters:ITYPEITYPE is INTEGER = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); = 2 or 3: compute U*A*U**T or L**T*A*L.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored and B is factored as U**T*U; = 'L': Lower triangle of A is stored and B is factored as L*L**T.NN is INTEGER The order of the matrices A and B. N >= 0.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 INFO = 0, the transformed matrix, stored in the same format as A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).BB is DOUBLE PRECISION array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by DPOTRF.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsyrfs(characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldaf,*)AF,integerLDAF,integer,dimension(*)IPIV,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(ldx,*)X,integerLDX,doubleprecision,dimension(*)FERR,doubleprecision,dimension(*)BERR,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DSYRFSPurpose:DSYRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and provides error bounds and backward error estimates for the solution.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN 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 B and X. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF is DOUBLE PRECISION array, dimension (LDAF,N) The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) The right hand side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).XX is DOUBLE PRECISION array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by DSYTRS. On exit, the improved solution matrix X.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).FERRFERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.BERRBERR is DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).WORKWORK is DOUBLE PRECISION array, dimension (3*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueInternalParameters:ITMAX is the maximum number of steps of iterative refinement.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsyrfsx(characterUPLO,characterEQUED,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldaf,*)AF,integerLDAF,integer,dimension(*)IPIV,doubleprecision,dimension(*)S,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(ldx,*)X,integerLDX,doubleprecisionRCOND,doubleprecision,dimension(*)BERR,integerN_ERR_BNDS,doubleprecision,dimension(nrhs,*)ERR_BNDS_NORM,doubleprecision,dimension(nrhs,*)ERR_BNDS_COMP,integerNPARAMS,doubleprecision,dimension(*)PARAMS,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DSYRFSXPurpose:DSYRFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and provides error bounds and backward error estimates for the solution. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In this case, the solution and error bounds returned are for the original unequilibrated system. Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.EQUEDEQUED is CHARACTER*1 Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly. = 'N': No equilibration = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). The right hand side B has been changed accordingly.NN 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 B and X. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF is DOUBLE PRECISION array, dimension (LDAF,N) The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.SS is DOUBLE PRECISION array, dimension (N) The scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. If S is output, each element of S is a power of the radix. If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) The right hand side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).XX is DOUBLE PRECISION array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix X.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).RCONDRCOND is DOUBLE PRECISION Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned.BERRBERR is DOUBLE PRECISION array, dimension (NRHS) Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).N_ERR_BNDSN_ERR_BNDS is INTEGER Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below.ERR_BNDS_NORMERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i)) The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions.ERR_BNDS_COMPERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i)) The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions.NPARAMSNPARAMS is INTEGER Specifies the number of parameters set in PARAMS. If .LE. 0, the PARAMS array is never referenced and default values are used.PARAMSPARAMS is DOUBLE PRECISION array, dimension (NPARAMS) Specifies algorithm parameters. If an entry is .LT. 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not. Default: 1.0D+0 = 0.0 : No refinement is performed, and no error bounds are computed. = 1.0 : Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. (other values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement. Default: 10 Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)WORKWORK is DOUBLE PRECISION array, dimension (4*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: Successful exit. The solution to every right-hand side is guaranteed. < 0: If INFO = -i, the i-th argument had an illegal value > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right- hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:April 2012subroutinedsytd2(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)TAU,integerINFO)DSYTD2reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).Purpose:DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.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.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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 >= max(1,N).DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).EE is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.TAUTAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.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-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).subroutinedsytf2(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,integerINFO)DSYTF2computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).Purpose:DSYTF2 computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**T or A = L*D*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS.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.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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).Contributors:09-29-06 - patch from Bobby Cheng, MathWorks Replace l.204 and l.372 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN by IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA 1-96 - Based on modifications by J. Lewis, Boeing Computer Services Companysubroutinedsytf2_rk(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,integerINFO)DSYTF2_RKcomputes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).Purpose:DSYTF2_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS. For more information see Further Details section.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.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, contains: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred. c) In both cases a) and b), always ABS( IPIV(k) ) <= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred. c) In both cases a) and b), always ABS( IPIV(k) ) >= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output.INFOINFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:TODO: put further detailsContributors:December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USAsubroutinedsytf2_rook(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,integerINFO)DSYTF2_ROOKcomputes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).Purpose:DSYTF2_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method: A = U*D*U**T or A = L*D*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS.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.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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:November 2013FurtherDetails:If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).Contributors:November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USAsubroutinedsytrd(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)TAU,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRDPurpose:DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 >= max(1,N).DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).EE is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.TAUTAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor: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-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).subroutinedsytrd_2stage(characterVECT,characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(*)TAU,doubleprecision,dimension(*)HOUS2,integerLHOUS2,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRD_2STAGEPurpose:DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric tridiagonal form T by a orthogonal similarity transformation: Q1**T Q2**T* A * Q2 * Q1 = T.Parameters:VECTVECT is CHARACTER*1 = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N); = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.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 band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over- written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors. See Further Details.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).DD is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T.EE is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T.TAUTAU is DOUBLE PRECISION array, dimension (N-KD) The scalar factors of the elementary reflectors of the first stage (see Further Details).HOUS2HOUS2 is DOUBLE PRECISION array, dimension LHOUS2, that store the Householder representation of the stage2 band to tridiagonal.LHOUS2LHOUS2 is INTEGER The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension) If LWORK = -1, or LHOUS2=-1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA. LHOUS2 = MAX(1, dimension) where dimension = 4*N if VECT='N' not available now if VECT='H'WORKWORK is DOUBLE PRECISION array, dimension (LWORK)LWORKLWORK is INTEGER The dimension of the array WORK. LWORK = MAX(1, dimension) If LWORK = -1, or LHOUS2=-1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. LWORK = MAX(1, dimension) where dimension = max(stage1,stage2) + (KD+1)*N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017FurtherDetails:Implemented by Azzam Haidar. All details are available on technical report, SC11, SC13 papers. Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196subroutinedsytrd_sy2sb(characterUPLO,integerN,integerKD,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)TAU,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRD_SY2SBPurpose:DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by a orthogonal similarity transformation: Q**T * A * Q = AB.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. The reduced matrix is stored in the array AB.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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 >= max(1,N).ABAB is DOUBLE PRECISION array, dimension (LDAB,N) On exit, 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).LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.TAUTAU is DOUBLE PRECISION array, dimension (N-KD) The scalar factors of the elementary reflectors (see Further Details).WORKWORK is DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, or if LWORK=-1, WORK(1) returns the size of LWORK.LWORKLWORK is INTEGER The dimension of the array WORK which should be calculated by a workspace query. LWORK = MAX(1, LWORK_QUERY) If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting LWORK=-1 will provide the size of WORK.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017FurtherDetails:Implemented by Azzam Haidar. All details are available on technical report, SC11, SC13 papers. Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196 If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd. 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+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = n-kd. 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(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( ab ab/v1 v1 v1 v1 ) ( ab ) ( ab ab/v2 v2 v2 ) ( ab/v1 ab ) ( ab ab/v3 v3 ) ( v1 ab/v2 ab ) ( ab ab/v4 ) ( v1 v2 ab/v3 ab ) ( ab ) ( v1 v2 v3 ab/v4 ab ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)..fisubroutinedsytrf(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRFPurpose:DSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = U*D*U**T or A = L*D*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).subroutinedsytrf_aa(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRF_AAPurpose:DSYTRF_AA computes the factorization of a real symmetric matrix A using the Aasen's algorithm. The form of the factorization is A = U*T*U**T or A = L*T*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a symmetric tridiagonal matrix. This is the blocked version of the algorithm, calling Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.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, the tridiagonal matrix is stored in the diagonals and the subdiagonals of A just below (or above) the diagonals, and L is stored below (or above) the subdiaonals, when UPLO is 'L' (or 'U').LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of WORK. LWORK >= MAX(1,2*N). For optimum performance LWORK >= N*(1+NB), where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsytrf_rk(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRF_RKcomputes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).Purpose:DSYTRF_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.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.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, contains: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred. c) In both cases a) and b), always ABS( IPIV(k) ) <= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred. c) In both cases a) and b), always ABS( IPIV(k) ) >= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output.WORKWORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails:TODO: put correct descriptionContributors:December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsytrf_rook(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRF_ROOKPurpose:DSYTRF_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The form of the factorization is A = U*D*U**T or A = L*D*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:April 2012FurtherDetails:If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).Contributors:April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsytri(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerINFO)DSYTRIPurpose:DSYTRI computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.WORKWORK is DOUBLE PRECISION array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsytri2(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRI2Purpose:DSYTRI2 computes the inverse of a DOUBLE PRECISION symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. DSYTRI2 sets the LEADING DIMENSION of the workspace before calling DSYTRI2X that actually computes the inverse.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the NB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the NB structure of D as determined by DSYTRF.WORKWORK is DOUBLE PRECISION array, dimension (N+NB+1)*(NB+3)LWORKLWORK is INTEGER The dimension of the array WORK. WORK is size >= (N+NB+1)*(NB+3) If LDWORK = -1, then a workspace query is assumed; the routine calculates: - the optimal size of the WORK array, returns this value as the first entry of the WORK array, - and no error message related to LDWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsytri2x(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(n+nb+1,*)WORK,integerNB,integerINFO)DSYTRI2XPurpose:DSYTRI2X computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the NNB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the NNB structure of D as determined by DSYTRF.WORKWORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3)NBNB is INTEGER Block sizeINFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017subroutinedsytri_3(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRI_3Purpose:DSYTRI_3 computes the inverse of a real symmetric indefinite matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK: A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. DSYTRI_3 sets the leading dimension of the workspace before calling DSYTRI_3X that actually computes the inverse. This is the blocked version of the algorithm, calling Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A. On exit, if INFO = 0, the symmetric inverse of the original matrix. If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.WORKWORK is DOUBLE PRECISION array, dimension (N+NB+1)*(NB+3). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of WORK. LWORK >= (N+NB+1)*(NB+3). If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeleysubroutinedsytri_3x(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,doubleprecision,dimension(n+nb+1,*)WORK,integerNB,integerINFO)DSYTRI_3XPurpose:DSYTRI_3X computes the inverse of a real symmetric indefinite matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK: A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A. On exit, if INFO = 0, the symmetric inverse of the original matrix. If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.WORKWORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3).NBNB is INTEGER Block size.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeleysubroutinedsytri_rook(characterUPLO,integerN,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(*)WORK,integerINFO)DSYTRI_ROOKPurpose:DSYTRI_ROOK computes the inverse of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN is INTEGER The order of the matrix A. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_ROOK. On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_ROOK.WORKWORK is DOUBLE PRECISION array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:April 2012Contributors:April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsytrs(characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldb,*)B,integerLDB,integerINFO)DSYTRSPurpose:DSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN 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 matrix B. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsytrs2(characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(*)WORK,integerINFO)DSYTRS2Purpose:DSYTRS2 solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF and converted by DSYCONV.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN 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 matrix B. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. Note that A is input / output. This might be counter-intuitive, and one may think that A is input only. A is input / output. This is because, at the start of the subroutine, we permute A in a "better" form and then we permute A back to its original form at the end.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinedsytrs_3(characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)E,integer,dimension(*)IPIV,doubleprecision,dimension(ldb,*)B,integerLDB,integerINFO)DSYTRS_3Purpose:DSYTRS_3 solves a system of linear equations A * X = B with a real symmetric matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK: A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This algorithm is using Level 3 BLAS.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).NN 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 matrix B. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).EE is DOUBLE PRECISION array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017Contributors:June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedsytrs_aa(characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSYTRS_AAPurpose:DSYTRS_AA solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*T*U**T or A = L*T*L**T computed by DSYTRF_AA.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*T*U**T; = 'L': Lower triangular, form is A = L*T*L**T.NN 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 matrix B. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) Details of factors computed by DSYTRF_AA.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges as computed by DSYTRF_AA.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).WORKWORK is DOUBLE array, dimension (MAX(1,LWORK))LWORKLWORK is INTEGER, LWORK >= MAX(1,3*N-2). aram[out] INFO batim INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017subroutinedsytrs_rook(characterUPLO,integerN,integerNRHS,doubleprecision,dimension(lda,*)A,integerLDA,integer,dimension(*)IPIV,doubleprecision,dimension(ldb,*)B,integerLDB,integerINFO)DSYTRS_ROOKPurpose:DSYTRS_ROOK solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK.Parameters:UPLOUPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.NN 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 matrix B. NRHS >= 0.AA is DOUBLE PRECISION array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_ROOK.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).IPIVIPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_ROOK.BB is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:April 2012Contributors:April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchestersubroutinedtgsyl(characterTRANS,integerIJOB,integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(ldc,*)C,integerLDC,doubleprecision,dimension(ldd,*)D,integerLDD,doubleprecision,dimension(lde,*)E,integerLDE,doubleprecision,dimension(ldf,*)F,integerLDF,doubleprecisionSCALE,doubleprecisionDIF,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerINFO)DTGSYLPurpose:DTGSYL solves the generalized Sylvester equation: A * R - L * B = scale * C (1) D * R - L * E = scale * F where R and L are unknown m-by-n matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n, respectively, with real entries. (A, D) and (B, E) must be in generalized (real) Schur canonical form, i.e. A, B are upper quasi triangular and D, E are upper triangular. The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow. In matrix notation (1) is equivalent to solve Zx = scale b, where Z is defined as Z = [ kron(In, A) -kron(B**T, Im) ] (2) [ kron(In, D) -kron(E**T, Im) ]. Here Ik is the identity matrix of size k and X**T is the transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b, which is equivalent to solve for R and L in A**T * R + D**T * L = scale * C (3) R * B**T + L * E**T = scale * -F This case (TRANS = 'T') is used to compute an one-norm-based estimate of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and (B,E), using DLACON. If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the reciprocal of the smallest singular value of Z. See [1-2] for more information. This is a level 3 BLAS algorithm.Parameters:TRANSTRANS is CHARACTER*1 = 'N', solve the generalized Sylvester equation (1). = 'T', solve the 'transposed' system (3).IJOBIJOB is INTEGER Specifies what kind of functionality to be performed. =0: solve (1) only. =1: The functionality of 0 and 3. =2: The functionality of 0 and 4. =3: Only an estimate of Dif[(A,D), (B,E)] is computed. (look ahead strategy IJOB = 1 is used). =4: Only an estimate of Dif[(A,D), (B,E)] is computed. ( DGECON on sub-systems is used ). Not referenced if TRANS = 'T'.MM is INTEGER The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.NN is INTEGER The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.AA is DOUBLE PRECISION array, dimension (LDA, M) The upper quasi triangular matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1, M).BB is DOUBLE PRECISION array, dimension (LDB, N) The upper quasi triangular matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1, N).CC is DOUBLE PRECISION array, dimension (LDC, N) On entry, C contains the right-hand-side of the first matrix equation in (1) or (3). On exit, if IJOB = 0, 1 or 2, C has been overwritten by the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R, the solution achieved during the computation of the Dif-estimate.LDCLDC is INTEGER The leading dimension of the array C. LDC >= max(1, M).DD is DOUBLE PRECISION array, dimension (LDD, M) The upper triangular matrix D.LDDLDD is INTEGER The leading dimension of the array D. LDD >= max(1, M).EE is DOUBLE PRECISION array, dimension (LDE, N) The upper triangular matrix E.LDELDE is INTEGER The leading dimension of the array E. LDE >= max(1, N).FF is DOUBLE PRECISION array, dimension (LDF, N) On entry, F contains the right-hand-side of the second matrix equation in (1) or (3). On exit, if IJOB = 0, 1 or 2, F has been overwritten by the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L, the solution achieved during the computation of the Dif-estimate.LDFLDF is INTEGER The leading dimension of the array F. LDF >= max(1, M).DIFDIF is DOUBLE PRECISION On exit DIF is the reciprocal of a lower bound of the reciprocal of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). IF IJOB = 0 or TRANS = 'T', DIF is not touched.SCALESCALE is DOUBLE PRECISION On exit SCALE is the scaling factor in (1) or (3). If 0 < SCALE < 1, C and F hold the solutions R and L, resp., to a slightly perturbed system but the input matrices A, B, D and E have not been changed. If SCALE = 0, C and F hold the solutions R and L, respectively, to the homogeneous system with C = F = 0. Normally, SCALE = 1.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. LWORK > = 1. If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.IWORKIWORK is INTEGER array, dimension (M+N+6)INFOINFO is INTEGER =0: successful exit <0: If INFO = -i, the i-th argument had an illegal value. >0: (A, D) and (B, E) have common or close eigenvalues.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.References:[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994 [3] B. Kagstrom and L. 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.subroutinedtrsyl(characterTRANA,characterTRANB,integerISGN,integerM,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(ldc,*)C,integerLDC,doubleprecisionSCALE,integerINFO)DTRSYLPurpose:DTRSYL solves the real Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or op(A)*X - X*op(B) = scale*C, where op(A) = A or A**T, and A and B are both upper quasi- triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. A and B must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign.Parameters:TRANATRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'T': op(A) = A**T (Transpose) = 'C': op(A) = A**H (Conjugate transpose = Transpose)TRANBTRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'T': op(B) = B**T (Transpose) = 'C': op(B) = B**H (Conjugate transpose = Transpose)ISGNISGN is INTEGER Specifies the sign in the equation: = +1: solve op(A)*X + X*op(B) = scale*C = -1: solve op(A)*X - X*op(B) = scale*CMM is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.NN is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.AA is DOUBLE PRECISION array, dimension (LDA,M) The upper quasi-triangular matrix A, in Schur canonical form.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).BB is DOUBLE PRECISION array, dimension (LDB,N) The upper quasi-triangular matrix B, in Schur canonical form.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).CC is DOUBLE PRECISION array, dimension (LDC,N) On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X.LDCLDC is INTEGER The leading dimension of the array C. LDC >= max(1,M)SCALESCALE is DOUBLE PRECISION The scale factor, scale, set <= 1 to avoid overflow in X.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016

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