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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZGBT05 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 * Definition: 00009 * =========== 00010 * 00011 * SUBROUTINE ZGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X, 00012 * LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER TRANS 00016 * INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS 00017 * .. 00018 * .. Array Arguments .. 00019 * DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) 00020 * COMPLEX*16 AB( LDAB, * ), B( LDB, * ), X( LDX, * ), 00021 * $ XACT( LDXACT, * ) 00022 * .. 00023 * 00024 * 00025 *> \par Purpose: 00026 * ============= 00027 *> 00028 *> \verbatim 00029 *> 00030 *> ZGBT05 tests the error bounds from iterative refinement for the 00031 *> computed solution to a system of equations op(A)*X = B, where A is a 00032 *> general band matrix of order n with kl subdiagonals and ku 00033 *> superdiagonals and op(A) = A or A**T, depending on TRANS. 00034 *> 00035 *> RESLTS(1) = test of the error bound 00036 *> = norm(X - XACT) / ( norm(X) * FERR ) 00037 *> 00038 *> A large value is returned if this ratio is not less than one. 00039 *> 00040 *> RESLTS(2) = residual from the iterative refinement routine 00041 *> = the maximum of BERR / ( NZ*EPS + (*) ), where 00042 *> (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 00043 *> and NZ = max. number of nonzeros in any row of A, plus 1 00044 *> \endverbatim 00045 * 00046 * Arguments: 00047 * ========== 00048 * 00049 *> \param[in] TRANS 00050 *> \verbatim 00051 *> TRANS is CHARACTER*1 00052 *> Specifies the form of the system of equations. 00053 *> = 'N': A * X = B (No transpose) 00054 *> = 'T': A**T * X = B (Transpose) 00055 *> = 'C': A**H * X = B (Conjugate transpose = Transpose) 00056 *> \endverbatim 00057 *> 00058 *> \param[in] N 00059 *> \verbatim 00060 *> N is INTEGER 00061 *> The number of rows of the matrices X, B, and XACT, and the 00062 *> order of the matrix A. N >= 0. 00063 *> \endverbatim 00064 *> 00065 *> \param[in] KL 00066 *> \verbatim 00067 *> KL is INTEGER 00068 *> The number of subdiagonals within the band of A. KL >= 0. 00069 *> \endverbatim 00070 *> 00071 *> \param[in] KU 00072 *> \verbatim 00073 *> KU is INTEGER 00074 *> The number of superdiagonals within the band of A. KU >= 0. 00075 *> \endverbatim 00076 *> 00077 *> \param[in] NRHS 00078 *> \verbatim 00079 *> NRHS is INTEGER 00080 *> The number of columns of the matrices X, B, and XACT. 00081 *> NRHS >= 0. 00082 *> \endverbatim 00083 *> 00084 *> \param[in] AB 00085 *> \verbatim 00086 *> AB is COMPLEX*16 array, dimension (LDAB,N) 00087 *> The original band matrix A, stored in rows 1 to KL+KU+1. 00088 *> The j-th column of A is stored in the j-th column of the 00089 *> array AB as follows: 00090 *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). 00091 *> \endverbatim 00092 *> 00093 *> \param[in] LDAB 00094 *> \verbatim 00095 *> LDAB is INTEGER 00096 *> The leading dimension of the array AB. LDAB >= KL+KU+1. 00097 *> \endverbatim 00098 *> 00099 *> \param[in] B 00100 *> \verbatim 00101 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00102 *> The right hand side vectors for the system of linear 00103 *> equations. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDB 00107 *> \verbatim 00108 *> LDB is INTEGER 00109 *> The leading dimension of the array B. LDB >= max(1,N). 00110 *> \endverbatim 00111 *> 00112 *> \param[in] X 00113 *> \verbatim 00114 *> X is COMPLEX*16 array, dimension (LDX,NRHS) 00115 *> The computed solution vectors. Each vector is stored as a 00116 *> column of the matrix X. 00117 *> \endverbatim 00118 *> 00119 *> \param[in] LDX 00120 *> \verbatim 00121 *> LDX is INTEGER 00122 *> The leading dimension of the array X. LDX >= max(1,N). 00123 *> \endverbatim 00124 *> 00125 *> \param[in] XACT 00126 *> \verbatim 00127 *> XACT is COMPLEX*16 array, dimension (LDX,NRHS) 00128 *> The exact solution vectors. Each vector is stored as a 00129 *> column of the matrix XACT. 00130 *> \endverbatim 00131 *> 00132 *> \param[in] LDXACT 00133 *> \verbatim 00134 *> LDXACT is INTEGER 00135 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00136 *> \endverbatim 00137 *> 00138 *> \param[in] FERR 00139 *> \verbatim 00140 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00141 *> The estimated forward error bounds for each solution vector 00142 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00143 *> of the largest entry in (X - XTRUE) divided by the magnitude 00144 *> of the largest entry in X. 00145 *> \endverbatim 00146 *> 00147 *> \param[in] BERR 00148 *> \verbatim 00149 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00150 *> The componentwise relative backward error of each solution 00151 *> vector (i.e., the smallest relative change in any entry of A 00152 *> or B that makes X an exact solution). 00153 *> \endverbatim 00154 *> 00155 *> \param[out] RESLTS 00156 *> \verbatim 00157 *> RESLTS is DOUBLE PRECISION array, dimension (2) 00158 *> The maximum over the NRHS solution vectors of the ratios: 00159 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00160 *> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00161 *> \endverbatim 00162 * 00163 * Authors: 00164 * ======== 00165 * 00166 *> \author Univ. of Tennessee 00167 *> \author Univ. of California Berkeley 00168 *> \author Univ. of Colorado Denver 00169 *> \author NAG Ltd. 00170 * 00171 *> \date November 2011 00172 * 00173 *> \ingroup complex16_lin 00174 * 00175 * ===================================================================== 00176 SUBROUTINE ZGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X, 00177 $ LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00178 * 00179 * -- LAPACK test routine (version 3.4.0) -- 00180 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00181 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00182 * November 2011 00183 * 00184 * .. Scalar Arguments .. 00185 CHARACTER TRANS 00186 INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS 00187 * .. 00188 * .. Array Arguments .. 00189 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) 00190 COMPLEX*16 AB( LDAB, * ), B( LDB, * ), X( LDX, * ), 00191 $ XACT( LDXACT, * ) 00192 * .. 00193 * 00194 * ===================================================================== 00195 * 00196 * .. Parameters .. 00197 DOUBLE PRECISION ZERO, ONE 00198 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00199 * .. 00200 * .. Local Scalars .. 00201 LOGICAL NOTRAN 00202 INTEGER I, IMAX, J, K, NZ 00203 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00204 COMPLEX*16 ZDUM 00205 * .. 00206 * .. External Functions .. 00207 LOGICAL LSAME 00208 INTEGER IZAMAX 00209 DOUBLE PRECISION DLAMCH 00210 EXTERNAL LSAME, IZAMAX, DLAMCH 00211 * .. 00212 * .. Intrinsic Functions .. 00213 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 00214 * .. 00215 * .. Statement Functions .. 00216 DOUBLE PRECISION CABS1 00217 * .. 00218 * .. Statement Function definitions .. 00219 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00220 * .. 00221 * .. Executable Statements .. 00222 * 00223 * Quick exit if N = 0 or NRHS = 0. 00224 * 00225 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00226 RESLTS( 1 ) = ZERO 00227 RESLTS( 2 ) = ZERO 00228 RETURN 00229 END IF 00230 * 00231 EPS = DLAMCH( 'Epsilon' ) 00232 UNFL = DLAMCH( 'Safe minimum' ) 00233 OVFL = ONE / UNFL 00234 NOTRAN = LSAME( TRANS, 'N' ) 00235 NZ = MIN( KL+KU+2, N+1 ) 00236 * 00237 * Test 1: Compute the maximum of 00238 * norm(X - XACT) / ( norm(X) * FERR ) 00239 * over all the vectors X and XACT using the infinity-norm. 00240 * 00241 ERRBND = ZERO 00242 DO 30 J = 1, NRHS 00243 IMAX = IZAMAX( N, X( 1, J ), 1 ) 00244 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00245 DIFF = ZERO 00246 DO 10 I = 1, N 00247 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00248 10 CONTINUE 00249 * 00250 IF( XNORM.GT.ONE ) THEN 00251 GO TO 20 00252 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00253 GO TO 20 00254 ELSE 00255 ERRBND = ONE / EPS 00256 GO TO 30 00257 END IF 00258 * 00259 20 CONTINUE 00260 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00261 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00262 ELSE 00263 ERRBND = ONE / EPS 00264 END IF 00265 30 CONTINUE 00266 RESLTS( 1 ) = ERRBND 00267 * 00268 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00269 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 00270 * 00271 DO 70 K = 1, NRHS 00272 DO 60 I = 1, N 00273 TMP = CABS1( B( I, K ) ) 00274 IF( NOTRAN ) THEN 00275 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N ) 00276 TMP = TMP + CABS1( AB( KU+1+I-J, J ) )* 00277 $ CABS1( X( J, K ) ) 00278 40 CONTINUE 00279 ELSE 00280 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N ) 00281 TMP = TMP + CABS1( AB( KU+1+J-I, I ) )* 00282 $ CABS1( X( J, K ) ) 00283 50 CONTINUE 00284 END IF 00285 IF( I.EQ.1 ) THEN 00286 AXBI = TMP 00287 ELSE 00288 AXBI = MIN( AXBI, TMP ) 00289 END IF 00290 60 CONTINUE 00291 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 00292 IF( K.EQ.1 ) THEN 00293 RESLTS( 2 ) = TMP 00294 ELSE 00295 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00296 END IF 00297 70 CONTINUE 00298 * 00299 RETURN 00300 * 00301 * End of ZGBT05 00302 * 00303 END