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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZGTSVX 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZGTSVX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgtsvx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgtsvx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgtsvx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, 00022 * DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, 00023 * WORK, RWORK, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER FACT, TRANS 00027 * INTEGER INFO, LDB, LDX, N, NRHS 00028 * DOUBLE PRECISION RCOND 00029 * .. 00030 * .. Array Arguments .. 00031 * INTEGER IPIV( * ) 00032 * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00033 * COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ), 00034 * $ DLF( * ), DU( * ), DU2( * ), DUF( * ), 00035 * $ WORK( * ), X( LDX, * ) 00036 * .. 00037 * 00038 * 00039 *> \par Purpose: 00040 * ============= 00041 *> 00042 *> \verbatim 00043 *> 00044 *> ZGTSVX uses the LU factorization to compute the solution to a complex 00045 *> system of linear equations A * X = B, A**T * X = B, or A**H * X = B, 00046 *> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS 00047 *> matrices. 00048 *> 00049 *> Error bounds on the solution and a condition estimate are also 00050 *> provided. 00051 *> \endverbatim 00052 * 00053 *> \par Description: 00054 * ================= 00055 *> 00056 *> \verbatim 00057 *> 00058 *> The following steps are performed: 00059 *> 00060 *> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A 00061 *> as A = L * U, where L is a product of permutation and unit lower 00062 *> bidiagonal matrices and U is upper triangular with nonzeros in 00063 *> only the main diagonal and first two superdiagonals. 00064 *> 00065 *> 2. If some U(i,i)=0, so that U is exactly singular, then the routine 00066 *> returns with INFO = i. Otherwise, the factored form of A is used 00067 *> to estimate the condition number of the matrix A. If the 00068 *> reciprocal of the condition number is less than machine precision, 00069 *> INFO = N+1 is returned as a warning, but the routine still goes on 00070 *> to solve for X and compute error bounds as described below. 00071 *> 00072 *> 3. The system of equations is solved for X using the factored form 00073 *> of A. 00074 *> 00075 *> 4. Iterative refinement is applied to improve the computed solution 00076 *> matrix and calculate error bounds and backward error estimates 00077 *> for it. 00078 *> \endverbatim 00079 * 00080 * Arguments: 00081 * ========== 00082 * 00083 *> \param[in] FACT 00084 *> \verbatim 00085 *> FACT is CHARACTER*1 00086 *> Specifies whether or not the factored form of A has been 00087 *> supplied on entry. 00088 *> = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form 00089 *> of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not 00090 *> be modified. 00091 *> = 'N': The matrix will be copied to DLF, DF, and DUF 00092 *> and factored. 00093 *> \endverbatim 00094 *> 00095 *> \param[in] TRANS 00096 *> \verbatim 00097 *> TRANS is CHARACTER*1 00098 *> Specifies the form of the system of equations: 00099 *> = 'N': A * X = B (No transpose) 00100 *> = 'T': A**T * X = B (Transpose) 00101 *> = 'C': A**H * X = B (Conjugate transpose) 00102 *> \endverbatim 00103 *> 00104 *> \param[in] N 00105 *> \verbatim 00106 *> N is INTEGER 00107 *> The order of the matrix A. N >= 0. 00108 *> \endverbatim 00109 *> 00110 *> \param[in] NRHS 00111 *> \verbatim 00112 *> NRHS is INTEGER 00113 *> The number of right hand sides, i.e., the number of columns 00114 *> of the matrix B. NRHS >= 0. 00115 *> \endverbatim 00116 *> 00117 *> \param[in] DL 00118 *> \verbatim 00119 *> DL is COMPLEX*16 array, dimension (N-1) 00120 *> The (n-1) subdiagonal elements of A. 00121 *> \endverbatim 00122 *> 00123 *> \param[in] D 00124 *> \verbatim 00125 *> D is COMPLEX*16 array, dimension (N) 00126 *> The n diagonal elements of A. 00127 *> \endverbatim 00128 *> 00129 *> \param[in] DU 00130 *> \verbatim 00131 *> DU is COMPLEX*16 array, dimension (N-1) 00132 *> The (n-1) superdiagonal elements of A. 00133 *> \endverbatim 00134 *> 00135 *> \param[in,out] DLF 00136 *> \verbatim 00137 *> DLF is COMPLEX*16 array, dimension (N-1) 00138 *> If FACT = 'F', then DLF is an input argument and on entry 00139 *> contains the (n-1) multipliers that define the matrix L from 00140 *> the LU factorization of A as computed by ZGTTRF. 00141 *> 00142 *> If FACT = 'N', then DLF is an output argument and on exit 00143 *> contains the (n-1) multipliers that define the matrix L from 00144 *> the LU factorization of A. 00145 *> \endverbatim 00146 *> 00147 *> \param[in,out] DF 00148 *> \verbatim 00149 *> DF is COMPLEX*16 array, dimension (N) 00150 *> If FACT = 'F', then DF is an input argument and on entry 00151 *> contains the n diagonal elements of the upper triangular 00152 *> matrix U from the LU factorization of A. 00153 *> 00154 *> If FACT = 'N', then DF is an output argument and on exit 00155 *> contains the n diagonal elements of the upper triangular 00156 *> matrix U from the LU factorization of A. 00157 *> \endverbatim 00158 *> 00159 *> \param[in,out] DUF 00160 *> \verbatim 00161 *> DUF is COMPLEX*16 array, dimension (N-1) 00162 *> If FACT = 'F', then DUF is an input argument and on entry 00163 *> contains the (n-1) elements of the first superdiagonal of U. 00164 *> 00165 *> If FACT = 'N', then DUF is an output argument and on exit 00166 *> contains the (n-1) elements of the first superdiagonal of U. 00167 *> \endverbatim 00168 *> 00169 *> \param[in,out] DU2 00170 *> \verbatim 00171 *> DU2 is COMPLEX*16 array, dimension (N-2) 00172 *> If FACT = 'F', then DU2 is an input argument and on entry 00173 *> contains the (n-2) elements of the second superdiagonal of 00174 *> U. 00175 *> 00176 *> If FACT = 'N', then DU2 is an output argument and on exit 00177 *> contains the (n-2) elements of the second superdiagonal of 00178 *> U. 00179 *> \endverbatim 00180 *> 00181 *> \param[in,out] IPIV 00182 *> \verbatim 00183 *> IPIV is INTEGER array, dimension (N) 00184 *> If FACT = 'F', then IPIV is an input argument and on entry 00185 *> contains the pivot indices from the LU factorization of A as 00186 *> computed by ZGTTRF. 00187 *> 00188 *> If FACT = 'N', then IPIV is an output argument and on exit 00189 *> contains the pivot indices from the LU factorization of A; 00190 *> row i of the matrix was interchanged with row IPIV(i). 00191 *> IPIV(i) will always be either i or i+1; IPIV(i) = i indicates 00192 *> a row interchange was not required. 00193 *> \endverbatim 00194 *> 00195 *> \param[in] B 00196 *> \verbatim 00197 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00198 *> The N-by-NRHS right hand side matrix B. 00199 *> \endverbatim 00200 *> 00201 *> \param[in] LDB 00202 *> \verbatim 00203 *> LDB is INTEGER 00204 *> The leading dimension of the array B. LDB >= max(1,N). 00205 *> \endverbatim 00206 *> 00207 *> \param[out] X 00208 *> \verbatim 00209 *> X is COMPLEX*16 array, dimension (LDX,NRHS) 00210 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 00211 *> \endverbatim 00212 *> 00213 *> \param[in] LDX 00214 *> \verbatim 00215 *> LDX is INTEGER 00216 *> The leading dimension of the array X. LDX >= max(1,N). 00217 *> \endverbatim 00218 *> 00219 *> \param[out] RCOND 00220 *> \verbatim 00221 *> RCOND is DOUBLE PRECISION 00222 *> The estimate of the reciprocal condition number of the matrix 00223 *> A. If RCOND is less than the machine precision (in 00224 *> particular, if RCOND = 0), the matrix is singular to working 00225 *> precision. This condition is indicated by a return code of 00226 *> INFO > 0. 00227 *> \endverbatim 00228 *> 00229 *> \param[out] FERR 00230 *> \verbatim 00231 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00232 *> The estimated forward error bound for each solution vector 00233 *> X(j) (the j-th column of the solution matrix X). 00234 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00235 *> is an estimated upper bound for the magnitude of the largest 00236 *> element in (X(j) - XTRUE) divided by the magnitude of the 00237 *> largest element in X(j). The estimate is as reliable as 00238 *> the estimate for RCOND, and is almost always a slight 00239 *> overestimate of the true error. 00240 *> \endverbatim 00241 *> 00242 *> \param[out] BERR 00243 *> \verbatim 00244 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00245 *> The componentwise relative backward error of each solution 00246 *> vector X(j) (i.e., the smallest relative change in 00247 *> any element of A or B that makes X(j) an exact solution). 00248 *> \endverbatim 00249 *> 00250 *> \param[out] WORK 00251 *> \verbatim 00252 *> WORK is COMPLEX*16 array, dimension (2*N) 00253 *> \endverbatim 00254 *> 00255 *> \param[out] RWORK 00256 *> \verbatim 00257 *> RWORK is DOUBLE PRECISION array, dimension (N) 00258 *> \endverbatim 00259 *> 00260 *> \param[out] INFO 00261 *> \verbatim 00262 *> INFO is INTEGER 00263 *> = 0: successful exit 00264 *> < 0: if INFO = -i, the i-th argument had an illegal value 00265 *> > 0: if INFO = i, and i is 00266 *> <= N: U(i,i) is exactly zero. The factorization 00267 *> has not been completed unless i = N, but the 00268 *> factor U is exactly singular, so the solution 00269 *> and error bounds could not be computed. 00270 *> RCOND = 0 is returned. 00271 *> = N+1: U is nonsingular, but RCOND is less than machine 00272 *> precision, meaning that the matrix is singular 00273 *> to working precision. Nevertheless, the 00274 *> solution and error bounds are computed because 00275 *> there are a number of situations where the 00276 *> computed solution can be more accurate than the 00277 *> value of RCOND would suggest. 00278 *> \endverbatim 00279 * 00280 * Authors: 00281 * ======== 00282 * 00283 *> \author Univ. of Tennessee 00284 *> \author Univ. of California Berkeley 00285 *> \author Univ. of Colorado Denver 00286 *> \author NAG Ltd. 00287 * 00288 *> \date April 2012 00289 * 00290 *> \ingroup complex16OTHERcomputational 00291 * 00292 * ===================================================================== 00293 SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, 00294 $ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, 00295 $ WORK, RWORK, INFO ) 00296 * 00297 * -- LAPACK computational routine (version 3.4.1) -- 00298 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00299 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00300 * April 2012 00301 * 00302 * .. Scalar Arguments .. 00303 CHARACTER FACT, TRANS 00304 INTEGER INFO, LDB, LDX, N, NRHS 00305 DOUBLE PRECISION RCOND 00306 * .. 00307 * .. Array Arguments .. 00308 INTEGER IPIV( * ) 00309 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00310 COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ), 00311 $ DLF( * ), DU( * ), DU2( * ), DUF( * ), 00312 $ WORK( * ), X( LDX, * ) 00313 * .. 00314 * 00315 * ===================================================================== 00316 * 00317 * .. Parameters .. 00318 DOUBLE PRECISION ZERO 00319 PARAMETER ( ZERO = 0.0D+0 ) 00320 * .. 00321 * .. Local Scalars .. 00322 LOGICAL NOFACT, NOTRAN 00323 CHARACTER NORM 00324 DOUBLE PRECISION ANORM 00325 * .. 00326 * .. External Functions .. 00327 LOGICAL LSAME 00328 DOUBLE PRECISION DLAMCH, ZLANGT 00329 EXTERNAL LSAME, DLAMCH, ZLANGT 00330 * .. 00331 * .. External Subroutines .. 00332 EXTERNAL XERBLA, ZCOPY, ZGTCON, ZGTRFS, ZGTTRF, ZGTTRS, 00333 $ ZLACPY 00334 * .. 00335 * .. Intrinsic Functions .. 00336 INTRINSIC MAX 00337 * .. 00338 * .. Executable Statements .. 00339 * 00340 INFO = 0 00341 NOFACT = LSAME( FACT, 'N' ) 00342 NOTRAN = LSAME( TRANS, 'N' ) 00343 IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN 00344 INFO = -1 00345 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00346 $ LSAME( TRANS, 'C' ) ) THEN 00347 INFO = -2 00348 ELSE IF( N.LT.0 ) THEN 00349 INFO = -3 00350 ELSE IF( NRHS.LT.0 ) THEN 00351 INFO = -4 00352 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00353 INFO = -14 00354 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00355 INFO = -16 00356 END IF 00357 IF( INFO.NE.0 ) THEN 00358 CALL XERBLA( 'ZGTSVX', -INFO ) 00359 RETURN 00360 END IF 00361 * 00362 IF( NOFACT ) THEN 00363 * 00364 * Compute the LU factorization of A. 00365 * 00366 CALL ZCOPY( N, D, 1, DF, 1 ) 00367 IF( N.GT.1 ) THEN 00368 CALL ZCOPY( N-1, DL, 1, DLF, 1 ) 00369 CALL ZCOPY( N-1, DU, 1, DUF, 1 ) 00370 END IF 00371 CALL ZGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO ) 00372 * 00373 * Return if INFO is non-zero. 00374 * 00375 IF( INFO.GT.0 )THEN 00376 RCOND = ZERO 00377 RETURN 00378 END IF 00379 END IF 00380 * 00381 * Compute the norm of the matrix A. 00382 * 00383 IF( NOTRAN ) THEN 00384 NORM = '1' 00385 ELSE 00386 NORM = 'I' 00387 END IF 00388 ANORM = ZLANGT( NORM, N, DL, D, DU ) 00389 * 00390 * Compute the reciprocal of the condition number of A. 00391 * 00392 CALL ZGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK, 00393 $ INFO ) 00394 * 00395 * Compute the solution vectors X. 00396 * 00397 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00398 CALL ZGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX, 00399 $ INFO ) 00400 * 00401 * Use iterative refinement to improve the computed solutions and 00402 * compute error bounds and backward error estimates for them. 00403 * 00404 CALL ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, 00405 $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO ) 00406 * 00407 * Set INFO = N+1 if the matrix is singular to working precision. 00408 * 00409 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 00410 $ INFO = N + 1 00411 * 00412 RETURN 00413 * 00414 * End of ZGTSVX 00415 * 00416 END