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dgebrd.f(3)				      LAPACK				      dgebrd.f(3)

NAME
       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

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

	   N

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

	   A

		     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

		     D is DOUBLE PRECISION array, dimension (min(M,N))
		     The diagonal elements of the bidiagonal matrix B:
		     D(i) = A(i,i).

	   E

		     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
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dgebrd.f(3)
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