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# zsytrf

```
NAME
ZSYTRF - compute the factorization of a complex symmetric
matrix A using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS
SUBROUTINE ZSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO
)

CHARACTER      UPLO

INTEGER        INFO, LDA, LWORK, N

INTEGER        IPIV( * )

COMPLEX*16     A( LDA, * ), WORK( LWORK )

PURPOSE
ZSYTRF computes the factorization of a complex symmetric
matrix A using the Bunch-Kaufman diagonal pivoting method.
The form of the factorization is

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper
(lower) triangular matrices, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level
3 BLAS.

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

N       (input) INTEGER
The order of the matrix A.  N >= 0.

A       (input/output) 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 con-
tains 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 tri-
angular part of A is not referenced.

On exit, the block diagonal matrix D and the multi-
pliers used to obtain the factor U or L (see below
for further details).

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

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

LWORK   (input) INTEGER
The length of WORK.  LWORK >=1.  For best perfor-
mance LWORK >= N*NB, where NB is the block size
returned by ILAENV.

INFO    (output) 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 fac-
torization has been completed, but the block diago-
nal matrix D is exactly singular, and division by
zero will occur if it is used to solve a system of
equations.

FURTHER DETAILS
If UPLO = 'U', then A = U*D*U', 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', 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).
```