00001 SUBROUTINE ZPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, 00002 $ 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 UPLO 00010 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) 00014 COMPLEX*16 AB( LDAB, * ), B( LDB, * ), X( LDX, * ), 00015 $ XACT( LDXACT, * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * ZPBT05 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 * Hermitian band 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 / ( NZ*EPS + (*) ), where 00032 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00033 * and NZ = max. number of nonzeros in any row of A, plus 1 00034 * 00035 * Arguments 00036 * ========= 00037 * 00038 * UPLO (input) CHARACTER*1 00039 * Specifies whether the upper or lower triangular part of the 00040 * Hermitian matrix A is stored. 00041 * = 'U': Upper triangular 00042 * = 'L': Lower triangular 00043 * 00044 * N (input) INTEGER 00045 * The number of rows of the matrices X, B, and XACT, and the 00046 * order of the matrix A. N >= 0. 00047 * 00048 * KD (input) INTEGER 00049 * The number of super-diagonals of the matrix A if UPLO = 'U', 00050 * or the number of sub-diagonals if UPLO = 'L'. KD >= 0. 00051 * 00052 * NRHS (input) INTEGER 00053 * The number of columns of the matrices X, B, and XACT. 00054 * NRHS >= 0. 00055 * 00056 * AB (input) COMPLEX*16 array, dimension (LDAB,N) 00057 * The upper or lower triangle of the Hermitian band matrix A, 00058 * stored in the first KD+1 rows of the array. The j-th column 00059 * of A is stored in the j-th column of the array AB as follows: 00060 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00061 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00062 * 00063 * LDAB (input) INTEGER 00064 * The leading dimension of the array AB. LDAB >= KD+1. 00065 * 00066 * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 00067 * The right hand side vectors for the system of linear 00068 * equations. 00069 * 00070 * LDB (input) INTEGER 00071 * The leading dimension of the array B. LDB >= max(1,N). 00072 * 00073 * X (input) COMPLEX*16 array, dimension (LDX,NRHS) 00074 * The computed solution vectors. Each vector is stored as a 00075 * column of the matrix X. 00076 * 00077 * LDX (input) INTEGER 00078 * The leading dimension of the array X. LDX >= max(1,N). 00079 * 00080 * XACT (input) COMPLEX*16 array, dimension (LDX,NRHS) 00081 * The exact solution vectors. Each vector is stored as a 00082 * column of the matrix XACT. 00083 * 00084 * LDXACT (input) INTEGER 00085 * The leading dimension of the array XACT. LDXACT >= max(1,N). 00086 * 00087 * FERR (input) DOUBLE PRECISION array, dimension (NRHS) 00088 * The estimated forward error bounds for each solution vector 00089 * X. If XTRUE is the true solution, FERR bounds the magnitude 00090 * of the largest entry in (X - XTRUE) divided by the magnitude 00091 * of the largest entry in X. 00092 * 00093 * BERR (input) DOUBLE PRECISION array, dimension (NRHS) 00094 * The componentwise relative backward error of each solution 00095 * vector (i.e., the smallest relative change in any entry of A 00096 * or B that makes X an exact solution). 00097 * 00098 * RESLTS (output) DOUBLE PRECISION array, dimension (2) 00099 * The maximum over the NRHS solution vectors of the ratios: 00100 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00101 * RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00102 * 00103 * ===================================================================== 00104 * 00105 * .. Parameters .. 00106 DOUBLE PRECISION ZERO, ONE 00107 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00108 * .. 00109 * .. Local Scalars .. 00110 LOGICAL UPPER 00111 INTEGER I, IMAX, J, K, NZ 00112 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00113 COMPLEX*16 ZDUM 00114 * .. 00115 * .. External Functions .. 00116 LOGICAL LSAME 00117 INTEGER IZAMAX 00118 DOUBLE PRECISION DLAMCH 00119 EXTERNAL LSAME, IZAMAX, DLAMCH 00120 * .. 00121 * .. Intrinsic Functions .. 00122 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 00123 * .. 00124 * .. Statement Functions .. 00125 DOUBLE PRECISION CABS1 00126 * .. 00127 * .. Statement Function definitions .. 00128 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 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 NZ = 2*MAX( KD, N-1 ) + 1 00145 * 00146 * Test 1: Compute the maximum of 00147 * norm(X - XACT) / ( norm(X) * FERR ) 00148 * over all the vectors X and XACT using the infinity-norm. 00149 * 00150 ERRBND = ZERO 00151 DO 30 J = 1, NRHS 00152 IMAX = IZAMAX( N, X( 1, J ), 1 ) 00153 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00154 DIFF = ZERO 00155 DO 10 I = 1, N 00156 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00157 10 CONTINUE 00158 * 00159 IF( XNORM.GT.ONE ) THEN 00160 GO TO 20 00161 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00162 GO TO 20 00163 ELSE 00164 ERRBND = ONE / EPS 00165 GO TO 30 00166 END IF 00167 * 00168 20 CONTINUE 00169 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00170 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00171 ELSE 00172 ERRBND = ONE / EPS 00173 END IF 00174 30 CONTINUE 00175 RESLTS( 1 ) = ERRBND 00176 * 00177 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00178 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00179 * 00180 DO 90 K = 1, NRHS 00181 DO 80 I = 1, N 00182 TMP = CABS1( B( I, K ) ) 00183 IF( UPPER ) THEN 00184 DO 40 J = MAX( I-KD, 1 ), I - 1 00185 TMP = TMP + CABS1( AB( KD+1-I+J, I ) )* 00186 $ CABS1( X( J, K ) ) 00187 40 CONTINUE 00188 TMP = TMP + ABS( DBLE( AB( KD+1, I ) ) )* 00189 $ CABS1( X( I, K ) ) 00190 DO 50 J = I + 1, MIN( I+KD, N ) 00191 TMP = TMP + CABS1( AB( KD+1+I-J, J ) )* 00192 $ CABS1( X( J, K ) ) 00193 50 CONTINUE 00194 ELSE 00195 DO 60 J = MAX( I-KD, 1 ), I - 1 00196 TMP = TMP + CABS1( AB( 1+I-J, J ) )*CABS1( X( J, K ) ) 00197 60 CONTINUE 00198 TMP = TMP + ABS( DBLE( AB( 1, I ) ) )*CABS1( X( I, K ) ) 00199 DO 70 J = I + 1, MIN( I+KD, N ) 00200 TMP = TMP + CABS1( AB( 1+J-I, I ) )*CABS1( X( J, K ) ) 00201 70 CONTINUE 00202 END IF 00203 IF( I.EQ.1 ) THEN 00204 AXBI = TMP 00205 ELSE 00206 AXBI = MIN( AXBI, TMP ) 00207 END IF 00208 80 CONTINUE 00209 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 00210 IF( K.EQ.1 ) THEN 00211 RESLTS( 2 ) = TMP 00212 ELSE 00213 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00214 END IF 00215 90 CONTINUE 00216 * 00217 RETURN 00218 * 00219 * End of ZPBT05 00220 * 00221 END
1.7.2