csytf2 man page on Scientific

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```CSYTF2(1)		 LAPACK routine (version 3.2)		     CSYTF2(1)

NAME
CSYTF2  -  computes  the	 factorization of a complex symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS
SUBROUTINE CSYTF2( UPLO, N, A, LDA, IPIV, INFO )

CHARACTER	  UPLO

INTEGER	  INFO, LDA, N

INTEGER	  IPIV( * )

COMPLEX	  A( LDA, * )

PURPOSE
CSYTF2 computes the factorization of a complex symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method:
A = U*D*U'  or  A = L*D*L'
where  U (or L) is a product of permutation and unit upper (lower) tri‐
angular matrices, U' 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.

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

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

A       (input/output) COMPLEX 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  mul‐
tipliers	 used  to obtain the factor U or L (see below for fur‐
ther 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 inter‐
changed 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 diag‐
onal  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.

INFO    (output) 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.

FURTHER DETAILS
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. SISNAN(ABSAKK)  )  THEN
1-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company
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).

LAPACK routine (version 3.2)	 November 2008			     CSYTF2(1)
```
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