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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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

```
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.```

WORK

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

Author

Univ. of Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

NAG Ltd.

References:

Livne, O.E. and Golub, G.H., 'Scaling by Binormalization',
Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
DOI 10.1023/B:NUMA.0000016606.32820.69
Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

## 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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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

```
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.```

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 Tennessee

Univ. of California Berkeley

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.

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.```

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

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 Tennessee

Univ. of California Berkeley

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

```
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.```

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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

```
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 Tennessee

Univ. of California Berkeley

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.

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.```

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

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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:

```
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

```
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)

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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

```
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)

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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

```
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)

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 Tennessee

Univ. of California Berkeley

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

```
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 Tennessee

Univ. of California Berkeley

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

```
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 Tennessee

Univ. of California Berkeley

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

```
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)

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

```
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 Tennessee

Univ. of California Berkeley

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 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

```
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.```

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 Tennessee

Univ. of California Berkeley

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

```
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.```

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.

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

```
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSYTRF.```

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 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 Tennessee

Univ. of California Berkeley

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

```
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)

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.```

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 Tennessee

Univ. of California Berkeley

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

```
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 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

```
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 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 Tennessee

Univ. of California Berkeley

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

```
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 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

```
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 Tennessee

Univ. of California Berkeley

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

```
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.```

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_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

```
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 Tennessee

Univ. of California Berkeley

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 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.

=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 Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

 B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
 B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.
 B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

## 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

= +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 Tennessee

Univ. of California Berkeley