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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DLATDF 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DLATDF + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatdf.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatdf.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatdf.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DLATDF( 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 * DOUBLE PRECISION RHS( * ), Z( LDZ, * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> DLATDF uses the LU factorization of the n-by-n matrix Z computed by 00040 *> DGETC2 and computes a contribution to the reciprocal Dif-estimate 00041 *> by solving Z * x = b for x, and choosing the r.h.s. b such that 00042 *> the norm of x is as large as possible. On entry RHS = b holds the 00043 *> contribution from earlier solved sub-systems, and on return RHS = x. 00044 *> 00045 *> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, 00046 *> where P and Q are permutation matrices. L is lower triangular with 00047 *> 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 DGECON, e is normalized and solve for 00058 *> Zx = +-e - f with the sign giving the greater value 00059 *> of 2-norm(x). About 5 times as expensive as Default. 00060 *> IJOB .ne. 2: Local look ahead strategy where all entries of 00061 *> the r.h.s. b is choosen as either +1 or -1 (Default). 00062 *> \endverbatim 00063 *> 00064 *> \param[in] N 00065 *> \verbatim 00066 *> N is INTEGER 00067 *> The number of columns of the matrix Z. 00068 *> \endverbatim 00069 *> 00070 *> \param[in] Z 00071 *> \verbatim 00072 *> Z is DOUBLE PRECISION array, dimension (LDZ, N) 00073 *> On entry, the LU part of the factorization of the n-by-n 00074 *> matrix Z computed by DGETC2: Z = P * L * U * Q 00075 *> \endverbatim 00076 *> 00077 *> \param[in] LDZ 00078 *> \verbatim 00079 *> LDZ is INTEGER 00080 *> The leading dimension of the array Z. LDA >= max(1, N). 00081 *> \endverbatim 00082 *> 00083 *> \param[in,out] RHS 00084 *> \verbatim 00085 *> RHS is DOUBLE PRECISION array, dimension (N) 00086 *> On entry, RHS contains contributions from other subsystems. 00087 *> On exit, RHS contains the solution of the subsystem with 00088 *> entries acoording to the value of IJOB (see above). 00089 *> \endverbatim 00090 *> 00091 *> \param[in,out] RDSUM 00092 *> \verbatim 00093 *> RDSUM is DOUBLE PRECISION 00094 *> On entry, the sum of squares of computed contributions to 00095 *> the Dif-estimate under computation by DTGSYL, where the 00096 *> scaling factor RDSCAL (see below) has been factored out. 00097 *> On exit, the corresponding sum of squares updated with the 00098 *> contributions from the current sub-system. 00099 *> If TRANS = 'T' RDSUM is not touched. 00100 *> NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL. 00101 *> \endverbatim 00102 *> 00103 *> \param[in,out] RDSCAL 00104 *> \verbatim 00105 *> RDSCAL is DOUBLE PRECISION 00106 *> On entry, scaling factor used to prevent overflow in RDSUM. 00107 *> On exit, RDSCAL is updated w.r.t. the current contributions 00108 *> in RDSUM. 00109 *> If TRANS = 'T', RDSCAL is not touched. 00110 *> NOTE: RDSCAL only makes sense when DTGSY2 is called by 00111 *> DTGSYL. 00112 *> \endverbatim 00113 *> 00114 *> \param[in] IPIV 00115 *> \verbatim 00116 *> IPIV is INTEGER array, dimension (N). 00117 *> The pivot indices; for 1 <= i <= N, row i of the 00118 *> matrix has been interchanged with row IPIV(i). 00119 *> \endverbatim 00120 *> 00121 *> \param[in] JPIV 00122 *> \verbatim 00123 *> JPIV is INTEGER array, dimension (N). 00124 *> The pivot indices; for 1 <= j <= N, column j of the 00125 *> matrix has been interchanged with column JPIV(j). 00126 *> \endverbatim 00127 * 00128 * Authors: 00129 * ======== 00130 * 00131 *> \author Univ. of Tennessee 00132 *> \author Univ. of California Berkeley 00133 *> \author Univ. of Colorado Denver 00134 *> \author NAG Ltd. 00135 * 00136 *> \date November 2011 00137 * 00138 *> \ingroup doubleOTHERauxiliary 00139 * 00140 *> \par Further Details: 00141 * ===================== 00142 *> 00143 *> This routine is a further developed implementation of algorithm 00144 *> BSOLVE in [1] using complete pivoting in the LU factorization. 00145 * 00146 *> \par Contributors: 00147 * ================== 00148 *> 00149 *> Bo Kagstrom and Peter Poromaa, Department of Computing Science, 00150 *> Umea University, S-901 87 Umea, Sweden. 00151 * 00152 *> \par References: 00153 * ================ 00154 *> 00155 *> \verbatim 00156 *> 00157 *> 00158 *> [1] Bo Kagstrom and Lars Westin, 00159 *> Generalized Schur Methods with Condition Estimators for 00160 *> Solving the Generalized Sylvester Equation, IEEE Transactions 00161 *> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. 00162 *> 00163 *> [2] Peter Poromaa, 00164 *> On Efficient and Robust Estimators for the Separation 00165 *> between two Regular Matrix Pairs with Applications in 00166 *> Condition Estimation. Report IMINF-95.05, Departement of 00167 *> Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. 00168 *> \endverbatim 00169 *> 00170 * ===================================================================== 00171 SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, 00172 $ JPIV ) 00173 * 00174 * -- LAPACK auxiliary routine (version 3.4.0) -- 00175 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00177 * November 2011 00178 * 00179 * .. Scalar Arguments .. 00180 INTEGER IJOB, LDZ, N 00181 DOUBLE PRECISION RDSCAL, RDSUM 00182 * .. 00183 * .. Array Arguments .. 00184 INTEGER IPIV( * ), JPIV( * ) 00185 DOUBLE PRECISION RHS( * ), Z( LDZ, * ) 00186 * .. 00187 * 00188 * ===================================================================== 00189 * 00190 * .. Parameters .. 00191 INTEGER MAXDIM 00192 PARAMETER ( MAXDIM = 8 ) 00193 DOUBLE PRECISION ZERO, ONE 00194 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00195 * .. 00196 * .. Local Scalars .. 00197 INTEGER I, INFO, J, K 00198 DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP 00199 * .. 00200 * .. Local Arrays .. 00201 INTEGER IWORK( MAXDIM ) 00202 DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM ) 00203 * .. 00204 * .. External Subroutines .. 00205 EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP, 00206 $ DSCAL 00207 * .. 00208 * .. External Functions .. 00209 DOUBLE PRECISION DASUM, DDOT 00210 EXTERNAL DASUM, DDOT 00211 * .. 00212 * .. Intrinsic Functions .. 00213 INTRINSIC ABS, SQRT 00214 * .. 00215 * .. Executable Statements .. 00216 * 00217 IF( IJOB.NE.2 ) THEN 00218 * 00219 * Apply permutations IPIV to RHS 00220 * 00221 CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 ) 00222 * 00223 * Solve for L-part choosing RHS either to +1 or -1. 00224 * 00225 PMONE = -ONE 00226 * 00227 DO 10 J = 1, N - 1 00228 BP = RHS( J ) + ONE 00229 BM = RHS( J ) - ONE 00230 SPLUS = ONE 00231 * 00232 * Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and 00233 * SMIN computed more efficiently than in BSOLVE [1]. 00234 * 00235 SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 ) 00236 SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) 00237 SPLUS = SPLUS*RHS( J ) 00238 IF( SPLUS.GT.SMINU ) THEN 00239 RHS( J ) = BP 00240 ELSE IF( SMINU.GT.SPLUS ) THEN 00241 RHS( J ) = BM 00242 ELSE 00243 * 00244 * In this case the updating sums are equal and we can 00245 * choose RHS(J) +1 or -1. The first time this happens 00246 * we choose -1, thereafter +1. This is a simple way to 00247 * get good estimates of matrices like Byers well-known 00248 * example (see [1]). (Not done in BSOLVE.) 00249 * 00250 RHS( J ) = RHS( J ) + PMONE 00251 PMONE = ONE 00252 END IF 00253 * 00254 * Compute the remaining r.h.s. 00255 * 00256 TEMP = -RHS( J ) 00257 CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 ) 00258 * 00259 10 CONTINUE 00260 * 00261 * Solve for U-part, look-ahead for RHS(N) = +-1. This is not done 00262 * in BSOLVE and will hopefully give us a better estimate because 00263 * any ill-conditioning of the original matrix is transfered to U 00264 * and not to L. U(N, N) is an approximation to sigma_min(LU). 00265 * 00266 CALL DCOPY( N-1, RHS, 1, XP, 1 ) 00267 XP( N ) = RHS( N ) + ONE 00268 RHS( N ) = RHS( N ) - ONE 00269 SPLUS = ZERO 00270 SMINU = ZERO 00271 DO 30 I = N, 1, -1 00272 TEMP = ONE / Z( I, I ) 00273 XP( I ) = XP( I )*TEMP 00274 RHS( I ) = RHS( I )*TEMP 00275 DO 20 K = I + 1, N 00276 XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP ) 00277 RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP ) 00278 20 CONTINUE 00279 SPLUS = SPLUS + ABS( XP( I ) ) 00280 SMINU = SMINU + ABS( RHS( I ) ) 00281 30 CONTINUE 00282 IF( SPLUS.GT.SMINU ) 00283 $ CALL DCOPY( N, XP, 1, RHS, 1 ) 00284 * 00285 * Apply the permutations JPIV to the computed solution (RHS) 00286 * 00287 CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 ) 00288 * 00289 * Compute the sum of squares 00290 * 00291 CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM ) 00292 * 00293 ELSE 00294 * 00295 * IJOB = 2, Compute approximate nullvector XM of Z 00296 * 00297 CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO ) 00298 CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 ) 00299 * 00300 * Compute RHS 00301 * 00302 CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 ) 00303 TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) ) 00304 CALL DSCAL( N, TEMP, XM, 1 ) 00305 CALL DCOPY( N, XM, 1, XP, 1 ) 00306 CALL DAXPY( N, ONE, RHS, 1, XP, 1 ) 00307 CALL DAXPY( N, -ONE, XM, 1, RHS, 1 ) 00308 CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP ) 00309 CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP ) 00310 IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) ) 00311 $ CALL DCOPY( N, XP, 1, RHS, 1 ) 00312 * 00313 * Compute the sum of squares 00314 * 00315 CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM ) 00316 * 00317 END IF 00318 * 00319 RETURN 00320 * 00321 * End of DLATDF 00322 * 00323 END