zlatrd (l) - Linux Manuals

zlatrd: reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Qaq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A

Command to display zlatrd manual in Linux: $ man l zlatrd

NAME

ZLATRD - reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Qaq * 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 Qaq * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = aqUaq, ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied;
if UPLO = aqLaq, 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*1: Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
= aqUaq: Upper triangular
= aqLaq: 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 = aqUaq, 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 = aqLaq, 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 = aqUaq, 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 = aqLaq, 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 elementary 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 = aqUaq, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = aqLaq, 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 = aqUaq, and in TAU(1:nb) if UPLO = aqLaq. See Further 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 = aqUaq, 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 * vaq
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 = aqLaq, 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 * vaq
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*Waq - W*Vaq.
The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:
if UPLO = aqUaq: if UPLO = aqLaq:

  (  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).