zgebd2.f(3)

NAME
zgebd2.f -

SYNOPSIS

  Functions/Subroutines

subroutine zgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Function/Subroutine Documentation

  subroutine zgebd2 (integerM, integerN, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( * )TAUQ, complex*16, dimension( * )TAUP, complex*16, dimension( * )WORK, integerINFO)

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.

Date:
    September 2012

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).

Definition at line 190 of file zgebd2.f.

Author

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