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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, 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 DIAG, TRANS, UPLO 00013 INTEGER INFO, LDA, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IWORK( * ) 00017 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 00018 $ WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DTRRFS provides error bounds and backward error estimates for the 00025 * solution to a system of linear equations with a triangular 00026 * coefficient matrix. 00027 * 00028 * The solution matrix X must be computed by DTRTRS or some other 00029 * means before entering this routine. DTRRFS does not do iterative 00030 * refinement because doing so cannot improve the backward error. 00031 * 00032 * Arguments 00033 * ========= 00034 * 00035 * UPLO (input) CHARACTER*1 00036 * = 'U': A is upper triangular; 00037 * = 'L': A is lower triangular. 00038 * 00039 * TRANS (input) CHARACTER*1 00040 * Specifies the form of the system of equations: 00041 * = 'N': A * X = B (No transpose) 00042 * = 'T': A**T * X = B (Transpose) 00043 * = 'C': A**H * X = B (Conjugate transpose = Transpose) 00044 * 00045 * DIAG (input) CHARACTER*1 00046 * = 'N': A is non-unit triangular; 00047 * = 'U': A is unit triangular. 00048 * 00049 * N (input) INTEGER 00050 * The order of the matrix A. N >= 0. 00051 * 00052 * NRHS (input) INTEGER 00053 * The number of right hand sides, i.e., the number of columns 00054 * of the matrices B and X. NRHS >= 0. 00055 * 00056 * A (input) DOUBLE PRECISION array, dimension (LDA,N) 00057 * The triangular matrix A. If UPLO = 'U', the leading N-by-N 00058 * upper triangular part of the array A contains the upper 00059 * triangular matrix, and the strictly lower triangular part of 00060 * A is not referenced. If UPLO = 'L', the leading N-by-N lower 00061 * triangular part of the array A contains the lower triangular 00062 * matrix, and the strictly upper triangular part of A is not 00063 * referenced. If DIAG = 'U', the diagonal elements of A are 00064 * also not referenced and are assumed to be 1. 00065 * 00066 * LDA (input) INTEGER 00067 * The leading dimension of the array A. LDA >= max(1,N). 00068 * 00069 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00070 * The right hand side matrix B. 00071 * 00072 * LDB (input) INTEGER 00073 * The leading dimension of the array B. LDB >= max(1,N). 00074 * 00075 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00076 * The solution matrix X. 00077 * 00078 * LDX (input) INTEGER 00079 * The leading dimension of the array X. LDX >= max(1,N). 00080 * 00081 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00082 * The estimated forward error bound for each solution vector 00083 * X(j) (the j-th column of the solution matrix X). 00084 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00085 * is an estimated upper bound for the magnitude of the largest 00086 * element in (X(j) - XTRUE) divided by the magnitude of the 00087 * largest element in X(j). The estimate is as reliable as 00088 * the estimate for RCOND, and is almost always a slight 00089 * overestimate of the true error. 00090 * 00091 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00092 * The componentwise relative backward error of each solution 00093 * vector X(j) (i.e., the smallest relative change in 00094 * any element of A or B that makes X(j) an exact solution). 00095 * 00096 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 00097 * 00098 * IWORK (workspace) INTEGER array, dimension (N) 00099 * 00100 * INFO (output) INTEGER 00101 * = 0: successful exit 00102 * < 0: if INFO = -i, the i-th argument had an illegal value 00103 * 00104 * ===================================================================== 00105 * 00106 * .. Parameters .. 00107 DOUBLE PRECISION ZERO 00108 PARAMETER ( ZERO = 0.0D+0 ) 00109 DOUBLE PRECISION ONE 00110 PARAMETER ( ONE = 1.0D+0 ) 00111 * .. 00112 * .. Local Scalars .. 00113 LOGICAL NOTRAN, NOUNIT, UPPER 00114 CHARACTER TRANST 00115 INTEGER I, J, K, KASE, NZ 00116 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00117 * .. 00118 * .. Local Arrays .. 00119 INTEGER ISAVE( 3 ) 00120 * .. 00121 * .. External Subroutines .. 00122 EXTERNAL DAXPY, DCOPY, DLACN2, DTRMV, DTRSV, XERBLA 00123 * .. 00124 * .. Intrinsic Functions .. 00125 INTRINSIC ABS, MAX 00126 * .. 00127 * .. External Functions .. 00128 LOGICAL LSAME 00129 DOUBLE PRECISION DLAMCH 00130 EXTERNAL LSAME, DLAMCH 00131 * .. 00132 * .. Executable Statements .. 00133 * 00134 * Test the input parameters. 00135 * 00136 INFO = 0 00137 UPPER = LSAME( UPLO, 'U' ) 00138 NOTRAN = LSAME( TRANS, 'N' ) 00139 NOUNIT = LSAME( DIAG, 'N' ) 00140 * 00141 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00142 INFO = -1 00143 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00144 $ LSAME( TRANS, 'C' ) ) THEN 00145 INFO = -2 00146 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN 00147 INFO = -3 00148 ELSE IF( N.LT.0 ) THEN 00149 INFO = -4 00150 ELSE IF( NRHS.LT.0 ) THEN 00151 INFO = -5 00152 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00153 INFO = -7 00154 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00155 INFO = -9 00156 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00157 INFO = -11 00158 END IF 00159 IF( INFO.NE.0 ) THEN 00160 CALL XERBLA( 'DTRRFS', -INFO ) 00161 RETURN 00162 END IF 00163 * 00164 * Quick return if possible 00165 * 00166 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00167 DO 10 J = 1, NRHS 00168 FERR( J ) = ZERO 00169 BERR( J ) = ZERO 00170 10 CONTINUE 00171 RETURN 00172 END IF 00173 * 00174 IF( NOTRAN ) THEN 00175 TRANST = 'T' 00176 ELSE 00177 TRANST = 'N' 00178 END IF 00179 * 00180 * NZ = maximum number of nonzero elements in each row of A, plus 1 00181 * 00182 NZ = N + 1 00183 EPS = DLAMCH( 'Epsilon' ) 00184 SAFMIN = DLAMCH( 'Safe minimum' ) 00185 SAFE1 = NZ*SAFMIN 00186 SAFE2 = SAFE1 / EPS 00187 * 00188 * Do for each right hand side 00189 * 00190 DO 250 J = 1, NRHS 00191 * 00192 * Compute residual R = B - op(A) * X, 00193 * where op(A) = A or A', depending on TRANS. 00194 * 00195 CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) 00196 CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 ) 00197 CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) 00198 * 00199 * Compute componentwise relative backward error from formula 00200 * 00201 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00202 * 00203 * where abs(Z) is the componentwise absolute value of the matrix 00204 * or vector Z. If the i-th component of the denominator is less 00205 * than SAFE2, then SAFE1 is added to the i-th components of the 00206 * numerator and denominator before dividing. 00207 * 00208 DO 20 I = 1, N 00209 WORK( I ) = ABS( B( I, J ) ) 00210 20 CONTINUE 00211 * 00212 IF( NOTRAN ) THEN 00213 * 00214 * Compute abs(A)*abs(X) + abs(B). 00215 * 00216 IF( UPPER ) THEN 00217 IF( NOUNIT ) THEN 00218 DO 40 K = 1, N 00219 XK = ABS( X( K, J ) ) 00220 DO 30 I = 1, K 00221 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00222 30 CONTINUE 00223 40 CONTINUE 00224 ELSE 00225 DO 60 K = 1, N 00226 XK = ABS( X( K, J ) ) 00227 DO 50 I = 1, K - 1 00228 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00229 50 CONTINUE 00230 WORK( K ) = WORK( K ) + XK 00231 60 CONTINUE 00232 END IF 00233 ELSE 00234 IF( NOUNIT ) THEN 00235 DO 80 K = 1, N 00236 XK = ABS( X( K, J ) ) 00237 DO 70 I = K, N 00238 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00239 70 CONTINUE 00240 80 CONTINUE 00241 ELSE 00242 DO 100 K = 1, N 00243 XK = ABS( X( K, J ) ) 00244 DO 90 I = K + 1, N 00245 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00246 90 CONTINUE 00247 WORK( K ) = WORK( K ) + XK 00248 100 CONTINUE 00249 END IF 00250 END IF 00251 ELSE 00252 * 00253 * Compute abs(A')*abs(X) + abs(B). 00254 * 00255 IF( UPPER ) THEN 00256 IF( NOUNIT ) THEN 00257 DO 120 K = 1, N 00258 S = ZERO 00259 DO 110 I = 1, K 00260 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00261 110 CONTINUE 00262 WORK( K ) = WORK( K ) + S 00263 120 CONTINUE 00264 ELSE 00265 DO 140 K = 1, N 00266 S = ABS( X( K, J ) ) 00267 DO 130 I = 1, K - 1 00268 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00269 130 CONTINUE 00270 WORK( K ) = WORK( K ) + S 00271 140 CONTINUE 00272 END IF 00273 ELSE 00274 IF( NOUNIT ) THEN 00275 DO 160 K = 1, N 00276 S = ZERO 00277 DO 150 I = K, N 00278 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00279 150 CONTINUE 00280 WORK( K ) = WORK( K ) + S 00281 160 CONTINUE 00282 ELSE 00283 DO 180 K = 1, N 00284 S = ABS( X( K, J ) ) 00285 DO 170 I = K + 1, N 00286 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00287 170 CONTINUE 00288 WORK( K ) = WORK( K ) + S 00289 180 CONTINUE 00290 END IF 00291 END IF 00292 END IF 00293 S = ZERO 00294 DO 190 I = 1, N 00295 IF( WORK( I ).GT.SAFE2 ) THEN 00296 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00297 ELSE 00298 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00299 $ ( WORK( I )+SAFE1 ) ) 00300 END IF 00301 190 CONTINUE 00302 BERR( J ) = S 00303 * 00304 * Bound error from formula 00305 * 00306 * norm(X - XTRUE) / norm(X) .le. FERR = 00307 * norm( abs(inv(op(A)))* 00308 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00309 * 00310 * where 00311 * norm(Z) is the magnitude of the largest component of Z 00312 * inv(op(A)) is the inverse of op(A) 00313 * abs(Z) is the componentwise absolute value of the matrix or 00314 * vector Z 00315 * NZ is the maximum number of nonzeros in any row of A, plus 1 00316 * EPS is machine epsilon 00317 * 00318 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00319 * is incremented by SAFE1 if the i-th component of 00320 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00321 * 00322 * Use DLACN2 to estimate the infinity-norm of the matrix 00323 * inv(op(A)) * diag(W), 00324 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00325 * 00326 DO 200 I = 1, N 00327 IF( WORK( I ).GT.SAFE2 ) THEN 00328 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00329 ELSE 00330 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00331 END IF 00332 200 CONTINUE 00333 * 00334 KASE = 0 00335 210 CONTINUE 00336 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00337 $ KASE, ISAVE ) 00338 IF( KASE.NE.0 ) THEN 00339 IF( KASE.EQ.1 ) THEN 00340 * 00341 * Multiply by diag(W)*inv(op(A)'). 00342 * 00343 CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ), 00344 $ 1 ) 00345 DO 220 I = 1, N 00346 WORK( N+I ) = WORK( I )*WORK( N+I ) 00347 220 CONTINUE 00348 ELSE 00349 * 00350 * Multiply by inv(op(A))*diag(W). 00351 * 00352 DO 230 I = 1, N 00353 WORK( N+I ) = WORK( I )*WORK( N+I ) 00354 230 CONTINUE 00355 CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 00356 $ 1 ) 00357 END IF 00358 GO TO 210 00359 END IF 00360 * 00361 * Normalize error. 00362 * 00363 LSTRES = ZERO 00364 DO 240 I = 1, N 00365 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00366 240 CONTINUE 00367 IF( LSTRES.NE.ZERO ) 00368 $ FERR( J ) = FERR( J ) / LSTRES 00369 * 00370 250 CONTINUE 00371 * 00372 RETURN 00373 * 00374 * End of DTRRFS 00375 * 00376 END
1.7.3 
