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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b SPTT05 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 SPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT, 00012 * FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * INTEGER LDB, LDX, LDXACT, N, NRHS 00016 * .. 00017 * .. Array Arguments .. 00018 * REAL B( LDB, * ), BERR( * ), D( * ), E( * ), 00019 * $ FERR( * ), RESLTS( * ), X( LDX, * ), 00020 * $ XACT( LDXACT, * ) 00021 * .. 00022 * 00023 * 00024 *> \par Purpose: 00025 * ============= 00026 *> 00027 *> \verbatim 00028 *> 00029 *> SPTT05 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 *> symmetric tridiagonal matrix of order n. 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 / ( NZ*EPS + (*) ), where 00040 *> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00041 *> and NZ = max. number of nonzeros in any row of A, plus 1 00042 *> \endverbatim 00043 * 00044 * Arguments: 00045 * ========== 00046 * 00047 *> \param[in] N 00048 *> \verbatim 00049 *> N is INTEGER 00050 *> The number of rows of the matrices X, B, and XACT, and the 00051 *> order of the matrix A. N >= 0. 00052 *> \endverbatim 00053 *> 00054 *> \param[in] NRHS 00055 *> \verbatim 00056 *> NRHS is INTEGER 00057 *> The number of columns of the matrices X, B, and XACT. 00058 *> NRHS >= 0. 00059 *> \endverbatim 00060 *> 00061 *> \param[in] D 00062 *> \verbatim 00063 *> D is REAL array, dimension (N) 00064 *> The n diagonal elements of the tridiagonal matrix A. 00065 *> \endverbatim 00066 *> 00067 *> \param[in] E 00068 *> \verbatim 00069 *> E is REAL array, dimension (N-1) 00070 *> The (n-1) subdiagonal elements of the tridiagonal matrix A. 00071 *> \endverbatim 00072 *> 00073 *> \param[in] B 00074 *> \verbatim 00075 *> B is REAL array, dimension (LDB,NRHS) 00076 *> The right hand side vectors for the system of linear 00077 *> equations. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] LDB 00081 *> \verbatim 00082 *> LDB is INTEGER 00083 *> The leading dimension of the array B. LDB >= max(1,N). 00084 *> \endverbatim 00085 *> 00086 *> \param[in] X 00087 *> \verbatim 00088 *> X is REAL array, dimension (LDX,NRHS) 00089 *> The computed solution vectors. Each vector is stored as a 00090 *> column of the matrix X. 00091 *> \endverbatim 00092 *> 00093 *> \param[in] LDX 00094 *> \verbatim 00095 *> LDX is INTEGER 00096 *> The leading dimension of the array X. LDX >= max(1,N). 00097 *> \endverbatim 00098 *> 00099 *> \param[in] XACT 00100 *> \verbatim 00101 *> XACT is REAL array, dimension (LDX,NRHS) 00102 *> The exact solution vectors. Each vector is stored as a 00103 *> column of the matrix XACT. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDXACT 00107 *> \verbatim 00108 *> LDXACT is INTEGER 00109 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00110 *> \endverbatim 00111 *> 00112 *> \param[in] FERR 00113 *> \verbatim 00114 *> FERR is REAL array, dimension (NRHS) 00115 *> The estimated forward error bounds for each solution vector 00116 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00117 *> of the largest entry in (X - XTRUE) divided by the magnitude 00118 *> of the largest entry in X. 00119 *> \endverbatim 00120 *> 00121 *> \param[in] BERR 00122 *> \verbatim 00123 *> BERR is REAL array, dimension (NRHS) 00124 *> The componentwise relative backward error of each solution 00125 *> vector (i.e., the smallest relative change in any entry of A 00126 *> or B that makes X an exact solution). 00127 *> \endverbatim 00128 *> 00129 *> \param[out] RESLTS 00130 *> \verbatim 00131 *> RESLTS is REAL array, dimension (2) 00132 *> The maximum over the NRHS solution vectors of the ratios: 00133 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00134 *> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00135 *> \endverbatim 00136 * 00137 * Authors: 00138 * ======== 00139 * 00140 *> \author Univ. of Tennessee 00141 *> \author Univ. of California Berkeley 00142 *> \author Univ. of Colorado Denver 00143 *> \author NAG Ltd. 00144 * 00145 *> \date November 2011 00146 * 00147 *> \ingroup single_lin 00148 * 00149 * ===================================================================== 00150 SUBROUTINE SPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT, 00151 $ FERR, BERR, RESLTS ) 00152 * 00153 * -- LAPACK test routine (version 3.4.0) -- 00154 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00155 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00156 * November 2011 00157 * 00158 * .. Scalar Arguments .. 00159 INTEGER LDB, LDX, LDXACT, N, NRHS 00160 * .. 00161 * .. Array Arguments .. 00162 REAL B( LDB, * ), BERR( * ), D( * ), E( * ), 00163 $ FERR( * ), RESLTS( * ), X( LDX, * ), 00164 $ XACT( LDXACT, * ) 00165 * .. 00166 * 00167 * ===================================================================== 00168 * 00169 * .. Parameters .. 00170 REAL ZERO, ONE 00171 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00172 * .. 00173 * .. Local Scalars .. 00174 INTEGER I, IMAX, J, K, NZ 00175 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00176 * .. 00177 * .. External Functions .. 00178 INTEGER ISAMAX 00179 REAL SLAMCH 00180 EXTERNAL ISAMAX, SLAMCH 00181 * .. 00182 * .. Intrinsic Functions .. 00183 INTRINSIC ABS, MAX, MIN 00184 * .. 00185 * .. Executable Statements .. 00186 * 00187 * Quick exit if N = 0 or NRHS = 0. 00188 * 00189 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00190 RESLTS( 1 ) = ZERO 00191 RESLTS( 2 ) = ZERO 00192 RETURN 00193 END IF 00194 * 00195 EPS = SLAMCH( 'Epsilon' ) 00196 UNFL = SLAMCH( 'Safe minimum' ) 00197 OVFL = ONE / UNFL 00198 NZ = 4 00199 * 00200 * Test 1: Compute the maximum of 00201 * norm(X - XACT) / ( norm(X) * FERR ) 00202 * over all the vectors X and XACT using the infinity-norm. 00203 * 00204 ERRBND = ZERO 00205 DO 30 J = 1, NRHS 00206 IMAX = ISAMAX( N, X( 1, J ), 1 ) 00207 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00208 DIFF = ZERO 00209 DO 10 I = 1, N 00210 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00211 10 CONTINUE 00212 * 00213 IF( XNORM.GT.ONE ) THEN 00214 GO TO 20 00215 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00216 GO TO 20 00217 ELSE 00218 ERRBND = ONE / EPS 00219 GO TO 30 00220 END IF 00221 * 00222 20 CONTINUE 00223 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00224 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00225 ELSE 00226 ERRBND = ONE / EPS 00227 END IF 00228 30 CONTINUE 00229 RESLTS( 1 ) = ERRBND 00230 * 00231 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00232 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00233 * 00234 DO 50 K = 1, NRHS 00235 IF( N.EQ.1 ) THEN 00236 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) 00237 ELSE 00238 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) + 00239 $ ABS( E( 1 )*X( 2, K ) ) 00240 DO 40 I = 2, N - 1 00241 TMP = ABS( B( I, K ) ) + ABS( E( I-1 )*X( I-1, K ) ) + 00242 $ ABS( D( I )*X( I, K ) ) + ABS( E( I )*X( I+1, K ) ) 00243 AXBI = MIN( AXBI, TMP ) 00244 40 CONTINUE 00245 TMP = ABS( B( N, K ) ) + ABS( E( N-1 )*X( N-1, K ) ) + 00246 $ ABS( D( N )*X( N, K ) ) 00247 AXBI = MIN( AXBI, TMP ) 00248 END IF 00249 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 00250 IF( K.EQ.1 ) THEN 00251 RESLTS( 2 ) = TMP 00252 ELSE 00253 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00254 END IF 00255 50 CONTINUE 00256 * 00257 RETURN 00258 * 00259 * End of SPTT05 00260 * 00261 END