sgebd2.f

          SUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, M, N
    *     ..
    *     .. Array Arguments ..
          REAL               A( LDA, * ), D( * ), E( * ), TAUP( * ),
         $                   TAUQ( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGEBD2 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) REAL 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) REAL array, dimension (min(M,N))
    *          The diagonal elements of the bidiagonal matrix B:
    *          D(i) = A(i,i).
    *
    *  E       (output) 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    (output) REAL array dimension (min(M,N))
    *          The scalar factors of the elementary reflectors which
    *          represent the orthogonal matrix Q. See Further Details.
    *
    *  TAUP    (output) REAL array, dimension (min(M,N))
    *          The scalar factors of the elementary reflectors which
    *          represent the orthogonal matrix P. See Further Details.
    *
    *  WORK    (workspace) REAL 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 ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SLARF, SLARFG, 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( 'SGEBD2', -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 SLARFG( 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 SLARF( '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 SLARFG( 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 SLARF( '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 SLARFG( 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 SLARF( '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 SLARFG( 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 SLARF( '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 SGEBD2
    *
          END