CGEBRD man page on IRIX

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CGEBRD(3F)							    CGEBRD(3F)

NAME
     CGEBRD - reduce a general complex M-by-N matrix A to upper or lower
     bidiagonal form B by a unitary transformation

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

	 INTEGER	INFO, LDA, LWORK, M, N

	 REAL		D( * ), E( * )

	 COMPLEX	A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( LWORK )

PURPOSE
     CGEBRD reduces a general complex M-by-N matrix A to upper or lower
     bidiagonal form B by a unitary transformation: Q**H * 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) COMPLEX 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 unitary matrix Q as
	     a product of elementary reflectors, and the elements above the
	     first superdiagonal, with the array TAUP, represent the unitary
	     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 unitary matrix Q as a product
	     of elementary reflectors, and the elements above the diagonal,
	     with the array TAUP, represent the unitary 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) REAL array, dimension (min(M,N))
	     The diagonal elements of the bidiagonal matrix B:	D(i) = A(i,i).

     E	     (output) REAL 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

CGEBRD(3F)							    CGEBRD(3F)

     TAUQ    (output) COMPLEX array dimension (min(M,N))
	     The scalar factors of the elementary reflectors which represent
	     the unitary matrix Q. See Further Details.	 TAUP	 (output)
	     COMPLEX array, dimension (min(M,N)) The scalar factors of the
	     elementary reflectors which represent the unitary matrix P. See
	     Further Details.  WORK    (workspace/output) COMPLEX 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 complex scalars, and v and u are complex 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 complex scalars, and v and u are complex 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

CGEBRD(3F)							    CGEBRD(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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