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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 transforma-
     tion 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, * )



     #include <sunperf.h>

     void clabrd(int m, int n, int  nb,  complex  *ca,  int  lda,
               float  *d, float *e, complex *tauq, complex *taup,
               complex *x, int ldx, complex *cy, int *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 transforma-
     tion 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,  dimen-
               sion  (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  unre-
               duced 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  elemen-
     tary 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).

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