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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CTPT05 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 CTPT05( 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 * REAL BERR( * ), FERR( * ), RESLTS( * ) 00020 * COMPLEX AP( * ), B( LDB, * ), X( LDX, * ), 00021 * $ XACT( LDXACT, * ) 00022 * .. 00023 * 00024 * 00025 *> \par Purpose: 00026 * ============= 00027 *> 00028 *> \verbatim 00029 *> 00030 *> CTPT05 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 *> triangular 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 matrix A is upper or lower triangular. 00051 *> = 'U': Upper triangular 00052 *> = 'L': Lower triangular 00053 *> \endverbatim 00054 *> 00055 *> \param[in] TRANS 00056 *> \verbatim 00057 *> TRANS is CHARACTER*1 00058 *> Specifies the form of the system of equations. 00059 *> = 'N': A * X = B (No transpose) 00060 *> = 'T': A'* X = B (Transpose) 00061 *> = 'C': A'* X = B (Conjugate transpose = Transpose) 00062 *> \endverbatim 00063 *> 00064 *> \param[in] DIAG 00065 *> \verbatim 00066 *> DIAG is CHARACTER*1 00067 *> Specifies whether or not the matrix A is unit triangular. 00068 *> = 'N': Non-unit triangular 00069 *> = 'U': Unit triangular 00070 *> \endverbatim 00071 *> 00072 *> \param[in] N 00073 *> \verbatim 00074 *> N is INTEGER 00075 *> The number of rows of the matrices X, B, and XACT, and the 00076 *> order of the matrix A. N >= 0. 00077 *> \endverbatim 00078 *> 00079 *> \param[in] NRHS 00080 *> \verbatim 00081 *> NRHS is INTEGER 00082 *> The number of columns of the matrices X, B, and XACT. 00083 *> NRHS >= 0. 00084 *> \endverbatim 00085 *> 00086 *> \param[in] AP 00087 *> \verbatim 00088 *> AP is COMPLEX array, dimension (N*(N+1)/2) 00089 *> The upper or lower triangular matrix A, packed columnwise in 00090 *> a linear array. The j-th column of A is stored in the array 00091 *> AP as follows: 00092 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00093 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00094 *> If DIAG = 'U', the diagonal elements of A are not referenced 00095 *> and are assumed to be 1. 00096 *> \endverbatim 00097 *> 00098 *> \param[in] B 00099 *> \verbatim 00100 *> B is COMPLEX array, dimension (LDB,NRHS) 00101 *> The right hand side vectors for the system of linear 00102 *> equations. 00103 *> \endverbatim 00104 *> 00105 *> \param[in] LDB 00106 *> \verbatim 00107 *> LDB is INTEGER 00108 *> The leading dimension of the array B. LDB >= max(1,N). 00109 *> \endverbatim 00110 *> 00111 *> \param[in] X 00112 *> \verbatim 00113 *> X is COMPLEX array, dimension (LDX,NRHS) 00114 *> The computed solution vectors. Each vector is stored as a 00115 *> column of the matrix X. 00116 *> \endverbatim 00117 *> 00118 *> \param[in] LDX 00119 *> \verbatim 00120 *> LDX is INTEGER 00121 *> The leading dimension of the array X. LDX >= max(1,N). 00122 *> \endverbatim 00123 *> 00124 *> \param[in] XACT 00125 *> \verbatim 00126 *> XACT is COMPLEX array, dimension (LDX,NRHS) 00127 *> The exact solution vectors. Each vector is stored as a 00128 *> column of the matrix XACT. 00129 *> \endverbatim 00130 *> 00131 *> \param[in] LDXACT 00132 *> \verbatim 00133 *> LDXACT is INTEGER 00134 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00135 *> \endverbatim 00136 *> 00137 *> \param[in] FERR 00138 *> \verbatim 00139 *> FERR is REAL array, dimension (NRHS) 00140 *> The estimated forward error bounds for each solution vector 00141 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00142 *> of the largest entry in (X - XTRUE) divided by the magnitude 00143 *> of the largest entry in X. 00144 *> \endverbatim 00145 *> 00146 *> \param[in] BERR 00147 *> \verbatim 00148 *> BERR is REAL array, dimension (NRHS) 00149 *> The componentwise relative backward error of each solution 00150 *> vector (i.e., the smallest relative change in any entry of A 00151 *> or B that makes X an exact solution). 00152 *> \endverbatim 00153 *> 00154 *> \param[out] RESLTS 00155 *> \verbatim 00156 *> RESLTS is REAL array, dimension (2) 00157 *> The maximum over the NRHS solution vectors of the ratios: 00158 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00159 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00160 *> \endverbatim 00161 * 00162 * Authors: 00163 * ======== 00164 * 00165 *> \author Univ. of Tennessee 00166 *> \author Univ. of California Berkeley 00167 *> \author Univ. of Colorado Denver 00168 *> \author NAG Ltd. 00169 * 00170 *> \date November 2011 00171 * 00172 *> \ingroup complex_lin 00173 * 00174 * ===================================================================== 00175 SUBROUTINE CTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 00176 $ XACT, LDXACT, FERR, BERR, RESLTS ) 00177 * 00178 * -- LAPACK test routine (version 3.4.0) -- 00179 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00180 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00181 * November 2011 00182 * 00183 * .. Scalar Arguments .. 00184 CHARACTER DIAG, TRANS, UPLO 00185 INTEGER LDB, LDX, LDXACT, N, NRHS 00186 * .. 00187 * .. Array Arguments .. 00188 REAL BERR( * ), FERR( * ), RESLTS( * ) 00189 COMPLEX AP( * ), B( LDB, * ), X( LDX, * ), 00190 $ XACT( LDXACT, * ) 00191 * .. 00192 * 00193 * ===================================================================== 00194 * 00195 * .. Parameters .. 00196 REAL ZERO, ONE 00197 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00198 * .. 00199 * .. Local Scalars .. 00200 LOGICAL NOTRAN, UNIT, UPPER 00201 INTEGER I, IFU, IMAX, J, JC, K 00202 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00203 COMPLEX ZDUM 00204 * .. 00205 * .. External Functions .. 00206 LOGICAL LSAME 00207 INTEGER ICAMAX 00208 REAL SLAMCH 00209 EXTERNAL LSAME, ICAMAX, SLAMCH 00210 * .. 00211 * .. Intrinsic Functions .. 00212 INTRINSIC ABS, AIMAG, MAX, MIN, REAL 00213 * .. 00214 * .. Statement Functions .. 00215 REAL CABS1 00216 * .. 00217 * .. Statement Function definitions .. 00218 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) ) 00219 * .. 00220 * .. Executable Statements .. 00221 * 00222 * Quick exit if N = 0 or NRHS = 0. 00223 * 00224 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00225 RESLTS( 1 ) = ZERO 00226 RESLTS( 2 ) = ZERO 00227 RETURN 00228 END IF 00229 * 00230 EPS = SLAMCH( 'Epsilon' ) 00231 UNFL = SLAMCH( 'Safe minimum' ) 00232 OVFL = ONE / UNFL 00233 UPPER = LSAME( UPLO, 'U' ) 00234 NOTRAN = LSAME( TRANS, 'N' ) 00235 UNIT = LSAME( DIAG, 'U' ) 00236 * 00237 * Test 1: Compute the maximum of 00238 * norm(X - XACT) / ( norm(X) * FERR ) 00239 * over all the vectors X and XACT using the infinity-norm. 00240 * 00241 ERRBND = ZERO 00242 DO 30 J = 1, NRHS 00243 IMAX = ICAMAX( N, X( 1, J ), 1 ) 00244 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00245 DIFF = ZERO 00246 DO 10 I = 1, N 00247 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00248 10 CONTINUE 00249 * 00250 IF( XNORM.GT.ONE ) THEN 00251 GO TO 20 00252 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00253 GO TO 20 00254 ELSE 00255 ERRBND = ONE / EPS 00256 GO TO 30 00257 END IF 00258 * 00259 20 CONTINUE 00260 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00261 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00262 ELSE 00263 ERRBND = ONE / EPS 00264 END IF 00265 30 CONTINUE 00266 RESLTS( 1 ) = ERRBND 00267 * 00268 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00269 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00270 * 00271 IFU = 0 00272 IF( UNIT ) 00273 $ IFU = 1 00274 DO 90 K = 1, NRHS 00275 DO 80 I = 1, N 00276 TMP = CABS1( B( I, K ) ) 00277 IF( UPPER ) THEN 00278 JC = ( ( I-1 )*I ) / 2 00279 IF( .NOT.NOTRAN ) THEN 00280 DO 40 J = 1, I - IFU 00281 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) ) 00282 40 CONTINUE 00283 IF( UNIT ) 00284 $ TMP = TMP + CABS1( X( I, K ) ) 00285 ELSE 00286 JC = JC + I 00287 IF( UNIT ) THEN 00288 TMP = TMP + CABS1( X( I, K ) ) 00289 JC = JC + I 00290 END IF 00291 DO 50 J = I + IFU, N 00292 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) 00293 JC = JC + J 00294 50 CONTINUE 00295 END IF 00296 ELSE 00297 IF( NOTRAN ) THEN 00298 JC = I 00299 DO 60 J = 1, I - IFU 00300 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) 00301 JC = JC + N - J 00302 60 CONTINUE 00303 IF( UNIT ) 00304 $ TMP = TMP + CABS1( X( I, K ) ) 00305 ELSE 00306 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 00307 IF( UNIT ) 00308 $ TMP = TMP + CABS1( X( I, K ) ) 00309 DO 70 J = I + IFU, N 00310 TMP = TMP + CABS1( AP( JC+J-I ) )* 00311 $ CABS1( X( J, K ) ) 00312 70 CONTINUE 00313 END IF 00314 END IF 00315 IF( I.EQ.1 ) THEN 00316 AXBI = TMP 00317 ELSE 00318 AXBI = MIN( AXBI, TMP ) 00319 END IF 00320 80 CONTINUE 00321 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00322 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00323 IF( K.EQ.1 ) THEN 00324 RESLTS( 2 ) = TMP 00325 ELSE 00326 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00327 END IF 00328 90 CONTINUE 00329 * 00330 RETURN 00331 * 00332 * End of CTPT05 00333 * 00334 END