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## dgebd2()

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.

Download DGEBD2 + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  M

      !>          M is INTEGER
      !>          The number of rows in the matrix A.  M >= 0.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The number of columns in the matrix A.  N >= 0.
      !>

  [in,out]

  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.
      !>

  [in]

  LDA

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

  [out]

  D

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

  [out]

  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.
      !>

  [out]

  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.
      !>

  [out]

  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.
      !>

  [out]

  WORK

      !>          WORK is DOUBLE PRECISION array, dimension (max(M,N))
      !>

  [out]

  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).
      !>