Chimie Théorique » Opt'n Path

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

*

*  -- LAPACK routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     November 2006

*

*     .. Scalar Arguments ..

      INTEGER            INFO, LDA, M, N

*     ..

*     .. Array Arguments ..

      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),

     $                   TAUQ( * ), WORK( * )

*     ..

*

*  Purpose

*  =======

*

*  DGEBD2 reduces a real general m by n matrix A to upper or lower

*  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.

*

*  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

*

*  Arguments

*  =========

*

*  M       (input) INTEGER

*          The number of rows in the matrix A.  M >= 0.

*

*  N       (input) INTEGER

*          The number of columns in the matrix A.  N >= 0.

*

*  A       (input/output) 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) INTEGER

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

*

*  D       (output) DOUBLE PRECISION array, dimension (min(M,N))

*          The diagonal elements of the bidiagonal matrix B:

*          D(i) = A(i,i).

*

*  E       (output) 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) 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) DOUBLE PRECISION array, dimension (min(M,N))

*          The scalar factors of the elementary reflectors which

*          represent the orthogonal matrix P. See Further Details.

*

*  WORK    (workspace) DOUBLE PRECISION array, dimension (max(M,N))

*

*  INFO    (output) 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'  and G(i) = I - taup * u * u'

*

*  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'  and G(i) = I - taup * u * u'

*

*  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).

*

*  =====================================================================

*

*     .. Parameters ..

      DOUBLE PRECISION   ZERO, ONE

      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )

*     ..

*     .. Local Scalars ..

      INTEGER            I

*     ..

*     .. External Subroutines ..

      EXTERNAL           DLARF, DLARFG, XERBLA

*     ..

*     .. Intrinsic Functions ..

      INTRINSIC          MAX, MIN

*     ..

*     .. Executable Statements ..

*

*     Test the input parameters

*

      INFO = 0

      IF( M.LT.0 ) THEN

         INFO = -1

      ELSE IF( N.LT.0 ) THEN

         INFO = -2

      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN

         INFO = -4

      END IF

      IF( INFO.LT.0 ) THEN

         CALL XERBLA( 'DGEBD2', -INFO )

         RETURN

      END IF

*

      IF( M.GE.N ) THEN

*

*        Reduce to upper bidiagonal form

*

         DO 10 I = 1, N

*

*           Generate elementary reflector H(i) to annihilate A(i+1:m,i)

*

            CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,

     $                   TAUQ( I ) )

            D( I ) = A( I, I )

            A( I, I ) = ONE

*

*           Apply H(i) to A(i:m,i+1:n) from the left

*

            IF( I.LT.N )

     $         CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),

     $                     A( I, I+1 ), LDA, WORK )

            A( I, I ) = D( I )

*

            IF( I.LT.N ) THEN

*

*              Generate elementary reflector G(i) to annihilate

*              A(i,i+2:n)

*

               CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),

     $                      LDA, TAUP( I ) )

               E( I ) = A( I, I+1 )

               A( I, I+1 ) = ONE

*

*              Apply G(i) to A(i+1:m,i+1:n) from the right

*

               CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,

     $                     TAUP( I ), A( I+1, I+1 ), LDA, WORK )

               A( I, I+1 ) = E( I )

            ELSE

               TAUP( I ) = ZERO

            END IF

   10    CONTINUE

      ELSE

*

*        Reduce to lower bidiagonal form

*

         DO 20 I = 1, M

*

*           Generate elementary reflector G(i) to annihilate A(i,i+1:n)

*

            CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,

     $                   TAUP( I ) )

            D( I ) = A( I, I )

            A( I, I ) = ONE

*

*           Apply G(i) to A(i+1:m,i:n) from the right

*

            IF( I.LT.M )

     $         CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,

     $                     TAUP( I ), A( I+1, I ), LDA, WORK )

            A( I, I ) = D( I )

*

            IF( I.LT.M ) THEN

*

*              Generate elementary reflector H(i) to annihilate

*              A(i+2:m,i)

*

               CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,

     $                      TAUQ( I ) )

               E( I ) = A( I+1, I )

               A( I+1, I ) = ONE

*

*              Apply H(i) to A(i+1:m,i+1:n) from the left

*

               CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),

     $                     A( I+1, I+1 ), LDA, WORK )

               A( I+1, I ) = E( I )

            ELSE

               TAUQ( I ) = ZERO

            END IF

   20    CONTINUE

      END IF

      RETURN

*

*     End of DGEBD2

*

      END