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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DTPT05 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 DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 00012 * XACT, LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER DIAG, TRANS, UPLO 00016 * INTEGER LDB, LDX, LDXACT, N, NRHS 00017 * .. 00018 * .. Array Arguments .. 00019 * DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00020 * $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00021 * .. 00022 * 00023 * 00024 *> \par Purpose: 00025 * ============= 00026 *> 00027 *> \verbatim 00028 *> 00029 *> DTPT05 tests the error bounds from iterative refinement for the 00030 *> computed solution to a system of equations A*X = B, where A is a 00031 *> triangular matrix in packed storage format. 00032 *> 00033 *> RESLTS(1) = test of the error bound 00034 *> = norm(X - XACT) / ( norm(X) * FERR ) 00035 *> 00036 *> A large value is returned if this ratio is not less than one. 00037 *> 00038 *> RESLTS(2) = residual from the iterative refinement routine 00039 *> = the maximum of BERR / ( (n+1)*EPS + (*) ), where 00040 *> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00041 *> \endverbatim 00042 * 00043 * Arguments: 00044 * ========== 00045 * 00046 *> \param[in] UPLO 00047 *> \verbatim 00048 *> UPLO is CHARACTER*1 00049 *> Specifies whether the matrix A is upper or lower triangular. 00050 *> = 'U': Upper triangular 00051 *> = 'L': Lower triangular 00052 *> \endverbatim 00053 *> 00054 *> \param[in] TRANS 00055 *> \verbatim 00056 *> TRANS is CHARACTER*1 00057 *> Specifies the form of the system of equations. 00058 *> = 'N': A * X = B (No transpose) 00059 *> = 'T': A'* X = B (Transpose) 00060 *> = 'C': A'* X = B (Conjugate transpose = Transpose) 00061 *> \endverbatim 00062 *> 00063 *> \param[in] DIAG 00064 *> \verbatim 00065 *> DIAG is CHARACTER*1 00066 *> Specifies whether or not the matrix A is unit triangular. 00067 *> = 'N': Non-unit triangular 00068 *> = 'U': Unit triangular 00069 *> \endverbatim 00070 *> 00071 *> \param[in] N 00072 *> \verbatim 00073 *> N is INTEGER 00074 *> The number of rows of the matrices X, B, and XACT, and the 00075 *> order of the matrix A. N >= 0. 00076 *> \endverbatim 00077 *> 00078 *> \param[in] NRHS 00079 *> \verbatim 00080 *> NRHS is INTEGER 00081 *> The number of columns of the matrices X, B, and XACT. 00082 *> NRHS >= 0. 00083 *> \endverbatim 00084 *> 00085 *> \param[in] AP 00086 *> \verbatim 00087 *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 00088 *> The upper or lower triangular matrix A, packed columnwise in 00089 *> a linear array. The j-th column of A is stored in the array 00090 *> AP as follows: 00091 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00092 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00093 *> If DIAG = 'U', the diagonal elements of A are not referenced 00094 *> and are assumed to be 1. 00095 *> \endverbatim 00096 *> 00097 *> \param[in] B 00098 *> \verbatim 00099 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00100 *> The right hand side vectors for the system of linear 00101 *> equations. 00102 *> \endverbatim 00103 *> 00104 *> \param[in] LDB 00105 *> \verbatim 00106 *> LDB is INTEGER 00107 *> The leading dimension of the array B. LDB >= max(1,N). 00108 *> \endverbatim 00109 *> 00110 *> \param[in] X 00111 *> \verbatim 00112 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00113 *> The computed solution vectors. Each vector is stored as a 00114 *> column of the matrix X. 00115 *> \endverbatim 00116 *> 00117 *> \param[in] LDX 00118 *> \verbatim 00119 *> LDX is INTEGER 00120 *> The leading dimension of the array X. LDX >= max(1,N). 00121 *> \endverbatim 00122 *> 00123 *> \param[in] XACT 00124 *> \verbatim 00125 *> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 00126 *> The exact solution vectors. Each vector is stored as a 00127 *> column of the matrix XACT. 00128 *> \endverbatim 00129 *> 00130 *> \param[in] LDXACT 00131 *> \verbatim 00132 *> LDXACT is INTEGER 00133 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00134 *> \endverbatim 00135 *> 00136 *> \param[in] FERR 00137 *> \verbatim 00138 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00139 *> The estimated forward error bounds for each solution vector 00140 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00141 *> of the largest entry in (X - XTRUE) divided by the magnitude 00142 *> of the largest entry in X. 00143 *> \endverbatim 00144 *> 00145 *> \param[in] BERR 00146 *> \verbatim 00147 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00148 *> The componentwise relative backward error of each solution 00149 *> vector (i.e., the smallest relative change in any entry of A 00150 *> or B that makes X an exact solution). 00151 *> \endverbatim 00152 *> 00153 *> \param[out] RESLTS 00154 *> \verbatim 00155 *> RESLTS is DOUBLE PRECISION array, dimension (2) 00156 *> The maximum over the NRHS solution vectors of the ratios: 00157 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00158 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00159 *> \endverbatim 00160 * 00161 * Authors: 00162 * ======== 00163 * 00164 *> \author Univ. of Tennessee 00165 *> \author Univ. of California Berkeley 00166 *> \author Univ. of Colorado Denver 00167 *> \author NAG Ltd. 00168 * 00169 *> \date November 2011 00170 * 00171 *> \ingroup double_lin 00172 * 00173 * ===================================================================== 00174 SUBROUTINE DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 00175 $ XACT, LDXACT, FERR, BERR, RESLTS ) 00176 * 00177 * -- LAPACK test routine (version 3.4.0) -- 00178 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00179 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00180 * November 2011 00181 * 00182 * .. Scalar Arguments .. 00183 CHARACTER DIAG, TRANS, UPLO 00184 INTEGER LDB, LDX, LDXACT, N, NRHS 00185 * .. 00186 * .. Array Arguments .. 00187 DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00188 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00189 * .. 00190 * 00191 * ===================================================================== 00192 * 00193 * .. Parameters .. 00194 DOUBLE PRECISION ZERO, ONE 00195 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00196 * .. 00197 * .. Local Scalars .. 00198 LOGICAL NOTRAN, UNIT, UPPER 00199 INTEGER I, IFU, IMAX, J, JC, K 00200 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00201 * .. 00202 * .. External Functions .. 00203 LOGICAL LSAME 00204 INTEGER IDAMAX 00205 DOUBLE PRECISION DLAMCH 00206 EXTERNAL LSAME, IDAMAX, DLAMCH 00207 * .. 00208 * .. Intrinsic Functions .. 00209 INTRINSIC ABS, MAX, MIN 00210 * .. 00211 * .. Executable Statements .. 00212 * 00213 * Quick exit if N = 0 or NRHS = 0. 00214 * 00215 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00216 RESLTS( 1 ) = ZERO 00217 RESLTS( 2 ) = ZERO 00218 RETURN 00219 END IF 00220 * 00221 EPS = DLAMCH( 'Epsilon' ) 00222 UNFL = DLAMCH( 'Safe minimum' ) 00223 OVFL = ONE / UNFL 00224 UPPER = LSAME( UPLO, 'U' ) 00225 NOTRAN = LSAME( TRANS, 'N' ) 00226 UNIT = LSAME( DIAG, 'U' ) 00227 * 00228 * Test 1: Compute the maximum of 00229 * norm(X - XACT) / ( norm(X) * FERR ) 00230 * over all the vectors X and XACT using the infinity-norm. 00231 * 00232 ERRBND = ZERO 00233 DO 30 J = 1, NRHS 00234 IMAX = IDAMAX( N, X( 1, J ), 1 ) 00235 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00236 DIFF = ZERO 00237 DO 10 I = 1, N 00238 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00239 10 CONTINUE 00240 * 00241 IF( XNORM.GT.ONE ) THEN 00242 GO TO 20 00243 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00244 GO TO 20 00245 ELSE 00246 ERRBND = ONE / EPS 00247 GO TO 30 00248 END IF 00249 * 00250 20 CONTINUE 00251 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00252 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00253 ELSE 00254 ERRBND = ONE / EPS 00255 END IF 00256 30 CONTINUE 00257 RESLTS( 1 ) = ERRBND 00258 * 00259 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00260 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00261 * 00262 IFU = 0 00263 IF( UNIT ) 00264 $ IFU = 1 00265 DO 90 K = 1, NRHS 00266 DO 80 I = 1, N 00267 TMP = ABS( B( I, K ) ) 00268 IF( UPPER ) THEN 00269 JC = ( ( I-1 )*I ) / 2 00270 IF( .NOT.NOTRAN ) THEN 00271 DO 40 J = 1, I - IFU 00272 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) ) 00273 40 CONTINUE 00274 IF( UNIT ) 00275 $ TMP = TMP + ABS( X( I, K ) ) 00276 ELSE 00277 JC = JC + I 00278 IF( UNIT ) THEN 00279 TMP = TMP + ABS( X( I, K ) ) 00280 JC = JC + I 00281 END IF 00282 DO 50 J = I + IFU, N 00283 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 00284 JC = JC + J 00285 50 CONTINUE 00286 END IF 00287 ELSE 00288 IF( NOTRAN ) THEN 00289 JC = I 00290 DO 60 J = 1, I - IFU 00291 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 00292 JC = JC + N - J 00293 60 CONTINUE 00294 IF( UNIT ) 00295 $ TMP = TMP + ABS( X( I, K ) ) 00296 ELSE 00297 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 00298 IF( UNIT ) 00299 $ TMP = TMP + ABS( X( I, K ) ) 00300 DO 70 J = I + IFU, N 00301 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) ) 00302 70 CONTINUE 00303 END IF 00304 END IF 00305 IF( I.EQ.1 ) THEN 00306 AXBI = TMP 00307 ELSE 00308 AXBI = MIN( AXBI, TMP ) 00309 END IF 00310 80 CONTINUE 00311 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00312 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00313 IF( K.EQ.1 ) THEN 00314 RESLTS( 2 ) = TMP 00315 ELSE 00316 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00317 END IF 00318 90 CONTINUE 00319 * 00320 RETURN 00321 * 00322 * End of DTPT05 00323 * 00324 END