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RedHat 9 (Linux i386) - man page for clabrd (redhat section l)

CLABRD(l)					)					CLABRD(l)

NAME
       CLABRD  -  reduce  the  first  NB rows and columns of a complex general m by n matrix A to
       upper or lower real bidiagonal form by a unitary transformation Q' * A *  P,  and  returns
       the matrices X and Y which are needed to apply the transformation to the unreduced part of
       A

SYNOPSIS
       SUBROUTINE CLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

	   INTEGER	  LDA, LDX, LDY, M, N, NB

	   REAL 	  D( * ), E( * )

	   COMPLEX	  A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y( LDY, * )

PURPOSE
       CLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper
       or  lower  real	bidiagonal  form  by a unitary transformation Q' * 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 CGEBRD

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

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

       NB      (input) INTEGER
	       The number of leading rows and columns of A to be reduced.

       A       (input/output) COMPLEX 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 unitary matrix Q as a product of  elementary  reflectors;  and
	       elements  above	the diagonal in the first NB rows, with the array TAUP, represent
	       the unitary 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
	       unitary 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 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 (NB)
	       The  diagonal  elements	of  the  first NB rows and columns of the reduced matrix.
	       D(i) = A(i,i).

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

       TAUQ    (output) COMPLEX array dimension (NB)
	       The scalar factors of the elementary reflectors which represent the unitary matrix
	       Q. See Further Details.	TAUP	(output) COMPLEX array, dimension (NB) The scalar
	       factors of the elementary reflectors which represent the  unitary  matrix  P.  See
	       Further	Details.   X	   (output) COMPLEX array, dimension (LDX,NB) The m-by-nb
	       matrix X required to update the unreduced part of A.

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

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

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

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

       where tauq and taup are complex scalars, and v and u are complex 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'  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' - X*U'.

       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  ele-
       ment of the vector defining H(i), and ui an element of the vector defining G(i).

LAPACK version 3.0			   15 June 2000 				CLABRD(l)


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