 LAPACK 3.3.1 Linear Algebra PACKage

# clatdf.f

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```00001       SUBROUTINE CLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
00002      \$                   JPIV )
00003 *
00004 *  -- LAPACK auxiliary 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            IJOB, LDZ, N
00011       REAL               RDSCAL, RDSUM
00012 *     ..
00013 *     .. Array Arguments ..
00014       INTEGER            IPIV( * ), JPIV( * )
00015       COMPLEX            RHS( * ), Z( LDZ, * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  CLATDF computes the contribution to the reciprocal Dif-estimate
00022 *  by solving for x in Z * x = b, where b is chosen such that the norm
00023 *  of x is as large as possible. It is assumed that LU decomposition
00024 *  of Z has been computed by CGETC2. On entry RHS = f holds the
00025 *  contribution from earlier solved sub-systems, and on return RHS = x.
00026 *
00027 *  The factorization of Z returned by CGETC2 has the form
00028 *  Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
00029 *  triangular with unit diagonal elements and U is upper triangular.
00030 *
00031 *  Arguments
00032 *  =========
00033 *
00034 *  IJOB    (input) INTEGER
00035 *          IJOB = 2: First compute an approximative null-vector e
00036 *              of Z using CGECON, e is normalized and solve for
00037 *              Zx = +-e - f with the sign giving the greater value of
00038 *              2-norm(x).  About 5 times as expensive as Default.
00039 *          IJOB .ne. 2: Local look ahead strategy where
00040 *              all entries of the r.h.s. b is choosen as either +1 or
00041 *              -1.  Default.
00042 *
00043 *  N       (input) INTEGER
00044 *          The number of columns of the matrix Z.
00045 *
00046 *  Z       (input) REAL array, dimension (LDZ, N)
00047 *          On entry, the LU part of the factorization of the n-by-n
00048 *          matrix Z computed by CGETC2:  Z = P * L * U * Q
00049 *
00050 *  LDZ     (input) INTEGER
00051 *          The leading dimension of the array Z.  LDA >= max(1, N).
00052 *
00053 *  RHS     (input/output) REAL array, dimension (N).
00054 *          On entry, RHS contains contributions from other subsystems.
00055 *          On exit, RHS contains the solution of the subsystem with
00056 *          entries according to the value of IJOB (see above).
00057 *
00058 *  RDSUM   (input/output) REAL
00059 *          On entry, the sum of squares of computed contributions to
00060 *          the Dif-estimate under computation by CTGSYL, where the
00061 *          scaling factor RDSCAL (see below) has been factored out.
00062 *          On exit, the corresponding sum of squares updated with the
00063 *          contributions from the current sub-system.
00064 *          If TRANS = 'T' RDSUM is not touched.
00065 *          NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.
00066 *
00067 *  RDSCAL  (input/output) REAL
00068 *          On entry, scaling factor used to prevent overflow in RDSUM.
00069 *          On exit, RDSCAL is updated w.r.t. the current contributions
00070 *          in RDSUM.
00071 *          If TRANS = 'T', RDSCAL is not touched.
00072 *          NOTE: RDSCAL only makes sense when CTGSY2 is called by
00073 *          CTGSYL.
00074 *
00075 *  IPIV    (input) INTEGER array, dimension (N).
00076 *          The pivot indices; for 1 <= i <= N, row i of the
00077 *          matrix has been interchanged with row IPIV(i).
00078 *
00079 *  JPIV    (input) INTEGER array, dimension (N).
00080 *          The pivot indices; for 1 <= j <= N, column j of the
00081 *          matrix has been interchanged with column JPIV(j).
00082 *
00083 *  Further Details
00084 *  ===============
00085 *
00086 *  Based on contributions by
00087 *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
00088 *     Umea University, S-901 87 Umea, Sweden.
00089 *
00090 *  This routine is a further developed implementation of algorithm
00091 *  BSOLVE in  using complete pivoting in the LU factorization.
00092 *
00093 *      Bo Kagstrom and Lars Westin,
00094 *         Generalized Schur Methods with Condition Estimators for
00095 *         Solving the Generalized Sylvester Equation, IEEE Transactions
00096 *         on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
00097 *
00098 *      Peter Poromaa,
00099 *         On Efficient and Robust Estimators for the Separation
00100 *         between two Regular Matrix Pairs with Applications in
00101 *         Condition Estimation. Report UMINF-95.05, Department of
00102 *         Computing Science, Umea University, S-901 87 Umea, Sweden,
00103 *         1995.
00104 *
00105 *  =====================================================================
00106 *
00107 *     .. Parameters ..
00108       INTEGER            MAXDIM
00109       PARAMETER          ( MAXDIM = 2 )
00110       REAL               ZERO, ONE
00111       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00112       COMPLEX            CONE
00113       PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
00114 *     ..
00115 *     .. Local Scalars ..
00116       INTEGER            I, INFO, J, K
00117       REAL               RTEMP, SCALE, SMINU, SPLUS
00118       COMPLEX            BM, BP, PMONE, TEMP
00119 *     ..
00120 *     .. Local Arrays ..
00121       REAL               RWORK( MAXDIM )
00122       COMPLEX            WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
00123 *     ..
00124 *     .. External Subroutines ..
00125       EXTERNAL           CAXPY, CCOPY, CGECON, CGESC2, CLASSQ, CLASWP,
00126      \$                   CSCAL
00127 *     ..
00128 *     .. External Functions ..
00129       REAL               SCASUM
00130       COMPLEX            CDOTC
00131       EXTERNAL           SCASUM, CDOTC
00132 *     ..
00133 *     .. Intrinsic Functions ..
00134       INTRINSIC          ABS, REAL, SQRT
00135 *     ..
00136 *     .. Executable Statements ..
00137 *
00138       IF( IJOB.NE.2 ) THEN
00139 *
00140 *        Apply permutations IPIV to RHS
00141 *
00142          CALL CLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
00143 *
00144 *        Solve for L-part choosing RHS either to +1 or -1.
00145 *
00146          PMONE = -CONE
00147          DO 10 J = 1, N - 1
00148             BP = RHS( J ) + CONE
00149             BM = RHS( J ) - CONE
00150             SPLUS = ONE
00151 *
00152 *           Lockahead for L- part RHS(1:N-1) = +-1
00153 *           SPLUS and SMIN computed more efficiently than in BSOLVE.
00154 *
00155             SPLUS = SPLUS + REAL( CDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
\$              J ), 1 ) )
00156             SMINU = REAL( CDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
00157             SPLUS = SPLUS*REAL( RHS( J ) )
00158             IF( SPLUS.GT.SMINU ) THEN
00159                RHS( J ) = BP
00160             ELSE IF( SMINU.GT.SPLUS ) THEN
00161                RHS( J ) = BM
00162             ELSE
00163 *
00164 *              In this case the updating sums are equal and we can
00165 *              choose RHS(J) +1 or -1. The first time this happens we
00166 *              choose -1, thereafter +1. This is a simple way to get
00167 *              good estimates of matrices like Byers well-known example
00168 *              (see ). (Not done in BSOLVE.)
00169 *
00170                RHS( J ) = RHS( J ) + PMONE
00171                PMONE = CONE
00172             END IF
00173 *
00174 *           Compute the remaining r.h.s.
00175 *
00176             TEMP = -RHS( J )
00177             CALL CAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
00178    10    CONTINUE
00179 *
00180 *        Solve for U- part, lockahead for RHS(N) = +-1. This is not done
00181 *        In BSOLVE and will hopefully give us a better estimate because
00182 *        any ill-conditioning of the original matrix is transfered to U
00183 *        and not to L. U(N, N) is an approximation to sigma_min(LU).
00184 *
00185          CALL CCOPY( N-1, RHS, 1, WORK, 1 )
00186          WORK( N ) = RHS( N ) + CONE
00187          RHS( N ) = RHS( N ) - CONE
00188          SPLUS = ZERO
00189          SMINU = ZERO
00190          DO 30 I = N, 1, -1
00191             TEMP = CONE / Z( I, I )
00192             WORK( I ) = WORK( I )*TEMP
00193             RHS( I ) = RHS( I )*TEMP
00194             DO 20 K = I + 1, N
00195                WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
00196                RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
00197    20       CONTINUE
00198             SPLUS = SPLUS + ABS( WORK( I ) )
00199             SMINU = SMINU + ABS( RHS( I ) )
00200    30    CONTINUE
00201          IF( SPLUS.GT.SMINU )
00202      \$      CALL CCOPY( N, WORK, 1, RHS, 1 )
00203 *
00204 *        Apply the permutations JPIV to the computed solution (RHS)
00205 *
00206          CALL CLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
00207 *
00208 *        Compute the sum of squares
00209 *
00210          CALL CLASSQ( N, RHS, 1, RDSCAL, RDSUM )
00211          RETURN
00212       END IF
00213 *
00214 *     ENTRY IJOB = 2
00215 *
00216 *     Compute approximate nullvector XM of Z
00217 *
00218       CALL CGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
00219       CALL CCOPY( N, WORK( N+1 ), 1, XM, 1 )
00220 *
00221 *     Compute RHS
00222 *
00223       CALL CLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
00224       TEMP = CONE / SQRT( CDOTC( N, XM, 1, XM, 1 ) )
00225       CALL CSCAL( N, TEMP, XM, 1 )
00226       CALL CCOPY( N, XM, 1, XP, 1 )
00227       CALL CAXPY( N, CONE, RHS, 1, XP, 1 )
00228       CALL CAXPY( N, -CONE, XM, 1, RHS, 1 )
00229       CALL CGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
00230       CALL CGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
00231       IF( SCASUM( N, XP, 1 ).GT.SCASUM( N, RHS, 1 ) )
00232      \$   CALL CCOPY( N, XP, 1, RHS, 1 )
00233 *
00234 *     Compute the sum of squares
00235 *
00236       CALL CLASSQ( N, RHS, 1, RDSCAL, RDSUM )
00237       RETURN
00238 *
00239 *     End of CLATDF
00240 *
00241       END
00242 ```