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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, 00002 $ LDX, FERR, BERR, WORK, IWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. 00010 * 00011 * .. Scalar Arguments .. 00012 CHARACTER UPLO 00013 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IWORK( * ) 00017 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00018 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DPORFS improves the computed solution to a system of linear 00025 * equations when the coefficient matrix is symmetric positive definite, 00026 * and provides error bounds and backward error estimates for the 00027 * solution. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * UPLO (input) CHARACTER*1 00033 * = 'U': Upper triangle of A is stored; 00034 * = 'L': Lower triangle of A is stored. 00035 * 00036 * N (input) INTEGER 00037 * The order of the matrix A. N >= 0. 00038 * 00039 * NRHS (input) INTEGER 00040 * The number of right hand sides, i.e., the number of columns 00041 * of the matrices B and X. NRHS >= 0. 00042 * 00043 * A (input) DOUBLE PRECISION array, dimension (LDA,N) 00044 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00045 * upper triangular part of A contains the upper triangular part 00046 * of the matrix A, and the strictly lower triangular part of A 00047 * is not referenced. If UPLO = 'L', the leading N-by-N lower 00048 * triangular part of A contains the lower triangular part of 00049 * the matrix A, and the strictly upper triangular part of A is 00050 * not referenced. 00051 * 00052 * LDA (input) INTEGER 00053 * The leading dimension of the array A. LDA >= max(1,N). 00054 * 00055 * AF (input) DOUBLE PRECISION array, dimension (LDAF,N) 00056 * The triangular factor U or L from the Cholesky factorization 00057 * A = U**T*U or A = L*L**T, as computed by DPOTRF. 00058 * 00059 * LDAF (input) INTEGER 00060 * The leading dimension of the array AF. LDAF >= max(1,N). 00061 * 00062 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00063 * The right hand side matrix B. 00064 * 00065 * LDB (input) INTEGER 00066 * The leading dimension of the array B. LDB >= max(1,N). 00067 * 00068 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) 00069 * On entry, the solution matrix X, as computed by DPOTRS. 00070 * On exit, the improved solution matrix X. 00071 * 00072 * LDX (input) INTEGER 00073 * The leading dimension of the array X. LDX >= max(1,N). 00074 * 00075 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00076 * The estimated forward error bound for each solution vector 00077 * X(j) (the j-th column of the solution matrix X). 00078 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00079 * is an estimated upper bound for the magnitude of the largest 00080 * element in (X(j) - XTRUE) divided by the magnitude of the 00081 * largest element in X(j). The estimate is as reliable as 00082 * the estimate for RCOND, and is almost always a slight 00083 * overestimate of the true error. 00084 * 00085 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00086 * The componentwise relative backward error of each solution 00087 * vector X(j) (i.e., the smallest relative change in 00088 * any element of A or B that makes X(j) an exact solution). 00089 * 00090 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 00091 * 00092 * IWORK (workspace) INTEGER array, dimension (N) 00093 * 00094 * INFO (output) INTEGER 00095 * = 0: successful exit 00096 * < 0: if INFO = -i, the i-th argument had an illegal value 00097 * 00098 * Internal Parameters 00099 * =================== 00100 * 00101 * ITMAX is the maximum number of steps of iterative refinement. 00102 * 00103 * ===================================================================== 00104 * 00105 * .. Parameters .. 00106 INTEGER ITMAX 00107 PARAMETER ( ITMAX = 5 ) 00108 DOUBLE PRECISION ZERO 00109 PARAMETER ( ZERO = 0.0D+0 ) 00110 DOUBLE PRECISION ONE 00111 PARAMETER ( ONE = 1.0D+0 ) 00112 DOUBLE PRECISION TWO 00113 PARAMETER ( TWO = 2.0D+0 ) 00114 DOUBLE PRECISION THREE 00115 PARAMETER ( THREE = 3.0D+0 ) 00116 * .. 00117 * .. Local Scalars .. 00118 LOGICAL UPPER 00119 INTEGER COUNT, I, J, K, KASE, NZ 00120 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00121 * .. 00122 * .. Local Arrays .. 00123 INTEGER ISAVE( 3 ) 00124 * .. 00125 * .. External Subroutines .. 00126 EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV, XERBLA 00127 * .. 00128 * .. Intrinsic Functions .. 00129 INTRINSIC ABS, MAX 00130 * .. 00131 * .. External Functions .. 00132 LOGICAL LSAME 00133 DOUBLE PRECISION DLAMCH 00134 EXTERNAL LSAME, DLAMCH 00135 * .. 00136 * .. Executable Statements .. 00137 * 00138 * Test the input parameters. 00139 * 00140 INFO = 0 00141 UPPER = LSAME( UPLO, 'U' ) 00142 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00143 INFO = -1 00144 ELSE IF( N.LT.0 ) THEN 00145 INFO = -2 00146 ELSE IF( NRHS.LT.0 ) THEN 00147 INFO = -3 00148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00149 INFO = -5 00150 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00151 INFO = -7 00152 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00153 INFO = -9 00154 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00155 INFO = -11 00156 END IF 00157 IF( INFO.NE.0 ) THEN 00158 CALL XERBLA( 'DPORFS', -INFO ) 00159 RETURN 00160 END IF 00161 * 00162 * Quick return if possible 00163 * 00164 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00165 DO 10 J = 1, NRHS 00166 FERR( J ) = ZERO 00167 BERR( J ) = ZERO 00168 10 CONTINUE 00169 RETURN 00170 END IF 00171 * 00172 * NZ = maximum number of nonzero elements in each row of A, plus 1 00173 * 00174 NZ = N + 1 00175 EPS = DLAMCH( 'Epsilon' ) 00176 SAFMIN = DLAMCH( 'Safe minimum' ) 00177 SAFE1 = NZ*SAFMIN 00178 SAFE2 = SAFE1 / EPS 00179 * 00180 * Do for each right hand side 00181 * 00182 DO 140 J = 1, NRHS 00183 * 00184 COUNT = 1 00185 LSTRES = THREE 00186 20 CONTINUE 00187 * 00188 * Loop until stopping criterion is satisfied. 00189 * 00190 * Compute residual R = B - A * X 00191 * 00192 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00193 CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, 00194 $ WORK( N+1 ), 1 ) 00195 * 00196 * Compute componentwise relative backward error from formula 00197 * 00198 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00199 * 00200 * where abs(Z) is the componentwise absolute value of the matrix 00201 * or vector Z. If the i-th component of the denominator is less 00202 * than SAFE2, then SAFE1 is added to the i-th components of the 00203 * numerator and denominator before dividing. 00204 * 00205 DO 30 I = 1, N 00206 WORK( I ) = ABS( B( I, J ) ) 00207 30 CONTINUE 00208 * 00209 * Compute abs(A)*abs(X) + abs(B). 00210 * 00211 IF( UPPER ) THEN 00212 DO 50 K = 1, N 00213 S = ZERO 00214 XK = ABS( X( K, J ) ) 00215 DO 40 I = 1, K - 1 00216 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00217 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00218 40 CONTINUE 00219 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S 00220 50 CONTINUE 00221 ELSE 00222 DO 70 K = 1, N 00223 S = ZERO 00224 XK = ABS( X( K, J ) ) 00225 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK 00226 DO 60 I = K + 1, N 00227 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00228 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00229 60 CONTINUE 00230 WORK( K ) = WORK( K ) + S 00231 70 CONTINUE 00232 END IF 00233 S = ZERO 00234 DO 80 I = 1, N 00235 IF( WORK( I ).GT.SAFE2 ) THEN 00236 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00237 ELSE 00238 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00239 $ ( WORK( I )+SAFE1 ) ) 00240 END IF 00241 80 CONTINUE 00242 BERR( J ) = S 00243 * 00244 * Test stopping criterion. Continue iterating if 00245 * 1) The residual BERR(J) is larger than machine epsilon, and 00246 * 2) BERR(J) decreased by at least a factor of 2 during the 00247 * last iteration, and 00248 * 3) At most ITMAX iterations tried. 00249 * 00250 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00251 $ COUNT.LE.ITMAX ) THEN 00252 * 00253 * Update solution and try again. 00254 * 00255 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) 00256 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00257 LSTRES = BERR( J ) 00258 COUNT = COUNT + 1 00259 GO TO 20 00260 END IF 00261 * 00262 * Bound error from formula 00263 * 00264 * norm(X - XTRUE) / norm(X) .le. FERR = 00265 * norm( abs(inv(A))* 00266 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00267 * 00268 * where 00269 * norm(Z) is the magnitude of the largest component of Z 00270 * inv(A) is the inverse of A 00271 * abs(Z) is the componentwise absolute value of the matrix or 00272 * vector Z 00273 * NZ is the maximum number of nonzeros in any row of A, plus 1 00274 * EPS is machine epsilon 00275 * 00276 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00277 * is incremented by SAFE1 if the i-th component of 00278 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00279 * 00280 * Use DLACN2 to estimate the infinity-norm of the matrix 00281 * inv(A) * diag(W), 00282 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00283 * 00284 DO 90 I = 1, N 00285 IF( WORK( I ).GT.SAFE2 ) THEN 00286 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00287 ELSE 00288 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00289 END IF 00290 90 CONTINUE 00291 * 00292 KASE = 0 00293 100 CONTINUE 00294 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00295 $ KASE, ISAVE ) 00296 IF( KASE.NE.0 ) THEN 00297 IF( KASE.EQ.1 ) THEN 00298 * 00299 * Multiply by diag(W)*inv(A**T). 00300 * 00301 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) 00302 DO 110 I = 1, N 00303 WORK( N+I ) = WORK( I )*WORK( N+I ) 00304 110 CONTINUE 00305 ELSE IF( KASE.EQ.2 ) THEN 00306 * 00307 * Multiply by inv(A)*diag(W). 00308 * 00309 DO 120 I = 1, N 00310 WORK( N+I ) = WORK( I )*WORK( N+I ) 00311 120 CONTINUE 00312 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) 00313 END IF 00314 GO TO 100 00315 END IF 00316 * 00317 * Normalize error. 00318 * 00319 LSTRES = ZERO 00320 DO 130 I = 1, N 00321 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00322 130 CONTINUE 00323 IF( LSTRES.NE.ZERO ) 00324 $ FERR( J ) = FERR( J ) / LSTRES 00325 * 00326 140 CONTINUE 00327 * 00328 RETURN 00329 * 00330 * End of DPORFS 00331 * 00332 END
1.7.3 
