NAME

       cgebd2.f -

SYNOPSIS

   Functions/Subroutines
       subroutine cgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
           CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Function/Subroutine Documentation

   subroutine cgebd2 (integerM, integerN, complex, dimension( lda, * )A, integerLDA, real,
       dimension( * )D, real, dimension( * )E, complex, dimension( * )TAUQ, complex, dimension( *
       )TAUP, complex, dimension( * )WORK, integerINFO)
       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.

       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 191 of file cgebd2.f.

Author

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