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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