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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DPPRFS 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DPPRFS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpprfs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpprfs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpprfs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, 00022 * BERR, WORK, IWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER UPLO 00026 * INTEGER INFO, LDB, LDX, N, NRHS 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER IWORK( * ) 00030 * DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00031 * $ FERR( * ), WORK( * ), X( LDX, * ) 00032 * .. 00033 * 00034 * 00035 *> \par Purpose: 00036 * ============= 00037 *> 00038 *> \verbatim 00039 *> 00040 *> DPPRFS improves the computed solution to a system of linear 00041 *> equations when the coefficient matrix is symmetric positive definite 00042 *> and packed, and provides error bounds and backward error estimates 00043 *> for the solution. 00044 *> \endverbatim 00045 * 00046 * Arguments: 00047 * ========== 00048 * 00049 *> \param[in] UPLO 00050 *> \verbatim 00051 *> UPLO is CHARACTER*1 00052 *> = 'U': Upper triangle of A is stored; 00053 *> = 'L': Lower triangle of A is stored. 00054 *> \endverbatim 00055 *> 00056 *> \param[in] N 00057 *> \verbatim 00058 *> N is INTEGER 00059 *> The order of the matrix A. N >= 0. 00060 *> \endverbatim 00061 *> 00062 *> \param[in] NRHS 00063 *> \verbatim 00064 *> NRHS is INTEGER 00065 *> The number of right hand sides, i.e., the number of columns 00066 *> of the matrices B and X. NRHS >= 0. 00067 *> \endverbatim 00068 *> 00069 *> \param[in] AP 00070 *> \verbatim 00071 *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 00072 *> The upper or lower triangle of the symmetric matrix A, packed 00073 *> columnwise in a linear array. The j-th column of A is stored 00074 *> in the array AP as follows: 00075 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00076 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00077 *> \endverbatim 00078 *> 00079 *> \param[in] AFP 00080 *> \verbatim 00081 *> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 00082 *> The triangular factor U or L from the Cholesky factorization 00083 *> A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF, 00084 *> packed columnwise in a linear array in the same format as A 00085 *> (see AP). 00086 *> \endverbatim 00087 *> 00088 *> \param[in] B 00089 *> \verbatim 00090 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00091 *> The right hand side matrix B. 00092 *> \endverbatim 00093 *> 00094 *> \param[in] LDB 00095 *> \verbatim 00096 *> LDB is INTEGER 00097 *> The leading dimension of the array B. LDB >= max(1,N). 00098 *> \endverbatim 00099 *> 00100 *> \param[in,out] X 00101 *> \verbatim 00102 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00103 *> On entry, the solution matrix X, as computed by DPPTRS. 00104 *> On exit, the improved solution matrix X. 00105 *> \endverbatim 00106 *> 00107 *> \param[in] LDX 00108 *> \verbatim 00109 *> LDX is INTEGER 00110 *> The leading dimension of the array X. LDX >= max(1,N). 00111 *> \endverbatim 00112 *> 00113 *> \param[out] FERR 00114 *> \verbatim 00115 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00116 *> The estimated forward error bound for each solution vector 00117 *> X(j) (the j-th column of the solution matrix X). 00118 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00119 *> is an estimated upper bound for the magnitude of the largest 00120 *> element in (X(j) - XTRUE) divided by the magnitude of the 00121 *> largest element in X(j). The estimate is as reliable as 00122 *> the estimate for RCOND, and is almost always a slight 00123 *> overestimate of the true error. 00124 *> \endverbatim 00125 *> 00126 *> \param[out] BERR 00127 *> \verbatim 00128 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00129 *> The componentwise relative backward error of each solution 00130 *> vector X(j) (i.e., the smallest relative change in 00131 *> any element of A or B that makes X(j) an exact solution). 00132 *> \endverbatim 00133 *> 00134 *> \param[out] WORK 00135 *> \verbatim 00136 *> WORK is DOUBLE PRECISION array, dimension (3*N) 00137 *> \endverbatim 00138 *> 00139 *> \param[out] IWORK 00140 *> \verbatim 00141 *> IWORK is INTEGER array, dimension (N) 00142 *> \endverbatim 00143 *> 00144 *> \param[out] INFO 00145 *> \verbatim 00146 *> INFO is INTEGER 00147 *> = 0: successful exit 00148 *> < 0: if INFO = -i, the i-th argument had an illegal value 00149 *> \endverbatim 00150 * 00151 *> \par Internal Parameters: 00152 * ========================= 00153 *> 00154 *> \verbatim 00155 *> ITMAX is the maximum number of steps of iterative refinement. 00156 *> \endverbatim 00157 * 00158 * Authors: 00159 * ======== 00160 * 00161 *> \author Univ. of Tennessee 00162 *> \author Univ. of California Berkeley 00163 *> \author Univ. of Colorado Denver 00164 *> \author NAG Ltd. 00165 * 00166 *> \date November 2011 00167 * 00168 *> \ingroup doubleOTHERcomputational 00169 * 00170 * ===================================================================== 00171 SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, 00172 $ BERR, WORK, IWORK, INFO ) 00173 * 00174 * -- LAPACK computational 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 CHARACTER UPLO 00181 INTEGER INFO, LDB, LDX, N, NRHS 00182 * .. 00183 * .. Array Arguments .. 00184 INTEGER IWORK( * ) 00185 DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00186 $ FERR( * ), WORK( * ), X( LDX, * ) 00187 * .. 00188 * 00189 * ===================================================================== 00190 * 00191 * .. Parameters .. 00192 INTEGER ITMAX 00193 PARAMETER ( ITMAX = 5 ) 00194 DOUBLE PRECISION ZERO 00195 PARAMETER ( ZERO = 0.0D+0 ) 00196 DOUBLE PRECISION ONE 00197 PARAMETER ( ONE = 1.0D+0 ) 00198 DOUBLE PRECISION TWO 00199 PARAMETER ( TWO = 2.0D+0 ) 00200 DOUBLE PRECISION THREE 00201 PARAMETER ( THREE = 3.0D+0 ) 00202 * .. 00203 * .. Local Scalars .. 00204 LOGICAL UPPER 00205 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ 00206 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00207 * .. 00208 * .. Local Arrays .. 00209 INTEGER ISAVE( 3 ) 00210 * .. 00211 * .. External Subroutines .. 00212 EXTERNAL DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV, XERBLA 00213 * .. 00214 * .. Intrinsic Functions .. 00215 INTRINSIC ABS, MAX 00216 * .. 00217 * .. External Functions .. 00218 LOGICAL LSAME 00219 DOUBLE PRECISION DLAMCH 00220 EXTERNAL LSAME, DLAMCH 00221 * .. 00222 * .. Executable Statements .. 00223 * 00224 * Test the input parameters. 00225 * 00226 INFO = 0 00227 UPPER = LSAME( UPLO, 'U' ) 00228 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00229 INFO = -1 00230 ELSE IF( N.LT.0 ) THEN 00231 INFO = -2 00232 ELSE IF( NRHS.LT.0 ) THEN 00233 INFO = -3 00234 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00235 INFO = -7 00236 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00237 INFO = -9 00238 END IF 00239 IF( INFO.NE.0 ) THEN 00240 CALL XERBLA( 'DPPRFS', -INFO ) 00241 RETURN 00242 END IF 00243 * 00244 * Quick return if possible 00245 * 00246 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00247 DO 10 J = 1, NRHS 00248 FERR( J ) = ZERO 00249 BERR( J ) = ZERO 00250 10 CONTINUE 00251 RETURN 00252 END IF 00253 * 00254 * NZ = maximum number of nonzero elements in each row of A, plus 1 00255 * 00256 NZ = N + 1 00257 EPS = DLAMCH( 'Epsilon' ) 00258 SAFMIN = DLAMCH( 'Safe minimum' ) 00259 SAFE1 = NZ*SAFMIN 00260 SAFE2 = SAFE1 / EPS 00261 * 00262 * Do for each right hand side 00263 * 00264 DO 140 J = 1, NRHS 00265 * 00266 COUNT = 1 00267 LSTRES = THREE 00268 20 CONTINUE 00269 * 00270 * Loop until stopping criterion is satisfied. 00271 * 00272 * Compute residual R = B - A * X 00273 * 00274 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00275 CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ), 00276 $ 1 ) 00277 * 00278 * Compute componentwise relative backward error from formula 00279 * 00280 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00281 * 00282 * where abs(Z) is the componentwise absolute value of the matrix 00283 * or vector Z. If the i-th component of the denominator is less 00284 * than SAFE2, then SAFE1 is added to the i-th components of the 00285 * numerator and denominator before dividing. 00286 * 00287 DO 30 I = 1, N 00288 WORK( I ) = ABS( B( I, J ) ) 00289 30 CONTINUE 00290 * 00291 * Compute abs(A)*abs(X) + abs(B). 00292 * 00293 KK = 1 00294 IF( UPPER ) THEN 00295 DO 50 K = 1, N 00296 S = ZERO 00297 XK = ABS( X( K, J ) ) 00298 IK = KK 00299 DO 40 I = 1, K - 1 00300 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK 00301 S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) 00302 IK = IK + 1 00303 40 CONTINUE 00304 WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S 00305 KK = KK + K 00306 50 CONTINUE 00307 ELSE 00308 DO 70 K = 1, N 00309 S = ZERO 00310 XK = ABS( X( K, J ) ) 00311 WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK 00312 IK = KK + 1 00313 DO 60 I = K + 1, N 00314 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK 00315 S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) 00316 IK = IK + 1 00317 60 CONTINUE 00318 WORK( K ) = WORK( K ) + S 00319 KK = KK + ( N-K+1 ) 00320 70 CONTINUE 00321 END IF 00322 S = ZERO 00323 DO 80 I = 1, N 00324 IF( WORK( I ).GT.SAFE2 ) THEN 00325 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00326 ELSE 00327 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00328 $ ( WORK( I )+SAFE1 ) ) 00329 END IF 00330 80 CONTINUE 00331 BERR( J ) = S 00332 * 00333 * Test stopping criterion. Continue iterating if 00334 * 1) The residual BERR(J) is larger than machine epsilon, and 00335 * 2) BERR(J) decreased by at least a factor of 2 during the 00336 * last iteration, and 00337 * 3) At most ITMAX iterations tried. 00338 * 00339 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00340 $ COUNT.LE.ITMAX ) THEN 00341 * 00342 * Update solution and try again. 00343 * 00344 CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) 00345 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00346 LSTRES = BERR( J ) 00347 COUNT = COUNT + 1 00348 GO TO 20 00349 END IF 00350 * 00351 * Bound error from formula 00352 * 00353 * norm(X - XTRUE) / norm(X) .le. FERR = 00354 * norm( abs(inv(A))* 00355 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00356 * 00357 * where 00358 * norm(Z) is the magnitude of the largest component of Z 00359 * inv(A) is the inverse of A 00360 * abs(Z) is the componentwise absolute value of the matrix or 00361 * vector Z 00362 * NZ is the maximum number of nonzeros in any row of A, plus 1 00363 * EPS is machine epsilon 00364 * 00365 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00366 * is incremented by SAFE1 if the i-th component of 00367 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00368 * 00369 * Use DLACN2 to estimate the infinity-norm of the matrix 00370 * inv(A) * diag(W), 00371 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00372 * 00373 DO 90 I = 1, N 00374 IF( WORK( I ).GT.SAFE2 ) THEN 00375 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00376 ELSE 00377 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00378 END IF 00379 90 CONTINUE 00380 * 00381 KASE = 0 00382 100 CONTINUE 00383 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00384 $ KASE, ISAVE ) 00385 IF( KASE.NE.0 ) THEN 00386 IF( KASE.EQ.1 ) THEN 00387 * 00388 * Multiply by diag(W)*inv(A**T). 00389 * 00390 CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) 00391 DO 110 I = 1, N 00392 WORK( N+I ) = WORK( I )*WORK( N+I ) 00393 110 CONTINUE 00394 ELSE IF( KASE.EQ.2 ) THEN 00395 * 00396 * Multiply by inv(A)*diag(W). 00397 * 00398 DO 120 I = 1, N 00399 WORK( N+I ) = WORK( I )*WORK( N+I ) 00400 120 CONTINUE 00401 CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) 00402 END IF 00403 GO TO 100 00404 END IF 00405 * 00406 * Normalize error. 00407 * 00408 LSTRES = ZERO 00409 DO 130 I = 1, N 00410 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00411 130 CONTINUE 00412 IF( LSTRES.NE.ZERO ) 00413 $ FERR( J ) = FERR( J ) / LSTRES 00414 * 00415 140 CONTINUE 00416 * 00417 RETURN 00418 * 00419 * End of DPPRFS 00420 * 00421 END