dgebrd-f(3)

dgebrd.f –

Synopsis

Functions/Subroutines

subroutine dgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
DGEBRD

Function/Subroutine Documentation

subroutine dgebrd (integerM, integerN, 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( * )WORK, integerLWORK, integerINFO)

DGEBRD

Purpose:

 DGEBRD reduces a general real M-by-N matrix A to upper or lower
 bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
 If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters:

 M is INTEGER
 The number of rows in the matrix A. M >= 0.

 N is INTEGER
 The number of columns in the matrix A. N >= 0.

 A is DOUBLE PRECISION array, dimension (LDA,N)
 On entry, the M-by-N general matrix to be reduced.
 On exit,
 if m >= n, the diagonal and the first superdiagonal are
 overwritten with the upper bidiagonal matrix B; the
 elements below the diagonal, with the array TAUQ, represent
 the orthogonal matrix Q as a product of elementary
 reflectors, and the elements above the first superdiagonal,
 with the array TAUP, represent the orthogonal matrix P as
 a product of elementary reflectors;
 if m < n, the diagonal and the first subdiagonal are
 overwritten with the lower bidiagonal matrix B; the
 elements below the first subdiagonal, with the array TAUQ,
 represent the orthogonal matrix Q as a product of
 elementary reflectors, and the elements above the diagonal,
 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 (min(M,N))
 The diagonal elements of the bidiagonal matrix B:
 D(i) = A(i,i).

 E is DOUBLE PRECISION array, dimension (min(M,N)-1)
 The off-diagonal elements of the bidiagonal matrix B:
 if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
 if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

TAUQ

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

TAUP

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

WORK

 WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

 LWORK is INTEGER
 The length of the array WORK. LWORK >= max(1,M,N).
 For optimum performance LWORK >= (M+N)*NB, where NB
 is the optimal blocksize.
 If LWORK = -1, then a workspace query is assumed; the routine
 only calculates the optimal size of the WORK array, returns
 this value as the first entry of the WORK array, and no error
 message related to LWORK is issued by XERBLA.

INFO

 INFO is INTEGER
 = 0: successful exit
 < 0: if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

 The matrices Q and P are represented as products of elementary
 reflectors:
 If m >= n,
 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
 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;
 v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
 u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
 tauq is stored in TAUQ(i) and taup in TAUP(i).
 If m < n,
 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
 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;
 v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
 u(1:i-1) = 0, u(i) = 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).
 The contents of A on exit are illustrated by the following examples:
 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
 ( v1 v2 v3 v4 v5 )
 where d and e denote diagonal and off-diagonal elements of B, vi
 denotes an element of the vector defining H(i), and ui an element of
 the vector defining G(i).

Definition at line 205 of file dgebrd.f.

Author

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