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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b SPBT05 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 SPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, 00012 * XACT, LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER UPLO 00016 * INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 00017 * .. 00018 * .. Array Arguments .. 00019 * REAL AB( LDAB, * ), B( LDB, * ), BERR( * ), 00020 * $ FERR( * ), RESLTS( * ), X( LDX, * ), 00021 * $ XACT( LDXACT, * ) 00022 * .. 00023 * 00024 * 00025 *> \par Purpose: 00026 * ============= 00027 *> 00028 *> \verbatim 00029 *> 00030 *> SPBT05 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 *> symmetric band matrix. 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 / ( NZ*EPS + (*) ), where 00041 *> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00042 *> and NZ = max. number of nonzeros in any row of A, plus 1 00043 *> \endverbatim 00044 * 00045 * Arguments: 00046 * ========== 00047 * 00048 *> \param[in] UPLO 00049 *> \verbatim 00050 *> UPLO is CHARACTER*1 00051 *> Specifies whether the upper or lower triangular part of the 00052 *> symmetric matrix A is stored. 00053 *> = 'U': Upper triangular 00054 *> = 'L': Lower triangular 00055 *> \endverbatim 00056 *> 00057 *> \param[in] N 00058 *> \verbatim 00059 *> N is INTEGER 00060 *> The number of rows of the matrices X, B, and XACT, and the 00061 *> order of the matrix A. N >= 0. 00062 *> \endverbatim 00063 *> 00064 *> \param[in] KD 00065 *> \verbatim 00066 *> KD is INTEGER 00067 *> The number of super-diagonals of the matrix A if UPLO = 'U', 00068 *> or the number of sub-diagonals if UPLO = 'L'. KD >= 0. 00069 *> \endverbatim 00070 *> 00071 *> \param[in] NRHS 00072 *> \verbatim 00073 *> NRHS is INTEGER 00074 *> The number of columns of the matrices X, B, and XACT. 00075 *> NRHS >= 0. 00076 *> \endverbatim 00077 *> 00078 *> \param[in] AB 00079 *> \verbatim 00080 *> AB is REAL array, dimension (LDAB,N) 00081 *> The upper or lower triangle of the symmetric band matrix A, 00082 *> stored in the first KD+1 rows of the array. The j-th column 00083 *> of A is stored in the j-th column of the array AB as follows: 00084 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00085 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00086 *> \endverbatim 00087 *> 00088 *> \param[in] LDAB 00089 *> \verbatim 00090 *> LDAB is INTEGER 00091 *> The leading dimension of the array AB. LDAB >= KD+1. 00092 *> \endverbatim 00093 *> 00094 *> \param[in] B 00095 *> \verbatim 00096 *> B is REAL array, dimension (LDB,NRHS) 00097 *> The right hand side vectors for the system of linear 00098 *> equations. 00099 *> \endverbatim 00100 *> 00101 *> \param[in] LDB 00102 *> \verbatim 00103 *> LDB is INTEGER 00104 *> The leading dimension of the array B. LDB >= max(1,N). 00105 *> \endverbatim 00106 *> 00107 *> \param[in] X 00108 *> \verbatim 00109 *> X is REAL array, dimension (LDX,NRHS) 00110 *> The computed solution vectors. Each vector is stored as a 00111 *> column of the matrix X. 00112 *> \endverbatim 00113 *> 00114 *> \param[in] LDX 00115 *> \verbatim 00116 *> LDX is INTEGER 00117 *> The leading dimension of the array X. LDX >= max(1,N). 00118 *> \endverbatim 00119 *> 00120 *> \param[in] XACT 00121 *> \verbatim 00122 *> XACT is REAL array, dimension (LDX,NRHS) 00123 *> The exact solution vectors. Each vector is stored as a 00124 *> column of the matrix XACT. 00125 *> \endverbatim 00126 *> 00127 *> \param[in] LDXACT 00128 *> \verbatim 00129 *> LDXACT is INTEGER 00130 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00131 *> \endverbatim 00132 *> 00133 *> \param[in] FERR 00134 *> \verbatim 00135 *> FERR is REAL array, dimension (NRHS) 00136 *> The estimated forward error bounds for each solution vector 00137 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00138 *> of the largest entry in (X - XTRUE) divided by the magnitude 00139 *> of the largest entry in X. 00140 *> \endverbatim 00141 *> 00142 *> \param[in] BERR 00143 *> \verbatim 00144 *> BERR is REAL array, dimension (NRHS) 00145 *> The componentwise relative backward error of each solution 00146 *> vector (i.e., the smallest relative change in any entry of A 00147 *> or B that makes X an exact solution). 00148 *> \endverbatim 00149 *> 00150 *> \param[out] RESLTS 00151 *> \verbatim 00152 *> RESLTS is REAL array, dimension (2) 00153 *> The maximum over the NRHS solution vectors of the ratios: 00154 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00155 *> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00156 *> \endverbatim 00157 * 00158 * Authors: 00159 * ======== 00160 * 00161 *> \author Univ. of Tennessee 00162 *> \author Univ. of California Berkeley 00163 *> \author Univ. of Colorado Denver 00164 *> \author NAG Ltd. 00165 * 00166 *> \date November 2011 00167 * 00168 *> \ingroup single_lin 00169 * 00170 * ===================================================================== 00171 SUBROUTINE SPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, 00172 $ XACT, LDXACT, FERR, BERR, RESLTS ) 00173 * 00174 * -- LAPACK test routine (version 3.4.0) -- 00175 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00177 * November 2011 00178 * 00179 * .. Scalar Arguments .. 00180 CHARACTER UPLO 00181 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 00182 * .. 00183 * .. Array Arguments .. 00184 REAL AB( LDAB, * ), B( LDB, * ), BERR( * ), 00185 $ FERR( * ), RESLTS( * ), X( LDX, * ), 00186 $ XACT( LDXACT, * ) 00187 * .. 00188 * 00189 * ===================================================================== 00190 * 00191 * .. Parameters .. 00192 REAL ZERO, ONE 00193 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00194 * .. 00195 * .. Local Scalars .. 00196 LOGICAL UPPER 00197 INTEGER I, IMAX, J, K, NZ 00198 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00199 * .. 00200 * .. External Functions .. 00201 LOGICAL LSAME 00202 INTEGER ISAMAX 00203 REAL SLAMCH 00204 EXTERNAL LSAME, ISAMAX, SLAMCH 00205 * .. 00206 * .. Intrinsic Functions .. 00207 INTRINSIC ABS, MAX, MIN 00208 * .. 00209 * .. Executable Statements .. 00210 * 00211 * Quick exit if N = 0 or NRHS = 0. 00212 * 00213 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00214 RESLTS( 1 ) = ZERO 00215 RESLTS( 2 ) = ZERO 00216 RETURN 00217 END IF 00218 * 00219 EPS = SLAMCH( 'Epsilon' ) 00220 UNFL = SLAMCH( 'Safe minimum' ) 00221 OVFL = ONE / UNFL 00222 UPPER = LSAME( UPLO, 'U' ) 00223 NZ = 2*MAX( KD, N-1 ) + 1 00224 * 00225 * Test 1: Compute the maximum of 00226 * norm(X - XACT) / ( norm(X) * FERR ) 00227 * over all the vectors X and XACT using the infinity-norm. 00228 * 00229 ERRBND = ZERO 00230 DO 30 J = 1, NRHS 00231 IMAX = ISAMAX( N, X( 1, J ), 1 ) 00232 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00233 DIFF = ZERO 00234 DO 10 I = 1, N 00235 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00236 10 CONTINUE 00237 * 00238 IF( XNORM.GT.ONE ) THEN 00239 GO TO 20 00240 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00241 GO TO 20 00242 ELSE 00243 ERRBND = ONE / EPS 00244 GO TO 30 00245 END IF 00246 * 00247 20 CONTINUE 00248 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00249 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00250 ELSE 00251 ERRBND = ONE / EPS 00252 END IF 00253 30 CONTINUE 00254 RESLTS( 1 ) = ERRBND 00255 * 00256 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00257 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00258 * 00259 DO 90 K = 1, NRHS 00260 DO 80 I = 1, N 00261 TMP = ABS( B( I, K ) ) 00262 IF( UPPER ) THEN 00263 DO 40 J = MAX( I-KD, 1 ), I 00264 TMP = TMP + ABS( AB( KD+1-I+J, I ) )*ABS( X( J, K ) ) 00265 40 CONTINUE 00266 DO 50 J = I + 1, MIN( I+KD, N ) 00267 TMP = TMP + ABS( AB( KD+1+I-J, J ) )*ABS( X( J, K ) ) 00268 50 CONTINUE 00269 ELSE 00270 DO 60 J = MAX( I-KD, 1 ), I - 1 00271 TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) ) 00272 60 CONTINUE 00273 DO 70 J = I, MIN( I+KD, N ) 00274 TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) ) 00275 70 CONTINUE 00276 END IF 00277 IF( I.EQ.1 ) THEN 00278 AXBI = TMP 00279 ELSE 00280 AXBI = MIN( AXBI, TMP ) 00281 END IF 00282 80 CONTINUE 00283 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*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 SPBT05 00294 * 00295 END