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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 DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, 00002 $ LDB, X, LDX, 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, KD, LDAB, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IWORK( * ) 00017 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), 00018 $ FERR( * ), WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DTBRFS provides error bounds and backward error estimates for the 00025 * solution to a system of linear equations with a triangular band 00026 * coefficient matrix. 00027 * 00028 * The solution matrix X must be computed by DTBTRS or some other 00029 * means before entering this routine. DTBRFS 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 * KD (input) INTEGER 00053 * The number of superdiagonals or subdiagonals of the 00054 * triangular band matrix A. KD >= 0. 00055 * 00056 * NRHS (input) INTEGER 00057 * The number of right hand sides, i.e., the number of columns 00058 * of the matrices B and X. NRHS >= 0. 00059 * 00060 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N) 00061 * The upper or lower triangular band matrix A, stored in the 00062 * first kd+1 rows of the array. The j-th column of A is stored 00063 * in the j-th column of the array AB as follows: 00064 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00065 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00066 * If DIAG = 'U', the diagonal elements of A are not referenced 00067 * and are assumed to be 1. 00068 * 00069 * LDAB (input) INTEGER 00070 * The leading dimension of the array AB. LDAB >= KD+1. 00071 * 00072 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00073 * The right hand side matrix B. 00074 * 00075 * LDB (input) INTEGER 00076 * The leading dimension of the array B. LDB >= max(1,N). 00077 * 00078 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00079 * The solution matrix X. 00080 * 00081 * LDX (input) INTEGER 00082 * The leading dimension of the array X. LDX >= max(1,N). 00083 * 00084 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00085 * The estimated forward error bound for each solution vector 00086 * X(j) (the j-th column of the solution matrix X). 00087 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00088 * is an estimated upper bound for the magnitude of the largest 00089 * element in (X(j) - XTRUE) divided by the magnitude of the 00090 * largest element in X(j). The estimate is as reliable as 00091 * the estimate for RCOND, and is almost always a slight 00092 * overestimate of the true error. 00093 * 00094 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00095 * The componentwise relative backward error of each solution 00096 * vector X(j) (i.e., the smallest relative change in 00097 * any element of A or B that makes X(j) an exact solution). 00098 * 00099 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 00100 * 00101 * IWORK (workspace) INTEGER array, dimension (N) 00102 * 00103 * INFO (output) INTEGER 00104 * = 0: successful exit 00105 * < 0: if INFO = -i, the i-th argument had an illegal value 00106 * 00107 * ===================================================================== 00108 * 00109 * .. Parameters .. 00110 DOUBLE PRECISION ZERO 00111 PARAMETER ( ZERO = 0.0D+0 ) 00112 DOUBLE PRECISION ONE 00113 PARAMETER ( ONE = 1.0D+0 ) 00114 * .. 00115 * .. Local Scalars .. 00116 LOGICAL NOTRAN, NOUNIT, UPPER 00117 CHARACTER TRANST 00118 INTEGER I, J, K, KASE, NZ 00119 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00120 * .. 00121 * .. Local Arrays .. 00122 INTEGER ISAVE( 3 ) 00123 * .. 00124 * .. External Subroutines .. 00125 EXTERNAL DAXPY, DCOPY, DLACN2, DTBMV, DTBSV, XERBLA 00126 * .. 00127 * .. Intrinsic Functions .. 00128 INTRINSIC ABS, MAX, MIN 00129 * .. 00130 * .. External Functions .. 00131 LOGICAL LSAME 00132 DOUBLE PRECISION DLAMCH 00133 EXTERNAL LSAME, DLAMCH 00134 * .. 00135 * .. Executable Statements .. 00136 * 00137 * Test the input parameters. 00138 * 00139 INFO = 0 00140 UPPER = LSAME( UPLO, 'U' ) 00141 NOTRAN = LSAME( TRANS, 'N' ) 00142 NOUNIT = LSAME( DIAG, 'N' ) 00143 * 00144 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00145 INFO = -1 00146 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00147 $ LSAME( TRANS, 'C' ) ) THEN 00148 INFO = -2 00149 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN 00150 INFO = -3 00151 ELSE IF( N.LT.0 ) THEN 00152 INFO = -4 00153 ELSE IF( KD.LT.0 ) THEN 00154 INFO = -5 00155 ELSE IF( NRHS.LT.0 ) THEN 00156 INFO = -6 00157 ELSE IF( LDAB.LT.KD+1 ) THEN 00158 INFO = -8 00159 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00160 INFO = -10 00161 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00162 INFO = -12 00163 END IF 00164 IF( INFO.NE.0 ) THEN 00165 CALL XERBLA( 'DTBRFS', -INFO ) 00166 RETURN 00167 END IF 00168 * 00169 * Quick return if possible 00170 * 00171 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00172 DO 10 J = 1, NRHS 00173 FERR( J ) = ZERO 00174 BERR( J ) = ZERO 00175 10 CONTINUE 00176 RETURN 00177 END IF 00178 * 00179 IF( NOTRAN ) THEN 00180 TRANST = 'T' 00181 ELSE 00182 TRANST = 'N' 00183 END IF 00184 * 00185 * NZ = maximum number of nonzero elements in each row of A, plus 1 00186 * 00187 NZ = KD + 2 00188 EPS = DLAMCH( 'Epsilon' ) 00189 SAFMIN = DLAMCH( 'Safe minimum' ) 00190 SAFE1 = NZ*SAFMIN 00191 SAFE2 = SAFE1 / EPS 00192 * 00193 * Do for each right hand side 00194 * 00195 DO 250 J = 1, NRHS 00196 * 00197 * Compute residual R = B - op(A) * X, 00198 * where op(A) = A or A**T, depending on TRANS. 00199 * 00200 CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) 00201 CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), 00202 $ 1 ) 00203 CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) 00204 * 00205 * Compute componentwise relative backward error from formula 00206 * 00207 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00208 * 00209 * where abs(Z) is the componentwise absolute value of the matrix 00210 * or vector Z. If the i-th component of the denominator is less 00211 * than SAFE2, then SAFE1 is added to the i-th components of the 00212 * numerator and denominator before dividing. 00213 * 00214 DO 20 I = 1, N 00215 WORK( I ) = ABS( B( I, J ) ) 00216 20 CONTINUE 00217 * 00218 IF( NOTRAN ) THEN 00219 * 00220 * Compute abs(A)*abs(X) + abs(B). 00221 * 00222 IF( UPPER ) THEN 00223 IF( NOUNIT ) THEN 00224 DO 40 K = 1, N 00225 XK = ABS( X( K, J ) ) 00226 DO 30 I = MAX( 1, K-KD ), K 00227 WORK( I ) = WORK( I ) + 00228 $ ABS( AB( KD+1+I-K, K ) )*XK 00229 30 CONTINUE 00230 40 CONTINUE 00231 ELSE 00232 DO 60 K = 1, N 00233 XK = ABS( X( K, J ) ) 00234 DO 50 I = MAX( 1, K-KD ), K - 1 00235 WORK( I ) = WORK( I ) + 00236 $ ABS( AB( KD+1+I-K, K ) )*XK 00237 50 CONTINUE 00238 WORK( K ) = WORK( K ) + XK 00239 60 CONTINUE 00240 END IF 00241 ELSE 00242 IF( NOUNIT ) THEN 00243 DO 80 K = 1, N 00244 XK = ABS( X( K, J ) ) 00245 DO 70 I = K, MIN( N, K+KD ) 00246 WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK 00247 70 CONTINUE 00248 80 CONTINUE 00249 ELSE 00250 DO 100 K = 1, N 00251 XK = ABS( X( K, J ) ) 00252 DO 90 I = K + 1, MIN( N, K+KD ) 00253 WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK 00254 90 CONTINUE 00255 WORK( K ) = WORK( K ) + XK 00256 100 CONTINUE 00257 END IF 00258 END IF 00259 ELSE 00260 * 00261 * Compute abs(A**T)*abs(X) + abs(B). 00262 * 00263 IF( UPPER ) THEN 00264 IF( NOUNIT ) THEN 00265 DO 120 K = 1, N 00266 S = ZERO 00267 DO 110 I = MAX( 1, K-KD ), K 00268 S = S + ABS( AB( KD+1+I-K, K ) )* 00269 $ ABS( X( I, J ) ) 00270 110 CONTINUE 00271 WORK( K ) = WORK( K ) + S 00272 120 CONTINUE 00273 ELSE 00274 DO 140 K = 1, N 00275 S = ABS( X( K, J ) ) 00276 DO 130 I = MAX( 1, K-KD ), K - 1 00277 S = S + ABS( AB( KD+1+I-K, K ) )* 00278 $ ABS( X( I, J ) ) 00279 130 CONTINUE 00280 WORK( K ) = WORK( K ) + S 00281 140 CONTINUE 00282 END IF 00283 ELSE 00284 IF( NOUNIT ) THEN 00285 DO 160 K = 1, N 00286 S = ZERO 00287 DO 150 I = K, MIN( N, K+KD ) 00288 S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) ) 00289 150 CONTINUE 00290 WORK( K ) = WORK( K ) + S 00291 160 CONTINUE 00292 ELSE 00293 DO 180 K = 1, N 00294 S = ABS( X( K, J ) ) 00295 DO 170 I = K + 1, MIN( N, K+KD ) 00296 S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) ) 00297 170 CONTINUE 00298 WORK( K ) = WORK( K ) + S 00299 180 CONTINUE 00300 END IF 00301 END IF 00302 END IF 00303 S = ZERO 00304 DO 190 I = 1, N 00305 IF( WORK( I ).GT.SAFE2 ) THEN 00306 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00307 ELSE 00308 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00309 $ ( WORK( I )+SAFE1 ) ) 00310 END IF 00311 190 CONTINUE 00312 BERR( J ) = S 00313 * 00314 * Bound error from formula 00315 * 00316 * norm(X - XTRUE) / norm(X) .le. FERR = 00317 * norm( abs(inv(op(A)))* 00318 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00319 * 00320 * where 00321 * norm(Z) is the magnitude of the largest component of Z 00322 * inv(op(A)) is the inverse of op(A) 00323 * abs(Z) is the componentwise absolute value of the matrix or 00324 * vector Z 00325 * NZ is the maximum number of nonzeros in any row of A, plus 1 00326 * EPS is machine epsilon 00327 * 00328 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00329 * is incremented by SAFE1 if the i-th component of 00330 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00331 * 00332 * Use DLACN2 to estimate the infinity-norm of the matrix 00333 * inv(op(A)) * diag(W), 00334 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00335 * 00336 DO 200 I = 1, N 00337 IF( WORK( I ).GT.SAFE2 ) THEN 00338 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00339 ELSE 00340 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00341 END IF 00342 200 CONTINUE 00343 * 00344 KASE = 0 00345 210 CONTINUE 00346 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00347 $ KASE, ISAVE ) 00348 IF( KASE.NE.0 ) THEN 00349 IF( KASE.EQ.1 ) THEN 00350 * 00351 * Multiply by diag(W)*inv(op(A)**T). 00352 * 00353 CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, 00354 $ WORK( N+1 ), 1 ) 00355 DO 220 I = 1, N 00356 WORK( N+I ) = WORK( I )*WORK( N+I ) 00357 220 CONTINUE 00358 ELSE 00359 * 00360 * Multiply by inv(op(A))*diag(W). 00361 * 00362 DO 230 I = 1, N 00363 WORK( N+I ) = WORK( I )*WORK( N+I ) 00364 230 CONTINUE 00365 CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, 00366 $ WORK( N+1 ), 1 ) 00367 END IF 00368 GO TO 210 00369 END IF 00370 * 00371 * Normalize error. 00372 * 00373 LSTRES = ZERO 00374 DO 240 I = 1, N 00375 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00376 240 CONTINUE 00377 IF( LSTRES.NE.ZERO ) 00378 $ FERR( J ) = FERR( J ) / LSTRES 00379 * 00380 250 CONTINUE 00381 * 00382 RETURN 00383 * 00384 * End of DTBRFS 00385 * 00386 END
1.7.3 
