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**T*(inv(T)*R)


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


 transpose of the matrix Z.

N

          N is INTEGER


          The number of rows of the matrices A and B. N >= 0.

M

          M is INTEGER


          The number of columns of the matrix A.  M >= 0.

P

          P is INTEGER


          The number of columns of the matrix B.  P >= 0.

A

          A is 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

          LDA is INTEGER


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

TAUA

          TAUA is REAL array, dimension (min(N,M))


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix Q (see Further Details).

B

          B is 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

          LDB is INTEGER


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

TAUB

          TAUB is REAL array, dimension (min(N,P))


          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 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.


          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 INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

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Univ. of Colorado Denver

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(n,m).


  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(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**T


  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.