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zlatrd.f(3) LAPACK zlatrd.f(3)

zlatrd.f -


subroutine zlatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
 
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
Purpose:
 ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
 Hermitian tridiagonal form by a unitary similarity
 transformation Q**H * 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.
Parameters:
UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U': Upper triangular
          = 'L': Lower triangular
N
          N is INTEGER
          The order of the matrix A.
NB
          NB is INTEGER
          The number of rows and columns to be reduced.
A
          A is 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 elementary reflectors.
          See Further Details.
LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
E
          E is 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
          TAU is 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 Further Details.
W
          W is COMPLEX*16 array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.
LDW
          LDW is INTEGER
          The leading dimension of the array W. LDW >= max(1,N).
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
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**H
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**H
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**H - W*V**H.
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).
Definition at line 200 of file zlatrd.f.

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Sat Nov 16 2013 Version 3.4.2

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