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## sggqrf()

subroutine sggqrf | ( | integer | n,
|  | integer | m,
|  | integer | p,
|  | real, dimension( lda, * ) | a,
|  | integer | lda,
|  | real, dimension( * ) | taua,
|  | real, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | real, dimension( * ) | taub,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer | info )

SGGQRF

Download SGGQRF + dependencies [TGZ] [ZIP] [TXT]

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

Parameters

  [in]

  N

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

  [in]

  M

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

  [in]

  P

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

  [in,out]

  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).
      !>

  [in]

  LDA

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

  [out]

  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).
      !>

  [in,out]

  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).
      !>

  [in]

  LDB

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

  [out]

  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).
      !>

  [out]

  WORK

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

  [in]

  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.
      !>

  [out]

  INFO

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

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

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**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.
      !>