00001 SUBROUTINE ZTRT05( 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 BERR( * ), FERR( * ), RESLTS( * ) 00014 COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * ), 00015 $ XACT( LDXACT, * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * ZTRT05 tests the error bounds from iterative refinement for the 00022 * computed solution to a system of equations A*X = B, where A is a 00023 * triangular n by n matrix. 00024 * 00025 * RESLTS(1) = test of the error bound 00026 * = norm(X - XACT) / ( norm(X) * FERR ) 00027 * 00028 * A large value is returned if this ratio is not less than one. 00029 * 00030 * RESLTS(2) = residual from the iterative refinement routine 00031 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where 00032 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00033 * 00034 * Arguments 00035 * ========= 00036 * 00037 * UPLO (input) CHARACTER*1 00038 * Specifies whether the matrix A is upper or lower triangular. 00039 * = 'U': Upper triangular 00040 * = 'L': Lower triangular 00041 * 00042 * TRANS (input) CHARACTER*1 00043 * Specifies the form of the system of equations. 00044 * = 'N': A * X = B (No transpose) 00045 * = 'T': A'* X = B (Transpose) 00046 * = 'C': A'* X = B (Conjugate transpose = Transpose) 00047 * 00048 * DIAG (input) CHARACTER*1 00049 * Specifies whether or not the matrix A is unit triangular. 00050 * = 'N': Non-unit triangular 00051 * = 'U': Unit triangular 00052 * 00053 * N (input) INTEGER 00054 * The number of rows of the matrices X, B, and XACT, and the 00055 * order of the matrix A. N >= 0. 00056 * 00057 * NRHS (input) INTEGER 00058 * The number of columns of the matrices X, B, and XACT. 00059 * NRHS >= 0. 00060 * 00061 * A (input) COMPLEX*16 array, dimension (LDA,N) 00062 * The triangular matrix A. If UPLO = 'U', the leading n by n 00063 * upper triangular part of the array A contains the upper 00064 * triangular matrix, and the strictly lower triangular part of 00065 * A is not referenced. If UPLO = 'L', the leading n by n lower 00066 * triangular part of the array A contains the lower triangular 00067 * matrix, and the strictly upper triangular part of A is not 00068 * referenced. If DIAG = 'U', the diagonal elements of A are 00069 * also not referenced and are assumed to be 1. 00070 * 00071 * LDA (input) INTEGER 00072 * The leading dimension of the array A. LDA >= max(1,N). 00073 * 00074 * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 00075 * The right hand side vectors for the system of linear 00076 * equations. 00077 * 00078 * LDB (input) INTEGER 00079 * The leading dimension of the array B. LDB >= max(1,N). 00080 * 00081 * X (input) COMPLEX*16 array, dimension (LDX,NRHS) 00082 * The computed solution vectors. Each vector is stored as a 00083 * column of the matrix X. 00084 * 00085 * LDX (input) INTEGER 00086 * The leading dimension of the array X. LDX >= max(1,N). 00087 * 00088 * XACT (input) COMPLEX*16 array, dimension (LDX,NRHS) 00089 * The exact solution vectors. Each vector is stored as a 00090 * column of the matrix XACT. 00091 * 00092 * LDXACT (input) INTEGER 00093 * The leading dimension of the array XACT. LDXACT >= max(1,N). 00094 * 00095 * FERR (input) DOUBLE PRECISION array, dimension (NRHS) 00096 * The estimated forward error bounds for each solution vector 00097 * X. If XTRUE is the true solution, FERR bounds the magnitude 00098 * of the largest entry in (X - XTRUE) divided by the magnitude 00099 * of the largest entry in X. 00100 * 00101 * BERR (input) DOUBLE PRECISION array, dimension (NRHS) 00102 * The componentwise relative backward error of each solution 00103 * vector (i.e., the smallest relative change in any entry of A 00104 * or B that makes X an exact solution). 00105 * 00106 * RESLTS (output) DOUBLE PRECISION array, dimension (2) 00107 * The maximum over the NRHS solution vectors of the ratios: 00108 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00109 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00110 * 00111 * ===================================================================== 00112 * 00113 * .. Parameters .. 00114 DOUBLE PRECISION ZERO, ONE 00115 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00116 * .. 00117 * .. Local Scalars .. 00118 LOGICAL NOTRAN, UNIT, UPPER 00119 INTEGER I, IFU, IMAX, J, K 00120 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00121 COMPLEX*16 ZDUM 00122 * .. 00123 * .. External Functions .. 00124 LOGICAL LSAME 00125 INTEGER IZAMAX 00126 DOUBLE PRECISION DLAMCH 00127 EXTERNAL LSAME, IZAMAX, DLAMCH 00128 * .. 00129 * .. Intrinsic Functions .. 00130 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 00131 * .. 00132 * .. Statement Functions .. 00133 DOUBLE PRECISION CABS1 00134 * .. 00135 * .. Statement Function definitions .. 00136 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00137 * .. 00138 * .. Executable Statements .. 00139 * 00140 * Quick exit if N = 0 or NRHS = 0. 00141 * 00142 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00143 RESLTS( 1 ) = ZERO 00144 RESLTS( 2 ) = ZERO 00145 RETURN 00146 END IF 00147 * 00148 EPS = DLAMCH( 'Epsilon' ) 00149 UNFL = DLAMCH( 'Safe minimum' ) 00150 OVFL = ONE / UNFL 00151 UPPER = LSAME( UPLO, 'U' ) 00152 NOTRAN = LSAME( TRANS, 'N' ) 00153 UNIT = LSAME( DIAG, 'U' ) 00154 * 00155 * Test 1: Compute the maximum of 00156 * norm(X - XACT) / ( norm(X) * FERR ) 00157 * over all the vectors X and XACT using the infinity-norm. 00158 * 00159 ERRBND = ZERO 00160 DO 30 J = 1, NRHS 00161 IMAX = IZAMAX( N, X( 1, J ), 1 ) 00162 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00163 DIFF = ZERO 00164 DO 10 I = 1, N 00165 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00166 10 CONTINUE 00167 * 00168 IF( XNORM.GT.ONE ) THEN 00169 GO TO 20 00170 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00171 GO TO 20 00172 ELSE 00173 ERRBND = ONE / EPS 00174 GO TO 30 00175 END IF 00176 * 00177 20 CONTINUE 00178 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00179 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00180 ELSE 00181 ERRBND = ONE / EPS 00182 END IF 00183 30 CONTINUE 00184 RESLTS( 1 ) = ERRBND 00185 * 00186 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00187 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00188 * 00189 IFU = 0 00190 IF( UNIT ) 00191 $ IFU = 1 00192 DO 90 K = 1, NRHS 00193 DO 80 I = 1, N 00194 TMP = CABS1( B( I, K ) ) 00195 IF( UPPER ) THEN 00196 IF( .NOT.NOTRAN ) THEN 00197 DO 40 J = 1, I - IFU 00198 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 00199 40 CONTINUE 00200 IF( UNIT ) 00201 $ TMP = TMP + CABS1( X( I, K ) ) 00202 ELSE 00203 IF( UNIT ) 00204 $ TMP = TMP + CABS1( X( I, K ) ) 00205 DO 50 J = I + IFU, N 00206 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 00207 50 CONTINUE 00208 END IF 00209 ELSE 00210 IF( NOTRAN ) THEN 00211 DO 60 J = 1, I - IFU 00212 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 00213 60 CONTINUE 00214 IF( UNIT ) 00215 $ TMP = TMP + CABS1( X( I, K ) ) 00216 ELSE 00217 IF( UNIT ) 00218 $ TMP = TMP + CABS1( X( I, K ) ) 00219 DO 70 J = I + IFU, N 00220 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 00221 70 CONTINUE 00222 END IF 00223 END IF 00224 IF( I.EQ.1 ) THEN 00225 AXBI = TMP 00226 ELSE 00227 AXBI = MIN( AXBI, TMP ) 00228 END IF 00229 80 CONTINUE 00230 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00231 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00232 IF( K.EQ.1 ) THEN 00233 RESLTS( 2 ) = TMP 00234 ELSE 00235 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00236 END IF 00237 90 CONTINUE 00238 * 00239 RETURN 00240 * 00241 * End of ZTRT05 00242 * 00243 END
1.7.2