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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 DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 00002 $ FERR, BERR, WORK, IWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.3.1) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * -- April 2011 -- 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, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IWORK( * ) 00017 DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00018 $ WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DTPRFS provides error bounds and backward error estimates for the 00025 * solution to a system of linear equations with a triangular packed 00026 * coefficient matrix. 00027 * 00028 * The solution matrix X must be computed by DTPTRS or some other 00029 * means before entering this routine. DTPRFS 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 * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) 00057 * The upper or lower triangular matrix A, packed columnwise in 00058 * a linear array. The j-th column of A is stored in the array 00059 * AP as follows: 00060 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00061 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 00062 * If DIAG = 'U', the diagonal elements of A are not referenced 00063 * and are assumed to be 1. 00064 * 00065 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00066 * The right hand side matrix B. 00067 * 00068 * LDB (input) INTEGER 00069 * The leading dimension of the array B. LDB >= max(1,N). 00070 * 00071 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00072 * The solution matrix X. 00073 * 00074 * LDX (input) INTEGER 00075 * The leading dimension of the array X. LDX >= max(1,N). 00076 * 00077 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00078 * The estimated forward error bound for each solution vector 00079 * X(j) (the j-th column of the solution matrix X). 00080 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00081 * is an estimated upper bound for the magnitude of the largest 00082 * element in (X(j) - XTRUE) divided by the magnitude of the 00083 * largest element in X(j). The estimate is as reliable as 00084 * the estimate for RCOND, and is almost always a slight 00085 * overestimate of the true error. 00086 * 00087 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00088 * The componentwise relative backward error of each solution 00089 * vector X(j) (i.e., the smallest relative change in 00090 * any element of A or B that makes X(j) an exact solution). 00091 * 00092 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 00093 * 00094 * IWORK (workspace) INTEGER array, dimension (N) 00095 * 00096 * INFO (output) INTEGER 00097 * = 0: successful exit 00098 * < 0: if INFO = -i, the i-th argument had an illegal value 00099 * 00100 * ===================================================================== 00101 * 00102 * .. Parameters .. 00103 DOUBLE PRECISION ZERO 00104 PARAMETER ( ZERO = 0.0D+0 ) 00105 DOUBLE PRECISION ONE 00106 PARAMETER ( ONE = 1.0D+0 ) 00107 * .. 00108 * .. Local Scalars .. 00109 LOGICAL NOTRAN, NOUNIT, UPPER 00110 CHARACTER TRANST 00111 INTEGER I, J, K, KASE, KC, NZ 00112 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00113 * .. 00114 * .. Local Arrays .. 00115 INTEGER ISAVE( 3 ) 00116 * .. 00117 * .. External Subroutines .. 00118 EXTERNAL DAXPY, DCOPY, DLACN2, DTPMV, DTPSV, XERBLA 00119 * .. 00120 * .. Intrinsic Functions .. 00121 INTRINSIC ABS, MAX 00122 * .. 00123 * .. External Functions .. 00124 LOGICAL LSAME 00125 DOUBLE PRECISION DLAMCH 00126 EXTERNAL LSAME, DLAMCH 00127 * .. 00128 * .. Executable Statements .. 00129 * 00130 * Test the input parameters. 00131 * 00132 INFO = 0 00133 UPPER = LSAME( UPLO, 'U' ) 00134 NOTRAN = LSAME( TRANS, 'N' ) 00135 NOUNIT = LSAME( DIAG, 'N' ) 00136 * 00137 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00138 INFO = -1 00139 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00140 $ LSAME( TRANS, 'C' ) ) THEN 00141 INFO = -2 00142 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN 00143 INFO = -3 00144 ELSE IF( N.LT.0 ) THEN 00145 INFO = -4 00146 ELSE IF( NRHS.LT.0 ) THEN 00147 INFO = -5 00148 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00149 INFO = -8 00150 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00151 INFO = -10 00152 END IF 00153 IF( INFO.NE.0 ) THEN 00154 CALL XERBLA( 'DTPRFS', -INFO ) 00155 RETURN 00156 END IF 00157 * 00158 * Quick return if possible 00159 * 00160 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00161 DO 10 J = 1, NRHS 00162 FERR( J ) = ZERO 00163 BERR( J ) = ZERO 00164 10 CONTINUE 00165 RETURN 00166 END IF 00167 * 00168 IF( NOTRAN ) THEN 00169 TRANST = 'T' 00170 ELSE 00171 TRANST = 'N' 00172 END IF 00173 * 00174 * NZ = maximum number of nonzero elements in each row of A, plus 1 00175 * 00176 NZ = N + 1 00177 EPS = DLAMCH( 'Epsilon' ) 00178 SAFMIN = DLAMCH( 'Safe minimum' ) 00179 SAFE1 = NZ*SAFMIN 00180 SAFE2 = SAFE1 / EPS 00181 * 00182 * Do for each right hand side 00183 * 00184 DO 250 J = 1, NRHS 00185 * 00186 * Compute residual R = B - op(A) * X, 00187 * where op(A) = A or A**T, depending on TRANS. 00188 * 00189 CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) 00190 CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) 00191 CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) 00192 * 00193 * Compute componentwise relative backward error from formula 00194 * 00195 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00196 * 00197 * where abs(Z) is the componentwise absolute value of the matrix 00198 * or vector Z. If the i-th component of the denominator is less 00199 * than SAFE2, then SAFE1 is added to the i-th components of the 00200 * numerator and denominator before dividing. 00201 * 00202 DO 20 I = 1, N 00203 WORK( I ) = ABS( B( I, J ) ) 00204 20 CONTINUE 00205 * 00206 IF( NOTRAN ) THEN 00207 * 00208 * Compute abs(A)*abs(X) + abs(B). 00209 * 00210 IF( UPPER ) THEN 00211 KC = 1 00212 IF( NOUNIT ) THEN 00213 DO 40 K = 1, N 00214 XK = ABS( X( K, J ) ) 00215 DO 30 I = 1, K 00216 WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK 00217 30 CONTINUE 00218 KC = KC + K 00219 40 CONTINUE 00220 ELSE 00221 DO 60 K = 1, N 00222 XK = ABS( X( K, J ) ) 00223 DO 50 I = 1, K - 1 00224 WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK 00225 50 CONTINUE 00226 WORK( K ) = WORK( K ) + XK 00227 KC = KC + K 00228 60 CONTINUE 00229 END IF 00230 ELSE 00231 KC = 1 00232 IF( NOUNIT ) THEN 00233 DO 80 K = 1, N 00234 XK = ABS( X( K, J ) ) 00235 DO 70 I = K, N 00236 WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK 00237 70 CONTINUE 00238 KC = KC + N - K + 1 00239 80 CONTINUE 00240 ELSE 00241 DO 100 K = 1, N 00242 XK = ABS( X( K, J ) ) 00243 DO 90 I = K + 1, N 00244 WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK 00245 90 CONTINUE 00246 WORK( K ) = WORK( K ) + XK 00247 KC = KC + N - K + 1 00248 100 CONTINUE 00249 END IF 00250 END IF 00251 ELSE 00252 * 00253 * Compute abs(A**T)*abs(X) + abs(B). 00254 * 00255 IF( UPPER ) THEN 00256 KC = 1 00257 IF( NOUNIT ) THEN 00258 DO 120 K = 1, N 00259 S = ZERO 00260 DO 110 I = 1, K 00261 S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) ) 00262 110 CONTINUE 00263 WORK( K ) = WORK( K ) + S 00264 KC = KC + K 00265 120 CONTINUE 00266 ELSE 00267 DO 140 K = 1, N 00268 S = ABS( X( K, J ) ) 00269 DO 130 I = 1, K - 1 00270 S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) ) 00271 130 CONTINUE 00272 WORK( K ) = WORK( K ) + S 00273 KC = KC + K 00274 140 CONTINUE 00275 END IF 00276 ELSE 00277 KC = 1 00278 IF( NOUNIT ) THEN 00279 DO 160 K = 1, N 00280 S = ZERO 00281 DO 150 I = K, N 00282 S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) ) 00283 150 CONTINUE 00284 WORK( K ) = WORK( K ) + S 00285 KC = KC + N - K + 1 00286 160 CONTINUE 00287 ELSE 00288 DO 180 K = 1, N 00289 S = ABS( X( K, J ) ) 00290 DO 170 I = K + 1, N 00291 S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) ) 00292 170 CONTINUE 00293 WORK( K ) = WORK( K ) + S 00294 KC = KC + N - K + 1 00295 180 CONTINUE 00296 END IF 00297 END IF 00298 END IF 00299 S = ZERO 00300 DO 190 I = 1, N 00301 IF( WORK( I ).GT.SAFE2 ) THEN 00302 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00303 ELSE 00304 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00305 $ ( WORK( I )+SAFE1 ) ) 00306 END IF 00307 190 CONTINUE 00308 BERR( J ) = S 00309 * 00310 * Bound error from formula 00311 * 00312 * norm(X - XTRUE) / norm(X) .le. FERR = 00313 * norm( abs(inv(op(A)))* 00314 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00315 * 00316 * where 00317 * norm(Z) is the magnitude of the largest component of Z 00318 * inv(op(A)) is the inverse of op(A) 00319 * abs(Z) is the componentwise absolute value of the matrix or 00320 * vector Z 00321 * NZ is the maximum number of nonzeros in any row of A, plus 1 00322 * EPS is machine epsilon 00323 * 00324 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00325 * is incremented by SAFE1 if the i-th component of 00326 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00327 * 00328 * Use DLACN2 to estimate the infinity-norm of the matrix 00329 * inv(op(A)) * diag(W), 00330 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00331 * 00332 DO 200 I = 1, N 00333 IF( WORK( I ).GT.SAFE2 ) THEN 00334 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00335 ELSE 00336 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00337 END IF 00338 200 CONTINUE 00339 * 00340 KASE = 0 00341 210 CONTINUE 00342 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00343 $ KASE, ISAVE ) 00344 IF( KASE.NE.0 ) THEN 00345 IF( KASE.EQ.1 ) THEN 00346 * 00347 * Multiply by diag(W)*inv(op(A)**T). 00348 * 00349 CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 ) 00350 DO 220 I = 1, N 00351 WORK( N+I ) = WORK( I )*WORK( N+I ) 00352 220 CONTINUE 00353 ELSE 00354 * 00355 * Multiply by inv(op(A))*diag(W). 00356 * 00357 DO 230 I = 1, N 00358 WORK( N+I ) = WORK( I )*WORK( N+I ) 00359 230 CONTINUE 00360 CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) 00361 END IF 00362 GO TO 210 00363 END IF 00364 * 00365 * Normalize error. 00366 * 00367 LSTRES = ZERO 00368 DO 240 I = 1, N 00369 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00370 240 CONTINUE 00371 IF( LSTRES.NE.ZERO ) 00372 $ FERR( J ) = FERR( J ) / LSTRES 00373 * 00374 250 CONTINUE 00375 * 00376 RETURN 00377 * 00378 * End of DTPRFS 00379 * 00380 END
1.7.3 
