.TH ZSYTF2 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME ZSYTF2 - the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method .SH SYNOPSIS .TP 19 SUBROUTINE ZSYTF2( UPLO, N, A, LDA, IPIV, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER INFO, LDA, N .TP 19 .ti +4 INTEGER IPIV( * ) .TP 19 .ti +4 COMPLEX*16 A( LDA, * ) .SH PURPOSE ZSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U\(aq or A = L*D*L\(aq .br where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U\(aq is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. .br This is the unblocked version of the algorithm, calling Level 2 BLAS. .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: .br = \(aqU\(aq: Upper triangular .br = \(aqL\(aq: Lower triangular .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 A (input/output) COMPLEX*16 array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = \(aqU\(aq, 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 = \(aqL\(aq, 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). .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). .TP 8 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 = \(aqU\(aq 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 = \(aqL\(aq 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. .TP 8 INFO (output) INTEGER .br = 0: successful exit .br < 0: if INFO = -k, the k-th argument had an illegal value .br > 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. .SH FURTHER DETAILS 09-29-06 - patch from .br Bobby Cheng, MathWorks .br Replace l.209 and l.377 .br IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN .br by .br IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 1-96 - Based on modifications by J. Lewis, Boeing Computer Services Company .br If UPLO = \(aqU\(aq, then A = U*D*U\(aq, where .br U = P(n)*U(n)* ... *P(k)U(k)* ..., .br 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 .br U(k) = ( 0 I 0 ) s .br ( 0 0 I ) n-k .br k-s s n-k .br 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). .br If UPLO = \(aqL\(aq, then A = L*D*L\(aq, where .br L = P(1)*L(1)* ... *P(k)*L(k)* ..., .br 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 .br L(k) = ( 0 I 0 ) s .br ( 0 v I ) n-k-s+1 .br k-1 s n-k-s+1 .br 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). .br