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    1:       SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            INFO, LDA, M, N
   10: *     ..
   11: *     .. Array Arguments ..
   12:       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
   13:      $                   TAUQ( * ), WORK( * )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  DGEBD2 reduces a real general m by n matrix A to upper or lower
   20: *  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.
   21: *
   22: *  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
   23: *
   24: *  Arguments
   25: *  =========
   26: *
   27: *  M       (input) INTEGER
   28: *          The number of rows in the matrix A.  M >= 0.
   29: *
   30: *  N       (input) INTEGER
   31: *          The number of columns in the matrix A.  N >= 0.
   32: *
   33: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
   34: *          On entry, the m by n general matrix to be reduced.
   35: *          On exit,
   36: *          if m >= n, the diagonal and the first superdiagonal are
   37: *            overwritten with the upper bidiagonal matrix B; the
   38: *            elements below the diagonal, with the array TAUQ, represent
   39: *            the orthogonal matrix Q as a product of elementary
   40: *            reflectors, and the elements above the first superdiagonal,
   41: *            with the array TAUP, represent the orthogonal matrix P as
   42: *            a product of elementary reflectors;
   43: *          if m < n, the diagonal and the first subdiagonal are
   44: *            overwritten with the lower bidiagonal matrix B; the
   45: *            elements below the first subdiagonal, with the array TAUQ,
   46: *            represent the orthogonal matrix Q as a product of
   47: *            elementary reflectors, and the elements above the diagonal,
   48: *            with the array TAUP, represent the orthogonal matrix P as
   49: *            a product of elementary reflectors.
   50: *          See Further Details.
   51: *
   52: *  LDA     (input) INTEGER
   53: *          The leading dimension of the array A.  LDA >= max(1,M).
   54: *
   55: *  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
   56: *          The diagonal elements of the bidiagonal matrix B:
   57: *          D(i) = A(i,i).
   58: *
   59: *  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
   60: *          The off-diagonal elements of the bidiagonal matrix B:
   61: *          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
   62: *          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
   63: *
   64: *  TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
   65: *          The scalar factors of the elementary reflectors which
   66: *          represent the orthogonal matrix Q. See Further Details.
   67: *
   68: *  TAUP    (output) DOUBLE PRECISION array, dimension (min(M,N))
   69: *          The scalar factors of the elementary reflectors which
   70: *          represent the orthogonal matrix P. See Further Details.
   71: *
   72: *  WORK    (workspace) DOUBLE PRECISION array, dimension (max(M,N))
   73: *
   74: *  INFO    (output) INTEGER
   75: *          = 0: successful exit.
   76: *          < 0: if INFO = -i, the i-th argument had an illegal value.
   77: *
   78: *  Further Details
   79: *  ===============
   80: *
   81: *  The matrices Q and P are represented as products of elementary
   82: *  reflectors:
   83: *
   84: *  If m >= n,
   85: *
   86: *     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
   87: *
   88: *  Each H(i) and G(i) has the form:
   89: *
   90: *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
   91: *
   92: *  where tauq and taup are real scalars, and v and u are real vectors;
   93: *  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
   94: *  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
   95: *  tauq is stored in TAUQ(i) and taup in TAUP(i).
   96: *
   97: *  If m < n,
   98: *
   99: *     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
  100: *
  101: *  Each H(i) and G(i) has the form:
  102: *
  103: *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
  104: *
  105: *  where tauq and taup are real scalars, and v and u are real vectors;
  106: *  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
  107: *  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
  108: *  tauq is stored in TAUQ(i) and taup in TAUP(i).
  109: *
  110: *  The contents of A on exit are illustrated by the following examples:
  111: *
  112: *  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
  113: *
  114: *    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
  115: *    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
  116: *    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
  117: *    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
  118: *    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
  119: *    (  v1  v2  v3  v4  v5 )
  120: *
  121: *  where d and e denote diagonal and off-diagonal elements of B, vi
  122: *  denotes an element of the vector defining H(i), and ui an element of
  123: *  the vector defining G(i).
  124: *
  125: *  =====================================================================
  126: *
  127: *     .. Parameters ..
  128:       DOUBLE PRECISION   ZERO, ONE
  129:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  130: *     ..
  131: *     .. Local Scalars ..
  132:       INTEGER            I
  133: *     ..
  134: *     .. External Subroutines ..
  135:       EXTERNAL           DLARF, DLARFG, XERBLA
  136: *     ..
  137: *     .. Intrinsic Functions ..
  138:       INTRINSIC          MAX, MIN
  139: *     ..
  140: *     .. Executable Statements ..
  141: *
  142: *     Test the input parameters
  143: *
  144:       INFO = 0
  145:       IF( M.LT.0 ) THEN
  146:          INFO = -1
  147:       ELSE IF( N.LT.0 ) THEN
  148:          INFO = -2
  149:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  150:          INFO = -4
  151:       END IF
  152:       IF( INFO.LT.0 ) THEN
  153:          CALL XERBLA( 'DGEBD2', -INFO )
  154:          RETURN
  155:       END IF
  156: *
  157:       IF( M.GE.N ) THEN
  158: *
  159: *        Reduce to upper bidiagonal form
  160: *
  161:          DO 10 I = 1, N
  162: *
  163: *           Generate elementary reflector H(i) to annihilate A(i+1:m,i)
  164: *
  165:             CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
  166:      $                   TAUQ( I ) )
  167:             D( I ) = A( I, I )
  168:             A( I, I ) = ONE
  169: *
  170: *           Apply H(i) to A(i:m,i+1:n) from the left
  171: *
  172:             IF( I.LT.N )
  173:      $         CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),
  174:      $                     A( I, I+1 ), LDA, WORK )
  175:             A( I, I ) = D( I )
  176: *
  177:             IF( I.LT.N ) THEN
  178: *
  179: *              Generate elementary reflector G(i) to annihilate
  180: *              A(i,i+2:n)
  181: *
  182:                CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
  183:      $                      LDA, TAUP( I ) )
  184:                E( I ) = A( I, I+1 )
  185:                A( I, I+1 ) = ONE
  186: *
  187: *              Apply G(i) to A(i+1:m,i+1:n) from the right
  188: *
  189:                CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
  190:      $                     TAUP( I ), A( I+1, I+1 ), LDA, WORK )
  191:                A( I, I+1 ) = E( I )
  192:             ELSE
  193:                TAUP( I ) = ZERO
  194:             END IF
  195:    10    CONTINUE
  196:       ELSE
  197: *
  198: *        Reduce to lower bidiagonal form
  199: *
  200:          DO 20 I = 1, M
  201: *
  202: *           Generate elementary reflector G(i) to annihilate A(i,i+1:n)
  203: *
  204:             CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
  205:      $                   TAUP( I ) )
  206:             D( I ) = A( I, I )
  207:             A( I, I ) = ONE
  208: *
  209: *           Apply G(i) to A(i+1:m,i:n) from the right
  210: *
  211:             IF( I.LT.M )
  212:      $         CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
  213:      $                     TAUP( I ), A( I+1, I ), LDA, WORK )
  214:             A( I, I ) = D( I )
  215: *
  216:             IF( I.LT.M ) THEN
  217: *
  218: *              Generate elementary reflector H(i) to annihilate
  219: *              A(i+2:m,i)
  220: *
  221:                CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
  222:      $                      TAUQ( I ) )
  223:                E( I ) = A( I+1, I )
  224:                A( I+1, I ) = ONE
  225: *
  226: *              Apply H(i) to A(i+1:m,i+1:n) from the left
  227: *
  228:                CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),
  229:      $                     A( I+1, I+1 ), LDA, WORK )
  230:                A( I+1, I ) = E( I )
  231:             ELSE
  232:                TAUQ( I ) = ZERO
  233:             END IF
  234:    20    CONTINUE
  235:       END IF
  236:       RETURN
  237: *
  238: *     End of DGEBD2
  239: *
  240:       END