![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief \b ZPPT05 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 * Definition: 00009 * =========== 00010 * 00011 * SUBROUTINE ZPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT, 00012 * LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER UPLO 00016 * INTEGER LDB, LDX, LDXACT, N, NRHS 00017 * .. 00018 * .. Array Arguments .. 00019 * DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) 00020 * COMPLEX*16 AP( * ), B( LDB, * ), X( LDX, * ), 00021 * $ XACT( LDXACT, * ) 00022 * .. 00023 * 00024 * 00025 *> \par Purpose: 00026 * ============= 00027 *> 00028 *> \verbatim 00029 *> 00030 *> ZPPT05 tests the error bounds from iterative refinement for the 00031 *> computed solution to a system of equations A*X = B, where A is a 00032 *> Hermitian matrix in packed storage format. 00033 *> 00034 *> RESLTS(1) = test of the error bound 00035 *> = norm(X - XACT) / ( norm(X) * FERR ) 00036 *> 00037 *> A large value is returned if this ratio is not less than one. 00038 *> 00039 *> RESLTS(2) = residual from the iterative refinement routine 00040 *> = the maximum of BERR / ( (n+1)*EPS + (*) ), where 00041 *> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00042 *> \endverbatim 00043 * 00044 * Arguments: 00045 * ========== 00046 * 00047 *> \param[in] UPLO 00048 *> \verbatim 00049 *> UPLO is CHARACTER*1 00050 *> Specifies whether the upper or lower triangular part of the 00051 *> Hermitian matrix A is stored. 00052 *> = 'U': Upper triangular 00053 *> = 'L': Lower triangular 00054 *> \endverbatim 00055 *> 00056 *> \param[in] N 00057 *> \verbatim 00058 *> N is INTEGER 00059 *> The number of rows of the matrices X, B, and XACT, and the 00060 *> order of the matrix A. N >= 0. 00061 *> \endverbatim 00062 *> 00063 *> \param[in] NRHS 00064 *> \verbatim 00065 *> NRHS is INTEGER 00066 *> The number of columns of the matrices X, B, and XACT. 00067 *> NRHS >= 0. 00068 *> \endverbatim 00069 *> 00070 *> \param[in] AP 00071 *> \verbatim 00072 *> AP is COMPLEX*16 array, dimension (N*(N+1)/2) 00073 *> The upper or lower triangle of the Hermitian matrix A, packed 00074 *> columnwise in a linear array. The j-th column of A is stored 00075 *> in the array AP as follows: 00076 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00077 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] B 00081 *> \verbatim 00082 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00083 *> The right hand side vectors for the system of linear 00084 *> equations. 00085 *> \endverbatim 00086 *> 00087 *> \param[in] LDB 00088 *> \verbatim 00089 *> LDB is INTEGER 00090 *> The leading dimension of the array B. LDB >= max(1,N). 00091 *> \endverbatim 00092 *> 00093 *> \param[in] X 00094 *> \verbatim 00095 *> X is COMPLEX*16 array, dimension (LDX,NRHS) 00096 *> The computed solution vectors. Each vector is stored as a 00097 *> column of the matrix X. 00098 *> \endverbatim 00099 *> 00100 *> \param[in] LDX 00101 *> \verbatim 00102 *> LDX is INTEGER 00103 *> The leading dimension of the array X. LDX >= max(1,N). 00104 *> \endverbatim 00105 *> 00106 *> \param[in] XACT 00107 *> \verbatim 00108 *> XACT is COMPLEX*16 array, dimension (LDX,NRHS) 00109 *> The exact solution vectors. Each vector is stored as a 00110 *> column of the matrix XACT. 00111 *> \endverbatim 00112 *> 00113 *> \param[in] LDXACT 00114 *> \verbatim 00115 *> LDXACT is INTEGER 00116 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00117 *> \endverbatim 00118 *> 00119 *> \param[in] FERR 00120 *> \verbatim 00121 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00122 *> The estimated forward error bounds for each solution vector 00123 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00124 *> of the largest entry in (X - XTRUE) divided by the magnitude 00125 *> of the largest entry in X. 00126 *> \endverbatim 00127 *> 00128 *> \param[in] BERR 00129 *> \verbatim 00130 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00131 *> The componentwise relative backward error of each solution 00132 *> vector (i.e., the smallest relative change in any entry of A 00133 *> or B that makes X an exact solution). 00134 *> \endverbatim 00135 *> 00136 *> \param[out] RESLTS 00137 *> \verbatim 00138 *> RESLTS is DOUBLE PRECISION array, dimension (2) 00139 *> The maximum over the NRHS solution vectors of the ratios: 00140 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00141 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00142 *> \endverbatim 00143 * 00144 * Authors: 00145 * ======== 00146 * 00147 *> \author Univ. of Tennessee 00148 *> \author Univ. of California Berkeley 00149 *> \author Univ. of Colorado Denver 00150 *> \author NAG Ltd. 00151 * 00152 *> \date November 2011 00153 * 00154 *> \ingroup complex16_lin 00155 * 00156 * ===================================================================== 00157 SUBROUTINE ZPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT, 00158 $ LDXACT, FERR, BERR, RESLTS ) 00159 * 00160 * -- LAPACK test routine (version 3.4.0) -- 00161 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00162 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00163 * November 2011 00164 * 00165 * .. Scalar Arguments .. 00166 CHARACTER UPLO 00167 INTEGER LDB, LDX, LDXACT, N, NRHS 00168 * .. 00169 * .. Array Arguments .. 00170 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) 00171 COMPLEX*16 AP( * ), B( LDB, * ), X( LDX, * ), 00172 $ XACT( LDXACT, * ) 00173 * .. 00174 * 00175 * ===================================================================== 00176 * 00177 * .. Parameters .. 00178 DOUBLE PRECISION ZERO, ONE 00179 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00180 * .. 00181 * .. Local Scalars .. 00182 LOGICAL UPPER 00183 INTEGER I, IMAX, J, JC, K 00184 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00185 COMPLEX*16 ZDUM 00186 * .. 00187 * .. External Functions .. 00188 LOGICAL LSAME 00189 INTEGER IZAMAX 00190 DOUBLE PRECISION DLAMCH 00191 EXTERNAL LSAME, IZAMAX, DLAMCH 00192 * .. 00193 * .. Intrinsic Functions .. 00194 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 00195 * .. 00196 * .. Statement Functions .. 00197 DOUBLE PRECISION CABS1 00198 * .. 00199 * .. Statement Function definitions .. 00200 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00201 * .. 00202 * .. Executable Statements .. 00203 * 00204 * Quick exit if N = 0 or NRHS = 0. 00205 * 00206 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00207 RESLTS( 1 ) = ZERO 00208 RESLTS( 2 ) = ZERO 00209 RETURN 00210 END IF 00211 * 00212 EPS = DLAMCH( 'Epsilon' ) 00213 UNFL = DLAMCH( 'Safe minimum' ) 00214 OVFL = ONE / UNFL 00215 UPPER = LSAME( UPLO, 'U' ) 00216 * 00217 * Test 1: Compute the maximum of 00218 * norm(X - XACT) / ( norm(X) * FERR ) 00219 * over all the vectors X and XACT using the infinity-norm. 00220 * 00221 ERRBND = ZERO 00222 DO 30 J = 1, NRHS 00223 IMAX = IZAMAX( N, X( 1, J ), 1 ) 00224 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00225 DIFF = ZERO 00226 DO 10 I = 1, N 00227 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00228 10 CONTINUE 00229 * 00230 IF( XNORM.GT.ONE ) THEN 00231 GO TO 20 00232 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00233 GO TO 20 00234 ELSE 00235 ERRBND = ONE / EPS 00236 GO TO 30 00237 END IF 00238 * 00239 20 CONTINUE 00240 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00241 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00242 ELSE 00243 ERRBND = ONE / EPS 00244 END IF 00245 30 CONTINUE 00246 RESLTS( 1 ) = ERRBND 00247 * 00248 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00249 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00250 * 00251 DO 90 K = 1, NRHS 00252 DO 80 I = 1, N 00253 TMP = CABS1( B( I, K ) ) 00254 IF( UPPER ) THEN 00255 JC = ( ( I-1 )*I ) / 2 00256 DO 40 J = 1, I - 1 00257 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) ) 00258 40 CONTINUE 00259 TMP = TMP + ABS( DBLE( AP( JC+I ) ) )*CABS1( X( I, K ) ) 00260 JC = JC + I + I 00261 DO 50 J = I + 1, N 00262 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) 00263 JC = JC + J 00264 50 CONTINUE 00265 ELSE 00266 JC = I 00267 DO 60 J = 1, I - 1 00268 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) 00269 JC = JC + N - J 00270 60 CONTINUE 00271 TMP = TMP + ABS( DBLE( AP( JC ) ) )*CABS1( X( I, K ) ) 00272 DO 70 J = I + 1, N 00273 TMP = TMP + CABS1( AP( JC+J-I ) )*CABS1( X( J, K ) ) 00274 70 CONTINUE 00275 END IF 00276 IF( I.EQ.1 ) THEN 00277 AXBI = TMP 00278 ELSE 00279 AXBI = MIN( AXBI, TMP ) 00280 END IF 00281 80 CONTINUE 00282 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00283 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00284 IF( K.EQ.1 ) THEN 00285 RESLTS( 2 ) = TMP 00286 ELSE 00287 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00288 END IF 00289 90 CONTINUE 00290 * 00291 RETURN 00292 * 00293 * End of ZPPT05 00294 * 00295 END