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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 SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, 00002 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, 00003 $ INFO ) 00004 * 00005 * -- LAPACK routine (version 3.2) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * November 2006 00009 * 00010 * Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH. 00011 * 00012 * .. Scalar Arguments .. 00013 CHARACTER TRANS 00014 INTEGER INFO, LDB, LDX, N, NRHS 00015 * .. 00016 * .. Array Arguments .. 00017 INTEGER IPIV( * ), IWORK( * ) 00018 REAL B( LDB, * ), BERR( * ), D( * ), DF( * ), 00019 $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), 00020 $ FERR( * ), WORK( * ), X( LDX, * ) 00021 * .. 00022 * 00023 * Purpose 00024 * ======= 00025 * 00026 * SGTRFS improves the computed solution to a system of linear 00027 * equations when the coefficient matrix is tridiagonal, and provides 00028 * error bounds and backward error estimates for the solution. 00029 * 00030 * Arguments 00031 * ========= 00032 * 00033 * TRANS (input) CHARACTER*1 00034 * Specifies the form of the system of equations: 00035 * = 'N': A * X = B (No transpose) 00036 * = 'T': A**T * X = B (Transpose) 00037 * = 'C': A**H * X = B (Conjugate transpose = Transpose) 00038 * 00039 * N (input) INTEGER 00040 * The order of the matrix A. N >= 0. 00041 * 00042 * NRHS (input) INTEGER 00043 * The number of right hand sides, i.e., the number of columns 00044 * of the matrix B. NRHS >= 0. 00045 * 00046 * DL (input) REAL array, dimension (N-1) 00047 * The (n-1) subdiagonal elements of A. 00048 * 00049 * D (input) REAL array, dimension (N) 00050 * The diagonal elements of A. 00051 * 00052 * DU (input) REAL array, dimension (N-1) 00053 * The (n-1) superdiagonal elements of A. 00054 * 00055 * DLF (input) REAL array, dimension (N-1) 00056 * The (n-1) multipliers that define the matrix L from the 00057 * LU factorization of A as computed by SGTTRF. 00058 * 00059 * DF (input) REAL array, dimension (N) 00060 * The n diagonal elements of the upper triangular matrix U from 00061 * the LU factorization of A. 00062 * 00063 * DUF (input) REAL array, dimension (N-1) 00064 * The (n-1) elements of the first superdiagonal of U. 00065 * 00066 * DU2 (input) REAL array, dimension (N-2) 00067 * The (n-2) elements of the second superdiagonal of U. 00068 * 00069 * IPIV (input) INTEGER array, dimension (N) 00070 * The pivot indices; for 1 <= i <= n, row i of the matrix was 00071 * interchanged with row IPIV(i). IPIV(i) will always be either 00072 * i or i+1; IPIV(i) = i indicates a row interchange was not 00073 * required. 00074 * 00075 * B (input) REAL array, dimension (LDB,NRHS) 00076 * The right hand side matrix B. 00077 * 00078 * LDB (input) INTEGER 00079 * The leading dimension of the array B. LDB >= max(1,N). 00080 * 00081 * X (input/output) REAL array, dimension (LDX,NRHS) 00082 * On entry, the solution matrix X, as computed by SGTTRS. 00083 * On exit, the improved solution matrix X. 00084 * 00085 * LDX (input) INTEGER 00086 * The leading dimension of the array X. LDX >= max(1,N). 00087 * 00088 * FERR (output) REAL array, dimension (NRHS) 00089 * The estimated forward error bound for each solution vector 00090 * X(j) (the j-th column of the solution matrix X). 00091 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00092 * is an estimated upper bound for the magnitude of the largest 00093 * element in (X(j) - XTRUE) divided by the magnitude of the 00094 * largest element in X(j). The estimate is as reliable as 00095 * the estimate for RCOND, and is almost always a slight 00096 * overestimate of the true error. 00097 * 00098 * BERR (output) REAL array, dimension (NRHS) 00099 * The componentwise relative backward error of each solution 00100 * vector X(j) (i.e., the smallest relative change in 00101 * any element of A or B that makes X(j) an exact solution). 00102 * 00103 * WORK (workspace) REAL array, dimension (3*N) 00104 * 00105 * IWORK (workspace) INTEGER array, dimension (N) 00106 * 00107 * INFO (output) INTEGER 00108 * = 0: successful exit 00109 * < 0: if INFO = -i, the i-th argument had an illegal value 00110 * 00111 * Internal Parameters 00112 * =================== 00113 * 00114 * ITMAX is the maximum number of steps of iterative refinement. 00115 * 00116 * ===================================================================== 00117 * 00118 * .. Parameters .. 00119 INTEGER ITMAX 00120 PARAMETER ( ITMAX = 5 ) 00121 REAL ZERO, ONE 00122 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00123 REAL TWO 00124 PARAMETER ( TWO = 2.0E+0 ) 00125 REAL THREE 00126 PARAMETER ( THREE = 3.0E+0 ) 00127 * .. 00128 * .. Local Scalars .. 00129 LOGICAL NOTRAN 00130 CHARACTER TRANSN, TRANST 00131 INTEGER COUNT, I, J, KASE, NZ 00132 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN 00133 * .. 00134 * .. Local Arrays .. 00135 INTEGER ISAVE( 3 ) 00136 * .. 00137 * .. External Subroutines .. 00138 EXTERNAL SAXPY, SCOPY, SGTTRS, SLACN2, SLAGTM, XERBLA 00139 * .. 00140 * .. Intrinsic Functions .. 00141 INTRINSIC ABS, MAX 00142 * .. 00143 * .. External Functions .. 00144 LOGICAL LSAME 00145 REAL SLAMCH 00146 EXTERNAL LSAME, SLAMCH 00147 * .. 00148 * .. Executable Statements .. 00149 * 00150 * Test the input parameters. 00151 * 00152 INFO = 0 00153 NOTRAN = LSAME( TRANS, 'N' ) 00154 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00155 $ LSAME( TRANS, 'C' ) ) THEN 00156 INFO = -1 00157 ELSE IF( N.LT.0 ) THEN 00158 INFO = -2 00159 ELSE IF( NRHS.LT.0 ) THEN 00160 INFO = -3 00161 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00162 INFO = -13 00163 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00164 INFO = -15 00165 END IF 00166 IF( INFO.NE.0 ) THEN 00167 CALL XERBLA( 'SGTRFS', -INFO ) 00168 RETURN 00169 END IF 00170 * 00171 * Quick return if possible 00172 * 00173 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00174 DO 10 J = 1, NRHS 00175 FERR( J ) = ZERO 00176 BERR( J ) = ZERO 00177 10 CONTINUE 00178 RETURN 00179 END IF 00180 * 00181 IF( NOTRAN ) THEN 00182 TRANSN = 'N' 00183 TRANST = 'T' 00184 ELSE 00185 TRANSN = 'T' 00186 TRANST = 'N' 00187 END IF 00188 * 00189 * NZ = maximum number of nonzero elements in each row of A, plus 1 00190 * 00191 NZ = 4 00192 EPS = SLAMCH( 'Epsilon' ) 00193 SAFMIN = SLAMCH( 'Safe minimum' ) 00194 SAFE1 = NZ*SAFMIN 00195 SAFE2 = SAFE1 / EPS 00196 * 00197 * Do for each right hand side 00198 * 00199 DO 110 J = 1, NRHS 00200 * 00201 COUNT = 1 00202 LSTRES = THREE 00203 20 CONTINUE 00204 * 00205 * Loop until stopping criterion is satisfied. 00206 * 00207 * Compute residual R = B - op(A) * X, 00208 * where op(A) = A, A**T, or A**H, depending on TRANS. 00209 * 00210 CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00211 CALL SLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE, 00212 $ WORK( N+1 ), N ) 00213 * 00214 * Compute abs(op(A))*abs(x) + abs(b) for use in the backward 00215 * error bound. 00216 * 00217 IF( NOTRAN ) THEN 00218 IF( N.EQ.1 ) THEN 00219 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) 00220 ELSE 00221 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + 00222 $ ABS( DU( 1 )*X( 2, J ) ) 00223 DO 30 I = 2, N - 1 00224 WORK( I ) = ABS( B( I, J ) ) + 00225 $ ABS( DL( I-1 )*X( I-1, J ) ) + 00226 $ ABS( D( I )*X( I, J ) ) + 00227 $ ABS( DU( I )*X( I+1, J ) ) 00228 30 CONTINUE 00229 WORK( N ) = ABS( B( N, J ) ) + 00230 $ ABS( DL( N-1 )*X( N-1, J ) ) + 00231 $ ABS( D( N )*X( N, J ) ) 00232 END IF 00233 ELSE 00234 IF( N.EQ.1 ) THEN 00235 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) 00236 ELSE 00237 WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + 00238 $ ABS( DL( 1 )*X( 2, J ) ) 00239 DO 40 I = 2, N - 1 00240 WORK( I ) = ABS( B( I, J ) ) + 00241 $ ABS( DU( I-1 )*X( I-1, J ) ) + 00242 $ ABS( D( I )*X( I, J ) ) + 00243 $ ABS( DL( I )*X( I+1, J ) ) 00244 40 CONTINUE 00245 WORK( N ) = ABS( B( N, J ) ) + 00246 $ ABS( DU( N-1 )*X( N-1, J ) ) + 00247 $ ABS( D( N )*X( N, J ) ) 00248 END IF 00249 END IF 00250 * 00251 * Compute componentwise relative backward error from formula 00252 * 00253 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00254 * 00255 * where abs(Z) is the componentwise absolute value of the matrix 00256 * or vector Z. If the i-th component of the denominator is less 00257 * than SAFE2, then SAFE1 is added to the i-th components of the 00258 * numerator and denominator before dividing. 00259 * 00260 S = ZERO 00261 DO 50 I = 1, N 00262 IF( WORK( I ).GT.SAFE2 ) THEN 00263 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00264 ELSE 00265 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00266 $ ( WORK( I )+SAFE1 ) ) 00267 END IF 00268 50 CONTINUE 00269 BERR( J ) = S 00270 * 00271 * Test stopping criterion. Continue iterating if 00272 * 1) The residual BERR(J) is larger than machine epsilon, and 00273 * 2) BERR(J) decreased by at least a factor of 2 during the 00274 * last iteration, and 00275 * 3) At most ITMAX iterations tried. 00276 * 00277 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00278 $ COUNT.LE.ITMAX ) THEN 00279 * 00280 * Update solution and try again. 00281 * 00282 CALL SGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, 00283 $ WORK( N+1 ), N, INFO ) 00284 CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00285 LSTRES = BERR( J ) 00286 COUNT = COUNT + 1 00287 GO TO 20 00288 END IF 00289 * 00290 * Bound error from formula 00291 * 00292 * norm(X - XTRUE) / norm(X) .le. FERR = 00293 * norm( abs(inv(op(A)))* 00294 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00295 * 00296 * where 00297 * norm(Z) is the magnitude of the largest component of Z 00298 * inv(op(A)) is the inverse of op(A) 00299 * abs(Z) is the componentwise absolute value of the matrix or 00300 * vector Z 00301 * NZ is the maximum number of nonzeros in any row of A, plus 1 00302 * EPS is machine epsilon 00303 * 00304 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00305 * is incremented by SAFE1 if the i-th component of 00306 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00307 * 00308 * Use SLACN2 to estimate the infinity-norm of the matrix 00309 * inv(op(A)) * diag(W), 00310 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00311 * 00312 DO 60 I = 1, N 00313 IF( WORK( I ).GT.SAFE2 ) THEN 00314 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00315 ELSE 00316 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00317 END IF 00318 60 CONTINUE 00319 * 00320 KASE = 0 00321 70 CONTINUE 00322 CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00323 $ KASE, ISAVE ) 00324 IF( KASE.NE.0 ) THEN 00325 IF( KASE.EQ.1 ) THEN 00326 * 00327 * Multiply by diag(W)*inv(op(A)**T). 00328 * 00329 CALL SGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, 00330 $ WORK( N+1 ), N, INFO ) 00331 DO 80 I = 1, N 00332 WORK( N+I ) = WORK( I )*WORK( N+I ) 00333 80 CONTINUE 00334 ELSE 00335 * 00336 * Multiply by inv(op(A))*diag(W). 00337 * 00338 DO 90 I = 1, N 00339 WORK( N+I ) = WORK( I )*WORK( N+I ) 00340 90 CONTINUE 00341 CALL SGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, 00342 $ WORK( N+1 ), N, INFO ) 00343 END IF 00344 GO TO 70 00345 END IF 00346 * 00347 * Normalize error. 00348 * 00349 LSTRES = ZERO 00350 DO 100 I = 1, N 00351 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00352 100 CONTINUE 00353 IF( LSTRES.NE.ZERO ) 00354 $ FERR( J ) = FERR( J ) / LSTRES 00355 * 00356 110 CONTINUE 00357 * 00358 RETURN 00359 * 00360 * End of SGTRFS 00361 * 00362 END
1.7.3 
