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clatrd-f(3)

clatrd.f –

Synopsis

Functions/Subroutines

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

Function/Subroutine Documentation

subroutine clatrd (characterUPLO, integerN, integerNB, complex, dimension( lda, * )A, integerLDA, real, dimension( * )E, complex, dimension( * )TAU, complex, dimension( ldw, * )W, integerLDW)

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

 CLATRD 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', CLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.
 This is an auxiliary routine called by CHETRD.

 

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 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 REAL 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 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 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 clatrd.f.

Author

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