# man csytf2 (Fonctions bibliothèques) - compute the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

## NAME

CSYTF2 - compute 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)
triangular 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 multipliers 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.
- 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

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).