 |
LAPACK 3.3.1
Linear Algebra PACKage
|
|
LAPACK 3.3.1
Linear Algebra PACKage
|
00001 SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, 00002 $ 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 UPLO 00013 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00017 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00018 $ WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * ZPORFS improves the computed solution to a system of linear 00025 * equations when the coefficient matrix is Hermitian positive definite, 00026 * and provides error bounds and backward error estimates for the 00027 * solution. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * UPLO (input) CHARACTER*1 00033 * = 'U': Upper triangle of A is stored; 00034 * = 'L': Lower triangle of A is stored. 00035 * 00036 * N (input) INTEGER 00037 * The order of the matrix A. N >= 0. 00038 * 00039 * NRHS (input) INTEGER 00040 * The number of right hand sides, i.e., the number of columns 00041 * of the matrices B and X. NRHS >= 0. 00042 * 00043 * A (input) COMPLEX*16 array, dimension (LDA,N) 00044 * The Hermitian matrix A. If UPLO = 'U', the leading N-by-N 00045 * upper triangular part of A contains the upper triangular part 00046 * of the matrix A, and the strictly lower triangular part of A 00047 * is not referenced. If UPLO = 'L', the leading N-by-N lower 00048 * triangular part of A contains the lower triangular part of 00049 * the matrix A, and the strictly upper triangular part of A is 00050 * not referenced. 00051 * 00052 * LDA (input) INTEGER 00053 * The leading dimension of the array A. LDA >= max(1,N). 00054 * 00055 * AF (input) COMPLEX*16 array, dimension (LDAF,N) 00056 * The triangular factor U or L from the Cholesky factorization 00057 * A = U**H*U or A = L*L**H, as computed by ZPOTRF. 00058 * 00059 * LDAF (input) INTEGER 00060 * The leading dimension of the array AF. LDAF >= max(1,N). 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 ZPOTRS. 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 UPPER 00119 INTEGER COUNT, I, J, K, KASE, NZ 00120 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00121 COMPLEX*16 ZDUM 00122 * .. 00123 * .. Local Arrays .. 00124 INTEGER ISAVE( 3 ) 00125 * .. 00126 * .. External Subroutines .. 00127 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS 00128 * .. 00129 * .. Intrinsic Functions .. 00130 INTRINSIC ABS, DBLE, DIMAG, MAX 00131 * .. 00132 * .. External Functions .. 00133 LOGICAL LSAME 00134 DOUBLE PRECISION DLAMCH 00135 EXTERNAL LSAME, DLAMCH 00136 * .. 00137 * .. Statement Functions .. 00138 DOUBLE PRECISION CABS1 00139 * .. 00140 * .. Statement Function definitions .. 00141 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00142 * .. 00143 * .. Executable Statements .. 00144 * 00145 * Test the input parameters. 00146 * 00147 INFO = 0 00148 UPPER = LSAME( UPLO, 'U' ) 00149 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00150 INFO = -1 00151 ELSE IF( N.LT.0 ) THEN 00152 INFO = -2 00153 ELSE IF( NRHS.LT.0 ) THEN 00154 INFO = -3 00155 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00156 INFO = -5 00157 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00158 INFO = -7 00159 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00160 INFO = -9 00161 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00162 INFO = -11 00163 END IF 00164 IF( INFO.NE.0 ) THEN 00165 CALL XERBLA( 'ZPORFS', -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 * NZ = maximum number of nonzero elements in each row of A, plus 1 00180 * 00181 NZ = N + 1 00182 EPS = DLAMCH( 'Epsilon' ) 00183 SAFMIN = DLAMCH( 'Safe minimum' ) 00184 SAFE1 = NZ*SAFMIN 00185 SAFE2 = SAFE1 / EPS 00186 * 00187 * Do for each right hand side 00188 * 00189 DO 140 J = 1, NRHS 00190 * 00191 COUNT = 1 00192 LSTRES = THREE 00193 20 CONTINUE 00194 * 00195 * Loop until stopping criterion is satisfied. 00196 * 00197 * Compute residual R = B - A * X 00198 * 00199 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 00200 CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 ) 00201 * 00202 * Compute componentwise relative backward error from formula 00203 * 00204 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00205 * 00206 * where abs(Z) is the componentwise absolute value of the matrix 00207 * or vector Z. If the i-th component of the denominator is less 00208 * than SAFE2, then SAFE1 is added to the i-th components of the 00209 * numerator and denominator before dividing. 00210 * 00211 DO 30 I = 1, N 00212 RWORK( I ) = CABS1( B( I, J ) ) 00213 30 CONTINUE 00214 * 00215 * Compute abs(A)*abs(X) + abs(B). 00216 * 00217 IF( UPPER ) THEN 00218 DO 50 K = 1, N 00219 S = ZERO 00220 XK = CABS1( X( K, J ) ) 00221 DO 40 I = 1, K - 1 00222 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 00223 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 00224 40 CONTINUE 00225 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S 00226 50 CONTINUE 00227 ELSE 00228 DO 70 K = 1, N 00229 S = ZERO 00230 XK = CABS1( X( K, J ) ) 00231 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK 00232 DO 60 I = K + 1, N 00233 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 00234 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 00235 60 CONTINUE 00236 RWORK( K ) = RWORK( K ) + S 00237 70 CONTINUE 00238 END IF 00239 S = ZERO 00240 DO 80 I = 1, N 00241 IF( RWORK( I ).GT.SAFE2 ) THEN 00242 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 00243 ELSE 00244 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 00245 $ ( RWORK( I )+SAFE1 ) ) 00246 END IF 00247 80 CONTINUE 00248 BERR( J ) = S 00249 * 00250 * Test stopping criterion. Continue iterating if 00251 * 1) The residual BERR(J) is larger than machine epsilon, and 00252 * 2) BERR(J) decreased by at least a factor of 2 during the 00253 * last iteration, and 00254 * 3) At most ITMAX iterations tried. 00255 * 00256 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00257 $ COUNT.LE.ITMAX ) THEN 00258 * 00259 * Update solution and try again. 00260 * 00261 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO ) 00262 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) 00263 LSTRES = BERR( J ) 00264 COUNT = COUNT + 1 00265 GO TO 20 00266 END IF 00267 * 00268 * Bound error from formula 00269 * 00270 * norm(X - XTRUE) / norm(X) .le. FERR = 00271 * norm( abs(inv(A))* 00272 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00273 * 00274 * where 00275 * norm(Z) is the magnitude of the largest component of Z 00276 * inv(A) is the inverse of A 00277 * abs(Z) is the componentwise absolute value of the matrix or 00278 * vector Z 00279 * NZ is the maximum number of nonzeros in any row of A, plus 1 00280 * EPS is machine epsilon 00281 * 00282 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00283 * is incremented by SAFE1 if the i-th component of 00284 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00285 * 00286 * Use ZLACN2 to estimate the infinity-norm of the matrix 00287 * inv(A) * diag(W), 00288 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00289 * 00290 DO 90 I = 1, N 00291 IF( RWORK( I ).GT.SAFE2 ) THEN 00292 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 00293 ELSE 00294 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 00295 $ SAFE1 00296 END IF 00297 90 CONTINUE 00298 * 00299 KASE = 0 00300 100 CONTINUE 00301 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 00302 IF( KASE.NE.0 ) THEN 00303 IF( KASE.EQ.1 ) THEN 00304 * 00305 * Multiply by diag(W)*inv(A**H). 00306 * 00307 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO ) 00308 DO 110 I = 1, N 00309 WORK( I ) = RWORK( I )*WORK( I ) 00310 110 CONTINUE 00311 ELSE IF( KASE.EQ.2 ) THEN 00312 * 00313 * Multiply by inv(A)*diag(W). 00314 * 00315 DO 120 I = 1, N 00316 WORK( I ) = RWORK( I )*WORK( I ) 00317 120 CONTINUE 00318 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO ) 00319 END IF 00320 GO TO 100 00321 END IF 00322 * 00323 * Normalize error. 00324 * 00325 LSTRES = ZERO 00326 DO 130 I = 1, N 00327 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 00328 130 CONTINUE 00329 IF( LSTRES.NE.ZERO ) 00330 $ FERR( J ) = FERR( J ) / LSTRES 00331 * 00332 140 CONTINUE 00333 * 00334 RETURN 00335 * 00336 * End of ZPORFS 00337 * 00338 END
1.7.3 
