00001 SUBROUTINE DPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT, 00002 $ 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 INTEGER LDB, LDX, LDXACT, N, NRHS 00010 * .. 00011 * .. Array Arguments .. 00012 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), E( * ), 00013 $ FERR( * ), RESLTS( * ), X( LDX, * ), 00014 $ XACT( LDXACT, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * DPTT05 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 tridiagonal matrix of order n. 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 / ( NZ*EPS + (*) ), where 00031 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00032 * and NZ = max. number of nonzeros in any row of A, plus 1 00033 * 00034 * Arguments 00035 * ========= 00036 * 00037 * N (input) INTEGER 00038 * The number of rows of the matrices X, B, and XACT, and the 00039 * order of the matrix A. N >= 0. 00040 * 00041 * NRHS (input) INTEGER 00042 * The number of columns of the matrices X, B, and XACT. 00043 * NRHS >= 0. 00044 * 00045 * D (input) DOUBLE PRECISION array, dimension (N) 00046 * The n diagonal elements of the tridiagonal matrix A. 00047 * 00048 * E (input) DOUBLE PRECISION array, dimension (N-1) 00049 * The (n-1) subdiagonal elements of the tridiagonal matrix A. 00050 * 00051 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00052 * The right hand side vectors for the system of linear 00053 * equations. 00054 * 00055 * LDB (input) INTEGER 00056 * The leading dimension of the array B. LDB >= max(1,N). 00057 * 00058 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00059 * The computed solution vectors. Each vector is stored as a 00060 * column of the matrix X. 00061 * 00062 * LDX (input) INTEGER 00063 * The leading dimension of the array X. LDX >= max(1,N). 00064 * 00065 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS) 00066 * The exact solution vectors. Each vector is stored as a 00067 * column of the matrix XACT. 00068 * 00069 * LDXACT (input) INTEGER 00070 * The leading dimension of the array XACT. LDXACT >= max(1,N). 00071 * 00072 * FERR (input) DOUBLE PRECISION array, dimension (NRHS) 00073 * The estimated forward error bounds for each solution vector 00074 * X. If XTRUE is the true solution, FERR bounds the magnitude 00075 * of the largest entry in (X - XTRUE) divided by the magnitude 00076 * of the largest entry in X. 00077 * 00078 * BERR (input) DOUBLE PRECISION array, dimension (NRHS) 00079 * The componentwise relative backward error of each solution 00080 * vector (i.e., the smallest relative change in any entry of A 00081 * or B that makes X an exact solution). 00082 * 00083 * RESLTS (output) DOUBLE PRECISION array, dimension (2) 00084 * The maximum over the NRHS solution vectors of the ratios: 00085 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00086 * RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00087 * 00088 * ===================================================================== 00089 * 00090 * .. Parameters .. 00091 DOUBLE PRECISION ZERO, ONE 00092 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00093 * .. 00094 * .. Local Scalars .. 00095 INTEGER I, IMAX, J, K, NZ 00096 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00097 * .. 00098 * .. External Functions .. 00099 INTEGER IDAMAX 00100 DOUBLE PRECISION DLAMCH 00101 EXTERNAL IDAMAX, DLAMCH 00102 * .. 00103 * .. Intrinsic Functions .. 00104 INTRINSIC ABS, MAX, MIN 00105 * .. 00106 * .. Executable Statements .. 00107 * 00108 * Quick exit if N = 0 or NRHS = 0. 00109 * 00110 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00111 RESLTS( 1 ) = ZERO 00112 RESLTS( 2 ) = ZERO 00113 RETURN 00114 END IF 00115 * 00116 EPS = DLAMCH( 'Epsilon' ) 00117 UNFL = DLAMCH( 'Safe minimum' ) 00118 OVFL = ONE / UNFL 00119 NZ = 4 00120 * 00121 * Test 1: Compute the maximum of 00122 * norm(X - XACT) / ( norm(X) * FERR ) 00123 * over all the vectors X and XACT using the infinity-norm. 00124 * 00125 ERRBND = ZERO 00126 DO 30 J = 1, NRHS 00127 IMAX = IDAMAX( N, X( 1, J ), 1 ) 00128 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00129 DIFF = ZERO 00130 DO 10 I = 1, N 00131 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00132 10 CONTINUE 00133 * 00134 IF( XNORM.GT.ONE ) THEN 00135 GO TO 20 00136 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00137 GO TO 20 00138 ELSE 00139 ERRBND = ONE / EPS 00140 GO TO 30 00141 END IF 00142 * 00143 20 CONTINUE 00144 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00145 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00146 ELSE 00147 ERRBND = ONE / EPS 00148 END IF 00149 30 CONTINUE 00150 RESLTS( 1 ) = ERRBND 00151 * 00152 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00153 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00154 * 00155 DO 50 K = 1, NRHS 00156 IF( N.EQ.1 ) THEN 00157 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) 00158 ELSE 00159 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) + 00160 $ ABS( E( 1 )*X( 2, K ) ) 00161 DO 40 I = 2, N - 1 00162 TMP = ABS( B( I, K ) ) + ABS( E( I-1 )*X( I-1, K ) ) + 00163 $ ABS( D( I )*X( I, K ) ) + ABS( E( I )*X( I+1, K ) ) 00164 AXBI = MIN( AXBI, TMP ) 00165 40 CONTINUE 00166 TMP = ABS( B( N, K ) ) + ABS( E( N-1 )*X( N-1, K ) ) + 00167 $ ABS( D( N )*X( N, K ) ) 00168 AXBI = MIN( AXBI, TMP ) 00169 END IF 00170 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 00171 IF( K.EQ.1 ) THEN 00172 RESLTS( 2 ) = TMP 00173 ELSE 00174 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00175 END IF 00176 50 CONTINUE 00177 * 00178 RETURN 00179 * 00180 * End of DPTT05 00181 * 00182 END
1.7.2