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## man page of LAPACK-3

### LAPACK-3: computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method

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## NAME

LAPACK-3 - computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method## SYNOPSIS

SUBROUTINE ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO ) CHARACTER UPLO INTEGER INFO, LDA, LWORK, N INTEGER IPIV( * ) COMPLEX*16 A( LDA, * ), WORK( * )## PURPOSE

ZHETRF computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = U*D*U**H or A = L*D*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian 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 Hermitian 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. WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The length of WORK. LWORK >=1. For best performance 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 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

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) March 2011 LAPACK-3(3)

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