00001 SUBROUTINE SGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, 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 TRANS 00010 INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS 00011 * .. 00012 * .. Array Arguments .. 00013 REAL AB( LDAB, * ), B( LDB, * ), BERR( * ), 00014 $ FERR( * ), RESLTS( * ), X( LDX, * ), 00015 $ XACT( LDXACT, * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * SGBT05 tests the error bounds from iterative refinement for the 00022 * computed solution to a system of equations op(A)*X = B, where A is a 00023 * general band matrix of order n with kl subdiagonals and ku 00024 * superdiagonals and op(A) = A or A**T, depending on TRANS. 00025 * 00026 * RESLTS(1) = test of the error bound 00027 * = norm(X - XACT) / ( norm(X) * FERR ) 00028 * 00029 * A large value is returned if this ratio is not less than one. 00030 * 00031 * RESLTS(2) = residual from the iterative refinement routine 00032 * = the maximum of BERR / ( NZ*EPS + (*) ), where 00033 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 00034 * and NZ = max. number of nonzeros in any row of A, plus 1 00035 * 00036 * Arguments 00037 * ========= 00038 * 00039 * TRANS (input) CHARACTER*1 00040 * Specifies the form of the system of equations. 00041 * = 'N': A * X = B (No transpose) 00042 * = 'T': A**T * X = B (Transpose) 00043 * = 'C': A**H * X = B (Conjugate transpose = Transpose) 00044 * 00045 * N (input) INTEGER 00046 * The number of rows of the matrices X, B, and XACT, and the 00047 * order of the matrix A. N >= 0. 00048 * 00049 * KL (input) INTEGER 00050 * The number of subdiagonals within the band of A. KL >= 0. 00051 * 00052 * KU (input) INTEGER 00053 * The number of superdiagonals within the band of A. KU >= 0. 00054 * 00055 * NRHS (input) INTEGER 00056 * The number of columns of the matrices X, B, and XACT. 00057 * NRHS >= 0. 00058 * 00059 * AB (input) REAL array, dimension (LDAB,N) 00060 * The original band matrix A, stored in rows 1 to KL+KU+1. 00061 * The j-th column of A is stored in the j-th column of the 00062 * array AB as follows: 00063 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). 00064 * 00065 * LDAB (input) INTEGER 00066 * The leading dimension of the array AB. LDAB >= KL+KU+1. 00067 * 00068 * B (input) REAL array, dimension (LDB,NRHS) 00069 * The right hand side vectors for the system of linear 00070 * equations. 00071 * 00072 * LDB (input) INTEGER 00073 * The leading dimension of the array B. LDB >= max(1,N). 00074 * 00075 * X (input) REAL array, dimension (LDX,NRHS) 00076 * The computed solution vectors. Each vector is stored as a 00077 * column of the matrix X. 00078 * 00079 * LDX (input) INTEGER 00080 * The leading dimension of the array X. LDX >= max(1,N). 00081 * 00082 * XACT (input) REAL array, dimension (LDX,NRHS) 00083 * The exact solution vectors. Each vector is stored as a 00084 * column of the matrix XACT. 00085 * 00086 * LDXACT (input) INTEGER 00087 * The leading dimension of the array XACT. LDXACT >= max(1,N). 00088 * 00089 * FERR (input) REAL array, dimension (NRHS) 00090 * The estimated forward error bounds for each solution vector 00091 * X. If XTRUE is the true solution, FERR bounds the magnitude 00092 * of the largest entry in (X - XTRUE) divided by the magnitude 00093 * of the largest entry in X. 00094 * 00095 * BERR (input) REAL array, dimension (NRHS) 00096 * The componentwise relative backward error of each solution 00097 * vector (i.e., the smallest relative change in any entry of A 00098 * or B that makes X an exact solution). 00099 * 00100 * RESLTS (output) REAL array, dimension (2) 00101 * The maximum over the NRHS solution vectors of the ratios: 00102 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00103 * RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00104 * 00105 * ===================================================================== 00106 * 00107 * .. Parameters .. 00108 REAL ZERO, ONE 00109 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00110 * .. 00111 * .. Local Scalars .. 00112 LOGICAL NOTRAN 00113 INTEGER I, IMAX, J, K, NZ 00114 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00115 * .. 00116 * .. External Functions .. 00117 LOGICAL LSAME 00118 INTEGER ISAMAX 00119 REAL SLAMCH 00120 EXTERNAL LSAME, ISAMAX, SLAMCH 00121 * .. 00122 * .. Intrinsic Functions .. 00123 INTRINSIC ABS, MAX, MIN 00124 * .. 00125 * .. Executable Statements .. 00126 * 00127 * Quick exit if N = 0 or NRHS = 0. 00128 * 00129 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00130 RESLTS( 1 ) = ZERO 00131 RESLTS( 2 ) = ZERO 00132 RETURN 00133 END IF 00134 * 00135 EPS = SLAMCH( 'Epsilon' ) 00136 UNFL = SLAMCH( 'Safe minimum' ) 00137 OVFL = ONE / UNFL 00138 NOTRAN = LSAME( TRANS, 'N' ) 00139 NZ = MIN( KL+KU+2, N+1 ) 00140 * 00141 * Test 1: Compute the maximum of 00142 * norm(X - XACT) / ( norm(X) * FERR ) 00143 * over all the vectors X and XACT using the infinity-norm. 00144 * 00145 ERRBND = ZERO 00146 DO 30 J = 1, NRHS 00147 IMAX = ISAMAX( N, X( 1, J ), 1 ) 00148 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00149 DIFF = ZERO 00150 DO 10 I = 1, N 00151 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00152 10 CONTINUE 00153 * 00154 IF( XNORM.GT.ONE ) THEN 00155 GO TO 20 00156 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00157 GO TO 20 00158 ELSE 00159 ERRBND = ONE / EPS 00160 GO TO 30 00161 END IF 00162 * 00163 20 CONTINUE 00164 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00165 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00166 ELSE 00167 ERRBND = ONE / EPS 00168 END IF 00169 30 CONTINUE 00170 RESLTS( 1 ) = ERRBND 00171 * 00172 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00173 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 00174 * 00175 DO 70 K = 1, NRHS 00176 DO 60 I = 1, N 00177 TMP = ABS( B( I, K ) ) 00178 IF( NOTRAN ) THEN 00179 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N ) 00180 TMP = TMP + ABS( AB( KU+1+I-J, J ) )*ABS( X( J, K ) ) 00181 40 CONTINUE 00182 ELSE 00183 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N ) 00184 TMP = TMP + ABS( AB( KU+1+J-I, I ) )*ABS( X( J, K ) ) 00185 50 CONTINUE 00186 END IF 00187 IF( I.EQ.1 ) THEN 00188 AXBI = TMP 00189 ELSE 00190 AXBI = MIN( AXBI, TMP ) 00191 END IF 00192 60 CONTINUE 00193 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 00194 IF( K.EQ.1 ) THEN 00195 RESLTS( 2 ) = TMP 00196 ELSE 00197 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00198 END IF 00199 70 CONTINUE 00200 * 00201 RETURN 00202 * 00203 * End of SGBT05 00204 * 00205 END
1.7.2