dlabrd-f(3)

dlabrd.f –

Synopsis

Functions/Subroutines

subroutine dlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Function/Subroutine Documentation

subroutine dlabrd (integerM, integerN, integerNB, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( * )TAUQ, double precision, dimension( * )TAUP, double precision, dimension( ldx, * )X, integerLDX, double precision, dimension( ldy, * )Y, integerLDY)

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

 DLABRD reduces the first NB rows and columns of a real general
 m by n matrix A to upper or lower bidiagonal form by an orthogonal
 transformation Q**T * A * P, and returns the matrices X and Y which
 are needed to apply the transformation to the unreduced part of A.
 If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
 bidiagonal form.
 This is an auxiliary routine called by DGEBRD

Parameters:

 M is INTEGER
 The number of rows in the matrix A.

 N is INTEGER
 The number of columns in the matrix A.

 NB is INTEGER
 The number of leading rows and columns of A to be reduced.

 A is DOUBLE PRECISION array, dimension (LDA,N)
 On entry, the m by n general matrix to be reduced.
 On exit, the first NB rows and columns of the matrix are
 overwritten; the rest of the array is unchanged.
 If m >= n, elements on and below the diagonal in the first NB
 columns, with the array TAUQ, represent the orthogonal
 matrix Q as a product of elementary reflectors; and
 elements above the diagonal in the first NB rows, with the
 array TAUP, represent the orthogonal matrix P as a product
 of elementary reflectors.
 If m < n, elements below the diagonal in the first NB
 columns, with the array TAUQ, represent the orthogonal
 matrix Q as a product of elementary reflectors, and
 elements on and above the diagonal in the first NB rows,
 with the array TAUP, represent the orthogonal matrix P as
 a product of elementary reflectors.
 See Further Details.

LDA

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max(1,M).

 D is DOUBLE PRECISION array, dimension (NB)
 The diagonal elements of the first NB rows and columns of
 the reduced matrix. D(i) = A(i,i).

 E is DOUBLE PRECISION array, dimension (NB)
 The off-diagonal elements of the first NB rows and columns of
 the reduced matrix.

TAUQ

 TAUQ is DOUBLE PRECISION array dimension (NB)
 The scalar factors of the elementary reflectors which
 represent the orthogonal matrix Q. See Further Details.

TAUP

 TAUP is DOUBLE PRECISION array, dimension (NB)
 The scalar factors of the elementary reflectors which
 represent the orthogonal matrix P. See Further Details.

 X is DOUBLE PRECISION array, dimension (LDX,NB)
 The m-by-nb matrix X required to update the unreduced part
 of A.

LDX

 LDX is INTEGER
 The leading dimension of the array X. LDX >= max(1,M).

 Y is DOUBLE PRECISION array, dimension (LDY,NB)
 The n-by-nb matrix Y required to update the unreduced part
 of A.

LDY

 LDY is INTEGER
 The leading dimension of the array Y. LDY >= max(1,N).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Further Details:

 The matrices Q and P are represented as products of elementary
 reflectors:
 Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
 Each H(i) and G(i) has the form:
 H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
 where tauq and taup are real scalars, and v and u are real vectors.
 If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
 A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
 A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
 If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
 A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
 A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
 The elements of the vectors v and u together form the m-by-nb matrix
 V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
 the transformation to the unreduced part of the matrix, using a block
 update of the form: A := A - V*Y**T - X*U**T.
 The contents of A on exit are illustrated by the following examples
 with nb = 2:
 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
 ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
 ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
 ( v1 v2 a a a ) ( v1 1 a a a a )
 ( v1 v2 a a a ) ( v1 v2 a a a a )
 ( v1 v2 a a a ) ( v1 v2 a a a a )
 ( v1 v2 a a a )
 where a denotes an element of the original matrix which is unchanged,
 vi denotes an element of the vector defining H(i), and ui an element
 of the vector defining G(i).

Definition at line 210 of file dlabrd.f.

Author

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