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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 ZHERFS( UPLO, 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 UPLO 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 * ZHERFS improves the computed solution to a system of linear 00026 * equations when the coefficient matrix is Hermitian indefinite, and 00027 * provides error bounds and backward error estimates for the 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 factored form of the matrix A. AF contains the block 00057 * diagonal matrix D and the multipliers used to obtain the 00058 * factor U or L from the factorization A = U*D*U**H or 00059 * A = L*D*L**H as computed by ZHETRF. 00060 * 00061 * LDAF (input) INTEGER 00062 * The leading dimension of the array AF. LDAF >= max(1,N). 00063 * 00064 * IPIV (input) INTEGER array, dimension (N) 00065 * Details of the interchanges and the block structure of D 00066 * as determined by ZHETRF. 00067 * 00068 * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 00069 * The right hand side matrix B. 00070 * 00071 * LDB (input) INTEGER 00072 * The leading dimension of the array B. LDB >= max(1,N). 00073 * 00074 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) 00075 * On entry, the solution matrix X, as computed by ZHETRS. 00076 * On exit, the improved solution matrix X. 00077 * 00078 * LDX (input) INTEGER 00079 * The leading dimension of the array X. LDX >= max(1,N). 00080 * 00081 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00082 * The estimated forward error bound for each solution vector 00083 * X(j) (the j-th column of the solution matrix X). 00084 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00085 * is an estimated upper bound for the magnitude of the largest 00086 * element in (X(j) - XTRUE) divided by the magnitude of the 00087 * largest element in X(j). The estimate is as reliable as 00088 * the estimate for RCOND, and is almost always a slight 00089 * overestimate of the true error. 00090 * 00091 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00092 * The componentwise relative backward error of each solution 00093 * vector X(j) (i.e., the smallest relative change in 00094 * any element of A or B that makes X(j) an exact solution). 00095 * 00096 * WORK (workspace) COMPLEX*16 array, dimension (2*N) 00097 * 00098 * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 00099 * 00100 * INFO (output) INTEGER 00101 * = 0: successful exit 00102 * < 0: if INFO = -i, the i-th argument had an illegal value 00103 * 00104 * Internal Parameters 00105 * =================== 00106 * 00107 * ITMAX is the maximum number of steps of iterative refinement. 00108 * 00109 * ===================================================================== 00110 * 00111 * .. Parameters .. 00112 INTEGER ITMAX 00113 PARAMETER ( ITMAX = 5 ) 00114 DOUBLE PRECISION ZERO 00115 PARAMETER ( ZERO = 0.0D+0 ) 00116 COMPLEX*16 ONE 00117 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 00118 DOUBLE PRECISION TWO 00119 PARAMETER ( TWO = 2.0D+0 ) 00120 DOUBLE PRECISION THREE 00121 PARAMETER ( THREE = 3.0D+0 ) 00122 * .. 00123 * .. Local Scalars .. 00124 LOGICAL UPPER 00125 INTEGER COUNT, I, J, K, KASE, NZ 00126 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00127 COMPLEX*16 ZDUM 00128 * .. 00129 * .. Local Arrays .. 00130 INTEGER ISAVE( 3 ) 00131 * .. 00132 * .. External Subroutines .. 00133 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZHETRS, ZLACN2 00134 * .. 00135 * .. Intrinsic Functions .. 00136 INTRINSIC ABS, DBLE, DIMAG, MAX 00137 * .. 00138 * .. External Functions .. 00139 LOGICAL LSAME 00140 DOUBLE PRECISION DLAMCH 00141 EXTERNAL LSAME, DLAMCH 00142 * .. 00143 * .. Statement Functions .. 00144 DOUBLE PRECISION CABS1 00145 * .. 00146 * .. Statement Function definitions .. 00147 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00148 * .. 00149 * .. Executable Statements .. 00150 * 00151 * Test the input parameters. 00152 * 00153 INFO = 0 00154 UPPER = LSAME( UPLO, 'U' ) 00155 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) 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( LDA.LT.MAX( 1, N ) ) THEN 00162 INFO = -5 00163 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00164 INFO = -7 00165 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00166 INFO = -10 00167 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00168 INFO = -12 00169 END IF 00170 IF( INFO.NE.0 ) THEN 00171 CALL XERBLA( 'ZHERFS', -INFO ) 00172 RETURN 00173 END IF 00174 * 00175 * Quick return if possible 00176 * 00177 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00178 DO 10 J = 1, NRHS 00179 FERR( J ) = ZERO 00180 BERR( J ) = ZERO 00181 10 CONTINUE 00182 RETURN 00183 END IF 00184 * 00185 * NZ = maximum number of nonzero elements in each row of A, plus 1 00186 * 00187 NZ = N + 1 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 140 J = 1, NRHS 00196 * 00197 COUNT = 1 00198 LSTRES = THREE 00199 20 CONTINUE 00200 * 00201 * Loop until stopping criterion is satisfied. 00202 * 00203 * Compute residual R = B - A * X 00204 * 00205 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 00206 CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 ) 00207 * 00208 * Compute componentwise relative backward error from formula 00209 * 00210 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00211 * 00212 * where abs(Z) is the componentwise absolute value of the matrix 00213 * or vector Z. If the i-th component of the denominator is less 00214 * than SAFE2, then SAFE1 is added to the i-th components of the 00215 * numerator and denominator before dividing. 00216 * 00217 DO 30 I = 1, N 00218 RWORK( I ) = CABS1( B( I, J ) ) 00219 30 CONTINUE 00220 * 00221 * Compute abs(A)*abs(X) + abs(B). 00222 * 00223 IF( UPPER ) THEN 00224 DO 50 K = 1, N 00225 S = ZERO 00226 XK = CABS1( X( K, J ) ) 00227 DO 40 I = 1, K - 1 00228 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 00229 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 00230 40 CONTINUE 00231 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S 00232 50 CONTINUE 00233 ELSE 00234 DO 70 K = 1, N 00235 S = ZERO 00236 XK = CABS1( X( K, J ) ) 00237 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK 00238 DO 60 I = K + 1, N 00239 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 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 ZHETRS( UPLO, 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(A))* 00278 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00279 * 00280 * where 00281 * norm(Z) is the magnitude of the largest component of Z 00282 * inv(A) is the inverse of 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(A)*abs(X)+abs(B)) 00289 * is incremented by SAFE1 if the i-th component of 00290 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00291 * 00292 * Use ZLACN2 to estimate the infinity-norm of the matrix 00293 * inv(A) * diag(W), 00294 * where W = abs(R) + NZ*EPS*( abs(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(A**H). 00312 * 00313 CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 00314 DO 110 I = 1, N 00315 WORK( I ) = RWORK( I )*WORK( I ) 00316 110 CONTINUE 00317 ELSE IF( KASE.EQ.2 ) THEN 00318 * 00319 * Multiply by inv(A)*diag(W). 00320 * 00321 DO 120 I = 1, N 00322 WORK( I ) = RWORK( I )*WORK( I ) 00323 120 CONTINUE 00324 CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 00325 END IF 00326 GO TO 100 00327 END IF 00328 * 00329 * Normalize error. 00330 * 00331 LSTRES = ZERO 00332 DO 130 I = 1, N 00333 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 00334 130 CONTINUE 00335 IF( LSTRES.NE.ZERO ) 00336 $ FERR( J ) = FERR( J ) / LSTRES 00337 * 00338 140 CONTINUE 00339 * 00340 RETURN 00341 * 00342 * End of ZHERFS 00343 * 00344 END
1.7.3 
