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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 DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 00002 $ X, 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 IPIV( * ), IWORK( * ) 00017 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00018 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DSYRFS improves the computed solution to a system of linear 00025 * equations when the coefficient matrix is symmetric indefinite, and 00026 * provides error bounds and backward error estimates for the solution. 00027 * 00028 * Arguments 00029 * ========= 00030 * 00031 * UPLO (input) CHARACTER*1 00032 * = 'U': Upper triangle of A is stored; 00033 * = 'L': Lower triangle of A is stored. 00034 * 00035 * N (input) INTEGER 00036 * The order of the matrix A. N >= 0. 00037 * 00038 * NRHS (input) INTEGER 00039 * The number of right hand sides, i.e., the number of columns 00040 * of the matrices B and X. NRHS >= 0. 00041 * 00042 * A (input) DOUBLE PRECISION array, dimension (LDA,N) 00043 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00044 * upper triangular part of A contains the upper triangular part 00045 * of the matrix A, and the strictly lower triangular part of A 00046 * is not referenced. If UPLO = 'L', the leading N-by-N lower 00047 * triangular part of A contains the lower triangular part of 00048 * the matrix A, and the strictly upper triangular part of A is 00049 * not referenced. 00050 * 00051 * LDA (input) INTEGER 00052 * The leading dimension of the array A. LDA >= max(1,N). 00053 * 00054 * AF (input) DOUBLE PRECISION array, dimension (LDAF,N) 00055 * The factored form of the matrix A. AF contains the block 00056 * diagonal matrix D and the multipliers used to obtain the 00057 * factor U or L from the factorization A = U*D*U**T or 00058 * A = L*D*L**T as computed by DSYTRF. 00059 * 00060 * LDAF (input) INTEGER 00061 * The leading dimension of the array AF. LDAF >= max(1,N). 00062 * 00063 * IPIV (input) INTEGER array, dimension (N) 00064 * Details of the interchanges and the block structure of D 00065 * as determined by DSYTRF. 00066 * 00067 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00068 * The right hand side matrix B. 00069 * 00070 * LDB (input) INTEGER 00071 * The leading dimension of the array B. LDB >= max(1,N). 00072 * 00073 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) 00074 * On entry, the solution matrix X, as computed by DSYTRS. 00075 * On exit, the improved solution matrix X. 00076 * 00077 * LDX (input) INTEGER 00078 * The leading dimension of the array X. LDX >= max(1,N). 00079 * 00080 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00081 * The estimated forward error bound for each solution vector 00082 * X(j) (the j-th column of the solution matrix X). 00083 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00084 * is an estimated upper bound for the magnitude of the largest 00085 * element in (X(j) - XTRUE) divided by the magnitude of the 00086 * largest element in X(j). The estimate is as reliable as 00087 * the estimate for RCOND, and is almost always a slight 00088 * overestimate of the true error. 00089 * 00090 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00091 * The componentwise relative backward error of each solution 00092 * vector X(j) (i.e., the smallest relative change in 00093 * any element of A or B that makes X(j) an exact solution). 00094 * 00095 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 00096 * 00097 * IWORK (workspace) INTEGER array, dimension (N) 00098 * 00099 * INFO (output) INTEGER 00100 * = 0: successful exit 00101 * < 0: if INFO = -i, the i-th argument had an illegal value 00102 * 00103 * Internal Parameters 00104 * =================== 00105 * 00106 * ITMAX is the maximum number of steps of iterative refinement. 00107 * 00108 * ===================================================================== 00109 * 00110 * .. Parameters .. 00111 INTEGER ITMAX 00112 PARAMETER ( ITMAX = 5 ) 00113 DOUBLE PRECISION ZERO 00114 PARAMETER ( ZERO = 0.0D+0 ) 00115 DOUBLE PRECISION ONE 00116 PARAMETER ( ONE = 1.0D+0 ) 00117 DOUBLE PRECISION TWO 00118 PARAMETER ( TWO = 2.0D+0 ) 00119 DOUBLE PRECISION THREE 00120 PARAMETER ( THREE = 3.0D+0 ) 00121 * .. 00122 * .. Local Scalars .. 00123 LOGICAL UPPER 00124 INTEGER COUNT, I, J, K, KASE, NZ 00125 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00126 * .. 00127 * .. Local Arrays .. 00128 INTEGER ISAVE( 3 ) 00129 * .. 00130 * .. External Subroutines .. 00131 EXTERNAL DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS, XERBLA 00132 * .. 00133 * .. Intrinsic Functions .. 00134 INTRINSIC ABS, MAX 00135 * .. 00136 * .. External Functions .. 00137 LOGICAL LSAME 00138 DOUBLE PRECISION DLAMCH 00139 EXTERNAL LSAME, DLAMCH 00140 * .. 00141 * .. Executable Statements .. 00142 * 00143 * Test the input parameters. 00144 * 00145 INFO = 0 00146 UPPER = LSAME( UPLO, 'U' ) 00147 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00148 INFO = -1 00149 ELSE IF( N.LT.0 ) THEN 00150 INFO = -2 00151 ELSE IF( NRHS.LT.0 ) THEN 00152 INFO = -3 00153 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00154 INFO = -5 00155 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00156 INFO = -7 00157 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00158 INFO = -10 00159 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00160 INFO = -12 00161 END IF 00162 IF( INFO.NE.0 ) THEN 00163 CALL XERBLA( 'DSYRFS', -INFO ) 00164 RETURN 00165 END IF 00166 * 00167 * Quick return if possible 00168 * 00169 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00170 DO 10 J = 1, NRHS 00171 FERR( J ) = ZERO 00172 BERR( J ) = ZERO 00173 10 CONTINUE 00174 RETURN 00175 END IF 00176 * 00177 * NZ = maximum number of nonzero elements in each row of A, plus 1 00178 * 00179 NZ = N + 1 00180 EPS = DLAMCH( 'Epsilon' ) 00181 SAFMIN = DLAMCH( 'Safe minimum' ) 00182 SAFE1 = NZ*SAFMIN 00183 SAFE2 = SAFE1 / EPS 00184 * 00185 * Do for each right hand side 00186 * 00187 DO 140 J = 1, NRHS 00188 * 00189 COUNT = 1 00190 LSTRES = THREE 00191 20 CONTINUE 00192 * 00193 * Loop until stopping criterion is satisfied. 00194 * 00195 * Compute residual R = B - A * X 00196 * 00197 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00198 CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, 00199 $ WORK( N+1 ), 1 ) 00200 * 00201 * Compute componentwise relative backward error from formula 00202 * 00203 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00204 * 00205 * where abs(Z) is the componentwise absolute value of the matrix 00206 * or vector Z. If the i-th component of the denominator is less 00207 * than SAFE2, then SAFE1 is added to the i-th components of the 00208 * numerator and denominator before dividing. 00209 * 00210 DO 30 I = 1, N 00211 WORK( I ) = ABS( B( I, J ) ) 00212 30 CONTINUE 00213 * 00214 * Compute abs(A)*abs(X) + abs(B). 00215 * 00216 IF( UPPER ) THEN 00217 DO 50 K = 1, N 00218 S = ZERO 00219 XK = ABS( X( K, J ) ) 00220 DO 40 I = 1, K - 1 00221 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00222 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00223 40 CONTINUE 00224 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S 00225 50 CONTINUE 00226 ELSE 00227 DO 70 K = 1, N 00228 S = ZERO 00229 XK = ABS( X( K, J ) ) 00230 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK 00231 DO 60 I = K + 1, N 00232 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00233 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00234 60 CONTINUE 00235 WORK( K ) = WORK( K ) + S 00236 70 CONTINUE 00237 END IF 00238 S = ZERO 00239 DO 80 I = 1, N 00240 IF( WORK( I ).GT.SAFE2 ) THEN 00241 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00242 ELSE 00243 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00244 $ ( WORK( I )+SAFE1 ) ) 00245 END IF 00246 80 CONTINUE 00247 BERR( J ) = S 00248 * 00249 * Test stopping criterion. Continue iterating if 00250 * 1) The residual BERR(J) is larger than machine epsilon, and 00251 * 2) BERR(J) decreased by at least a factor of 2 during the 00252 * last iteration, and 00253 * 3) At most ITMAX iterations tried. 00254 * 00255 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00256 $ COUNT.LE.ITMAX ) THEN 00257 * 00258 * Update solution and try again. 00259 * 00260 CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 00261 $ INFO ) 00262 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00263 LSTRES = BERR( J ) 00264 COUNT = COUNT + 1 00265 GO TO 20 00266 END IF 00267 * 00268 * Bound error from formula 00269 * 00270 * norm(X - XTRUE) / norm(X) .le. FERR = 00271 * norm( abs(inv(A))* 00272 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00273 * 00274 * where 00275 * norm(Z) is the magnitude of the largest component of Z 00276 * inv(A) is the inverse of A 00277 * abs(Z) is the componentwise absolute value of the matrix or 00278 * vector Z 00279 * NZ is the maximum number of nonzeros in any row of A, plus 1 00280 * EPS is machine epsilon 00281 * 00282 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00283 * is incremented by SAFE1 if the i-th component of 00284 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00285 * 00286 * Use DLACN2 to estimate the infinity-norm of the matrix 00287 * inv(A) * diag(W), 00288 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00289 * 00290 DO 90 I = 1, N 00291 IF( WORK( I ).GT.SAFE2 ) THEN 00292 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00293 ELSE 00294 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00295 END IF 00296 90 CONTINUE 00297 * 00298 KASE = 0 00299 100 CONTINUE 00300 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00301 $ KASE, ISAVE ) 00302 IF( KASE.NE.0 ) THEN 00303 IF( KASE.EQ.1 ) THEN 00304 * 00305 * Multiply by diag(W)*inv(A**T). 00306 * 00307 CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 00308 $ INFO ) 00309 DO 110 I = 1, N 00310 WORK( N+I ) = WORK( I )*WORK( N+I ) 00311 110 CONTINUE 00312 ELSE IF( KASE.EQ.2 ) THEN 00313 * 00314 * Multiply by inv(A)*diag(W). 00315 * 00316 DO 120 I = 1, N 00317 WORK( N+I ) = WORK( I )*WORK( N+I ) 00318 120 CONTINUE 00319 CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 00320 $ INFO ) 00321 END IF 00322 GO TO 100 00323 END IF 00324 * 00325 * Normalize error. 00326 * 00327 LSTRES = ZERO 00328 DO 130 I = 1, N 00329 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00330 130 CONTINUE 00331 IF( LSTRES.NE.ZERO ) 00332 $ FERR( J ) = FERR( J ) / LSTRES 00333 * 00334 140 CONTINUE 00335 * 00336 RETURN 00337 * 00338 * End of DSYRFS 00339 * 00340 END
1.7.3 
