clabrd(3S)

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, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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