CLABRD man page on IRIX

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CLABRD(3F)							    CLABRD(3F)

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

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CLABRD(3F)							    CLABRD(3F)

	     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'.

									Page 2

CLABRD(3F)							    CLABRD(3F)

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

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