# zsysv_rk.f - Man Page

SRC/zsysv_rk.f

## Synopsis

### Functions/Subroutines

subroutine zsysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

## Function/Subroutine Documentation

### subroutine zsysv_rk (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer lwork, integer info)

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

Purpose:

``` ZSYSV_RK computes the solution to a complex system of linear
equations A * X = B, where A is an N-by-N symmetric matrix
and X and B are N-by-NRHS matrices.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

ZSYTRF_RK is called to compute the factorization of a complex
symmetric matrix.  The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine ZSYTRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= '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.  NRHS >= 0.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.

If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.

On exit, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L  as computed by ZSYTRF_RK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

LDA

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

E

```          E is COMPLEX*16 array, dimension (N)
On exit, contains the output computed by the factorization
routine ZSYTRF_RK, i.e. the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV

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

B

```          B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.```

LDB

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

WORK

```          WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).
Work array used in the factorization stage.
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The length of WORK.  LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for ZSYTRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, returns this value as
the first entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0: successful exit

< 0: If INFO = -k, the k-th argument had an illegal value

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.

NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  December 2016,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 226 of file zsysv_rk.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zsysv_rk(3) is an alias of zsysv_rk.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK