complex16SYcomputational(3) complex16

## Functions

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_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_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
ZSYCON_ROOK
subroutine zsyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO)
ZSYCONV
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 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_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_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_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_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

## 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.

Date:

September 2012

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.

Date:

September 2012

## 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.

Date:

September 2012

## 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', '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 .LT. 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3  Reciprocal condition number: Estimated componentwise
reciprocal condition number.  Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
```

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 definte as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
```

IGNORE_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.

Date:

September 2012

## 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.

Date:

November 2015

## 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.

Date:

November 2013

Contributors:

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

## 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.

Date:

November 2013

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.

Date:

November 2011

## 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.

Date:

November 2015

Contributors:

November 2015, 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.

Date:

November 2015

## 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 and reduce its condition number
(with respect to the two-norm).  S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings.
```

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 N-by-N symmetric matrix whose scaling
factors are to be computed.  Only the diagonal elements of A
are referenced.
```

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
Absolute value of largest 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 (3*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.

Date:

November 2011

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

## 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.

Date:

November 2011

## 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 DSYTRF.
```

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 DSYTRF.
```

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 DGETRS.
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 .LT. 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3  Reciprocal condition number: Estimated componentwise
reciprocal condition number.  Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
```

NPARAMS

```          NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS.  If .LE. 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 .LT. 0.0, then
that entry will be filled with default value used for that
parameter.  Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0D+0
= 0.0 : No refinement is performed, and no error bounds are
computed.
= 1.0 : Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm.  Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)
```

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.

Date:

April 2012

## 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.

Date:

November 2013

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_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.

Date:

November 2013

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
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.

Date:

November 2011

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_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.

Date:

June 2016

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.

Date:

November 2011

## 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 NB 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 NB 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.

Date:

November 2015

## 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+NNB+1,NNB+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.

Date:

November 2011

## 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.

Date:

November 2015

Contributors:

```   November 2015, 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.

Date:

November 2011

## 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 real
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.

Date:

June 2016

## 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.

Date:

November 2015

Contributors:

```   November 2015, 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 homogenious 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.

Date:

November 2011

Contributors:

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

References:

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.

[3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

## 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