zhetrf man page on IRIX

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ZHETRF(3F)							    ZHETRF(3F)

NAME
     ZHETRF - compute 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( LWORK )

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).

									Page 1

ZHETRF(3F)							    ZHETRF(3F)

     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 (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

									Page 2

ZHETRF(3F)							    ZHETRF(3F)

	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).

									Page 3

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