LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zlatdf.f
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00001 *> \brief \b ZLATDF
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download ZLATDF + dependencies 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
00022 *                          JPIV )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       INTEGER            IJOB, LDZ, N
00026 *       DOUBLE PRECISION   RDSCAL, RDSUM
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       INTEGER            IPIV( * ), JPIV( * )
00030 *       COMPLEX*16         RHS( * ), Z( LDZ, * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *> ZLATDF computes the contribution to the reciprocal Dif-estimate
00040 *> by solving for x in Z * x = b, where b is chosen such that the norm
00041 *> of x is as large as possible. It is assumed that LU decomposition
00042 *> of Z has been computed by ZGETC2. On entry RHS = f holds the
00043 *> contribution from earlier solved sub-systems, and on return RHS = x.
00044 *>
00045 *> The factorization of Z returned by ZGETC2 has the form
00046 *> Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
00047 *> triangular with unit diagonal elements and U is upper triangular.
00048 *> \endverbatim
00049 *
00050 *  Arguments:
00051 *  ==========
00052 *
00053 *> \param[in] IJOB
00054 *> \verbatim
00055 *>          IJOB is INTEGER
00056 *>          IJOB = 2: First compute an approximative null-vector e
00057 *>              of Z using ZGECON, e is normalized and solve for
00058 *>              Zx = +-e - f with the sign giving the greater value of
00059 *>              2-norm(x).  About 5 times as expensive as Default.
00060 *>          IJOB .ne. 2: Local look ahead strategy where
00061 *>              all entries of the r.h.s. b is choosen as either +1 or
00062 *>              -1.  Default.
00063 *> \endverbatim
00064 *>
00065 *> \param[in] N
00066 *> \verbatim
00067 *>          N is INTEGER
00068 *>          The number of columns of the matrix Z.
00069 *> \endverbatim
00070 *>
00071 *> \param[in] Z
00072 *> \verbatim
00073 *>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
00074 *>          On entry, the LU part of the factorization of the n-by-n
00075 *>          matrix Z computed by ZGETC2:  Z = P * L * U * Q
00076 *> \endverbatim
00077 *>
00078 *> \param[in] LDZ
00079 *> \verbatim
00080 *>          LDZ is INTEGER
00081 *>          The leading dimension of the array Z.  LDA >= max(1, N).
00082 *> \endverbatim
00083 *>
00084 *> \param[in,out] RHS
00085 *> \verbatim
00086 *>          RHS is DOUBLE PRECISION array, dimension (N).
00087 *>          On entry, RHS contains contributions from other subsystems.
00088 *>          On exit, RHS contains the solution of the subsystem with
00089 *>          entries according to the value of IJOB (see above).
00090 *> \endverbatim
00091 *>
00092 *> \param[in,out] RDSUM
00093 *> \verbatim
00094 *>          RDSUM is DOUBLE PRECISION
00095 *>          On entry, the sum of squares of computed contributions to
00096 *>          the Dif-estimate under computation by ZTGSYL, where the
00097 *>          scaling factor RDSCAL (see below) has been factored out.
00098 *>          On exit, the corresponding sum of squares updated with the
00099 *>          contributions from the current sub-system.
00100 *>          If TRANS = 'T' RDSUM is not touched.
00101 *>          NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
00102 *> \endverbatim
00103 *>
00104 *> \param[in,out] RDSCAL
00105 *> \verbatim
00106 *>          RDSCAL is DOUBLE PRECISION
00107 *>          On entry, scaling factor used to prevent overflow in RDSUM.
00108 *>          On exit, RDSCAL is updated w.r.t. the current contributions
00109 *>          in RDSUM.
00110 *>          If TRANS = 'T', RDSCAL is not touched.
00111 *>          NOTE: RDSCAL only makes sense when ZTGSY2 is called by
00112 *>          ZTGSYL.
00113 *> \endverbatim
00114 *>
00115 *> \param[in] IPIV
00116 *> \verbatim
00117 *>          IPIV is INTEGER array, dimension (N).
00118 *>          The pivot indices; for 1 <= i <= N, row i of the
00119 *>          matrix has been interchanged with row IPIV(i).
00120 *> \endverbatim
00121 *>
00122 *> \param[in] JPIV
00123 *> \verbatim
00124 *>          JPIV is INTEGER array, dimension (N).
00125 *>          The pivot indices; for 1 <= j <= N, column j of the
00126 *>          matrix has been interchanged with column JPIV(j).
00127 *> \endverbatim
00128 *
00129 *  Authors:
00130 *  ========
00131 *
00132 *> \author Univ. of Tennessee 
00133 *> \author Univ. of California Berkeley 
00134 *> \author Univ. of Colorado Denver 
00135 *> \author NAG Ltd. 
00136 *
00137 *> \date November 2011
00138 *
00139 *> \ingroup complex16OTHERauxiliary
00140 *
00141 *> \par Further Details:
00142 *  =====================
00143 *>
00144 *>  This routine is a further developed implementation of algorithm
00145 *>  BSOLVE in [1] using complete pivoting in the LU factorization.
00146 *
00147 *> \par Contributors:
00148 *  ==================
00149 *>
00150 *>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
00151 *>     Umea University, S-901 87 Umea, Sweden.
00152 *
00153 *> \par References:
00154 *  ================
00155 *>
00156 *>   [1]   Bo Kagstrom and Lars Westin,
00157 *>         Generalized Schur Methods with Condition Estimators for
00158 *>         Solving the Generalized Sylvester Equation, IEEE Transactions
00159 *>         on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
00160 *>\n
00161 *>   [2]   Peter Poromaa,
00162 *>         On Efficient and Robust Estimators for the Separation
00163 *>         between two Regular Matrix Pairs with Applications in
00164 *>         Condition Estimation. Report UMINF-95.05, Department of
00165 *>         Computing Science, Umea University, S-901 87 Umea, Sweden,
00166 *>         1995.
00167 *
00168 *  =====================================================================
00169       SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
00170      $                   JPIV )
00171 *
00172 *  -- LAPACK auxiliary routine (version 3.4.0) --
00173 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00174 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00175 *     November 2011
00176 *
00177 *     .. Scalar Arguments ..
00178       INTEGER            IJOB, LDZ, N
00179       DOUBLE PRECISION   RDSCAL, RDSUM
00180 *     ..
00181 *     .. Array Arguments ..
00182       INTEGER            IPIV( * ), JPIV( * )
00183       COMPLEX*16         RHS( * ), Z( LDZ, * )
00184 *     ..
00185 *
00186 *  =====================================================================
00187 *
00188 *     .. Parameters ..
00189       INTEGER            MAXDIM
00190       PARAMETER          ( MAXDIM = 2 )
00191       DOUBLE PRECISION   ZERO, ONE
00192       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00193       COMPLEX*16         CONE
00194       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
00195 *     ..
00196 *     .. Local Scalars ..
00197       INTEGER            I, INFO, J, K
00198       DOUBLE PRECISION   RTEMP, SCALE, SMINU, SPLUS
00199       COMPLEX*16         BM, BP, PMONE, TEMP
00200 *     ..
00201 *     .. Local Arrays ..
00202       DOUBLE PRECISION   RWORK( MAXDIM )
00203       COMPLEX*16         WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
00204 *     ..
00205 *     .. External Subroutines ..
00206       EXTERNAL           ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
00207      $                   ZSCAL
00208 *     ..
00209 *     .. External Functions ..
00210       DOUBLE PRECISION   DZASUM
00211       COMPLEX*16         ZDOTC
00212       EXTERNAL           DZASUM, ZDOTC
00213 *     ..
00214 *     .. Intrinsic Functions ..
00215       INTRINSIC          ABS, DBLE, SQRT
00216 *     ..
00217 *     .. Executable Statements ..
00218 *
00219       IF( IJOB.NE.2 ) THEN
00220 *
00221 *        Apply permutations IPIV to RHS
00222 *
00223          CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
00224 *
00225 *        Solve for L-part choosing RHS either to +1 or -1.
00226 *
00227          PMONE = -CONE
00228          DO 10 J = 1, N - 1
00229             BP = RHS( J ) + CONE
00230             BM = RHS( J ) - CONE
00231             SPLUS = ONE
00232 *
00233 *           Lockahead for L- part RHS(1:N-1) = +-1
00234 *           SPLUS and SMIN computed more efficiently than in BSOLVE[1].
00235 *
00236             SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
00237      $              J ), 1 ) )
00238             SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
00239             SPLUS = SPLUS*DBLE( RHS( J ) )
00240             IF( SPLUS.GT.SMINU ) THEN
00241                RHS( J ) = BP
00242             ELSE IF( SMINU.GT.SPLUS ) THEN
00243                RHS( J ) = BM
00244             ELSE
00245 *
00246 *              In this case the updating sums are equal and we can
00247 *              choose RHS(J) +1 or -1. The first time this happens we
00248 *              choose -1, thereafter +1. This is a simple way to get
00249 *              good estimates of matrices like Byers well-known example
00250 *              (see [1]). (Not done in BSOLVE.)
00251 *
00252                RHS( J ) = RHS( J ) + PMONE
00253                PMONE = CONE
00254             END IF
00255 *
00256 *           Compute the remaining r.h.s.
00257 *
00258             TEMP = -RHS( J )
00259             CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
00260    10    CONTINUE
00261 *
00262 *        Solve for U- part, lockahead for RHS(N) = +-1. This is not done
00263 *        In BSOLVE and will hopefully give us a better estimate because
00264 *        any ill-conditioning of the original matrix is transfered to U
00265 *        and not to L. U(N, N) is an approximation to sigma_min(LU).
00266 *
00267          CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
00268          WORK( N ) = RHS( N ) + CONE
00269          RHS( N ) = RHS( N ) - CONE
00270          SPLUS = ZERO
00271          SMINU = ZERO
00272          DO 30 I = N, 1, -1
00273             TEMP = CONE / Z( I, I )
00274             WORK( I ) = WORK( I )*TEMP
00275             RHS( I ) = RHS( I )*TEMP
00276             DO 20 K = I + 1, N
00277                WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
00278                RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
00279    20       CONTINUE
00280             SPLUS = SPLUS + ABS( WORK( I ) )
00281             SMINU = SMINU + ABS( RHS( I ) )
00282    30    CONTINUE
00283          IF( SPLUS.GT.SMINU )
00284      $      CALL ZCOPY( N, WORK, 1, RHS, 1 )
00285 *
00286 *        Apply the permutations JPIV to the computed solution (RHS)
00287 *
00288          CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
00289 *
00290 *        Compute the sum of squares
00291 *
00292          CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
00293          RETURN
00294       END IF
00295 *
00296 *     ENTRY IJOB = 2
00297 *
00298 *     Compute approximate nullvector XM of Z
00299 *
00300       CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
00301       CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
00302 *
00303 *     Compute RHS
00304 *
00305       CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
00306       TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
00307       CALL ZSCAL( N, TEMP, XM, 1 )
00308       CALL ZCOPY( N, XM, 1, XP, 1 )
00309       CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
00310       CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
00311       CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
00312       CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
00313       IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
00314      $   CALL ZCOPY( N, XP, 1, RHS, 1 )
00315 *
00316 *     Compute the sum of squares
00317 *
00318       CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
00319       RETURN
00320 *
00321 *     End of ZLATDF
00322 *
00323       END
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