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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DGTRFS 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DGTRFS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtrfs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtrfs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtrfs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, 00022 * IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, 00023 * INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER TRANS 00027 * INTEGER INFO, LDB, LDX, N, NRHS 00028 * .. 00029 * .. Array Arguments .. 00030 * INTEGER IPIV( * ), IWORK( * ) 00031 * DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), 00032 * $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), 00033 * $ FERR( * ), WORK( * ), X( LDX, * ) 00034 * .. 00035 * 00036 * 00037 *> \par Purpose: 00038 * ============= 00039 *> 00040 *> \verbatim 00041 *> 00042 *> DGTRFS improves the computed solution to a system of linear 00043 *> equations when the coefficient matrix is tridiagonal, and provides 00044 *> error bounds and backward error estimates for the solution. 00045 *> \endverbatim 00046 * 00047 * Arguments: 00048 * ========== 00049 * 00050 *> \param[in] TRANS 00051 *> \verbatim 00052 *> TRANS is CHARACTER*1 00053 *> Specifies the form of the system of equations: 00054 *> = 'N': A * X = B (No transpose) 00055 *> = 'T': A**T * X = B (Transpose) 00056 *> = 'C': A**H * X = B (Conjugate transpose = Transpose) 00057 *> \endverbatim 00058 *> 00059 *> \param[in] N 00060 *> \verbatim 00061 *> N is INTEGER 00062 *> The order of the matrix A. N >= 0. 00063 *> \endverbatim 00064 *> 00065 *> \param[in] NRHS 00066 *> \verbatim 00067 *> NRHS is INTEGER 00068 *> The number of right hand sides, i.e., the number of columns 00069 *> of the matrix B. NRHS >= 0. 00070 *> \endverbatim 00071 *> 00072 *> \param[in] DL 00073 *> \verbatim 00074 *> DL is DOUBLE PRECISION array, dimension (N-1) 00075 *> The (n-1) subdiagonal elements of A. 00076 *> \endverbatim 00077 *> 00078 *> \param[in] D 00079 *> \verbatim 00080 *> D is DOUBLE PRECISION array, dimension (N) 00081 *> The diagonal elements of A. 00082 *> \endverbatim 00083 *> 00084 *> \param[in] DU 00085 *> \verbatim 00086 *> DU is DOUBLE PRECISION array, dimension (N-1) 00087 *> The (n-1) superdiagonal elements of A. 00088 *> \endverbatim 00089 *> 00090 *> \param[in] DLF 00091 *> \verbatim 00092 *> DLF is DOUBLE PRECISION array, dimension (N-1) 00093 *> The (n-1) multipliers that define the matrix L from the 00094 *> LU factorization of A as computed by DGTTRF. 00095 *> \endverbatim 00096 *> 00097 *> \param[in] DF 00098 *> \verbatim 00099 *> DF is DOUBLE PRECISION array, dimension (N) 00100 *> The n diagonal elements of the upper triangular matrix U from 00101 *> the LU factorization of A. 00102 *> \endverbatim 00103 *> 00104 *> \param[in] DUF 00105 *> \verbatim 00106 *> DUF is DOUBLE PRECISION array, dimension (N-1) 00107 *> The (n-1) elements of the first superdiagonal of U. 00108 *> \endverbatim 00109 *> 00110 *> \param[in] DU2 00111 *> \verbatim 00112 *> DU2 is DOUBLE PRECISION array, dimension (N-2) 00113 *> The (n-2) elements of the second superdiagonal of U. 00114 *> \endverbatim 00115 *> 00116 *> \param[in] IPIV 00117 *> \verbatim 00118 *> IPIV is INTEGER array, dimension (N) 00119 *> The pivot indices; for 1 <= i <= n, row i of the matrix was 00120 *> interchanged with row IPIV(i). IPIV(i) will always be either 00121 *> i or i+1; IPIV(i) = i indicates a row interchange was not 00122 *> required. 00123 *> \endverbatim 00124 *> 00125 *> \param[in] B 00126 *> \verbatim 00127 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00128 *> The right hand side matrix B. 00129 *> \endverbatim 00130 *> 00131 *> \param[in] LDB 00132 *> \verbatim 00133 *> LDB is INTEGER 00134 *> The leading dimension of the array B. LDB >= max(1,N). 00135 *> \endverbatim 00136 *> 00137 *> \param[in,out] X 00138 *> \verbatim 00139 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00140 *> On entry, the solution matrix X, as computed by DGTTRS. 00141 *> On exit, the improved solution matrix X. 00142 *> \endverbatim 00143 *> 00144 *> \param[in] LDX 00145 *> \verbatim 00146 *> LDX is INTEGER 00147 *> The leading dimension of the array X. LDX >= max(1,N). 00148 *> \endverbatim 00149 *> 00150 *> \param[out] FERR 00151 *> \verbatim 00152 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00153 *> The estimated forward error bound for each solution vector 00154 *> X(j) (the j-th column of the solution matrix X). 00155 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00156 *> is an estimated upper bound for the magnitude of the largest 00157 *> element in (X(j) - XTRUE) divided by the magnitude of the 00158 *> largest element in X(j). The estimate is as reliable as 00159 *> the estimate for RCOND, and is almost always a slight 00160 *> overestimate of the true error. 00161 *> \endverbatim 00162 *> 00163 *> \param[out] BERR 00164 *> \verbatim 00165 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00166 *> The componentwise relative backward error of each solution 00167 *> vector X(j) (i.e., the smallest relative change in 00168 *> any element of A or B that makes X(j) an exact solution). 00169 *> \endverbatim 00170 *> 00171 *> \param[out] WORK 00172 *> \verbatim 00173 *> WORK is DOUBLE PRECISION array, dimension (3*N) 00174 *> \endverbatim 00175 *> 00176 *> \param[out] IWORK 00177 *> \verbatim 00178 *> IWORK is INTEGER array, dimension (N) 00179 *> \endverbatim 00180 *> 00181 *> \param[out] INFO 00182 *> \verbatim 00183 *> INFO is INTEGER 00184 *> = 0: successful exit 00185 *> < 0: if INFO = -i, the i-th argument had an illegal value 00186 *> \endverbatim 00187 * 00188 *> \par Internal Parameters: 00189 * ========================= 00190 *> 00191 *> \verbatim 00192 *> ITMAX is the maximum number of steps of iterative refinement. 00193 *> \endverbatim 00194 * 00195 * Authors: 00196 * ======== 00197 * 00198 *> \author Univ. of Tennessee 00199 *> \author Univ. of California Berkeley 00200 *> \author Univ. of Colorado Denver 00201 *> \author NAG Ltd. 00202 * 00203 *> \date November 2011 00204 * 00205 *> \ingroup doubleOTHERcomputational 00206 * 00207 * ===================================================================== 00208 SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, 00209 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, 00210 $ INFO ) 00211 * 00212 * -- LAPACK computational routine (version 3.4.0) -- 00213 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00214 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00215 * November 2011 00216 * 00217 * .. Scalar Arguments .. 00218 CHARACTER TRANS 00219 INTEGER INFO, LDB, LDX, N, NRHS 00220 * .. 00221 * .. Array Arguments .. 00222 INTEGER IPIV( * ), IWORK( * ) 00223 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), 00224 $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), 00225 $ FERR( * ), WORK( * ), X( LDX, * ) 00226 * .. 00227 * 00228 * ===================================================================== 00229 * 00230 * .. Parameters .. 00231 INTEGER ITMAX 00232 PARAMETER ( ITMAX = 5 ) 00233 DOUBLE PRECISION ZERO, ONE 00234 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00235 DOUBLE PRECISION TWO 00236 PARAMETER ( TWO = 2.0D+0 ) 00237 DOUBLE PRECISION THREE 00238 PARAMETER ( THREE = 3.0D+0 ) 00239 * .. 00240 * .. Local Scalars .. 00241 LOGICAL NOTRAN 00242 CHARACTER TRANSN, TRANST 00243 INTEGER COUNT, I, J, KASE, NZ 00244 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN 00245 * .. 00246 * .. Local Arrays .. 00247 INTEGER ISAVE( 3 ) 00248 * .. 00249 * .. External Subroutines .. 00250 EXTERNAL DAXPY, DCOPY, DGTTRS, DLACN2, DLAGTM, XERBLA 00251 * .. 00252 * .. Intrinsic Functions .. 00253 INTRINSIC ABS, MAX 00254 * .. 00255 * .. External Functions .. 00256 LOGICAL LSAME 00257 DOUBLE PRECISION DLAMCH 00258 EXTERNAL LSAME, DLAMCH 00259 * .. 00260 * .. Executable Statements .. 00261 * 00262 * Test the input parameters. 00263 * 00264 INFO = 0 00265 NOTRAN = LSAME( TRANS, 'N' ) 00266 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00267 $ LSAME( TRANS, 'C' ) ) THEN 00268 INFO = -1 00269 ELSE IF( N.LT.0 ) THEN 00270 INFO = -2 00271 ELSE IF( NRHS.LT.0 ) THEN 00272 INFO = -3 00273 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00274 INFO = -13 00275 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00276 INFO = -15 00277 END IF 00278 IF( INFO.NE.0 ) THEN 00279 CALL XERBLA( 'DGTRFS', -INFO ) 00280 RETURN 00281 END IF 00282 * 00283 * Quick return if possible 00284 * 00285 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00286 DO 10 J = 1, NRHS 00287 FERR( J ) = ZERO 00288 BERR( J ) = ZERO 00289 10 CONTINUE 00290 RETURN 00291 END IF 00292 * 00293 IF( NOTRAN ) THEN 00294 TRANSN = 'N' 00295 TRANST = 'T' 00296 ELSE 00297 TRANSN = 'T' 00298 TRANST = 'N' 00299 END IF 00300 * 00301 * NZ = maximum number of nonzero elements in each row of A, plus 1 00302 * 00303 NZ = 4 00304 EPS = DLAMCH( 'Epsilon' ) 00305 SAFMIN = DLAMCH( 'Safe minimum' ) 00306 SAFE1 = NZ*SAFMIN 00307 SAFE2 = SAFE1 / EPS 00308 * 00309 * Do for each right hand side 00310 * 00311 DO 110 J = 1, NRHS 00312 * 00313 COUNT = 1 00314 LSTRES = THREE 00315 20 CONTINUE 00316 * 00317 * Loop until stopping criterion is satisfied. 00318 * 00319 * Compute residual R = B - op(A) * X, 00320 * where op(A) = A, A**T, or A**H, depending on TRANS. 00321 * 00322 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00323 CALL DLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE, 00324 $ WORK( N+1 ), N ) 00325 * 00326 * Compute abs(op(A))*abs(x) + abs(b) for use in the backward 00327 * error bound. 00328 * 00329 IF( NOTRAN ) THEN 00330 IF( N.EQ.1 ) THEN 00331 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) 00332 ELSE 00333 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + 00334 $ ABS( DU( 1 )*X( 2, J ) ) 00335 DO 30 I = 2, N - 1 00336 WORK( I ) = ABS( B( I, J ) ) + 00337 $ ABS( DL( I-1 )*X( I-1, J ) ) + 00338 $ ABS( D( I )*X( I, J ) ) + 00339 $ ABS( DU( I )*X( I+1, J ) ) 00340 30 CONTINUE 00341 WORK( N ) = ABS( B( N, J ) ) + 00342 $ ABS( DL( N-1 )*X( N-1, J ) ) + 00343 $ ABS( D( N )*X( N, J ) ) 00344 END IF 00345 ELSE 00346 IF( N.EQ.1 ) THEN 00347 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) 00348 ELSE 00349 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + 00350 $ ABS( DL( 1 )*X( 2, J ) ) 00351 DO 40 I = 2, N - 1 00352 WORK( I ) = ABS( B( I, J ) ) + 00353 $ ABS( DU( I-1 )*X( I-1, J ) ) + 00354 $ ABS( D( I )*X( I, J ) ) + 00355 $ ABS( DL( I )*X( I+1, J ) ) 00356 40 CONTINUE 00357 WORK( N ) = ABS( B( N, J ) ) + 00358 $ ABS( DU( N-1 )*X( N-1, J ) ) + 00359 $ ABS( D( N )*X( N, J ) ) 00360 END IF 00361 END IF 00362 * 00363 * Compute componentwise relative backward error from formula 00364 * 00365 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00366 * 00367 * where abs(Z) is the componentwise absolute value of the matrix 00368 * or vector Z. If the i-th component of the denominator is less 00369 * than SAFE2, then SAFE1 is added to the i-th components of the 00370 * numerator and denominator before dividing. 00371 * 00372 S = ZERO 00373 DO 50 I = 1, N 00374 IF( WORK( I ).GT.SAFE2 ) THEN 00375 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00376 ELSE 00377 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00378 $ ( WORK( I )+SAFE1 ) ) 00379 END IF 00380 50 CONTINUE 00381 BERR( J ) = S 00382 * 00383 * Test stopping criterion. Continue iterating if 00384 * 1) The residual BERR(J) is larger than machine epsilon, and 00385 * 2) BERR(J) decreased by at least a factor of 2 during the 00386 * last iteration, and 00387 * 3) At most ITMAX iterations tried. 00388 * 00389 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00390 $ COUNT.LE.ITMAX ) THEN 00391 * 00392 * Update solution and try again. 00393 * 00394 CALL DGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, 00395 $ WORK( N+1 ), N, INFO ) 00396 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00397 LSTRES = BERR( J ) 00398 COUNT = COUNT + 1 00399 GO TO 20 00400 END IF 00401 * 00402 * Bound error from formula 00403 * 00404 * norm(X - XTRUE) / norm(X) .le. FERR = 00405 * norm( abs(inv(op(A)))* 00406 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00407 * 00408 * where 00409 * norm(Z) is the magnitude of the largest component of Z 00410 * inv(op(A)) is the inverse of op(A) 00411 * abs(Z) is the componentwise absolute value of the matrix or 00412 * vector Z 00413 * NZ is the maximum number of nonzeros in any row of A, plus 1 00414 * EPS is machine epsilon 00415 * 00416 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00417 * is incremented by SAFE1 if the i-th component of 00418 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00419 * 00420 * Use DLACN2 to estimate the infinity-norm of the matrix 00421 * inv(op(A)) * diag(W), 00422 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00423 * 00424 DO 60 I = 1, N 00425 IF( WORK( I ).GT.SAFE2 ) THEN 00426 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00427 ELSE 00428 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00429 END IF 00430 60 CONTINUE 00431 * 00432 KASE = 0 00433 70 CONTINUE 00434 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00435 $ KASE, ISAVE ) 00436 IF( KASE.NE.0 ) THEN 00437 IF( KASE.EQ.1 ) THEN 00438 * 00439 * Multiply by diag(W)*inv(op(A)**T). 00440 * 00441 CALL DGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, 00442 $ WORK( N+1 ), N, INFO ) 00443 DO 80 I = 1, N 00444 WORK( N+I ) = WORK( I )*WORK( N+I ) 00445 80 CONTINUE 00446 ELSE 00447 * 00448 * Multiply by inv(op(A))*diag(W). 00449 * 00450 DO 90 I = 1, N 00451 WORK( N+I ) = WORK( I )*WORK( N+I ) 00452 90 CONTINUE 00453 CALL DGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, 00454 $ WORK( N+1 ), N, INFO ) 00455 END IF 00456 GO TO 70 00457 END IF 00458 * 00459 * Normalize error. 00460 * 00461 LSTRES = ZERO 00462 DO 100 I = 1, N 00463 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00464 100 CONTINUE 00465 IF( LSTRES.NE.ZERO ) 00466 $ FERR( J ) = FERR( J ) / LSTRES 00467 * 00468 110 CONTINUE 00469 * 00470 RETURN 00471 * 00472 * End of DGTRFS 00473 * 00474 END