dgebd2.f

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      SUBROUTINE dgebd2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. 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
*
            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
*
            CALL dlarf( 'Right', m-i, n-i+1, a( i, i ), lda, taup( i ),
     $                  a( min( i+1, m ), 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