00001 SUBROUTINE DPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 00002 $ LDXACT, FERR, BERR, RESLTS ) 00003 * 00004 * -- LAPACK test routine (version 3.1) -- 00005 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 00014 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * DPOT05 tests the error bounds from iterative refinement for the 00021 * computed solution to a system of equations A*X = B, where A is a 00022 * symmetric n by n matrix. 00023 * 00024 * RESLTS(1) = test of the error bound 00025 * = norm(X - XACT) / ( norm(X) * FERR ) 00026 * 00027 * A large value is returned if this ratio is not less than one. 00028 * 00029 * RESLTS(2) = residual from the iterative refinement routine 00030 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where 00031 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00032 * 00033 * Arguments 00034 * ========= 00035 * 00036 * UPLO (input) CHARACTER*1 00037 * Specifies whether the upper or lower triangular part of the 00038 * symmetric matrix A is stored. 00039 * = 'U': Upper triangular 00040 * = 'L': Lower triangular 00041 * 00042 * N (input) INTEGER 00043 * The number of rows of the matrices X, B, and XACT, and the 00044 * order of the matrix A. N >= 0. 00045 * 00046 * NRHS (input) INTEGER 00047 * The number of columns of the matrices X, B, and XACT. 00048 * NRHS >= 0. 00049 * 00050 * A (input) DOUBLE PRECISION array, dimension (LDA,N) 00051 * The symmetric matrix A. If UPLO = 'U', the leading n by n 00052 * upper triangular part of A contains the upper triangular part 00053 * of the matrix A, and the strictly lower triangular part of A 00054 * is not referenced. If UPLO = 'L', the leading n by n lower 00055 * triangular part of A contains the lower triangular part of 00056 * the matrix A, and the strictly upper triangular part of A is 00057 * not referenced. 00058 * 00059 * LDA (input) INTEGER 00060 * The leading dimension of the array A. LDA >= max(1,N). 00061 * 00062 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00063 * The right hand side vectors for the system of linear 00064 * equations. 00065 * 00066 * LDB (input) INTEGER 00067 * The leading dimension of the array B. LDB >= max(1,N). 00068 * 00069 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00070 * The computed solution vectors. Each vector is stored as a 00071 * column of the matrix X. 00072 * 00073 * LDX (input) INTEGER 00074 * The leading dimension of the array X. LDX >= max(1,N). 00075 * 00076 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00077 * The exact solution vectors. Each vector is stored as a 00078 * column of the matrix XACT. 00079 * 00080 * LDXACT (input) INTEGER 00081 * The leading dimension of the array XACT. LDXACT >= max(1,N). 00082 * 00083 * FERR (input) DOUBLE PRECISION array, dimension (NRHS) 00084 * The estimated forward error bounds for each solution vector 00085 * X. If XTRUE is the true solution, FERR bounds the magnitude 00086 * of the largest entry in (X - XTRUE) divided by the magnitude 00087 * of the largest entry in X. 00088 * 00089 * BERR (input) DOUBLE PRECISION array, dimension (NRHS) 00090 * The componentwise relative backward error of each solution 00091 * vector (i.e., the smallest relative change in any entry of A 00092 * or B that makes X an exact solution). 00093 * 00094 * RESLTS (output) DOUBLE PRECISION array, dimension (2) 00095 * The maximum over the NRHS solution vectors of the ratios: 00096 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00097 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00098 * 00099 * ===================================================================== 00100 * 00101 * .. Parameters .. 00102 DOUBLE PRECISION ZERO, ONE 00103 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00104 * .. 00105 * .. Local Scalars .. 00106 LOGICAL UPPER 00107 INTEGER I, IMAX, J, K 00108 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00109 * .. 00110 * .. External Functions .. 00111 LOGICAL LSAME 00112 INTEGER IDAMAX 00113 DOUBLE PRECISION DLAMCH 00114 EXTERNAL LSAME, IDAMAX, DLAMCH 00115 * .. 00116 * .. Intrinsic Functions .. 00117 INTRINSIC ABS, MAX, MIN 00118 * .. 00119 * .. Executable Statements .. 00120 * 00121 * Quick exit if N = 0 or NRHS = 0. 00122 * 00123 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00124 RESLTS( 1 ) = ZERO 00125 RESLTS( 2 ) = ZERO 00126 RETURN 00127 END IF 00128 * 00129 EPS = DLAMCH( 'Epsilon' ) 00130 UNFL = DLAMCH( 'Safe minimum' ) 00131 OVFL = ONE / UNFL 00132 UPPER = LSAME( UPLO, 'U' ) 00133 * 00134 * Test 1: Compute the maximum of 00135 * norm(X - XACT) / ( norm(X) * FERR ) 00136 * over all the vectors X and XACT using the infinity-norm. 00137 * 00138 ERRBND = ZERO 00139 DO 30 J = 1, NRHS 00140 IMAX = IDAMAX( N, X( 1, J ), 1 ) 00141 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00142 DIFF = ZERO 00143 DO 10 I = 1, N 00144 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00145 10 CONTINUE 00146 * 00147 IF( XNORM.GT.ONE ) THEN 00148 GO TO 20 00149 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00150 GO TO 20 00151 ELSE 00152 ERRBND = ONE / EPS 00153 GO TO 30 00154 END IF 00155 * 00156 20 CONTINUE 00157 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00158 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00159 ELSE 00160 ERRBND = ONE / EPS 00161 END IF 00162 30 CONTINUE 00163 RESLTS( 1 ) = ERRBND 00164 * 00165 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00166 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00167 * 00168 DO 90 K = 1, NRHS 00169 DO 80 I = 1, N 00170 TMP = ABS( B( I, K ) ) 00171 IF( UPPER ) THEN 00172 DO 40 J = 1, I 00173 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00174 40 CONTINUE 00175 DO 50 J = I + 1, N 00176 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00177 50 CONTINUE 00178 ELSE 00179 DO 60 J = 1, I - 1 00180 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00181 60 CONTINUE 00182 DO 70 J = I, N 00183 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00184 70 CONTINUE 00185 END IF 00186 IF( I.EQ.1 ) THEN 00187 AXBI = TMP 00188 ELSE 00189 AXBI = MIN( AXBI, TMP ) 00190 END IF 00191 80 CONTINUE 00192 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00193 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00194 IF( K.EQ.1 ) THEN 00195 RESLTS( 2 ) = TMP 00196 ELSE 00197 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00198 END IF 00199 90 CONTINUE 00200 * 00201 RETURN 00202 * 00203 * End of DPOT05 00204 * 00205 END
1.7.2