LAPACK 3.3.1
Linear Algebra PACKage

slasd6.f

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00001       SUBROUTINE SLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
00002      $                   IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
00003      $                   LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
00004      $                   IWORK, INFO )
00005 *
00006 *  -- LAPACK auxiliary routine (version 3.3.0) --
00007 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00008 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00009 *     November 2010
00010 *
00011 *     .. Scalar Arguments ..
00012       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
00013      $                   NR, SQRE
00014       REAL               ALPHA, BETA, C, S
00015 *     ..
00016 *     .. Array Arguments ..
00017       INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
00018      $                   PERM( * )
00019       REAL               D( * ), DIFL( * ), DIFR( * ),
00020      $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
00021      $                   VF( * ), VL( * ), WORK( * ), Z( * )
00022 *     ..
00023 *
00024 *  Purpose
00025 *  =======
00026 *
00027 *  SLASD6 computes the SVD of an updated upper bidiagonal matrix B
00028 *  obtained by merging two smaller ones by appending a row. This
00029 *  routine is used only for the problem which requires all singular
00030 *  values and optionally singular vector matrices in factored form.
00031 *  B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
00032 *  A related subroutine, SLASD1, handles the case in which all singular
00033 *  values and singular vectors of the bidiagonal matrix are desired.
00034 *
00035 *  SLASD6 computes the SVD as follows:
00036 *
00037 *                ( D1(in)    0    0       0 )
00038 *    B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
00039 *                (   0       0   D2(in)   0 )
00040 *
00041 *      = U(out) * ( D(out) 0) * VT(out)
00042 *
00043 *  where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
00044 *  with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
00045 *  elsewhere; and the entry b is empty if SQRE = 0.
00046 *
00047 *  The singular values of B can be computed using D1, D2, the first
00048 *  components of all the right singular vectors of the lower block, and
00049 *  the last components of all the right singular vectors of the upper
00050 *  block. These components are stored and updated in VF and VL,
00051 *  respectively, in SLASD6. Hence U and VT are not explicitly
00052 *  referenced.
00053 *
00054 *  The singular values are stored in D. The algorithm consists of two
00055 *  stages:
00056 *
00057 *        The first stage consists of deflating the size of the problem
00058 *        when there are multiple singular values or if there is a zero
00059 *        in the Z vector. For each such occurence the dimension of the
00060 *        secular equation problem is reduced by one. This stage is
00061 *        performed by the routine SLASD7.
00062 *
00063 *        The second stage consists of calculating the updated
00064 *        singular values. This is done by finding the roots of the
00065 *        secular equation via the routine SLASD4 (as called by SLASD8).
00066 *        This routine also updates VF and VL and computes the distances
00067 *        between the updated singular values and the old singular
00068 *        values.
00069 *
00070 *  SLASD6 is called from SLASDA.
00071 *
00072 *  Arguments
00073 *  =========
00074 *
00075 *  ICOMPQ (input) INTEGER
00076 *         Specifies whether singular vectors are to be computed in
00077 *         factored form:
00078 *         = 0: Compute singular values only.
00079 *         = 1: Compute singular vectors in factored form as well.
00080 *
00081 *  NL     (input) INTEGER
00082 *         The row dimension of the upper block.  NL >= 1.
00083 *
00084 *  NR     (input) INTEGER
00085 *         The row dimension of the lower block.  NR >= 1.
00086 *
00087 *  SQRE   (input) INTEGER
00088 *         = 0: the lower block is an NR-by-NR square matrix.
00089 *         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
00090 *
00091 *         The bidiagonal matrix has row dimension N = NL + NR + 1,
00092 *         and column dimension M = N + SQRE.
00093 *
00094 *  D      (input/output) REAL array, dimension (NL+NR+1).
00095 *         On entry D(1:NL,1:NL) contains the singular values of the
00096 *         upper block, and D(NL+2:N) contains the singular values
00097 *         of the lower block. On exit D(1:N) contains the singular
00098 *         values of the modified matrix.
00099 *
00100 *  VF     (input/output) REAL array, dimension (M)
00101 *         On entry, VF(1:NL+1) contains the first components of all
00102 *         right singular vectors of the upper block; and VF(NL+2:M)
00103 *         contains the first components of all right singular vectors
00104 *         of the lower block. On exit, VF contains the first components
00105 *         of all right singular vectors of the bidiagonal matrix.
00106 *
00107 *  VL     (input/output) REAL array, dimension (M)
00108 *         On entry, VL(1:NL+1) contains the  last components of all
00109 *         right singular vectors of the upper block; and VL(NL+2:M)
00110 *         contains the last components of all right singular vectors of
00111 *         the lower block. On exit, VL contains the last components of
00112 *         all right singular vectors of the bidiagonal matrix.
00113 *
00114 *  ALPHA  (input/output) REAL
00115 *         Contains the diagonal element associated with the added row.
00116 *
00117 *  BETA   (input/output) REAL
00118 *         Contains the off-diagonal element associated with the added
00119 *         row.
00120 *
00121 *  IDXQ   (output) INTEGER array, dimension (N)
00122 *         This contains the permutation which will reintegrate the
00123 *         subproblem just solved back into sorted order, i.e.
00124 *         D( IDXQ( I = 1, N ) ) will be in ascending order.
00125 *
00126 *  PERM   (output) INTEGER array, dimension ( N )
00127 *         The permutations (from deflation and sorting) to be applied
00128 *         to each block. Not referenced if ICOMPQ = 0.
00129 *
00130 *  GIVPTR (output) INTEGER
00131 *         The number of Givens rotations which took place in this
00132 *         subproblem. Not referenced if ICOMPQ = 0.
00133 *
00134 *  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
00135 *         Each pair of numbers indicates a pair of columns to take place
00136 *         in a Givens rotation. Not referenced if ICOMPQ = 0.
00137 *
00138 *  LDGCOL (input) INTEGER
00139 *         leading dimension of GIVCOL, must be at least N.
00140 *
00141 *  GIVNUM (output) REAL array, dimension ( LDGNUM, 2 )
00142 *         Each number indicates the C or S value to be used in the
00143 *         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
00144 *
00145 *  LDGNUM (input) INTEGER
00146 *         The leading dimension of GIVNUM and POLES, must be at least N.
00147 *
00148 *  POLES  (output) REAL array, dimension ( LDGNUM, 2 )
00149 *         On exit, POLES(1,*) is an array containing the new singular
00150 *         values obtained from solving the secular equation, and
00151 *         POLES(2,*) is an array containing the poles in the secular
00152 *         equation. Not referenced if ICOMPQ = 0.
00153 *
00154 *  DIFL   (output) REAL array, dimension ( N )
00155 *         On exit, DIFL(I) is the distance between I-th updated
00156 *         (undeflated) singular value and the I-th (undeflated) old
00157 *         singular value.
00158 *
00159 *  DIFR   (output) REAL array,
00160 *                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
00161 *                  dimension ( N ) if ICOMPQ = 0.
00162 *         On exit, DIFR(I, 1) is the distance between I-th updated
00163 *         (undeflated) singular value and the I+1-th (undeflated) old
00164 *         singular value.
00165 *
00166 *         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
00167 *         normalizing factors for the right singular vector matrix.
00168 *
00169 *         See SLASD8 for details on DIFL and DIFR.
00170 *
00171 *  Z      (output) REAL array, dimension ( M )
00172 *         The first elements of this array contain the components
00173 *         of the deflation-adjusted updating row vector.
00174 *
00175 *  K      (output) INTEGER
00176 *         Contains the dimension of the non-deflated matrix,
00177 *         This is the order of the related secular equation. 1 <= K <=N.
00178 *
00179 *  C      (output) REAL
00180 *         C contains garbage if SQRE =0 and the C-value of a Givens
00181 *         rotation related to the right null space if SQRE = 1.
00182 *
00183 *  S      (output) REAL
00184 *         S contains garbage if SQRE =0 and the S-value of a Givens
00185 *         rotation related to the right null space if SQRE = 1.
00186 *
00187 *  WORK   (workspace) REAL array, dimension ( 4 * M )
00188 *
00189 *  IWORK  (workspace) INTEGER array, dimension ( 3 * N )
00190 *
00191 *  INFO   (output) INTEGER
00192 *          = 0:  successful exit.
00193 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00194 *          > 0:  if INFO = 1, a singular value did not converge
00195 *
00196 *  Further Details
00197 *  ===============
00198 *
00199 *  Based on contributions by
00200 *     Ming Gu and Huan Ren, Computer Science Division, University of
00201 *     California at Berkeley, USA
00202 *
00203 *  =====================================================================
00204 *
00205 *     .. Parameters ..
00206       REAL               ONE, ZERO
00207       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00208 *     ..
00209 *     .. Local Scalars ..
00210       INTEGER            I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
00211      $                   N, N1, N2
00212       REAL               ORGNRM
00213 *     ..
00214 *     .. External Subroutines ..
00215       EXTERNAL           SCOPY, SLAMRG, SLASCL, SLASD7, SLASD8, XERBLA
00216 *     ..
00217 *     .. Intrinsic Functions ..
00218       INTRINSIC          ABS, MAX
00219 *     ..
00220 *     .. Executable Statements ..
00221 *
00222 *     Test the input parameters.
00223 *
00224       INFO = 0
00225       N = NL + NR + 1
00226       M = N + SQRE
00227 *
00228       IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
00229          INFO = -1
00230       ELSE IF( NL.LT.1 ) THEN
00231          INFO = -2
00232       ELSE IF( NR.LT.1 ) THEN
00233          INFO = -3
00234       ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
00235          INFO = -4
00236       ELSE IF( LDGCOL.LT.N ) THEN
00237          INFO = -14
00238       ELSE IF( LDGNUM.LT.N ) THEN
00239          INFO = -16
00240       END IF
00241       IF( INFO.NE.0 ) THEN
00242          CALL XERBLA( 'SLASD6', -INFO )
00243          RETURN
00244       END IF
00245 *
00246 *     The following values are for bookkeeping purposes only.  They are
00247 *     integer pointers which indicate the portion of the workspace
00248 *     used by a particular array in SLASD7 and SLASD8.
00249 *
00250       ISIGMA = 1
00251       IW = ISIGMA + N
00252       IVFW = IW + M
00253       IVLW = IVFW + M
00254 *
00255       IDX = 1
00256       IDXC = IDX + N
00257       IDXP = IDXC + N
00258 *
00259 *     Scale.
00260 *
00261       ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
00262       D( NL+1 ) = ZERO
00263       DO 10 I = 1, N
00264          IF( ABS( D( I ) ).GT.ORGNRM ) THEN
00265             ORGNRM = ABS( D( I ) )
00266          END IF
00267    10 CONTINUE
00268       CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
00269       ALPHA = ALPHA / ORGNRM
00270       BETA = BETA / ORGNRM
00271 *
00272 *     Sort and Deflate singular values.
00273 *
00274       CALL SLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
00275      $             WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
00276      $             WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
00277      $             PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
00278      $             INFO )
00279 *
00280 *     Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
00281 *
00282       CALL SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
00283      $             WORK( ISIGMA ), WORK( IW ), INFO )
00284 *
00285 *     Handle error returned
00286 *
00287       IF( INFO.NE.0 ) THEN
00288          CALL XERBLA( 'SLASD8', -INFO )
00289          RETURN
00290       END IF
00291 *
00292 *     Save the poles if ICOMPQ = 1.
00293 *
00294       IF( ICOMPQ.EQ.1 ) THEN
00295          CALL SCOPY( K, D, 1, POLES( 1, 1 ), 1 )
00296          CALL SCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
00297       END IF
00298 *
00299 *     Unscale.
00300 *
00301       CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
00302 *
00303 *     Prepare the IDXQ sorting permutation.
00304 *
00305       N1 = K
00306       N2 = N - K
00307       CALL SLAMRG( N1, N2, D, 1, -1, IDXQ )
00308 *
00309       RETURN
00310 *
00311 *     End of SLASD6
00312 *
00313       END
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