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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 ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 00002 $ X, LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH. 00010 * 00011 * .. Scalar Arguments .. 00012 CHARACTER TRANS 00013 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IPIV( * ) 00017 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00018 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00019 $ WORK( * ), X( LDX, * ) 00020 * .. 00021 * 00022 * Purpose 00023 * ======= 00024 * 00025 * ZGERFS improves the computed solution to a system of linear 00026 * equations and provides error bounds and backward error estimates for 00027 * 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) 00037 * 00038 * N (input) INTEGER 00039 * The order of the matrix A. N >= 0. 00040 * 00041 * NRHS (input) INTEGER 00042 * The number of right hand sides, i.e., the number of columns 00043 * of the matrices B and X. NRHS >= 0. 00044 * 00045 * A (input) COMPLEX*16 array, dimension (LDA,N) 00046 * The original N-by-N matrix A. 00047 * 00048 * LDA (input) INTEGER 00049 * The leading dimension of the array A. LDA >= max(1,N). 00050 * 00051 * AF (input) COMPLEX*16 array, dimension (LDAF,N) 00052 * The factors L and U from the factorization A = P*L*U 00053 * as computed by ZGETRF. 00054 * 00055 * LDAF (input) INTEGER 00056 * The leading dimension of the array AF. LDAF >= max(1,N). 00057 * 00058 * IPIV (input) INTEGER array, dimension (N) 00059 * The pivot indices from ZGETRF; for 1<=i<=N, row i of the 00060 * matrix was interchanged with row IPIV(i). 00061 * 00062 * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 00063 * The right hand side matrix B. 00064 * 00065 * LDB (input) INTEGER 00066 * The leading dimension of the array B. LDB >= max(1,N). 00067 * 00068 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) 00069 * On entry, the solution matrix X, as computed by ZGETRS. 00070 * On exit, the improved solution matrix X. 00071 * 00072 * LDX (input) INTEGER 00073 * The leading dimension of the array X. LDX >= max(1,N). 00074 * 00075 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00076 * The estimated forward error bound for each solution vector 00077 * X(j) (the j-th column of the solution matrix X). 00078 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00079 * is an estimated upper bound for the magnitude of the largest 00080 * element in (X(j) - XTRUE) divided by the magnitude of the 00081 * largest element in X(j). The estimate is as reliable as 00082 * the estimate for RCOND, and is almost always a slight 00083 * overestimate of the true error. 00084 * 00085 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00086 * The componentwise relative backward error of each solution 00087 * vector X(j) (i.e., the smallest relative change in 00088 * any element of A or B that makes X(j) an exact solution). 00089 * 00090 * WORK (workspace) COMPLEX*16 array, dimension (2*N) 00091 * 00092 * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 00093 * 00094 * INFO (output) INTEGER 00095 * = 0: successful exit 00096 * < 0: if INFO = -i, the i-th argument had an illegal value 00097 * 00098 * Internal Parameters 00099 * =================== 00100 * 00101 * ITMAX is the maximum number of steps of iterative refinement. 00102 * 00103 * ===================================================================== 00104 * 00105 * .. Parameters .. 00106 INTEGER ITMAX 00107 PARAMETER ( ITMAX = 5 ) 00108 DOUBLE PRECISION ZERO 00109 PARAMETER ( ZERO = 0.0D+0 ) 00110 COMPLEX*16 ONE 00111 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 00112 DOUBLE PRECISION TWO 00113 PARAMETER ( TWO = 2.0D+0 ) 00114 DOUBLE PRECISION THREE 00115 PARAMETER ( THREE = 3.0D+0 ) 00116 * .. 00117 * .. Local Scalars .. 00118 LOGICAL NOTRAN 00119 CHARACTER TRANSN, TRANST 00120 INTEGER COUNT, I, J, K, KASE, NZ 00121 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00122 COMPLEX*16 ZDUM 00123 * .. 00124 * .. Local Arrays .. 00125 INTEGER ISAVE( 3 ) 00126 * .. 00127 * .. External Functions .. 00128 LOGICAL LSAME 00129 DOUBLE PRECISION DLAMCH 00130 EXTERNAL LSAME, DLAMCH 00131 * .. 00132 * .. External Subroutines .. 00133 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGETRS, ZLACN2 00134 * .. 00135 * .. Intrinsic Functions .. 00136 INTRINSIC ABS, DBLE, DIMAG, MAX 00137 * .. 00138 * .. Statement Functions .. 00139 DOUBLE PRECISION CABS1 00140 * .. 00141 * .. Statement Function definitions .. 00142 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00143 * .. 00144 * .. Executable Statements .. 00145 * 00146 * Test the input parameters. 00147 * 00148 INFO = 0 00149 NOTRAN = LSAME( TRANS, 'N' ) 00150 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00151 $ LSAME( TRANS, 'C' ) ) THEN 00152 INFO = -1 00153 ELSE IF( N.LT.0 ) THEN 00154 INFO = -2 00155 ELSE IF( NRHS.LT.0 ) THEN 00156 INFO = -3 00157 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00158 INFO = -5 00159 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00160 INFO = -7 00161 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00162 INFO = -10 00163 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00164 INFO = -12 00165 END IF 00166 IF( INFO.NE.0 ) THEN 00167 CALL XERBLA( 'ZGERFS', -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 = 'C' 00184 ELSE 00185 TRANSN = 'C' 00186 TRANST = 'N' 00187 END IF 00188 * 00189 * NZ = maximum number of nonzero elements in each row of A, plus 1 00190 * 00191 NZ = N + 1 00192 EPS = DLAMCH( 'Epsilon' ) 00193 SAFMIN = DLAMCH( 'Safe minimum' ) 00194 SAFE1 = NZ*SAFMIN 00195 SAFE2 = SAFE1 / EPS 00196 * 00197 * Do for each right hand side 00198 * 00199 DO 140 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 ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 00211 CALL ZGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 00212 $ 1 ) 00213 * 00214 * Compute componentwise relative backward error from formula 00215 * 00216 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00217 * 00218 * where abs(Z) is the componentwise absolute value of the matrix 00219 * or vector Z. If the i-th component of the denominator is less 00220 * than SAFE2, then SAFE1 is added to the i-th components of the 00221 * numerator and denominator before dividing. 00222 * 00223 DO 30 I = 1, N 00224 RWORK( I ) = CABS1( B( I, J ) ) 00225 30 CONTINUE 00226 * 00227 * Compute abs(op(A))*abs(X) + abs(B). 00228 * 00229 IF( NOTRAN ) THEN 00230 DO 50 K = 1, N 00231 XK = CABS1( X( K, J ) ) 00232 DO 40 I = 1, N 00233 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 00234 40 CONTINUE 00235 50 CONTINUE 00236 ELSE 00237 DO 70 K = 1, N 00238 S = ZERO 00239 DO 60 I = 1, N 00240 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 00241 60 CONTINUE 00242 RWORK( K ) = RWORK( K ) + S 00243 70 CONTINUE 00244 END IF 00245 S = ZERO 00246 DO 80 I = 1, N 00247 IF( RWORK( I ).GT.SAFE2 ) THEN 00248 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 00249 ELSE 00250 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 00251 $ ( RWORK( I )+SAFE1 ) ) 00252 END IF 00253 80 CONTINUE 00254 BERR( J ) = S 00255 * 00256 * Test stopping criterion. Continue iterating if 00257 * 1) The residual BERR(J) is larger than machine epsilon, and 00258 * 2) BERR(J) decreased by at least a factor of 2 during the 00259 * last iteration, and 00260 * 3) At most ITMAX iterations tried. 00261 * 00262 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00263 $ COUNT.LE.ITMAX ) THEN 00264 * 00265 * Update solution and try again. 00266 * 00267 CALL ZGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 00268 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) 00269 LSTRES = BERR( J ) 00270 COUNT = COUNT + 1 00271 GO TO 20 00272 END IF 00273 * 00274 * Bound error from formula 00275 * 00276 * norm(X - XTRUE) / norm(X) .le. FERR = 00277 * norm( abs(inv(op(A)))* 00278 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00279 * 00280 * where 00281 * norm(Z) is the magnitude of the largest component of Z 00282 * inv(op(A)) is the inverse of op(A) 00283 * abs(Z) is the componentwise absolute value of the matrix or 00284 * vector Z 00285 * NZ is the maximum number of nonzeros in any row of A, plus 1 00286 * EPS is machine epsilon 00287 * 00288 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00289 * is incremented by SAFE1 if the i-th component of 00290 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00291 * 00292 * Use ZLACN2 to estimate the infinity-norm of the matrix 00293 * inv(op(A)) * diag(W), 00294 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00295 * 00296 DO 90 I = 1, N 00297 IF( RWORK( I ).GT.SAFE2 ) THEN 00298 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 00299 ELSE 00300 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 00301 $ SAFE1 00302 END IF 00303 90 CONTINUE 00304 * 00305 KASE = 0 00306 100 CONTINUE 00307 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 00308 IF( KASE.NE.0 ) THEN 00309 IF( KASE.EQ.1 ) THEN 00310 * 00311 * Multiply by diag(W)*inv(op(A)**H). 00312 * 00313 CALL ZGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK, N, 00314 $ INFO ) 00315 DO 110 I = 1, N 00316 WORK( I ) = RWORK( I )*WORK( I ) 00317 110 CONTINUE 00318 ELSE 00319 * 00320 * Multiply by inv(op(A))*diag(W). 00321 * 00322 DO 120 I = 1, N 00323 WORK( I ) = RWORK( I )*WORK( I ) 00324 120 CONTINUE 00325 CALL ZGETRS( TRANSN, N, 1, AF, LDAF, IPIV, WORK, N, 00326 $ INFO ) 00327 END IF 00328 GO TO 100 00329 END IF 00330 * 00331 * Normalize error. 00332 * 00333 LSTRES = ZERO 00334 DO 130 I = 1, N 00335 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 00336 130 CONTINUE 00337 IF( LSTRES.NE.ZERO ) 00338 $ FERR( J ) = FERR( J ) / LSTRES 00339 * 00340 140 CONTINUE 00341 * 00342 RETURN 00343 * 00344 * End of ZGERFS 00345 * 00346 END
1.7.3 
