gebrd(3)

NAME
gebrd - gebrd: reduction to bidiagonal

SYNOPSIS

  Functions

subroutine cgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
CGEBRD
subroutine dgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
DGEBRD
subroutine sgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
SGEBRD
subroutine zgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
ZGEBRD

Detailed Description

Function Documentation

  subroutine cgebrd (integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( * ) work, integer lwork, integer info)

CGEBRD

Purpose:

    !>
    !> CGEBRD reduces a general complex M-by-N matrix A to upper or lower
    !> bidiagonal form B by a unitary transformation: Q**H * A * P = B.
    !>
    !> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
    !>

Parameters
    M

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

    N

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

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the M-by-N general matrix to be reduced.
    !>          On exit,
    !>          if m >= n, the diagonal and the first superdiagonal are
    !>            overwritten with the upper bidiagonal matrix B; the
    !>            elements below the diagonal, with the array TAUQ, represent
    !>            the unitary matrix Q as a product of elementary
    !>            reflectors, and the elements above the first superdiagonal,
    !>            with the array TAUP, represent the unitary matrix P as
    !>            a product of elementary reflectors;
    !>          if m < n, the diagonal and the first subdiagonal are
    !>            overwritten with the lower bidiagonal matrix B; the
    !>            elements below the first subdiagonal, with the array TAUQ,
    !>            represent the unitary matrix Q as a product of
    !>            elementary reflectors, and the elements above the diagonal,
    !>            with the array TAUP, represent the unitary matrix P as
    !>            a product of elementary reflectors.
    !>          See Further Details.
    !>

    LDA

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

    D

    !>          D is REAL array, dimension (min(M,N))
    !>          The diagonal elements of the bidiagonal matrix B:
    !>          D(i) = A(i,i).
    !>

    E

    !>          E is REAL array, dimension (min(M,N)-1)
    !>          The off-diagonal elements of the bidiagonal matrix B:
    !>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
    !>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is COMPLEX array, dimension (min(M,N))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the unitary matrix P. 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 length of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
    !>          For optimum performance LWORK >= (M+N)*NB, where NB
    !>          is the optimal blocksize.
    !>
    !>          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 matrices Q and P are represented as products of elementary
    !>  reflectors:
    !>
    !>  If m >= n,
    !>
    !>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
    !>
    !>  where tauq and taup are complex scalars, and v and u are complex
    !>  vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
    !>  A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
    !>  A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  If m < n,
    !>
    !>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
    !>
    !>  where tauq and taup are complex scalars, and v and u are complex
    !>  vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
    !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
    !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The contents of A on exit are illustrated by the following examples:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    !>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    !>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    !>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    !>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    !>    (  v1  v2  v3  v4  v5 )
    !>
    !>  where d and e denote diagonal and off-diagonal elements of B, vi
    !>  denotes an element of the vector defining H(i), and ui an element of
    !>  the vector defining G(i).
    !>

  subroutine dgebrd (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tauq, double precision, dimension( * ) taup, double precision, dimension( * ) work, integer lwork, integer info)

DGEBRD

Purpose:

    !>
    !> DGEBRD reduces a general real M-by-N matrix A to upper or lower
    !> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
    !>
    !> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
    !>

Parameters
    M

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

    N

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

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the M-by-N general matrix to be reduced.
    !>          On exit,
    !>          if m >= n, the diagonal and the first superdiagonal are
    !>            overwritten with the upper bidiagonal matrix B; the
    !>            elements below the diagonal, with the array TAUQ, represent
    !>            the orthogonal matrix Q as a product of elementary
    !>            reflectors, and the elements above the first superdiagonal,
    !>            with the array TAUP, represent the orthogonal matrix P as
    !>            a product of elementary reflectors;
    !>          if m < n, the diagonal and the first subdiagonal are
    !>            overwritten with the lower bidiagonal matrix B; the
    !>            elements below the first subdiagonal, with the array TAUQ,
    !>            represent the orthogonal matrix Q as a product of
    !>            elementary reflectors, and the elements above the diagonal,
    !>            with the array TAUP, represent the orthogonal matrix P as
    !>            a product of elementary reflectors.
    !>          See Further Details.
    !>

    LDA

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

    D

    !>          D is DOUBLE PRECISION array, dimension (min(M,N))
    !>          The diagonal elements of the bidiagonal matrix B:
    !>          D(i) = A(i,i).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
    !>          The off-diagonal elements of the bidiagonal matrix B:
    !>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
    !>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is DOUBLE PRECISION array, dimension (min(M,N))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix P. 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 length of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
    !>          For optimum performance LWORK >= (M+N)*NB, where NB
    !>          is the optimal blocksize.
    !>
    !>          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 matrices Q and P are represented as products of elementary
    !>  reflectors:
    !>
    !>  If m >= n,
    !>
    !>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
    !>
    !>  where tauq and taup are real scalars, and v and u are real vectors;
    !>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
    !>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
    !>  tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  If m < n,
    !>
    !>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
    !>
    !>  where tauq and taup are real scalars, and v and u are real vectors;
    !>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
    !>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
    !>  tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The contents of A on exit are illustrated by the following examples:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    !>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    !>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    !>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    !>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    !>    (  v1  v2  v3  v4  v5 )
    !>
    !>  where d and e denote diagonal and off-diagonal elements of B, vi
    !>  denotes an element of the vector defining H(i), and ui an element of
    !>  the vector defining G(i).
    !>

  subroutine sgebrd (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tauq, real, dimension( * ) taup, real, dimension( * ) work, integer lwork, integer info)

SGEBRD

Purpose:

    !>
    !> SGEBRD reduces a general real M-by-N matrix A to upper or lower
    !> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
    !>
    !> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
    !>

Parameters
    M

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

    N

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

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the M-by-N general matrix to be reduced.
    !>          On exit,
    !>          if m >= n, the diagonal and the first superdiagonal are
    !>            overwritten with the upper bidiagonal matrix B; the
    !>            elements below the diagonal, with the array TAUQ, represent
    !>            the orthogonal matrix Q as a product of elementary
    !>            reflectors, and the elements above the first superdiagonal,
    !>            with the array TAUP, represent the orthogonal matrix P as
    !>            a product of elementary reflectors;
    !>          if m < n, the diagonal and the first subdiagonal are
    !>            overwritten with the lower bidiagonal matrix B; the
    !>            elements below the first subdiagonal, with the array TAUQ,
    !>            represent the orthogonal matrix Q as a product of
    !>            elementary reflectors, and the elements above the diagonal,
    !>            with the array TAUP, represent the orthogonal matrix P as
    !>            a product of elementary reflectors.
    !>          See Further Details.
    !>

    LDA

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

    D

    !>          D is REAL array, dimension (min(M,N))
    !>          The diagonal elements of the bidiagonal matrix B:
    !>          D(i) = A(i,i).
    !>

    E

    !>          E is REAL array, dimension (min(M,N)-1)
    !>          The off-diagonal elements of the bidiagonal matrix B:
    !>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
    !>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
    !>

    TAUQ

    !>          TAUQ is REAL array, dimension (min(M,N))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix Q. See Further Details.
    !>

    TAUP

    !>          TAUP is REAL array, dimension (min(M,N))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix P. 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 length of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
    !>          For optimum performance LWORK >= (M+N)*NB, where NB
    !>          is the optimal blocksize.
    !>
    !>          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 matrices Q and P are represented as products of elementary
    !>  reflectors:
    !>
    !>  If m >= n,
    !>
    !>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
    !>
    !>  where tauq and taup are real scalars, and v and u are real vectors;
    !>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
    !>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
    !>  tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  If m < n,
    !>
    !>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
    !>
    !>  where tauq and taup are real scalars, and v and u are real vectors;
    !>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
    !>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
    !>  tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The contents of A on exit are illustrated by the following examples:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    !>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    !>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    !>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    !>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    !>    (  v1  v2  v3  v4  v5 )
    !>
    !>  where d and e denote diagonal and off-diagonal elements of B, vi
    !>  denotes an element of the vector defining H(i), and ui an element of
    !>  the vector defining G(i).
    !>

  subroutine zgebrd (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tauq, complex*16, dimension( * ) taup, complex*16, dimension( * ) work, integer lwork, integer info)

ZGEBRD

Purpose:

    !>
    !> ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
    !> bidiagonal form B by a unitary transformation: Q**H * A * P = B.
    !>
    !> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
    !>

Parameters
    M

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

    N

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

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the M-by-N general matrix to be reduced.
    !>          On exit,
    !>          if m >= n, the diagonal and the first superdiagonal are
    !>            overwritten with the upper bidiagonal matrix B; the
    !>            elements below the diagonal, with the array TAUQ, represent
    !>            the unitary matrix Q as a product of elementary
    !>            reflectors, and the elements above the first superdiagonal,
    !>            with the array TAUP, represent the unitary matrix P as
    !>            a product of elementary reflectors;
    !>          if m < n, the diagonal and the first subdiagonal are
    !>            overwritten with the lower bidiagonal matrix B; the
    !>            elements below the first subdiagonal, with the array TAUQ,
    !>            represent the unitary matrix Q as a product of
    !>            elementary reflectors, and the elements above the diagonal,
    !>            with the array TAUP, represent the unitary matrix P as
    !>            a product of elementary reflectors.
    !>          See Further Details.
    !>

    LDA

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

    D

    !>          D is DOUBLE PRECISION array, dimension (min(M,N))
    !>          The diagonal elements of the bidiagonal matrix B:
    !>          D(i) = A(i,i).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
    !>          The off-diagonal elements of the bidiagonal matrix B:
    !>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
    !>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is COMPLEX*16 array, dimension (min(M,N))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the unitary matrix P. 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 length of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
    !>          For optimum performance LWORK >= (M+N)*NB, where NB
    !>          is the optimal blocksize.
    !>
    !>          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 matrices Q and P are represented as products of elementary
    !>  reflectors:
    !>
    !>  If m >= n,
    !>
    !>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
    !>
    !>  where tauq and taup are complex scalars, and v and u are complex
    !>  vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
    !>  A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
    !>  A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  If m < n,
    !>
    !>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
    !>
    !>  Each H(i) and G(i) has the form:
    !>
    !>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
    !>
    !>  where tauq and taup are complex scalars, and v and u are complex
    !>  vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
    !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
    !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The contents of A on exit are illustrated by the following examples:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    !>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    !>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    !>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    !>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    !>    (  v1  v2  v3  v4  v5 )
    !>
    !>  where d and e denote diagonal and off-diagonal elements of B, vi
    !>  denotes an element of the vector defining H(i), and ui an element of
    !>  the vector defining G(i).
    !>

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