DGEBRD man page on IRIX

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DGEBRD(3F)							    DGEBRD(3F)

NAME
     DGEBRD - reduce a general real M-by-N matrix A to upper or lower
     bidiagonal form B by an orthogonal transformation

SYNOPSIS
     SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )

	 INTEGER	INFO, LDA, LWORK, M, N

	 DOUBLE		PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
			TAUQ( * ), WORK( LWORK )

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.

ARGUMENTS
     M	     (input) INTEGER
	     The number of rows in the matrix A.  M >= 0.

     N	     (input) INTEGER
	     The number of columns in the matrix A.  N >= 0.

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

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

     E	     (output) 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.

									Page 1

DGEBRD(3F)							    DGEBRD(3F)

     TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
	     The scalar factors of the elementary reflectors which represent
	     the orthogonal matrix Q. See Further Details.  TAUP    (output)
	     DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors
	     of the elementary reflectors which represent the orthogonal
	     matrix P. See Further Details.  WORK    (workspace/output) DOUBLE
	     PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1)
	     returns the optimal LWORK.

     LWORK   (input) 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.

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

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'  and G(i) = I - taup * u * u'

     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'  and G(i) = I - taup * u * u'

     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 )

									Page 2

DGEBRD(3F)							    DGEBRD(3F)

       (  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).

									Page 3

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