zhetrf (l)  Linux Manuals
zhetrf: computes the factorization of a complex Hermitian matrix A using the BunchKaufman diagonal pivoting method
NAME
ZHETRF  computes the factorization of a complex Hermitian matrix A using the BunchKaufman diagonal pivoting methodSYNOPSIS
 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 BunchKaufman diagonal pivoting method. The form of the factorization isA
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1by1 and 2by2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
ARGUMENTS
 UPLO (input) CHARACTER*1

= aqUaq: Upper triangle of A is stored;
= aqLaq: 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 = aqUaq, the leading NbyN 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 = aqLaq, the leading NbyN 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 1by1 diagonal block. If UPLO = aqUaq and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 diagonal block. If UPLO = aqLaq 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 2by2 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 ith 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 = aqUaq, then A = U*D*Uaq, whereU
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 1by1 and 2by2 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
U(k)
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper triangle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1:k).
If UPLO = aqLaq, then A = L*D*Laq, where
L
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 1by1 and 2by2 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
L(k)
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).