ggqrf(3)

NAME
ggqrf - ggqrf: Generalized QR factor

SYNOPSIS

  Functions
subroutine cggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
CGGQRF
subroutine dggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
DGGQRF
subroutine sggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGQRF
subroutine zggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
ZGGQRF

Detailed Description

Function Documentation

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

CGGQRF

Purpose:

    !>
    !> CGGQRF 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 unitary matrix, Z is a P-by-P unitary 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**H * (inv(T)*R)
    !>
    !> where inv(B) denotes the inverse of the matrix B, and Z' denotes the
    !> conjugate transpose of matrix Z.
    !>

Parameters
    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 COMPLEX 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 unitary 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 COMPLEX array, dimension (min(N,M))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the unitary matrix Q (see Further Details).
    !>

    B

    !>          B is COMPLEX 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 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,N).
    !>

    TAUB

    !>          TAUB is COMPLEX array, dimension (min(N,P))
    !>          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 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 CUNMQR.
    !>
    !>          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(n,m).
    !>
    !>  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(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 CUNGQR.
    !>  To use Q to update another matrix, use LAPACK subroutine CUNMQR.
    !>
    !>  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**H
    !>
    !>  where taub is a complex scalar, and v is a complex 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 CUNGRQ.
    !>  To use Z to update another matrix, use LAPACK subroutine CUNMRQ.
    !>

Definition at line 213 of file cggqrf.f.

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

DGGQRF

Purpose:

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

    B

    !>          B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (min(N,P))
    !>          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 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 DORMQR.
    !>
    !>          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(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 DORGQR.
    !>  To use Q to update another matrix, use LAPACK subroutine DORMQR.
    !>
    !>  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 DORGRQ.
    !>  To use Z to update another matrix, use LAPACK subroutine DORMRQ.
    !>

Definition at line 213 of file dggqrf.f.

  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

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
    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.
    !>

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.
    !>

Definition at line 213 of file sggqrf.f.

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

ZGGQRF

Purpose:

    !>
    !> ZGGQRF 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 unitary matrix, Z is a P-by-P unitary 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**H * (inv(T)*R)
    !>
    !> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
    !> conjugate transpose of matrix Z.
    !>

Parameters
    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 COMPLEX*16 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 unitary 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 COMPLEX*16 array, dimension (min(N,M))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the unitary matrix Q (see Further Details).
    !>

    B

    !>          B is COMPLEX*16 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 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,N).
    !>

    TAUB

    !>          TAUB is COMPLEX*16 array, dimension (min(N,P))
    !>          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 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 ZUNMQR.
    !>
    !>          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(n,m).
    !>
    !>  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(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 ZUNGQR.
    !>  To use Q to update another matrix, use LAPACK subroutine ZUNMQR.
    !>
    !>  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**H
    !>
    !>  where taub is a complex scalar, and v is a complex 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 ZUNGRQ.
    !>  To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.
    !>

Definition at line 213 of file zggqrf.f.

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