ggrqf(3)

NAME
ggrqf - ggrqf: Generalized RQ factor

SYNOPSIS

  Functions
subroutine cggrqf (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
CGGRQF
subroutine dggrqf (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
DGGRQF
subroutine sggrqf (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGRQF
subroutine zggrqf (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
ZGGRQF

Detailed Description

Function Documentation

  subroutine cggrqf (integer m, integer p, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) taua, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) taub, complex, dimension( * ) work, integer lwork, integer info)

CGGRQF

Purpose:

     CGGRQF 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 unitary matrix, Z is a P-by-P unitary
     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**H
     where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
     conjugate 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 COMPLEX 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 unitary
              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 COMPLEX array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix Q (see Further Details).

    B

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

    WORK

              WORK is COMPLEX 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 CUNMRQ.
              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.

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(m,n).
      Each H(i) has the form
         H(i) = I - taua * v * v**H
      where taua is a complex scalar, and v is a complex 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 CUNGRQ.
      To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
      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**H
      where taub is a complex scalar, and v is a complex 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 CUNGQR.
      To use Z to update another matrix, use LAPACK subroutine CUNMQR.

Definition at line 212 of file cggrqf.f.

  subroutine dggrqf (integer m, integer p, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) taua, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) taub, double precision, dimension( * ) work, integer lwork, integer info)

DGGRQF

Purpose:

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

    B

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

    WORK

              WORK is DOUBLE PRECISION 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 DORMRQ.
              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.

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 DORGRQ.
      To use Q to update another matrix, use LAPACK subroutine DORMRQ.
      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 DORGQR.
      To use Z to update another matrix, use LAPACK subroutine DORMQR.

Definition at line 212 of file dggrqf.f.

  subroutine sggrqf (integer m, integer p, integer n, 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)

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.

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 212 of file sggrqf.f.

  subroutine zggrqf (integer m, integer p, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) taua, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) taub, complex*16, dimension( * ) work, integer lwork, integer info)

ZGGRQF

Purpose:

     ZGGRQF 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 unitary matrix, Z is a P-by-P unitary
     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**H
     where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
     conjugate 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 COMPLEX*16 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 unitary
              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 COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix Q (see Further Details).

    B

              B is COMPLEX*16 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 unitary 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 COMPLEX*16 array, dimension (min(P,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix Z (see Further Details).

    WORK

              WORK is COMPLEX*16 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 ZUNMRQ.
              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.

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(m,n).
      Each H(i) has the form
         H(i) = I - taua * v * v**H
      where taua is a complex scalar, and v is a complex 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 ZUNGRQ.
      To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.
      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**H
      where taub is a complex scalar, and v is a complex 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 ZUNGQR.
      To use Z to update another matrix, use LAPACK subroutine ZUNMQR.

Definition at line 212 of file zggrqf.f.

Author

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