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NAME
     sggqrf - compute a generalized QR factorization of an N-by-M
     matrix A and an N-by-P matrix B

SYNOPSIS
     SUBROUTINE SGGQRF( N, M, P, A,  LDA,  TAUA,  B,  LDB,  TAUB,
               WORK, LWORK, INFO )

     INTEGER INFO, LDA, LDB, LWORK, M, N, P

     REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( *
               )



     #include <sunperf.h>

     void sggqrf(int n, int m, int p, float *sa, int  lda,  float
               *taua, float *sb, int ldb, float *taub, int *info)
               ;

PURPOSE
     SGGQRF computes a generalized QR factorization of an  N-by-M
     matrix A and an N-by-P matrix B:

                 A = Q*R,        B = Q*T*Z,

     where Q is an  N-by-N  orthogonal  matrix,  Z  is  a  P-by-P
     orthogonal matrix, and R and T assume one of the forms:

     if N>=M, R = ( R11 ) M  , or if N < M,  R = ( R11  R12 ) N,
                  (  0  ) N-M                       N   M-N
                     M

     where R11 is upper triangular, and

     if N<=P,  T = ( 0  T12 ) N, or if N > P,  T = ( T11 ) N-P,
                    P-N  N                         ( T21 ) P
                                                      P

     where T12 or T21 is upper triangular.

     In particular, if B is square and nonsingular, the GQR  fac-
     torization  of A and B implicitly gives the QR factorization
     of inv(B)*A:

                  inv(B)*A = Z'*(inv(T)*R)

     where inv(B) denotes the inverse of the  matrix  B,  and  Z'
     denotes the transpose of the matrix Z.


ARGUMENTS
     N         (input) INTEGER
               The number of rows of the matrices A and B.  N  >=
               0.

     M         (input) INTEGER
               The number of columns of the matrix A.  M >= 0.

     P         (input) INTEGER
               The number of columns of the matrix B.  P >= 0.

     A         (input/output) REAL array, dimension (LDA,M)
               On entry, the N-by-M matrix A.  On exit, the  ele-
               ments  on and above the diagonal of the array con-
               tain the min(N,M)-by-M upper trapezoidal matrix  R
               (R  is  upper  triangular if N >= M); the elements
               below the diagonal, with the array TAUA, represent
               the  orthogonal  matrix Q as a product of min(N,M)
               elementary reflectors (see Further Details).

     LDA       (input) INTEGER
               The leading dimension  of  the  array  A.  LDA  >=
               max(1,N).

     TAUA      (output) REAL array, dimension (min(N,M))
               The scalar factors of  the  elementary  reflectors
               which  represent  the  orthogonal  matrix  Q  (see
               Further  Details).   B        (input/output)  REAL
               array,  dimension  (LDB,P)  On  entry,  the N-by-P
               matrix B.  On exit, if N <= P, the upper  triangle
               of the subarray B(1:N,P-N+1:P) contains the N-by-N
               upper triangular matrix T; if N > P, the  elements
               on  and above the (N-P)-th subdiagonal contain the
               N-by-P upper trapezoidal matrix T;  the  remaining
               elements,  with  the  array  TAUB,  represent  the
               orthogonal matrix Z as  a  product  of  elementary
               reflectors (see Further Details).

     LDB       (input) INTEGER
               The leading dimension  of  the  array  B.  LDB  >=
               max(1,N).

     TAUB      (output) REAL array, dimension (min(N,P))
               The scalar factors of  the  elementary  reflectors
               which  represent  the  orthogonal  matrix  Z  (see
               Further Details).  WORK    (workspace/output) REAL
               array,  dimension  (LWORK)  On  exit, if INFO = 0,
               WORK(1) returns the optimal LWORK.

     LWORK     (input) INTEGER
               The  dimension  of  the  array  WORK.   LWORK   >=
               max(1,N,M,P).   For  optimum  performance LWORK >=
               max(N,M,P)*max(NB1,NB2,NB3),  where  NB1  is   the
               optimal  blocksize  for the QR factorization of an
               N-by-M matrix, NB2 is the  optimal  blocksize  for
               the  RQ factorization of an N-by-P matrix, and NB3
               is the optimal blocksize for a call of SORMQR.

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value.

FURTHER DETAILS
     The matrix Q is  represented  as  a  product  of  elementary
     reflectors

        Q = H(1) H(2) . . . H(k), where k = min(n,m).

     Each H(i) has the form

        H(i) = I - taua * v * v'

     where taua is a real scalar, and v is a real vector with
     v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is  stored  on  exit  in
     A(i+1:n,i), and taua in TAUA(i).
     To form Q explicitly, use LAPACK subroutine SORGQR.
     To use Q to update another  matrix,  use  LAPACK  subroutine
     SORMQR.

     The matrix Z is  represented  as  a  product  of  elementary
     reflectors

        Z = H(1) H(2) . . . H(k), where k = min(n,p).

     Each H(i) has the form

        H(i) = I - taub * v * v'

     where taub is a real scalar, and v is a real vector with
     v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on
     exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
     To form Z explicitly, use LAPACK subroutine SORGRQ.
     To use Z to update another  matrix,  use  LAPACK  subroutine
     SORMRQ.

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