DGEBD2(3)      LAPACK routine of NEC Numeric Library Collection      DGEBD2(3)



NAME
       DGEBD2

SYNOPSIS
       SUBROUTINE DGEBD2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)



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.




ARGUMENTS
           M         (input)
                     M is INTEGER
                     The number of rows in the matrix A.  M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns in the matrix A.  N >= 0.

           A         (input/output)
                     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       (input)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           D         (output)
                     D is DOUBLE PRECISION array, dimension (min(M,N))
                     The diagonal elements of the bidiagonal matrix B:
                     D(i) = A(i,i).

           E         (output)
                     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      (output)
                     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      (output)
                     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      (output)
                     WORK is DOUBLE PRECISION array, dimension (max(M,N))

           INFO      (output)
                     INFO is INTEGER
                     = 0: successful exit.
                     < 0: if INFO = -i, the i-th argument had an illegal value.






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).



LAPACK routine                  31 October 2017                      DGEBD2(3)