labrd(3)

NAME
labrd - labrd: step in gebrd

SYNOPSIS

  Functions

subroutine clabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine dlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine slabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine zlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Detailed Description

Function Documentation

  subroutine clabrd (integer m, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( ldx, * ) x, integer ldx, complex, dimension( ldy, * ) y, integer ldy)

CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

    !>
    !> CLABRD reduces the first NB rows and columns of a complex general
    !> m by n matrix A to upper or lower real bidiagonal form by a unitary
    !> transformation Q**H * A * P, and returns the matrices X and Y which
    !> are needed to apply the transformation to the unreduced part of A.
    !>
    !> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
    !> bidiagonal form.
    !>
    !> This is an auxiliary routine called by CGEBRD
    !>

Parameters
    M

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

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of leading rows and columns of A to be reduced.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the m by n general matrix to be reduced.
    !>          On exit, the first NB rows and columns of the matrix are
    !>          overwritten; the rest of the array is unchanged.
    !>          If m >= n, elements on and below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the unitary
    !>            matrix Q as a product of elementary reflectors; and
    !>            elements above the diagonal in the first NB rows, with the
    !>            array TAUP, represent the unitary matrix P as a product
    !>            of elementary reflectors.
    !>          If m < n, elements below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the unitary
    !>            matrix Q as a product of elementary reflectors, and
    !>            elements on and above the diagonal in the first NB rows,
    !>            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 (NB)
    !>          The diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.  D(i) = A(i,i).
    !>

    E

    !>          E is REAL array, dimension (NB)
    !>          The off-diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is COMPLEX array, dimension (NB)
    !>          The scalar factors of the elementary reflectors which
    !>          represent the unitary matrix P. See Further Details.
    !>

    X

    !>          X is COMPLEX array, dimension (LDX,NB)
    !>          The m-by-nb matrix X required to update the unreduced part
    !>          of A.
    !>

    LDX

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

    Y

    !>          Y is COMPLEX array, dimension (LDY,NB)
    !>          The n-by-nb matrix Y required to update the unreduced part
    !>          of A.
    !>

    LDY

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

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:
    !>
    !>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
    !>
    !>  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.
    !>
    !>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
    !>  A(i:m,i); u(1:i) = 0, u(i+1) = 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).
    !>
    !>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
    !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
    !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The elements of the vectors v and u together form the m-by-nb matrix
    !>  V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
    !>  the transformation to the unreduced part of the matrix, using a block
    !>  update of the form:  A := A - V*Y**H - X*U**H.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with nb = 2:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    !>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    !>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )
    !>
    !>  where a denotes an element of the original matrix which is unchanged,
    !>  vi denotes an element of the vector defining H(i), and ui an element
    !>  of the vector defining G(i).
    !>

  subroutine dlabrd (integer m, integer n, integer nb, 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( ldx, * ) x, integer ldx, double precision, dimension( ldy, * ) y, integer ldy)

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

    !>
    !> DLABRD reduces the first NB rows and columns of a real general
    !> m by n matrix A to upper or lower bidiagonal form by an orthogonal
    !> transformation Q**T * A * P, and returns the matrices X and Y which
    !> are needed to apply the transformation to the unreduced part of A.
    !>
    !> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
    !> bidiagonal form.
    !>
    !> This is an auxiliary routine called by DGEBRD
    !>

Parameters
    M

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

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of leading rows and columns of A to be reduced.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the m by n general matrix to be reduced.
    !>          On exit, the first NB rows and columns of the matrix are
    !>          overwritten; the rest of the array is unchanged.
    !>          If m >= n, elements on and below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the orthogonal
    !>            matrix Q as a product of elementary reflectors; and
    !>            elements above the diagonal in the first NB rows, with the
    !>            array TAUP, represent the orthogonal matrix P as a product
    !>            of elementary reflectors.
    !>          If m < n, elements below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the orthogonal
    !>            matrix Q as a product of elementary reflectors, and
    !>            elements on and above the diagonal in the first NB rows,
    !>            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 (NB)
    !>          The diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.  D(i) = A(i,i).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (NB)
    !>          The off-diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is DOUBLE PRECISION array, dimension (NB)
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix P. See Further Details.
    !>

    X

    !>          X is DOUBLE PRECISION array, dimension (LDX,NB)
    !>          The m-by-nb matrix X required to update the unreduced part
    !>          of A.
    !>

    LDX

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

    Y

    !>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
    !>          The n-by-nb matrix Y required to update the unreduced part
    !>          of A.
    !>

    LDY

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

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:
    !>
    !>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
    !>
    !>  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.
    !>
    !>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
    !>  A(i:m,i); u(1:i) = 0, u(i+1) = 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).
    !>
    !>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
    !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
    !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The elements of the vectors v and u together form the m-by-nb matrix
    !>  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
    !>  the transformation to the unreduced part of the matrix, using a block
    !>  update of the form:  A := A - V*Y**T - X*U**T.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with nb = 2:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    !>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    !>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )
    !>
    !>  where a denotes an element of the original matrix which is unchanged,
    !>  vi denotes an element of the vector defining H(i), and ui an element
    !>  of the vector defining G(i).
    !>

  subroutine slabrd (integer m, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tauq, real, dimension( * ) taup, real, dimension( ldx, * ) x, integer ldx, real, dimension( ldy, * ) y, integer ldy)

SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

    !>
    !> SLABRD reduces the first NB rows and columns of a real general
    !> m by n matrix A to upper or lower bidiagonal form by an orthogonal
    !> transformation Q**T * A * P, and returns the matrices X and Y which
    !> are needed to apply the transformation to the unreduced part of A.
    !>
    !> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
    !> bidiagonal form.
    !>
    !> This is an auxiliary routine called by SGEBRD
    !>

Parameters
    M

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

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of leading rows and columns of A to be reduced.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the m by n general matrix to be reduced.
    !>          On exit, the first NB rows and columns of the matrix are
    !>          overwritten; the rest of the array is unchanged.
    !>          If m >= n, elements on and below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the orthogonal
    !>            matrix Q as a product of elementary reflectors; and
    !>            elements above the diagonal in the first NB rows, with the
    !>            array TAUP, represent the orthogonal matrix P as a product
    !>            of elementary reflectors.
    !>          If m < n, elements below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the orthogonal
    !>            matrix Q as a product of elementary reflectors, and
    !>            elements on and above the diagonal in the first NB rows,
    !>            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 (NB)
    !>          The diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.  D(i) = A(i,i).
    !>

    E

    !>          E is REAL array, dimension (NB)
    !>          The off-diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is REAL array, dimension (NB)
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix P. See Further Details.
    !>

    X

    !>          X is REAL array, dimension (LDX,NB)
    !>          The m-by-nb matrix X required to update the unreduced part
    !>          of A.
    !>

    LDX

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

    Y

    !>          Y is REAL array, dimension (LDY,NB)
    !>          The n-by-nb matrix Y required to update the unreduced part
    !>          of A.
    !>

    LDY

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

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:
    !>
    !>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
    !>
    !>  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.
    !>
    !>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
    !>  A(i:m,i); u(1:i) = 0, u(i+1) = 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).
    !>
    !>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
    !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
    !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The elements of the vectors v and u together form the m-by-nb matrix
    !>  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
    !>  the transformation to the unreduced part of the matrix, using a block
    !>  update of the form:  A := A - V*Y**T - X*U**T.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with nb = 2:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    !>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    !>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )
    !>
    !>  where a denotes an element of the original matrix which is unchanged,
    !>  vi denotes an element of the vector defining H(i), and ui an element
    !>  of the vector defining G(i).
    !>

  subroutine zlabrd (integer m, integer n, integer nb, 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( ldx, * ) x, integer ldx, complex*16, dimension( ldy, * ) y, integer ldy)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

    !>
    !> ZLABRD reduces the first NB rows and columns of a complex general
    !> m by n matrix A to upper or lower real bidiagonal form by a unitary
    !> transformation Q**H * A * P, and returns the matrices X and Y which
    !> are needed to apply the transformation to the unreduced part of A.
    !>
    !> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
    !> bidiagonal form.
    !>
    !> This is an auxiliary routine called by ZGEBRD
    !>

Parameters
    M

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

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of leading rows and columns of A to be reduced.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the m by n general matrix to be reduced.
    !>          On exit, the first NB rows and columns of the matrix are
    !>          overwritten; the rest of the array is unchanged.
    !>          If m >= n, elements on and below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the unitary
    !>            matrix Q as a product of elementary reflectors; and
    !>            elements above the diagonal in the first NB rows, with the
    !>            array TAUP, represent the unitary matrix P as a product
    !>            of elementary reflectors.
    !>          If m < n, elements below the diagonal in the first NB
    !>            columns, with the array TAUQ, represent the unitary
    !>            matrix Q as a product of elementary reflectors, and
    !>            elements on and above the diagonal in the first NB rows,
    !>            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 (NB)
    !>          The diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.  D(i) = A(i,i).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (NB)
    !>          The off-diagonal elements of the first NB rows and columns of
    !>          the reduced matrix.
    !>

    TAUQ

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

    TAUP

    !>          TAUP is COMPLEX*16 array, dimension (NB)
    !>          The scalar factors of the elementary reflectors which
    !>          represent the unitary matrix P. See Further Details.
    !>

    X

    !>          X is COMPLEX*16 array, dimension (LDX,NB)
    !>          The m-by-nb matrix X required to update the unreduced part
    !>          of A.
    !>

    LDX

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

    Y

    !>          Y is COMPLEX*16 array, dimension (LDY,NB)
    !>          The n-by-nb matrix Y required to update the unreduced part
    !>          of A.
    !>

    LDY

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

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:
    !>
    !>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
    !>
    !>  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.
    !>
    !>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
    !>  A(i:m,i); u(1:i) = 0, u(i+1) = 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).
    !>
    !>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
    !>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
    !>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
    !>
    !>  The elements of the vectors v and u together form the m-by-nb matrix
    !>  V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
    !>  the transformation to the unreduced part of the matrix, using a block
    !>  update of the form:  A := A - V*Y**H - X*U**H.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with nb = 2:
    !>
    !>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    !>
    !>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    !>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    !>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    !>    (  v1  v2  a   a   a  )
    !>
    !>  where a denotes an element of the original matrix which is unchanged,
    !>  vi denotes an element of the vector defining H(i), and ui an element
    !>  of the vector defining G(i).
    !>

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