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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 SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS 00015 * .. 00016 * .. Array Arguments .. 00017 INTEGER IPIV( * ), IWORK( * ) 00018 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 00019 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 00020 * .. 00021 * 00022 * Purpose 00023 * ======= 00024 * 00025 * SGBRFS improves the computed solution to a system of linear 00026 * equations when the coefficient matrix is banded, and provides 00027 * error bounds and backward error estimates for the solution. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * TRANS (input) CHARACTER*1 00033 * Specifies the form of the system of equations: 00034 * = 'N': A * X = B (No transpose) 00035 * = 'T': A**T * X = B (Transpose) 00036 * = 'C': A**H * X = B (Conjugate transpose = Transpose) 00037 * 00038 * N (input) INTEGER 00039 * The order of the matrix A. N >= 0. 00040 * 00041 * KL (input) INTEGER 00042 * The number of subdiagonals within the band of A. KL >= 0. 00043 * 00044 * KU (input) INTEGER 00045 * The number of superdiagonals within the band of A. KU >= 0. 00046 * 00047 * NRHS (input) INTEGER 00048 * The number of right hand sides, i.e., the number of columns 00049 * of the matrices B and X. NRHS >= 0. 00050 * 00051 * AB (input) REAL array, dimension (LDAB,N) 00052 * The original band matrix A, stored in rows 1 to KL+KU+1. 00053 * The j-th column of A is stored in the j-th column of the 00054 * array AB as follows: 00055 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). 00056 * 00057 * LDAB (input) INTEGER 00058 * The leading dimension of the array AB. LDAB >= KL+KU+1. 00059 * 00060 * AFB (input) REAL array, dimension (LDAFB,N) 00061 * Details of the LU factorization of the band matrix A, as 00062 * computed by SGBTRF. U is stored as an upper triangular band 00063 * matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and 00064 * the multipliers used during the factorization are stored in 00065 * rows KL+KU+2 to 2*KL+KU+1. 00066 * 00067 * LDAFB (input) INTEGER 00068 * The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. 00069 * 00070 * IPIV (input) INTEGER array, dimension (N) 00071 * The pivot indices from SGBTRF; for 1<=i<=N, row i of the 00072 * matrix was interchanged with row IPIV(i). 00073 * 00074 * B (input) REAL array, dimension (LDB,NRHS) 00075 * The right hand side matrix B. 00076 * 00077 * LDB (input) INTEGER 00078 * The leading dimension of the array B. LDB >= max(1,N). 00079 * 00080 * X (input/output) REAL array, dimension (LDX,NRHS) 00081 * On entry, the solution matrix X, as computed by SGBTRS. 00082 * On exit, the improved solution matrix X. 00083 * 00084 * LDX (input) INTEGER 00085 * The leading dimension of the array X. LDX >= max(1,N). 00086 * 00087 * FERR (output) REAL array, dimension (NRHS) 00088 * The estimated forward error bound for each solution vector 00089 * X(j) (the j-th column of the solution matrix X). 00090 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00091 * is an estimated upper bound for the magnitude of the largest 00092 * element in (X(j) - XTRUE) divided by the magnitude of the 00093 * largest element in X(j). The estimate is as reliable as 00094 * the estimate for RCOND, and is almost always a slight 00095 * overestimate of the true error. 00096 * 00097 * BERR (output) REAL array, dimension (NRHS) 00098 * The componentwise relative backward error of each solution 00099 * vector X(j) (i.e., the smallest relative change in 00100 * any element of A or B that makes X(j) an exact solution). 00101 * 00102 * WORK (workspace) REAL array, dimension (3*N) 00103 * 00104 * IWORK (workspace) INTEGER array, dimension (N) 00105 * 00106 * INFO (output) INTEGER 00107 * = 0: successful exit 00108 * < 0: if INFO = -i, the i-th argument had an illegal value 00109 * 00110 * Internal Parameters 00111 * =================== 00112 * 00113 * ITMAX is the maximum number of steps of iterative refinement. 00114 * 00115 * ===================================================================== 00116 * 00117 * .. Parameters .. 00118 INTEGER ITMAX 00119 PARAMETER ( ITMAX = 5 ) 00120 REAL ZERO 00121 PARAMETER ( ZERO = 0.0E+0 ) 00122 REAL ONE 00123 PARAMETER ( ONE = 1.0E+0 ) 00124 REAL TWO 00125 PARAMETER ( TWO = 2.0E+0 ) 00126 REAL THREE 00127 PARAMETER ( THREE = 3.0E+0 ) 00128 * .. 00129 * .. Local Scalars .. 00130 LOGICAL NOTRAN 00131 CHARACTER TRANST 00132 INTEGER COUNT, I, J, K, KASE, KK, NZ 00133 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00134 * .. 00135 * .. Local Arrays .. 00136 INTEGER ISAVE( 3 ) 00137 * .. 00138 * .. External Subroutines .. 00139 EXTERNAL SAXPY, SCOPY, SGBMV, SGBTRS, SLACN2, XERBLA 00140 * .. 00141 * .. Intrinsic Functions .. 00142 INTRINSIC ABS, MAX, MIN 00143 * .. 00144 * .. External Functions .. 00145 LOGICAL LSAME 00146 REAL SLAMCH 00147 EXTERNAL LSAME, SLAMCH 00148 * .. 00149 * .. Executable Statements .. 00150 * 00151 * Test the input parameters. 00152 * 00153 INFO = 0 00154 NOTRAN = LSAME( TRANS, 'N' ) 00155 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00156 $ LSAME( TRANS, 'C' ) ) THEN 00157 INFO = -1 00158 ELSE IF( N.LT.0 ) THEN 00159 INFO = -2 00160 ELSE IF( KL.LT.0 ) THEN 00161 INFO = -3 00162 ELSE IF( KU.LT.0 ) THEN 00163 INFO = -4 00164 ELSE IF( NRHS.LT.0 ) THEN 00165 INFO = -5 00166 ELSE IF( LDAB.LT.KL+KU+1 ) THEN 00167 INFO = -7 00168 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN 00169 INFO = -9 00170 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00171 INFO = -12 00172 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00173 INFO = -14 00174 END IF 00175 IF( INFO.NE.0 ) THEN 00176 CALL XERBLA( 'SGBRFS', -INFO ) 00177 RETURN 00178 END IF 00179 * 00180 * Quick return if possible 00181 * 00182 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00183 DO 10 J = 1, NRHS 00184 FERR( J ) = ZERO 00185 BERR( J ) = ZERO 00186 10 CONTINUE 00187 RETURN 00188 END IF 00189 * 00190 IF( NOTRAN ) THEN 00191 TRANST = 'T' 00192 ELSE 00193 TRANST = 'N' 00194 END IF 00195 * 00196 * NZ = maximum number of nonzero elements in each row of A, plus 1 00197 * 00198 NZ = MIN( KL+KU+2, N+1 ) 00199 EPS = SLAMCH( 'Epsilon' ) 00200 SAFMIN = SLAMCH( 'Safe minimum' ) 00201 SAFE1 = NZ*SAFMIN 00202 SAFE2 = SAFE1 / EPS 00203 * 00204 * Do for each right hand side 00205 * 00206 DO 140 J = 1, NRHS 00207 * 00208 COUNT = 1 00209 LSTRES = THREE 00210 20 CONTINUE 00211 * 00212 * Loop until stopping criterion is satisfied. 00213 * 00214 * Compute residual R = B - op(A) * X, 00215 * where op(A) = A, A**T, or A**H, depending on TRANS. 00216 * 00217 CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00218 CALL SGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1, 00219 $ ONE, WORK( N+1 ), 1 ) 00220 * 00221 * Compute componentwise relative backward error from formula 00222 * 00223 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00224 * 00225 * where abs(Z) is the componentwise absolute value of the matrix 00226 * or vector Z. If the i-th component of the denominator is less 00227 * than SAFE2, then SAFE1 is added to the i-th components of the 00228 * numerator and denominator before dividing. 00229 * 00230 DO 30 I = 1, N 00231 WORK( I ) = ABS( B( I, J ) ) 00232 30 CONTINUE 00233 * 00234 * Compute abs(op(A))*abs(X) + abs(B). 00235 * 00236 IF( NOTRAN ) THEN 00237 DO 50 K = 1, N 00238 KK = KU + 1 - K 00239 XK = ABS( X( K, J ) ) 00240 DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL ) 00241 WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK 00242 40 CONTINUE 00243 50 CONTINUE 00244 ELSE 00245 DO 70 K = 1, N 00246 S = ZERO 00247 KK = KU + 1 - K 00248 DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL ) 00249 S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) ) 00250 60 CONTINUE 00251 WORK( K ) = WORK( K ) + S 00252 70 CONTINUE 00253 END IF 00254 S = ZERO 00255 DO 80 I = 1, N 00256 IF( WORK( I ).GT.SAFE2 ) THEN 00257 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00258 ELSE 00259 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00260 $ ( WORK( I )+SAFE1 ) ) 00261 END IF 00262 80 CONTINUE 00263 BERR( J ) = S 00264 * 00265 * Test stopping criterion. Continue iterating if 00266 * 1) The residual BERR(J) is larger than machine epsilon, and 00267 * 2) BERR(J) decreased by at least a factor of 2 during the 00268 * last iteration, and 00269 * 3) At most ITMAX iterations tried. 00270 * 00271 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00272 $ COUNT.LE.ITMAX ) THEN 00273 * 00274 * Update solution and try again. 00275 * 00276 CALL SGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, 00277 $ WORK( N+1 ), N, INFO ) 00278 CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00279 LSTRES = BERR( J ) 00280 COUNT = COUNT + 1 00281 GO TO 20 00282 END IF 00283 * 00284 * Bound error from formula 00285 * 00286 * norm(X - XTRUE) / norm(X) .le. FERR = 00287 * norm( abs(inv(op(A)))* 00288 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00289 * 00290 * where 00291 * norm(Z) is the magnitude of the largest component of Z 00292 * inv(op(A)) is the inverse of op(A) 00293 * abs(Z) is the componentwise absolute value of the matrix or 00294 * vector Z 00295 * NZ is the maximum number of nonzeros in any row of A, plus 1 00296 * EPS is machine epsilon 00297 * 00298 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00299 * is incremented by SAFE1 if the i-th component of 00300 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00301 * 00302 * Use SLACN2 to estimate the infinity-norm of the matrix 00303 * inv(op(A)) * diag(W), 00304 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00305 * 00306 DO 90 I = 1, N 00307 IF( WORK( I ).GT.SAFE2 ) THEN 00308 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00309 ELSE 00310 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00311 END IF 00312 90 CONTINUE 00313 * 00314 KASE = 0 00315 100 CONTINUE 00316 CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00317 $ KASE, ISAVE ) 00318 IF( KASE.NE.0 ) THEN 00319 IF( KASE.EQ.1 ) THEN 00320 * 00321 * Multiply by diag(W)*inv(op(A)**T). 00322 * 00323 CALL SGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV, 00324 $ WORK( N+1 ), N, INFO ) 00325 DO 110 I = 1, N 00326 WORK( N+I ) = WORK( N+I )*WORK( I ) 00327 110 CONTINUE 00328 ELSE 00329 * 00330 * Multiply by inv(op(A))*diag(W). 00331 * 00332 DO 120 I = 1, N 00333 WORK( N+I ) = WORK( N+I )*WORK( I ) 00334 120 CONTINUE 00335 CALL SGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, 00336 $ WORK( N+1 ), N, INFO ) 00337 END IF 00338 GO TO 100 00339 END IF 00340 * 00341 * Normalize error. 00342 * 00343 LSTRES = ZERO 00344 DO 130 I = 1, N 00345 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00346 130 CONTINUE 00347 IF( LSTRES.NE.ZERO ) 00348 $ FERR( J ) = FERR( J ) / LSTRES 00349 * 00350 140 CONTINUE 00351 * 00352 RETURN 00353 * 00354 * End of SGBRFS 00355 * 00356 END
1.7.3 
