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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b STPRFS 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download STPRFS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/stprfs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/stprfs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/stprfs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE STPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 00022 * FERR, BERR, WORK, IWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER DIAG, TRANS, UPLO 00026 * INTEGER INFO, LDB, LDX, N, NRHS 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER IWORK( * ) 00030 * REAL AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00031 * $ WORK( * ), X( LDX, * ) 00032 * .. 00033 * 00034 * 00035 *> \par Purpose: 00036 * ============= 00037 *> 00038 *> \verbatim 00039 *> 00040 *> STPRFS provides error bounds and backward error estimates for the 00041 *> solution to a system of linear equations with a triangular packed 00042 *> coefficient matrix. 00043 *> 00044 *> The solution matrix X must be computed by STPTRS or some other 00045 *> means before entering this routine. STPRFS does not do iterative 00046 *> refinement because doing so cannot improve the backward error. 00047 *> \endverbatim 00048 * 00049 * Arguments: 00050 * ========== 00051 * 00052 *> \param[in] UPLO 00053 *> \verbatim 00054 *> UPLO is CHARACTER*1 00055 *> = 'U': A is upper triangular; 00056 *> = 'L': A is lower triangular. 00057 *> \endverbatim 00058 *> 00059 *> \param[in] TRANS 00060 *> \verbatim 00061 *> TRANS is CHARACTER*1 00062 *> Specifies the form of the system of equations: 00063 *> = 'N': A * X = B (No transpose) 00064 *> = 'T': A**T * X = B (Transpose) 00065 *> = 'C': A**H * X = B (Conjugate transpose = Transpose) 00066 *> \endverbatim 00067 *> 00068 *> \param[in] DIAG 00069 *> \verbatim 00070 *> DIAG is CHARACTER*1 00071 *> = 'N': A is non-unit triangular; 00072 *> = 'U': A is unit triangular. 00073 *> \endverbatim 00074 *> 00075 *> \param[in] N 00076 *> \verbatim 00077 *> N is INTEGER 00078 *> The order of the matrix A. N >= 0. 00079 *> \endverbatim 00080 *> 00081 *> \param[in] NRHS 00082 *> \verbatim 00083 *> NRHS is INTEGER 00084 *> The number of right hand sides, i.e., the number of columns 00085 *> of the matrices B and X. NRHS >= 0. 00086 *> \endverbatim 00087 *> 00088 *> \param[in] AP 00089 *> \verbatim 00090 *> AP is REAL array, dimension (N*(N+1)/2) 00091 *> The upper or lower triangular matrix A, packed columnwise in 00092 *> a linear array. The j-th column of A is stored in the array 00093 *> AP as follows: 00094 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00095 *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 00096 *> If DIAG = 'U', the diagonal elements of A are not referenced 00097 *> and are assumed to be 1. 00098 *> \endverbatim 00099 *> 00100 *> \param[in] B 00101 *> \verbatim 00102 *> B is REAL array, dimension (LDB,NRHS) 00103 *> The right hand side matrix B. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDB 00107 *> \verbatim 00108 *> LDB is INTEGER 00109 *> The leading dimension of the array B. LDB >= max(1,N). 00110 *> \endverbatim 00111 *> 00112 *> \param[in] X 00113 *> \verbatim 00114 *> X is REAL array, dimension (LDX,NRHS) 00115 *> The solution matrix X. 00116 *> \endverbatim 00117 *> 00118 *> \param[in] LDX 00119 *> \verbatim 00120 *> LDX is INTEGER 00121 *> The leading dimension of the array X. LDX >= max(1,N). 00122 *> \endverbatim 00123 *> 00124 *> \param[out] FERR 00125 *> \verbatim 00126 *> FERR is REAL array, dimension (NRHS) 00127 *> The estimated forward error bound for each solution vector 00128 *> X(j) (the j-th column of the solution matrix X). 00129 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00130 *> is an estimated upper bound for the magnitude of the largest 00131 *> element in (X(j) - XTRUE) divided by the magnitude of the 00132 *> largest element in X(j). The estimate is as reliable as 00133 *> the estimate for RCOND, and is almost always a slight 00134 *> overestimate of the true error. 00135 *> \endverbatim 00136 *> 00137 *> \param[out] BERR 00138 *> \verbatim 00139 *> BERR is REAL array, dimension (NRHS) 00140 *> The componentwise relative backward error of each solution 00141 *> vector X(j) (i.e., the smallest relative change in 00142 *> any element of A or B that makes X(j) an exact solution). 00143 *> \endverbatim 00144 *> 00145 *> \param[out] WORK 00146 *> \verbatim 00147 *> WORK is REAL array, dimension (3*N) 00148 *> \endverbatim 00149 *> 00150 *> \param[out] IWORK 00151 *> \verbatim 00152 *> IWORK is INTEGER array, dimension (N) 00153 *> \endverbatim 00154 *> 00155 *> \param[out] INFO 00156 *> \verbatim 00157 *> INFO is INTEGER 00158 *> = 0: successful exit 00159 *> < 0: if INFO = -i, the i-th argument had an illegal value 00160 *> \endverbatim 00161 * 00162 * Authors: 00163 * ======== 00164 * 00165 *> \author Univ. of Tennessee 00166 *> \author Univ. of California Berkeley 00167 *> \author Univ. of Colorado Denver 00168 *> \author NAG Ltd. 00169 * 00170 *> \date November 2011 00171 * 00172 *> \ingroup realOTHERcomputational 00173 * 00174 * ===================================================================== 00175 SUBROUTINE STPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 00176 $ FERR, BERR, WORK, IWORK, INFO ) 00177 * 00178 * -- LAPACK computational routine (version 3.4.0) -- 00179 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00180 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00181 * November 2011 00182 * 00183 * .. Scalar Arguments .. 00184 CHARACTER DIAG, TRANS, UPLO 00185 INTEGER INFO, LDB, LDX, N, NRHS 00186 * .. 00187 * .. Array Arguments .. 00188 INTEGER IWORK( * ) 00189 REAL AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00190 $ WORK( * ), X( LDX, * ) 00191 * .. 00192 * 00193 * ===================================================================== 00194 * 00195 * .. Parameters .. 00196 REAL ZERO 00197 PARAMETER ( ZERO = 0.0E+0 ) 00198 REAL ONE 00199 PARAMETER ( ONE = 1.0E+0 ) 00200 * .. 00201 * .. Local Scalars .. 00202 LOGICAL NOTRAN, NOUNIT, UPPER 00203 CHARACTER TRANST 00204 INTEGER I, J, K, KASE, KC, NZ 00205 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00206 * .. 00207 * .. Local Arrays .. 00208 INTEGER ISAVE( 3 ) 00209 * .. 00210 * .. External Subroutines .. 00211 EXTERNAL SAXPY, SCOPY, SLACN2, STPMV, STPSV, XERBLA 00212 * .. 00213 * .. Intrinsic Functions .. 00214 INTRINSIC ABS, MAX 00215 * .. 00216 * .. External Functions .. 00217 LOGICAL LSAME 00218 REAL SLAMCH 00219 EXTERNAL LSAME, SLAMCH 00220 * .. 00221 * .. Executable Statements .. 00222 * 00223 * Test the input parameters. 00224 * 00225 INFO = 0 00226 UPPER = LSAME( UPLO, 'U' ) 00227 NOTRAN = LSAME( TRANS, 'N' ) 00228 NOUNIT = LSAME( DIAG, 'N' ) 00229 * 00230 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00231 INFO = -1 00232 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00233 $ LSAME( TRANS, 'C' ) ) THEN 00234 INFO = -2 00235 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN 00236 INFO = -3 00237 ELSE IF( N.LT.0 ) THEN 00238 INFO = -4 00239 ELSE IF( NRHS.LT.0 ) THEN 00240 INFO = -5 00241 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00242 INFO = -8 00243 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00244 INFO = -10 00245 END IF 00246 IF( INFO.NE.0 ) THEN 00247 CALL XERBLA( 'STPRFS', -INFO ) 00248 RETURN 00249 END IF 00250 * 00251 * Quick return if possible 00252 * 00253 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00254 DO 10 J = 1, NRHS 00255 FERR( J ) = ZERO 00256 BERR( J ) = ZERO 00257 10 CONTINUE 00258 RETURN 00259 END IF 00260 * 00261 IF( NOTRAN ) THEN 00262 TRANST = 'T' 00263 ELSE 00264 TRANST = 'N' 00265 END IF 00266 * 00267 * NZ = maximum number of nonzero elements in each row of A, plus 1 00268 * 00269 NZ = N + 1 00270 EPS = SLAMCH( 'Epsilon' ) 00271 SAFMIN = SLAMCH( 'Safe minimum' ) 00272 SAFE1 = NZ*SAFMIN 00273 SAFE2 = SAFE1 / EPS 00274 * 00275 * Do for each right hand side 00276 * 00277 DO 250 J = 1, NRHS 00278 * 00279 * Compute residual R = B - op(A) * X, 00280 * where op(A) = A or A**T, depending on TRANS. 00281 * 00282 CALL SCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) 00283 CALL STPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) 00284 CALL SAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) 00285 * 00286 * Compute componentwise relative backward error from formula 00287 * 00288 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00289 * 00290 * where abs(Z) is the componentwise absolute value of the matrix 00291 * or vector Z. If the i-th component of the denominator is less 00292 * than SAFE2, then SAFE1 is added to the i-th components of the 00293 * numerator and denominator before dividing. 00294 * 00295 DO 20 I = 1, N 00296 WORK( I ) = ABS( B( I, J ) ) 00297 20 CONTINUE 00298 * 00299 IF( NOTRAN ) THEN 00300 * 00301 * Compute abs(A)*abs(X) + abs(B). 00302 * 00303 IF( UPPER ) THEN 00304 KC = 1 00305 IF( NOUNIT ) THEN 00306 DO 40 K = 1, N 00307 XK = ABS( X( K, J ) ) 00308 DO 30 I = 1, K 00309 WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK 00310 30 CONTINUE 00311 KC = KC + K 00312 40 CONTINUE 00313 ELSE 00314 DO 60 K = 1, N 00315 XK = ABS( X( K, J ) ) 00316 DO 50 I = 1, K - 1 00317 WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK 00318 50 CONTINUE 00319 WORK( K ) = WORK( K ) + XK 00320 KC = KC + K 00321 60 CONTINUE 00322 END IF 00323 ELSE 00324 KC = 1 00325 IF( NOUNIT ) THEN 00326 DO 80 K = 1, N 00327 XK = ABS( X( K, J ) ) 00328 DO 70 I = K, N 00329 WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK 00330 70 CONTINUE 00331 KC = KC + N - K + 1 00332 80 CONTINUE 00333 ELSE 00334 DO 100 K = 1, N 00335 XK = ABS( X( K, J ) ) 00336 DO 90 I = K + 1, N 00337 WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK 00338 90 CONTINUE 00339 WORK( K ) = WORK( K ) + XK 00340 KC = KC + N - K + 1 00341 100 CONTINUE 00342 END IF 00343 END IF 00344 ELSE 00345 * 00346 * Compute abs(A**T)*abs(X) + abs(B). 00347 * 00348 IF( UPPER ) THEN 00349 KC = 1 00350 IF( NOUNIT ) THEN 00351 DO 120 K = 1, N 00352 S = ZERO 00353 DO 110 I = 1, K 00354 S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) ) 00355 110 CONTINUE 00356 WORK( K ) = WORK( K ) + S 00357 KC = KC + K 00358 120 CONTINUE 00359 ELSE 00360 DO 140 K = 1, N 00361 S = ABS( X( K, J ) ) 00362 DO 130 I = 1, K - 1 00363 S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) ) 00364 130 CONTINUE 00365 WORK( K ) = WORK( K ) + S 00366 KC = KC + K 00367 140 CONTINUE 00368 END IF 00369 ELSE 00370 KC = 1 00371 IF( NOUNIT ) THEN 00372 DO 160 K = 1, N 00373 S = ZERO 00374 DO 150 I = K, N 00375 S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) ) 00376 150 CONTINUE 00377 WORK( K ) = WORK( K ) + S 00378 KC = KC + N - K + 1 00379 160 CONTINUE 00380 ELSE 00381 DO 180 K = 1, N 00382 S = ABS( X( K, J ) ) 00383 DO 170 I = K + 1, N 00384 S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) ) 00385 170 CONTINUE 00386 WORK( K ) = WORK( K ) + S 00387 KC = KC + N - K + 1 00388 180 CONTINUE 00389 END IF 00390 END IF 00391 END IF 00392 S = ZERO 00393 DO 190 I = 1, N 00394 IF( WORK( I ).GT.SAFE2 ) THEN 00395 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00396 ELSE 00397 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00398 $ ( WORK( I )+SAFE1 ) ) 00399 END IF 00400 190 CONTINUE 00401 BERR( J ) = S 00402 * 00403 * Bound error from formula 00404 * 00405 * norm(X - XTRUE) / norm(X) .le. FERR = 00406 * norm( abs(inv(op(A)))* 00407 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00408 * 00409 * where 00410 * norm(Z) is the magnitude of the largest component of Z 00411 * inv(op(A)) is the inverse of op(A) 00412 * abs(Z) is the componentwise absolute value of the matrix or 00413 * vector Z 00414 * NZ is the maximum number of nonzeros in any row of A, plus 1 00415 * EPS is machine epsilon 00416 * 00417 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00418 * is incremented by SAFE1 if the i-th component of 00419 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00420 * 00421 * Use SLACN2 to estimate the infinity-norm of the matrix 00422 * inv(op(A)) * diag(W), 00423 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00424 * 00425 DO 200 I = 1, N 00426 IF( WORK( I ).GT.SAFE2 ) THEN 00427 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00428 ELSE 00429 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00430 END IF 00431 200 CONTINUE 00432 * 00433 KASE = 0 00434 210 CONTINUE 00435 CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00436 $ KASE, ISAVE ) 00437 IF( KASE.NE.0 ) THEN 00438 IF( KASE.EQ.1 ) THEN 00439 * 00440 * Multiply by diag(W)*inv(op(A)**T). 00441 * 00442 CALL STPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 ) 00443 DO 220 I = 1, N 00444 WORK( N+I ) = WORK( I )*WORK( N+I ) 00445 220 CONTINUE 00446 ELSE 00447 * 00448 * Multiply by inv(op(A))*diag(W). 00449 * 00450 DO 230 I = 1, N 00451 WORK( N+I ) = WORK( I )*WORK( N+I ) 00452 230 CONTINUE 00453 CALL STPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) 00454 END IF 00455 GO TO 210 00456 END IF 00457 * 00458 * Normalize error. 00459 * 00460 LSTRES = ZERO 00461 DO 240 I = 1, N 00462 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00463 240 CONTINUE 00464 IF( LSTRES.NE.ZERO ) 00465 $ FERR( J ) = FERR( J ) / LSTRES 00466 * 00467 250 CONTINUE 00468 * 00469 RETURN 00470 * 00471 * End of STPRFS 00472 * 00473 END