gebd2(3)

NAME
gebd2 - gebd2: reduction to bidiagonal, level 2

SYNOPSIS

  Functions

subroutine cgebd2 (m, n, a, lda, d, e, tauq, taup, work, info)
CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine dgebd2 (m, n, a, lda, d, e, tauq, taup, work, info)
DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine sgebd2 (m, n, a, lda, d, e, tauq, taup, work, info)
SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine zgebd2 (m, n, a, lda, d, e, tauq, taup, work, info)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Detailed Description

Function Documentation

  subroutine cgebd2 (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 info)

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

    !>
    !> CGEBD2 reduces a complex general m by n matrix A to upper or lower
    !> real 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(M,N))
    !>

    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, 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 dgebd2 (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 info)

DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

    !>
    !> DGEBD2 reduces a real general 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(M,N))
    !>

    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 sgebd2 (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 info)

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

    !>
    !> SGEBD2 reduces a real general 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(M,N))
    !>

    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 zgebd2 (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 info)

ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

    !>
    !> ZGEBD2 reduces a complex general m by n matrix A to upper or lower
    !> real 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(M,N))
    !>

    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, 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).
    !>

Author

Generated automatically by Doxygen for LAPACK from the source code.