SGGRQF (3lapack)

compute a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B

SYNOPSIS

  SUBROUTINE SGGRQF( M, P, N, 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

  SGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a
  P-by-N matrix B:

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

  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 M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                   N-M  M                           ( R21 ) N
                                                       N

  where R12 or R21 is upper triangular, and

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

  where T11 is upper triangular.

  In particular, if B is square and nonsingular, the GRQ factorization of A
  and B implicitly gives the RQ factorization of A*inv(B):

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

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

ARGUMENTS

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

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

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

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.  On exit, if M <= N, the upper
          triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper
          triangular matrix R; if M > N, the elements on and above the (M-
          N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R;
          the remaining elements, with the array TAUA, represent the
          orthogonal matrix Q as a product of elementary reflectors (see
          Further Details).

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

  TAUA    (output) REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which represent the
          orthogonal matrix Q (see Further Details).  B       (input/output)
          REAL array, dimension (LDB,N) On entry, the P-by-N matrix B.  On
          exit, the elements on and above the diagonal of the array contain
          the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular
          if P >= N); the elements below the diagonal, 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,P).

  TAUB    (output) REAL array, dimension (min(P,N))
          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 RQ factorization of an M-by-N
          matrix, NB2 is the optimal blocksize for the QR factorization of a
          P-by-N matrix, and NB3 is the optimal blocksize for a call of
          SORMRQ.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INF0= -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(m,n).

  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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(m-
  k+i,1:n-k+i-1), and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine SORGRQ.
  To use Q to update another matrix, use LAPACK subroutine SORMRQ.

  The matrix Z is represented as a product of elementary reflectors

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

  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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), and
  taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine SORGQR.
  To use Z to update another matrix, use LAPACK subroutine SORMQR.