redhat man page for zlatrd

ZLATRD(l)								 )								 ZLATRD(l)

NAME
       ZLATRD - reduce NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q'
       * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A

SYNOPSIS
       SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

	   CHARACTER	  UPLO

	   INTEGER	  LDA, LDW, N, NB

	   DOUBLE	  PRECISION E( * )

	   COMPLEX*16	  A( LDA, * ), TAU( * ), W( LDW, * )

PURPOSE
       ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation	Q'
       *  A  * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.  If UPLO = 'U', ZLATRD
       reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied;
       if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

       This is an auxiliary routine called by ZHETRD.

ARGUMENTS
       UPLO    (input) CHARACTER
	       Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
	       = 'U': Upper triangular
	       = 'L': Lower triangular

       N       (input) INTEGER
	       The order of the matrix A.

       NB      (input) INTEGER
	       The number of rows and columns to be reduced.

       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: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal
	       elements of A; the elements above the diagonal with the array TAU, represent the unitary  matrix  Q  as	a  product  of	elementary
	       reflectors;  if	UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the
	       diagonal elements of A; the elements below the diagonal with the array TAU, represent the  unitary matrix Q as a product of elemen-
	       tary reflectors.  See Further Details.  LDA     (input) INTEGER The leading dimension of the array A.  LDA >= max(1,N).

       E       (output) DOUBLE PRECISION array, dimension (N-1)
	       If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb)
	       contains the subdiagonal elements of the first NB columns of the reduced matrix.

       TAU     (output) COMPLEX*16 array, dimension (N-1)
	       The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.	 See  Fur-
	       ther  Details.  W       (output) COMPLEX*16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of
	       A.

       LDW     (input) INTEGER
	       The leading dimension of the array W. LDW >= max(1,N).

FURTHER DETAILS
       If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

	  Q = H(n) H(n-1) . . . H(n-nb+1).

       Each H(i) has the form

	  H(i) = I - tau * v * v'

       where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and  tau
       in TAU(i-1).

       If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

	  Q = H(1) H(2) . . . H(nb).

       Each H(i) has the form

	  H(i) = I - tau * v * v'

       where  tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau
       in TAU(i).

       The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part
       of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

       The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

       if UPLO = 'U':			    if UPLO = 'L':

	 (  a	a   a	v4  v5 )	      (  d		    )
	 (	a   a	v4  v5 )	      (  1   d		    )
	 (	    a	1   v5 )	      (  v1  1	 a	    )
	 (		d   1  )	      (  v1  v2  a   a	    )
	 (		    d  )	      (  v1  v2  a   a	 a  )

       where  d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an
       element of the vector defining H(i).

LAPACK version 3.0						   15 June 2000 							 ZLATRD(l)