sggrqf.f - Man Page

subroutine sggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
SGGRQF

SGGRQF  

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**T

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

Parameters:

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B. N >= 0.

A

          A is 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

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

TAUA

          TAUA is REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

B

          B is 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

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

TAUB

          TAUB is REAL array, dimension (min(P,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is 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.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INF0= -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

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**T

  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**T

  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.

Definition at line 216 of file sggrqf.f.