00001 SUBROUTINE DTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, 00002 $ LDX, XACT, 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 DIAG, TRANS, 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 * DTRT05 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 * triangular 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 matrix A is upper or lower triangular. 00038 * = 'U': Upper triangular 00039 * = 'L': Lower triangular 00040 * 00041 * TRANS (input) CHARACTER*1 00042 * Specifies the form of the system of equations. 00043 * = 'N': A * X = B (No transpose) 00044 * = 'T': A'* X = B (Transpose) 00045 * = 'C': A'* X = B (Conjugate transpose = Transpose) 00046 * 00047 * DIAG (input) CHARACTER*1 00048 * Specifies whether or not the matrix A is unit triangular. 00049 * = 'N': Non-unit triangular 00050 * = 'U': Unit triangular 00051 * 00052 * N (input) INTEGER 00053 * The number of rows of the matrices X, B, and XACT, and the 00054 * order of the matrix A. N >= 0. 00055 * 00056 * NRHS (input) INTEGER 00057 * The number of columns of the matrices X, B, and XACT. 00058 * NRHS >= 0. 00059 * 00060 * A (input) DOUBLE PRECISION array, dimension (LDA,N) 00061 * The triangular matrix A. If UPLO = 'U', the leading n by n 00062 * upper triangular part of the array A contains the upper 00063 * triangular matrix, and the strictly lower triangular part of 00064 * A is not referenced. If UPLO = 'L', the leading n by n lower 00065 * triangular part of the array A contains the lower triangular 00066 * matrix, and the strictly upper triangular part of A is not 00067 * referenced. If DIAG = 'U', the diagonal elements of A are 00068 * also not referenced and are assumed to be 1. 00069 * 00070 * LDA (input) INTEGER 00071 * The leading dimension of the array A. LDA >= max(1,N). 00072 * 00073 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00074 * The right hand side vectors for the system of linear 00075 * equations. 00076 * 00077 * LDB (input) INTEGER 00078 * The leading dimension of the array B. LDB >= max(1,N). 00079 * 00080 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00081 * The computed solution vectors. Each vector is stored as a 00082 * column of the matrix X. 00083 * 00084 * LDX (input) INTEGER 00085 * The leading dimension of the array X. LDX >= max(1,N). 00086 * 00087 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00088 * The exact solution vectors. Each vector is stored as a 00089 * column of the matrix XACT. 00090 * 00091 * LDXACT (input) INTEGER 00092 * The leading dimension of the array XACT. LDXACT >= max(1,N). 00093 * 00094 * FERR (input) DOUBLE PRECISION array, dimension (NRHS) 00095 * The estimated forward error bounds for each solution vector 00096 * X. If XTRUE is the true solution, FERR bounds the magnitude 00097 * of the largest entry in (X - XTRUE) divided by the magnitude 00098 * of the largest entry in X. 00099 * 00100 * BERR (input) DOUBLE PRECISION array, dimension (NRHS) 00101 * The componentwise relative backward error of each solution 00102 * vector (i.e., the smallest relative change in any entry of A 00103 * or B that makes X an exact solution). 00104 * 00105 * RESLTS (output) DOUBLE PRECISION array, dimension (2) 00106 * The maximum over the NRHS solution vectors of the ratios: 00107 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00108 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00109 * 00110 * ===================================================================== 00111 * 00112 * .. Parameters .. 00113 DOUBLE PRECISION ZERO, ONE 00114 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00115 * .. 00116 * .. Local Scalars .. 00117 LOGICAL NOTRAN, UNIT, UPPER 00118 INTEGER I, IFU, IMAX, J, K 00119 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00120 * .. 00121 * .. External Functions .. 00122 LOGICAL LSAME 00123 INTEGER IDAMAX 00124 DOUBLE PRECISION DLAMCH 00125 EXTERNAL LSAME, IDAMAX, DLAMCH 00126 * .. 00127 * .. Intrinsic Functions .. 00128 INTRINSIC ABS, MAX, MIN 00129 * .. 00130 * .. Executable Statements .. 00131 * 00132 * Quick exit if N = 0 or NRHS = 0. 00133 * 00134 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00135 RESLTS( 1 ) = ZERO 00136 RESLTS( 2 ) = ZERO 00137 RETURN 00138 END IF 00139 * 00140 EPS = DLAMCH( 'Epsilon' ) 00141 UNFL = DLAMCH( 'Safe minimum' ) 00142 OVFL = ONE / UNFL 00143 UPPER = LSAME( UPLO, 'U' ) 00144 NOTRAN = LSAME( TRANS, 'N' ) 00145 UNIT = LSAME( DIAG, 'U' ) 00146 * 00147 * Test 1: Compute the maximum of 00148 * norm(X - XACT) / ( norm(X) * FERR ) 00149 * over all the vectors X and XACT using the infinity-norm. 00150 * 00151 ERRBND = ZERO 00152 DO 30 J = 1, NRHS 00153 IMAX = IDAMAX( N, X( 1, J ), 1 ) 00154 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00155 DIFF = ZERO 00156 DO 10 I = 1, N 00157 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00158 10 CONTINUE 00159 * 00160 IF( XNORM.GT.ONE ) THEN 00161 GO TO 20 00162 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00163 GO TO 20 00164 ELSE 00165 ERRBND = ONE / EPS 00166 GO TO 30 00167 END IF 00168 * 00169 20 CONTINUE 00170 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00171 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00172 ELSE 00173 ERRBND = ONE / EPS 00174 END IF 00175 30 CONTINUE 00176 RESLTS( 1 ) = ERRBND 00177 * 00178 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00179 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00180 * 00181 IFU = 0 00182 IF( UNIT ) 00183 $ IFU = 1 00184 DO 90 K = 1, NRHS 00185 DO 80 I = 1, N 00186 TMP = ABS( B( I, K ) ) 00187 IF( UPPER ) THEN 00188 IF( .NOT.NOTRAN ) THEN 00189 DO 40 J = 1, I - IFU 00190 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00191 40 CONTINUE 00192 IF( UNIT ) 00193 $ TMP = TMP + ABS( X( I, K ) ) 00194 ELSE 00195 IF( UNIT ) 00196 $ TMP = TMP + ABS( X( I, K ) ) 00197 DO 50 J = I + IFU, N 00198 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00199 50 CONTINUE 00200 END IF 00201 ELSE 00202 IF( NOTRAN ) THEN 00203 DO 60 J = 1, I - IFU 00204 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00205 60 CONTINUE 00206 IF( UNIT ) 00207 $ TMP = TMP + ABS( X( I, K ) ) 00208 ELSE 00209 IF( UNIT ) 00210 $ TMP = TMP + ABS( X( I, K ) ) 00211 DO 70 J = I + IFU, N 00212 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00213 70 CONTINUE 00214 END IF 00215 END IF 00216 IF( I.EQ.1 ) THEN 00217 AXBI = TMP 00218 ELSE 00219 AXBI = MIN( AXBI, TMP ) 00220 END IF 00221 80 CONTINUE 00222 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00223 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00224 IF( K.EQ.1 ) THEN 00225 RESLTS( 2 ) = TMP 00226 ELSE 00227 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00228 END IF 00229 90 CONTINUE 00230 * 00231 RETURN 00232 * 00233 * End of DTRT05 00234 * 00235 END
1.7.2