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.

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.

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NAG Ltd.

December 2016

  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.