SGGQRF (3lapack)

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( *
                     )

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 factorization 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 elements on and above
          the diagonal of the array contain 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 illegal 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.