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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZPPRFS 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZPPRFS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpprfs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpprfs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpprfs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, 00022 * BERR, WORK, RWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER UPLO 00026 * INTEGER INFO, LDB, LDX, N, NRHS 00027 * .. 00028 * .. Array Arguments .. 00029 * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00030 * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), 00031 * $ X( LDX, * ) 00032 * .. 00033 * 00034 * 00035 *> \par Purpose: 00036 * ============= 00037 *> 00038 *> \verbatim 00039 *> 00040 *> ZPPRFS improves the computed solution to a system of linear 00041 *> equations when the coefficient matrix is Hermitian 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 COMPLEX*16 array, dimension (N*(N+1)/2) 00072 *> The upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (N*(N+1)/2) 00082 *> The triangular factor U or L from the Cholesky factorization 00083 *> A = U**H*U or A = L*L**H, 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS) 00103 *> On entry, the solution matrix X, as computed by ZPPTRS. 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 COMPLEX*16 array, dimension (2*N) 00137 *> \endverbatim 00138 *> 00139 *> \param[out] RWORK 00140 *> \verbatim 00141 *> RWORK is DOUBLE PRECISION 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 complex16OTHERcomputational 00169 * 00170 * ===================================================================== 00171 SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, 00172 $ BERR, WORK, RWORK, 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 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00185 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), 00186 $ 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 COMPLEX*16 CONE 00197 PARAMETER ( CONE = ( 1.0D+0, 0.0D+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 COMPLEX*16 ZDUM 00208 * .. 00209 * .. Local Arrays .. 00210 INTEGER ISAVE( 3 ) 00211 * .. 00212 * .. External Subroutines .. 00213 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS 00214 * .. 00215 * .. Intrinsic Functions .. 00216 INTRINSIC ABS, DBLE, DIMAG, MAX 00217 * .. 00218 * .. External Functions .. 00219 LOGICAL LSAME 00220 DOUBLE PRECISION DLAMCH 00221 EXTERNAL LSAME, DLAMCH 00222 * .. 00223 * .. Statement Functions .. 00224 DOUBLE PRECISION CABS1 00225 * .. 00226 * .. Statement Function definitions .. 00227 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00228 * .. 00229 * .. Executable Statements .. 00230 * 00231 * Test the input parameters. 00232 * 00233 INFO = 0 00234 UPPER = LSAME( UPLO, 'U' ) 00235 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00236 INFO = -1 00237 ELSE IF( N.LT.0 ) THEN 00238 INFO = -2 00239 ELSE IF( NRHS.LT.0 ) THEN 00240 INFO = -3 00241 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00242 INFO = -7 00243 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00244 INFO = -9 00245 END IF 00246 IF( INFO.NE.0 ) THEN 00247 CALL XERBLA( 'ZPPRFS', -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 * NZ = maximum number of nonzero elements in each row of A, plus 1 00262 * 00263 NZ = N + 1 00264 EPS = DLAMCH( 'Epsilon' ) 00265 SAFMIN = DLAMCH( 'Safe minimum' ) 00266 SAFE1 = NZ*SAFMIN 00267 SAFE2 = SAFE1 / EPS 00268 * 00269 * Do for each right hand side 00270 * 00271 DO 140 J = 1, NRHS 00272 * 00273 COUNT = 1 00274 LSTRES = THREE 00275 20 CONTINUE 00276 * 00277 * Loop until stopping criterion is satisfied. 00278 * 00279 * Compute residual R = B - A * X 00280 * 00281 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 00282 CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 ) 00283 * 00284 * Compute componentwise relative backward error from formula 00285 * 00286 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00287 * 00288 * where abs(Z) is the componentwise absolute value of the matrix 00289 * or vector Z. If the i-th component of the denominator is less 00290 * than SAFE2, then SAFE1 is added to the i-th components of the 00291 * numerator and denominator before dividing. 00292 * 00293 DO 30 I = 1, N 00294 RWORK( I ) = CABS1( B( I, J ) ) 00295 30 CONTINUE 00296 * 00297 * Compute abs(A)*abs(X) + abs(B). 00298 * 00299 KK = 1 00300 IF( UPPER ) THEN 00301 DO 50 K = 1, N 00302 S = ZERO 00303 XK = CABS1( X( K, J ) ) 00304 IK = KK 00305 DO 40 I = 1, K - 1 00306 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK 00307 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) ) 00308 IK = IK + 1 00309 40 CONTINUE 00310 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )* 00311 $ XK + S 00312 KK = KK + K 00313 50 CONTINUE 00314 ELSE 00315 DO 70 K = 1, N 00316 S = ZERO 00317 XK = CABS1( X( K, J ) ) 00318 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK 00319 IK = KK + 1 00320 DO 60 I = K + 1, N 00321 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK 00322 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) ) 00323 IK = IK + 1 00324 60 CONTINUE 00325 RWORK( K ) = RWORK( K ) + S 00326 KK = KK + ( N-K+1 ) 00327 70 CONTINUE 00328 END IF 00329 S = ZERO 00330 DO 80 I = 1, N 00331 IF( RWORK( I ).GT.SAFE2 ) THEN 00332 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 00333 ELSE 00334 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 00335 $ ( RWORK( I )+SAFE1 ) ) 00336 END IF 00337 80 CONTINUE 00338 BERR( J ) = S 00339 * 00340 * Test stopping criterion. Continue iterating if 00341 * 1) The residual BERR(J) is larger than machine epsilon, and 00342 * 2) BERR(J) decreased by at least a factor of 2 during the 00343 * last iteration, and 00344 * 3) At most ITMAX iterations tried. 00345 * 00346 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00347 $ COUNT.LE.ITMAX ) THEN 00348 * 00349 * Update solution and try again. 00350 * 00351 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) 00352 CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 ) 00353 LSTRES = BERR( J ) 00354 COUNT = COUNT + 1 00355 GO TO 20 00356 END IF 00357 * 00358 * Bound error from formula 00359 * 00360 * norm(X - XTRUE) / norm(X) .le. FERR = 00361 * norm( abs(inv(A))* 00362 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00363 * 00364 * where 00365 * norm(Z) is the magnitude of the largest component of Z 00366 * inv(A) is the inverse of A 00367 * abs(Z) is the componentwise absolute value of the matrix or 00368 * vector Z 00369 * NZ is the maximum number of nonzeros in any row of A, plus 1 00370 * EPS is machine epsilon 00371 * 00372 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00373 * is incremented by SAFE1 if the i-th component of 00374 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00375 * 00376 * Use ZLACN2 to estimate the infinity-norm of the matrix 00377 * inv(A) * diag(W), 00378 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00379 * 00380 DO 90 I = 1, N 00381 IF( RWORK( I ).GT.SAFE2 ) THEN 00382 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 00383 ELSE 00384 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 00385 $ SAFE1 00386 END IF 00387 90 CONTINUE 00388 * 00389 KASE = 0 00390 100 CONTINUE 00391 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 00392 IF( KASE.NE.0 ) THEN 00393 IF( KASE.EQ.1 ) THEN 00394 * 00395 * Multiply by diag(W)*inv(A**H). 00396 * 00397 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) 00398 DO 110 I = 1, N 00399 WORK( I ) = RWORK( I )*WORK( I ) 00400 110 CONTINUE 00401 ELSE IF( KASE.EQ.2 ) THEN 00402 * 00403 * Multiply by inv(A)*diag(W). 00404 * 00405 DO 120 I = 1, N 00406 WORK( I ) = RWORK( I )*WORK( I ) 00407 120 CONTINUE 00408 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) 00409 END IF 00410 GO TO 100 00411 END IF 00412 * 00413 * Normalize error. 00414 * 00415 LSTRES = ZERO 00416 DO 130 I = 1, N 00417 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 00418 130 CONTINUE 00419 IF( LSTRES.NE.ZERO ) 00420 $ FERR( J ) = FERR( J ) / LSTRES 00421 * 00422 140 CONTINUE 00423 * 00424 RETURN 00425 * 00426 * End of ZPPRFS 00427 * 00428 END