LAPACK 3.3.0

dgebrd.f

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00001       SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
00002      $                   INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       INTEGER            INFO, LDA, LWORK, M, N
00011 *     ..
00012 *     .. Array Arguments ..
00013       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
00014      $                   TAUQ( * ), WORK( * )
00015 *     ..
00016 *
00017 *  Purpose
00018 *  =======
00019 *
00020 *  DGEBRD reduces a general real M-by-N matrix A to upper or lower
00021 *  bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
00022 *
00023 *  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
00024 *
00025 *  Arguments
00026 *  =========
00027 *
00028 *  M       (input) INTEGER
00029 *          The number of rows in the matrix A.  M >= 0.
00030 *
00031 *  N       (input) INTEGER
00032 *          The number of columns in the matrix A.  N >= 0.
00033 *
00034 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
00035 *          On entry, the M-by-N general matrix to be reduced.
00036 *          On exit,
00037 *          if m >= n, the diagonal and the first superdiagonal are
00038 *            overwritten with the upper bidiagonal matrix B; the
00039 *            elements below the diagonal, with the array TAUQ, represent
00040 *            the orthogonal matrix Q as a product of elementary
00041 *            reflectors, and the elements above the first superdiagonal,
00042 *            with the array TAUP, represent the orthogonal matrix P as
00043 *            a product of elementary reflectors;
00044 *          if m < n, the diagonal and the first subdiagonal are
00045 *            overwritten with the lower bidiagonal matrix B; the
00046 *            elements below the first subdiagonal, with the array TAUQ,
00047 *            represent the orthogonal matrix Q as a product of
00048 *            elementary reflectors, and the elements above the diagonal,
00049 *            with the array TAUP, represent the orthogonal matrix P as
00050 *            a product of elementary reflectors.
00051 *          See Further Details.
00052 *
00053 *  LDA     (input) INTEGER
00054 *          The leading dimension of the array A.  LDA >= max(1,M).
00055 *
00056 *  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
00057 *          The diagonal elements of the bidiagonal matrix B:
00058 *          D(i) = A(i,i).
00059 *
00060 *  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
00061 *          The off-diagonal elements of the bidiagonal matrix B:
00062 *          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
00063 *          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
00064 *
00065 *  TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
00066 *          The scalar factors of the elementary reflectors which
00067 *          represent the orthogonal matrix Q. See Further Details.
00068 *
00069 *  TAUP    (output) DOUBLE PRECISION array, dimension (min(M,N))
00070 *          The scalar factors of the elementary reflectors which
00071 *          represent the orthogonal matrix P. See Further Details.
00072 *
00073 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00074 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00075 *
00076 *  LWORK   (input) INTEGER
00077 *          The length of the array WORK.  LWORK >= max(1,M,N).
00078 *          For optimum performance LWORK >= (M+N)*NB, where NB
00079 *          is the optimal blocksize.
00080 *
00081 *          If LWORK = -1, then a workspace query is assumed; the routine
00082 *          only calculates the optimal size of the WORK array, returns
00083 *          this value as the first entry of the WORK array, and no error
00084 *          message related to LWORK is issued by XERBLA.
00085 *
00086 *  INFO    (output) INTEGER
00087 *          = 0:  successful exit
00088 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00089 *
00090 *  Further Details
00091 *  ===============
00092 *
00093 *  The matrices Q and P are represented as products of elementary
00094 *  reflectors:
00095 *
00096 *  If m >= n,
00097 *
00098 *     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
00099 *
00100 *  Each H(i) and G(i) has the form:
00101 *
00102 *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
00103 *
00104 *  where tauq and taup are real scalars, and v and u are real vectors;
00105 *  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
00106 *  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
00107 *  tauq is stored in TAUQ(i) and taup in TAUP(i).
00108 *
00109 *  If m < n,
00110 *
00111 *     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
00112 *
00113 *  Each H(i) and G(i) has the form:
00114 *
00115 *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
00116 *
00117 *  where tauq and taup are real scalars, and v and u are real vectors;
00118 *  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
00119 *  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
00120 *  tauq is stored in TAUQ(i) and taup in TAUP(i).
00121 *
00122 *  The contents of A on exit are illustrated by the following examples:
00123 *
00124 *  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
00125 *
00126 *    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
00127 *    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
00128 *    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
00129 *    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
00130 *    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
00131 *    (  v1  v2  v3  v4  v5 )
00132 *
00133 *  where d and e denote diagonal and off-diagonal elements of B, vi
00134 *  denotes an element of the vector defining H(i), and ui an element of
00135 *  the vector defining G(i).
00136 *
00137 *  =====================================================================
00138 *
00139 *     .. Parameters ..
00140       DOUBLE PRECISION   ONE
00141       PARAMETER          ( ONE = 1.0D+0 )
00142 *     ..
00143 *     .. Local Scalars ..
00144       LOGICAL            LQUERY
00145       INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
00146      $                   NBMIN, NX
00147       DOUBLE PRECISION   WS
00148 *     ..
00149 *     .. External Subroutines ..
00150       EXTERNAL           DGEBD2, DGEMM, DLABRD, XERBLA
00151 *     ..
00152 *     .. Intrinsic Functions ..
00153       INTRINSIC          DBLE, MAX, MIN
00154 *     ..
00155 *     .. External Functions ..
00156       INTEGER            ILAENV
00157       EXTERNAL           ILAENV
00158 *     ..
00159 *     .. Executable Statements ..
00160 *
00161 *     Test the input parameters
00162 *
00163       INFO = 0
00164       NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
00165       LWKOPT = ( M+N )*NB
00166       WORK( 1 ) = DBLE( LWKOPT )
00167       LQUERY = ( LWORK.EQ.-1 )
00168       IF( M.LT.0 ) THEN
00169          INFO = -1
00170       ELSE IF( N.LT.0 ) THEN
00171          INFO = -2
00172       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00173          INFO = -4
00174       ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
00175          INFO = -10
00176       END IF
00177       IF( INFO.LT.0 ) THEN
00178          CALL XERBLA( 'DGEBRD', -INFO )
00179          RETURN
00180       ELSE IF( LQUERY ) THEN
00181          RETURN
00182       END IF
00183 *
00184 *     Quick return if possible
00185 *
00186       MINMN = MIN( M, N )
00187       IF( MINMN.EQ.0 ) THEN
00188          WORK( 1 ) = 1
00189          RETURN
00190       END IF
00191 *
00192       WS = MAX( M, N )
00193       LDWRKX = M
00194       LDWRKY = N
00195 *
00196       IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
00197 *
00198 *        Set the crossover point NX.
00199 *
00200          NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
00201 *
00202 *        Determine when to switch from blocked to unblocked code.
00203 *
00204          IF( NX.LT.MINMN ) THEN
00205             WS = ( M+N )*NB
00206             IF( LWORK.LT.WS ) THEN
00207 *
00208 *              Not enough work space for the optimal NB, consider using
00209 *              a smaller block size.
00210 *
00211                NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
00212                IF( LWORK.GE.( M+N )*NBMIN ) THEN
00213                   NB = LWORK / ( M+N )
00214                ELSE
00215                   NB = 1
00216                   NX = MINMN
00217                END IF
00218             END IF
00219          END IF
00220       ELSE
00221          NX = MINMN
00222       END IF
00223 *
00224       DO 30 I = 1, MINMN - NX, NB
00225 *
00226 *        Reduce rows and columns i:i+nb-1 to bidiagonal form and return
00227 *        the matrices X and Y which are needed to update the unreduced
00228 *        part of the matrix
00229 *
00230          CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
00231      $                TAUQ( I ), TAUP( I ), WORK, LDWRKX,
00232      $                WORK( LDWRKX*NB+1 ), LDWRKY )
00233 *
00234 *        Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
00235 *        of the form  A := A - V*Y' - X*U'
00236 *
00237          CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
00238      $               NB, -ONE, A( I+NB, I ), LDA,
00239      $               WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
00240      $               A( I+NB, I+NB ), LDA )
00241          CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
00242      $               NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
00243      $               ONE, A( I+NB, I+NB ), LDA )
00244 *
00245 *        Copy diagonal and off-diagonal elements of B back into A
00246 *
00247          IF( M.GE.N ) THEN
00248             DO 10 J = I, I + NB - 1
00249                A( J, J ) = D( J )
00250                A( J, J+1 ) = E( J )
00251    10       CONTINUE
00252          ELSE
00253             DO 20 J = I, I + NB - 1
00254                A( J, J ) = D( J )
00255                A( J+1, J ) = E( J )
00256    20       CONTINUE
00257          END IF
00258    30 CONTINUE
00259 *
00260 *     Use unblocked code to reduce the remainder of the matrix
00261 *
00262       CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
00263      $             TAUQ( I ), TAUP( I ), WORK, IINFO )
00264       WORK( 1 ) = WS
00265       RETURN
00266 *
00267 *     End of DGEBRD
00268 *
00269       END
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