## table of contents

complex16SYcomputational(3) | LAPACK | complex16SYcomputational(3) |

# NAME¶

complex16SYcomputational - complex16# SYNOPSIS¶

## Functions¶

subroutine

**zhetrf_aa_2stage**(UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO)

**ZHETRF_AA_2STAGE**subroutine

**zhetrs_aa_2stage**(UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO)

**ZHETRS_AA_2STAGE**subroutine

**zla_syamv**(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)

**ZLA_SYAMV**computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. double precision function

**zla_syrcond_c**(UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK)

**ZLA_SYRCOND_C**computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices. double precision function

**zla_syrcond_x**(UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)

**ZLA_SYRCOND_X**computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices. subroutine

**zla_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)

**ZLA_SYRFSX_EXTENDED**improves 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 function

**zla_syrpvgrw**(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)

**ZLA_SYRPVGRW**computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. subroutine

**zlasyf**(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)

**ZLASYF**computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method. subroutine

**zlasyf_aa**(UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK)

**ZLASYF_AA**subroutine

**zlasyf_rk**(UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)

**ZLASYF_RK**computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. subroutine

**zlasyf_rook**(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)

**ZLASYF_ROOK**computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. subroutine

**zsycon**(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)

**ZSYCON**subroutine

**zsycon_3**(UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO)

**ZSYCON_3**subroutine

**zsycon_rook**(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)

**ZSYCON_ROOK**subroutine

**zsyconv**(UPLO, WAY, N, A, LDA, IPIV, E, INFO)

**ZSYCONV**subroutine

**zsyconvf**(UPLO, WAY, N, A, LDA, E, IPIV, INFO)

**ZSYCONVF**subroutine

**zsyconvf_rook**(UPLO, WAY, N, A, LDA, E, IPIV, INFO)

**ZSYCONVF_ROOK**subroutine

**zsyequb**(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)

**ZSYEQUB**subroutine

**zsyrfs**(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)

**ZSYRFS**subroutine

**zsyrfsx**(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, RWORK, INFO)

**ZSYRFSX**subroutine

**zsysv_aa_2stage**(UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO)

**ZSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices**subroutine

**zsytf2**(UPLO, N, A, LDA, IPIV, INFO)

**ZSYTF2**computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). subroutine

**zsytf2_rk**(UPLO, N, A, LDA, E, IPIV, INFO)

**ZSYTF2_RK**computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). subroutine

**zsytf2_rook**(UPLO, N, A, LDA, IPIV, INFO)

**ZSYTF2_ROOK**computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). subroutine

**zsytrf**(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)

**ZSYTRF**subroutine

**zsytrf_aa**(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)

**ZSYTRF_AA**subroutine

**zsytrf_aa_2stage**(UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO)

**ZSYTRF_AA_2STAGE**subroutine

**zsytrf_rk**(UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)

**ZSYTRF_RK**computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). subroutine

**zsytrf_rook**(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)

**ZSYTRF_ROOK**subroutine

**zsytri**(UPLO, N, A, LDA, IPIV, WORK, INFO)

**ZSYTRI**subroutine

**zsytri2**(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)

**ZSYTRI2**subroutine

**zsytri2x**(UPLO, N, A, LDA, IPIV, WORK, NB, INFO)

**ZSYTRI2X**subroutine

**zsytri_3**(UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)

**ZSYTRI_3**subroutine

**zsytri_3x**(UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO)

**ZSYTRI_3X**subroutine

**zsytri_rook**(UPLO, N, A, LDA, IPIV, WORK, INFO)

**ZSYTRI_ROOK**subroutine

**zsytrs**(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)

**ZSYTRS**subroutine

**zsytrs2**(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)

**ZSYTRS2**subroutine

**zsytrs_3**(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)

**ZSYTRS_3**subroutine

**zsytrs_aa**(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)

**ZSYTRS_AA**subroutine

**zsytrs_aa_2stage**(UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO)

**ZSYTRS_AA_2STAGE**subroutine

**zsytrs_rook**(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)

**ZSYTRS_ROOK**subroutine

**ztgsyl**(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)

**ZTGSYL**subroutine

**ztrsyl**(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)

**ZTRSYL**

# Detailed Description¶

This is the group of complex16 computational functions for SY matrices# Function Documentation¶

## subroutine zhetrf_aa_2stage (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZHETRF_AA_2STAGE**

**Purpose:**

ZHETRF_AA_2STAGE computes the factorization of a double hermitian matrix A using the Aasen's algorithm. The form of the factorization is A = U**H*T*U or A = L*T*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a hermitian band matrix with the bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is LU factorized with partial pivoting). This is the blocked version of the algorithm, calling Level 3 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the hermitian 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, L is stored below (or above) the subdiaonal blocks, when UPLO is 'L' (or 'U').

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TB*

TB is COMPLEX*16 array, dimension (LTB) On exit, details of the LU factorization of the band matrix.

*LTB*

LTB is INTEGER The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N. If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.

*IPIV*

IPIV 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).

*IPIV2*

IPIV2 is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k).

*WORK*

WORK is COMPLEX*16 workspace of size LWORK

*LWORK*

LWORK is INTEGER The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB. 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.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, band LU factorization failed on i-th column

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zhetrs_aa_2stage (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZHETRS_AA_2STAGE**

**Purpose:**

ZHETRS_AA_2STAGE solves a system of linear equations A*X = B with a hermitian matrix A using the factorization A = U**H*T*U or A = L*T*L**H computed by ZHETRF_AA_2STAGE.

**Parameters**

*UPLO*

UPLO 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**H*T*U; = 'L': Lower triangular, form is A = L*T*L**H.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) Details of factors computed by ZHETRF_AA_2STAGE.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TB*

TB is COMPLEX*16 array, dimension (LTB) Details of factors computed by ZHETRF_AA_2STAGE.

*LTB*

LTB is INTEGER The size of the array TB. LTB >= 4*N.

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges as computed by ZHETRF_AA_2STAGE.

*IPIV2*

IPIV2 is INTEGER array, dimension (N) Details of the interchanges as computed by ZHETRF_AA_2STAGE.

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zla_syamv (integer UPLO, integer N, double precision ALPHA, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)¶

**ZLA_SYAMV**computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.

**Purpose:**

ZLA_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**

*UPLO*

UPLO 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.

*N*

N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.

*ALPHA*

ALPHA is DOUBLE PRECISION . On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

*A*

A is COMPLEX*16 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.

*LDA*

LDA 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.

*X*

X is COMPLEX*16 array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ) Before entry, the incremented array X must contain the vector x. Unchanged on exit.

*INCX*

INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.

*BETA*

BETA 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.

*Y*

Y 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.

*INCY*

INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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 Berkeley

## double precision function zla_syrcond_c (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) C, logical CAPPLY, integer INFO, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK)¶

**ZLA_SYRCOND_C**computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.

**Purpose:**

ZLA_SYRCOND_C Computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.

*LDAF*

LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF.

*C*

C is DOUBLE PRECISION array, dimension (N) The vector C in the formula op(A) * inv(diag(C)).

*CAPPLY*

CAPPLY is LOGICAL If .TRUE. then access the vector C in the formula above.

*INFO*

INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N). Workspace.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (N). Workspace.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zla_syrcond_x (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, complex*16, dimension( * ) X, integer INFO, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK)¶

**ZLA_SYRCOND_X**computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.

**Purpose:**

ZLA_SYRCOND_X Computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX*16 vector.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.

*LDAF*

LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF.

*X*

X is COMPLEX*16 array, dimension (N) The vector X in the formula op(A) * diag(X).

*INFO*

INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N). Workspace.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (N). Workspace.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double precision, dimension( * ) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, complex*16, dimension( * ) RES, double precision, dimension( * ) AYB, complex*16, dimension( * ) DY, complex*16, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)¶

**ZLA_SYRFSX_EXTENDED**improves 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:**

ZLA_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 ZSYRFSX 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_TYPE*

PREC_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' or 'E': Extra

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right-hand-sides, i.e., the number of columns of the matrix B.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is COMPLEX*16 array, dimension (LDAF,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.

*LDAF*

LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF.

*COLEQU*

COLEQU 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.

*C*

C 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.

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) The right-hand-side matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*Y*

Y is COMPLEX*16 array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by ZSYTRS. On exit, the improved solution matrix Y.

*LDY*

LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).

*BERR_OUT*

BERR_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 ZLA_LIN_BERR.

*N_NORMS*

N_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_NORM*

ERR_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_COMP*

ERR_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 < 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.

*RES*

RES is COMPLEX*16 array, dimension (N) Workspace to hold the intermediate residual.

*AYB*

AYB is DOUBLE PRECISION array, dimension (N) Workspace.

*DY*

DY is COMPLEX*16 array, dimension (N) Workspace to hold the intermediate solution.

*Y_TAIL*

Y_TAIL is COMPLEX*16 array, dimension (N) Workspace to hold the trailing bits of the intermediate solution.

*RCOND*

RCOND 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.

*ITHRESH*

ITHRESH 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.

*RTHRESH*

RTHRESH 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_UB*

DZ_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 define 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_CWISE*

IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

*INFO*

INFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to ZLA_HERFSX_EXTENDED had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zla_syrpvgrw (character*1 UPLO, integer N, integer INFO, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK)¶

**ZLA_SYRPVGRW**computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.

**Purpose:**

ZLA_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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.

*INFO*

INFO is INTEGER The value of INFO returned from ZSYTRF, .i.e., the pivot in column INFO is exactly 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*AF*

*LDAF*

LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

*WORK*

WORK is DOUBLE PRECISION array, dimension (2*N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlasyf (character UPLO, integer N, integer NB, integer KB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldw, * ) W, integer LDW, integer INFO)¶

**ZLASYF**computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

**Purpose:**

ZLASYF computes a partial factorization of a complex 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. Note that U**T denotes the transpose of U. ZLASYF is an auxiliary routine called by ZSYTRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NB*

NB 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.

*KB*

KB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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.

*W*

W is COMPLEX*16 array, dimension (LDW,NB)

*LDW*

LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley

## subroutine zlasyf_aa (character UPLO, integer J1, integer M, integer NB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) WORK)¶

**ZLASYF_AA**

**Purpose:**

DLATRF_AA factorizes a panel of a complex 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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*J1*

J1 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 ZSYTRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2.

*M*

M is INTEGER The dimension of the submatrix. M >= 0.

*NB*

NB is INTEGER The dimension of the panel to be facotorized.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV 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).

*H*

H is COMPLEX*16 workspace, dimension (LDH,NB).

*LDH*

LDH is INTEGER The leading dimension of the workspace H. LDH >= max(1,M).

*WORK*

WORK is COMPLEX*16 workspace, dimension (M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlasyf_rk (character UPLO, integer N, integer NB, integer KB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( ldw, * ) W, integer LDW, integer INFO)¶

**ZLASYF_RK**computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

**Purpose:**

ZLASYF_RK computes a partial factorization of a complex 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. ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NB*

NB 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.

*KB*

KB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

IPIV 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.

*W*

W is COMPLEX*16 array, dimension (LDW,NB)

*LDW*

LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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

## subroutine zlasyf_rook (character UPLO, integer N, integer NB, integer KB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldw, * ) W, integer LDW, integer INFO)¶

**ZLASYF_ROOK**computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.

**Purpose:**

ZLASYF_ROOK computes a partial factorization of a complex 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. ZLASYF_ROOK is an auxiliary routine called by ZSYTRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NB*

NB 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.

*KB*

KB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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.

*W*

W is COMPLEX*16 array, dimension (LDW,NB)

*LDW*

LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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

## subroutine zsycon (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYCON**

**Purpose:**

ZSYCON estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. 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**

*UPLO*

UPLO 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.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

*ANORM*

ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.

*RCOND*

RCOND 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.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsycon_3 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYCON_3**

**Purpose:**

ZSYCON_3 estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_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 ZSYTRS_3.

**Parameters**

*UPLO*

UPLO 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).

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_RK or ZSYTRF_BK.

*ANORM*

ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.

*RCOND*

RCOND 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.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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 Manchester

## subroutine zsycon_rook (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYCON_ROOK**

**Purpose:**

ZSYCON_ROOK estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF_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**

*UPLO*

UPLO 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.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF_ROOK.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_ROOK.

*ANORM*

ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.

*RCOND*

RCOND 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.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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

## subroutine zsyconv (character UPLO, character WAY, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) E, integer INFO)¶

**ZSYCONV**

**Purpose:**

ZSYCONV converts A given by ZHETRF into L and D or vice-versa. Get nondiagonal elements of D (returned in workspace) and apply or reverse permutation done in TRF.

**Parameters**

*UPLO*

UPLO 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.

*WAY*

WAY is CHARACTER*1 = 'C': Convert = 'R': Revert

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

*E*

E is COMPLEX*16 array, dimension (N) E stores the supdiagonal/subdiagonal of the symmetric 1-by-1 or 2-by-2 block diagonal matrix D in LDLT.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsyconvf (character UPLO, character WAY, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)¶

**ZSYCONVF**

**Purpose:**

If parameter WAY = 'C': ZSYCONVF converts the factorization output format used in ZSYTRF provided on entry in parameter A into the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) that is stored on exit in parameters A and E. It also converts in place details of the intechanges stored in IPIV from the format used in ZSYTRF into the format used in ZSYTRF_RK (or ZSYTRF_BK). If parameter WAY = 'R': ZSYCONVF performs the conversion in reverse direction, i.e. converts the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in ZSYTRF that is stored on exit in parameter A. It also converts in place details of the intechanges stored in IPIV from the format used in ZSYTRF_RK (or ZSYTRF_BK) into the format used in ZSYTRF. ZSYCONVF can also convert in Hermitian matrix case, i.e. between formats used in ZHETRF and ZHETRF_RK (or ZHETRF_BK).

**Parameters**

*UPLO*

UPLO 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 triangular

*WAY*

WAY is CHARACTER*1 = 'C': Convert = 'R': Revert

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) 1) If WAY ='C': On entry, contains factorization details in format used in ZSYTRF: 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 ZSYTRF_RK or ZSYTRF_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 ZSYTRF_RK or ZSYTRF_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 ZSYTRF: 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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 changed

*IPIV*

IPIV 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 ZSYTRF. On exit, details of the interchanges and the block structure of D in the format used in ZSYTRF_RK ( or ZSYTRF_BK). 1) If WAY ='R': On entry, details of the interchanges and the block structure of D in the format used in ZSYTRF_RK ( or ZSYTRF_BK). On exit, details of the interchanges and the block structure of D in the format used in ZSYTRF.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

## subroutine zsyconvf_rook (character UPLO, character WAY, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)¶

**ZSYCONVF_ROOK**

**Purpose:**

If parameter WAY = 'C': ZSYCONVF_ROOK converts the factorization output format used in ZSYTRF_ROOK provided on entry in parameter A into the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) that is stored on exit in parameters A and E. IPIV format for ZSYTRF_ROOK and ZSYTRF_RK (or ZSYTRF_BK) is the same and is not converted. If parameter WAY = 'R': ZSYCONVF_ROOK performs the conversion in reverse direction, i.e. converts the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in ZSYTRF_ROOK that is stored on exit in parameter A. IPIV format for ZSYTRF_ROOK and ZSYTRF_RK (or ZSYTRF_BK) is the same and is not converted. ZSYCONVF_ROOK can also convert in Hermitian matrix case, i.e. between formats used in ZHETRF_ROOK and ZHETRF_RK (or ZHETRF_BK).

**Parameters**

*UPLO*

UPLO 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 triangular

*WAY*

WAY is CHARACTER*1 = 'C': Convert = 'R': Revert

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) 1) If WAY ='C': On entry, contains factorization details in format used in ZSYTRF_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 ZSYTRF_RK or ZSYTRF_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 ZSYTRF_RK or ZSYTRF_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 ZSYTRF_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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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 changed

*IPIV*

IPIV is INTEGER array, dimension (N) On entry, details of the interchanges and the block structure of D as determined: 1) by ZSYTRF_ROOK, if WAY ='C'; 2) by ZSYTRF_RK (or ZSYTRF_BK), if WAY ='R'. The IPIV format is the same for all these routines. On exit, is not changed.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

## subroutine zsyequb (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYEQUB**

**Purpose:**

ZSYEQUB 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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The N-by-N symmetric matrix whose scaling factors are to be computed.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*S*

S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A.

*SCOND*

SCOND 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.

*AMAX*

AMAX 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.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**References:**

## subroutine zsyrfs (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

**ZSYRFS**

**Purpose:**

ZSYRFS 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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is COMPLEX*16 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 ZSYTRF.

*LDAF*

LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X*

X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZSYTRS. On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).

*FERR*

FERR 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.

*BERR*

BERR 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).

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsyrfsx (character UPLO, character EQUED, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) S, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

**ZSYRFSX**

**Purpose:**

ZSYRFSX 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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*EQUED*

EQUED 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.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is COMPLEX*16 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 ZSYTRF.

*LDAF*

LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

*S*

S 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.

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X*

X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZGETRS. On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).

*RCOND*

RCOND 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.

*BERR*

BERR 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_BNDS*

N_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_NORM*

ERR_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_COMP*

ERR_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 < 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.

*NPARAMS*

NPARAMS is INTEGER Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.

*PARAMS*

PARAMS is DOUBLE PRECISION array, dimension NPARAMS Specifies algorithm parameters. If an entry is < 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)

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (2*N)

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsysv_aa_2stage (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices**

**Purpose:**

ZSYSV_AA_2STAGE computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. Aasen's 2-stage algorithm is used to factor A as A = U**T * T * U, if UPLO = 'U', or A = L * T * L**T, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is symmetric and band. The matrix T is then LU-factored with partial pivoting. The factored form of A is then used to solve the system of equations A * X = B. This is the blocked version of the algorithm, calling Level 3 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

*A*

A is COMPLEX*16 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, L is stored below (or above) the subdiaonal blocks, when UPLO is 'L' (or 'U').

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TB*

TB is COMPLEX*16 array, dimension (LTB) On exit, details of the LU factorization of the band matrix.

*LTB*

LTB is INTEGER The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N. If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.

*IPIV*

IPIV 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).

*IPIV2*

IPIV2 is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k).

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*WORK*

WORK is COMPLEX*16 workspace of size LWORK

*LWORK*

LWORK is INTEGER The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB. 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.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, band LU factorization failed on i-th column

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytf2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**ZSYTF2**computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

**Purpose:**

ZSYTF2 computes the factorization of a complex 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**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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).

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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.209 and l.377 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN by IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 1-96 - Based on modifications by J. Lewis, Boeing Computer Services Company

## subroutine zsytf2_rk (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)¶

**ZSYTF2_RK**computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

**Purpose:**

ZSYTF2_RK computes the factorization of a complex 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**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

IPIV 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

TODO: put further details

**Contributors:**

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 , USA

## subroutine zsytf2_rook (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**ZSYTF2_ROOK**computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).

**Purpose:**

ZSYTF2_ROOK computes the factorization of a complex 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**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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).

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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 , USA

## subroutine zsytrf (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRF**

**Purpose:**

ZSYTRF computes the factorization of a complex 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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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).

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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.

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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).

## subroutine zsytrf_aa (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRF_AA**

**Purpose:**

ZSYTRF_AA computes the factorization of a complex symmetric matrix A using the Aasen's algorithm. The form of the factorization is A = U**T*T*U 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 complex symmetric tridiagonal matrix. This is the blocked version of the algorithm, calling Level 3 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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').

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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).

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK 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.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytrf_aa_2stage (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRF_AA_2STAGE**

**Purpose:**

ZSYTRF_AA_2STAGE computes the factorization of a complex symmetric matrix A using the Aasen's algorithm. The form of the factorization is A = U**T*T*U 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 complex symmetric band matrix with the bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is LU factorized with partial pivoting). This is the blocked version of the algorithm, calling Level 3 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the hermitian 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, L is stored below (or above) the subdiaonal blocks, when UPLO is 'L' (or 'U').

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TB*

TB is COMPLEX*16 array, dimension (LTB) On exit, details of the LU factorization of the band matrix.

*LTB*

LTB is INTEGER The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N. If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.

*IPIV*

*IPIV2*

IPIV2 is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k).

*WORK*

WORK is COMPLEX*16 workspace of size LWORK

*LWORK*

LWORK is INTEGER The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB. 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.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, band LU factorization failed on i-th column

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytrf_rk (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRF_RK**computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

**Purpose:**

ZSYTRF_RK computes the factorization of a complex 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**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

IPIV 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.

*WORK*

WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

TODO: put correct description

**Contributors:**

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

## subroutine zsytrf_rook (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRF_ROOK**

**Purpose:**

ZSYTRF_ROOK computes the factorization of a complex 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**

*UPLO*

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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).

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV 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.

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

**Contributors:**

June 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

## subroutine zsytri (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYTRI**

**Purpose:**

ZSYTRI computes the inverse of a complex symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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 ZSYTRF. 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytri2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRI2**

**Purpose:**

ZSYTRI2 computes the inverse of a COMPLEX*16 symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. ZSYTRI2 sets the LEADING DIMENSION of the workspace before calling ZSYTRI2X that actually computes the inverse.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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 ZSYTRF. 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

*WORK*

WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3)

*LWORK*

LWORK 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.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytri2x (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( n+nb+1,* ) WORK, integer NB, integer INFO)¶

**ZSYTRI2X**

**Purpose:**

ZSYTRI2X computes the inverse of a complex symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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 ZSYTRF. 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the NNB structure of D as determined by ZSYTRF.

*WORK*

WORK is COMPLEX*16 array, dimension (N+NB+1,NB+3)

*NB*

NB is INTEGER Block size

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytri_3 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRI_3**

**Purpose:**

ZSYTRI_3 computes the inverse of a complex symmetric indefinite matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_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. ZSYTRI_3 sets the leading dimension of the workspace before calling ZSYTRI_3X that actually computes the inverse. This is the blocked version of the algorithm, calling Level 3 BLAS.

**Parameters**

*UPLO*

UPLO 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.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_RK or ZSYTRF_BK.

*WORK*

WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK 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.

*INFO*

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

## subroutine zsytri_3x (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)¶

**ZSYTRI_3X**

**Purpose:**

ZSYTRI_3X computes the inverse of a complex symmetric indefinite matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_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**

*UPLO*

UPLO 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.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_RK or ZSYTRF_BK.

*WORK*

WORK is COMPLEX*16 array, dimension (N+NB+1,NB+3).

*NB*

NB is INTEGER Block size.

*INFO*

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

## subroutine zsytri_rook (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYTRI_ROOK**

**Purpose:**

ZSYTRI_ROOK computes the inverse of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF_ROOK.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 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 ZSYTRF_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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_ROOK.

*WORK*

WORK is COMPLEX*16 array, dimension (N)

*INFO*

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine zsytrs (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZSYTRS**

**Purpose:**

ZSYTRS solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytrs2 (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSYTRS2**

**Purpose:**

ZSYTRS2 solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF and converted by ZSYCONV.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

*A*

A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF. 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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

*B*

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*WORK*

WORK is COMPLEX*16 array, dimension (N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytrs_3 (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZSYTRS_3**

**Purpose:**

ZSYTRS_3 solves a system of linear equations A * X = B with a complex symmetric matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_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**

*UPLO*

UPLO 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).

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

*A*

A is COMPLEX*16 array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is COMPLEX*16 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.

*IPIV*

*B*

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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 Manchester

## subroutine zsytrs_aa (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZSYTRS_AA**

**Purpose:**

ZSYTRS_AA solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U**T*T*U or A = L*T*L**T computed by ZSYTRF_AA.

**Parameters**

*UPLO*

UPLO 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*T*U; = 'L': Lower triangular, form is A = L*T*L**T.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

*A*

A is COMPLEX*16 array, dimension (LDA,N) Details of factors computed by ZSYTRF_AA.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges as computed by ZSYTRF_AA.

*B*

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,3*N-2).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytrs_aa_2stage (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZSYTRS_AA_2STAGE**

**Purpose:**

ZSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U**T*T*U or A = L*T*L**T computed by ZSYTRF_AA_2STAGE.

**Parameters**

*UPLO*

UPLO 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*T*U; = 'L': Lower triangular, form is A = L*T*L**T.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

*A*

A is COMPLEX*16 array, dimension (LDA,N) Details of factors computed by ZSYTRF_AA_2STAGE.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TB*

TB is COMPLEX*16 array, dimension (LTB) Details of factors computed by ZSYTRF_AA_2STAGE.

*LTB*

LTB is INTEGER The size of the array TB. LTB >= 4*N.

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges as computed by ZSYTRF_AA_2STAGE.

*IPIV2*

IPIV2 is INTEGER array, dimension (N) Details of the interchanges as computed by ZSYTRF_AA_2STAGE.

*B*

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsytrs_rook (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZSYTRS_ROOK**

**Purpose:**

ZSYTRS_ROOK solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF_ROOK.

**Parameters**

*UPLO*

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

*A*

A is COMPLEX*16 array, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF_ROOK.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by ZSYTRF_ROOK.

*B*

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine ztgsyl (character TRANS, integer IJOB, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldd, * ) D, integer LDD, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( ldf, * ) F, integer LDF, double precision SCALE, double precision DIF, complex*16, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

**ZTGSYL**

**Purpose:**

ZTGSYL 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 complex entries. A, B, D and E are upper triangular (i.e., (A,D) and (B,E) in generalized Schur form). 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**H, Im) ] (2) [ kron(In, D) -kron(E**H, Im) ], Here Ix is the identity matrix of size x and X**H is the conjugate transpose of X. Kron(X, Y) is the Kronecker product between the matrices X and Y. If TRANS = 'C', y in the conjugate transposed system Z**H *y = scale*b is solved for, which is equivalent to solve for R and L in A**H * R + D**H * L = scale * C (3) R * B**H + L * E**H = scale * -F This case (TRANS = 'C') 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 ZLACON. If IJOB >= 1, ZTGSYL 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. This is a level-3 BLAS algorithm.

**Parameters**

*TRANS*

TRANS is CHARACTER*1 = 'N': solve the generalized sylvester equation (1). = 'C': solve the "conjugate transposed" system (3).

*IJOB*

IJOB 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 is used). =4: Only an estimate of Dif[(A,D), (B,E)] is computed. (ZGECON on sub-systems is used). Not referenced if TRANS = 'C'.

*M*

M is INTEGER The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.

*N*

N is INTEGER The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.

*A*

A is COMPLEX*16 array, dimension (LDA, M) The upper triangular matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1, M).

*B*

B is COMPLEX*16 array, dimension (LDB, N) The upper triangular matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1, N).

*C*

C is COMPLEX*16 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.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1, M).

*D*

D is COMPLEX*16 array, dimension (LDD, M) The upper triangular matrix D.

*LDD*

LDD is INTEGER The leading dimension of the array D. LDD >= max(1, M).

*E*

E is COMPLEX*16 array, dimension (LDE, N) The upper triangular matrix E.

*LDE*

LDE is INTEGER The leading dimension of the array E. LDE >= max(1, N).

*F*

F is COMPLEX*16 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.

*LDF*

LDF is INTEGER The leading dimension of the array F. LDF >= max(1, M).

*DIF*

DIF 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 = 'C', DIF is not referenced.

*SCALE*

SCALE 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, R and L will hold the solutions to the homogeneous system with C = F = 0.

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK 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.

*IWORK*

IWORK is INTEGER array, dimension (M+N+2)

*INFO*

INFO 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 very close eigenvalues.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**References:**

## subroutine ztrsyl (character TRANA, character TRANB, integer ISGN, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, double precision SCALE, integer INFO)¶

**ZTRSYL**

**Purpose:**

ZTRSYL solves the complex 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**H, and A and B are both upper 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.

**Parameters**

*TRANA*

TRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'C': op(A) = A**H (Conjugate transpose)

*TRANB*

TRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'C': op(B) = B**H (Conjugate transpose)

*ISGN*

ISGN 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*C

*M*

M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.

*N*

N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,M) The upper triangular matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*B*

B is COMPLEX*16 array, dimension (LDB,N) The upper triangular matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*C*

C is COMPLEX*16 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.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M)

*SCALE*

SCALE is DOUBLE PRECISION The scale factor, scale, set <= 1 to avoid overflow in X.

*INFO*

INFO 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

# Author¶

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