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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZTBT05 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 ZTBT05( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, 00012 * LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER DIAG, TRANS, UPLO 00016 * INTEGER KD, 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 *> ZTBT05 tests the error bounds from iterative refinement for the 00031 *> computed solution to a system of equations A*X = B, where A is a 00032 *> triangular band matrix. 00033 *> 00034 *> RESLTS(1) = test of the error bound 00035 *> = norm(X - XACT) / ( norm(X) * FERR ) 00036 *> 00037 *> A large value is returned if this ratio is not less than one. 00038 *> 00039 *> RESLTS(2) = residual from the iterative refinement routine 00040 *> = the maximum of BERR / ( NZ*EPS + (*) ), where 00041 *> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00042 *> and NZ = max. number of nonzeros in any row of A, plus 1 00043 *> \endverbatim 00044 * 00045 * Arguments: 00046 * ========== 00047 * 00048 *> \param[in] UPLO 00049 *> \verbatim 00050 *> UPLO is CHARACTER*1 00051 *> Specifies whether the matrix A is upper or lower triangular. 00052 *> = 'U': Upper triangular 00053 *> = 'L': Lower triangular 00054 *> \endverbatim 00055 *> 00056 *> \param[in] TRANS 00057 *> \verbatim 00058 *> TRANS is CHARACTER*1 00059 *> Specifies the form of the system of equations. 00060 *> = 'N': A * X = B (No transpose) 00061 *> = 'T': A'* X = B (Transpose) 00062 *> = 'C': A'* X = B (Conjugate transpose = Transpose) 00063 *> \endverbatim 00064 *> 00065 *> \param[in] DIAG 00066 *> \verbatim 00067 *> DIAG is CHARACTER*1 00068 *> Specifies whether or not the matrix A is unit triangular. 00069 *> = 'N': Non-unit triangular 00070 *> = 'U': Unit triangular 00071 *> \endverbatim 00072 *> 00073 *> \param[in] N 00074 *> \verbatim 00075 *> N is INTEGER 00076 *> The number of rows of the matrices X, B, and XACT, and the 00077 *> order of the matrix A. N >= 0. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] KD 00081 *> \verbatim 00082 *> KD is INTEGER 00083 *> The number of super-diagonals of the matrix A if UPLO = 'U', 00084 *> or the number of sub-diagonals if UPLO = 'L'. KD >= 0. 00085 *> \endverbatim 00086 *> 00087 *> \param[in] NRHS 00088 *> \verbatim 00089 *> NRHS is INTEGER 00090 *> The number of columns of the matrices X, B, and XACT. 00091 *> NRHS >= 0. 00092 *> \endverbatim 00093 *> 00094 *> \param[in] AB 00095 *> \verbatim 00096 *> AB is COMPLEX*16 array, dimension (LDAB,N) 00097 *> The upper or lower triangular band matrix A, stored in the 00098 *> first kd+1 rows of the array. The j-th column of A is stored 00099 *> in the j-th column of the array AB as follows: 00100 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00101 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00102 *> If DIAG = 'U', the diagonal elements of A are not referenced 00103 *> and are assumed to be 1. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDAB 00107 *> \verbatim 00108 *> LDAB is INTEGER 00109 *> The leading dimension of the array AB. LDAB >= KD+1. 00110 *> \endverbatim 00111 *> 00112 *> \param[in] B 00113 *> \verbatim 00114 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00115 *> The right hand side vectors for the system of linear 00116 *> equations. 00117 *> \endverbatim 00118 *> 00119 *> \param[in] LDB 00120 *> \verbatim 00121 *> LDB is INTEGER 00122 *> The leading dimension of the array B. LDB >= max(1,N). 00123 *> \endverbatim 00124 *> 00125 *> \param[in] X 00126 *> \verbatim 00127 *> X is COMPLEX*16 array, dimension (LDX,NRHS) 00128 *> The computed solution vectors. Each vector is stored as a 00129 *> column of the matrix X. 00130 *> \endverbatim 00131 *> 00132 *> \param[in] LDX 00133 *> \verbatim 00134 *> LDX is INTEGER 00135 *> The leading dimension of the array X. LDX >= max(1,N). 00136 *> \endverbatim 00137 *> 00138 *> \param[in] XACT 00139 *> \verbatim 00140 *> XACT is COMPLEX*16 array, dimension (LDX,NRHS) 00141 *> The exact solution vectors. Each vector is stored as a 00142 *> column of the matrix XACT. 00143 *> \endverbatim 00144 *> 00145 *> \param[in] LDXACT 00146 *> \verbatim 00147 *> LDXACT is INTEGER 00148 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00149 *> \endverbatim 00150 *> 00151 *> \param[in] FERR 00152 *> \verbatim 00153 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00154 *> The estimated forward error bounds for each solution vector 00155 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00156 *> of the largest entry in (X - XTRUE) divided by the magnitude 00157 *> of the largest entry in X. 00158 *> \endverbatim 00159 *> 00160 *> \param[in] BERR 00161 *> \verbatim 00162 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00163 *> The componentwise relative backward error of each solution 00164 *> vector (i.e., the smallest relative change in any entry of A 00165 *> or B that makes X an exact solution). 00166 *> \endverbatim 00167 *> 00168 *> \param[out] RESLTS 00169 *> \verbatim 00170 *> RESLTS is DOUBLE PRECISION array, dimension (2) 00171 *> The maximum over the NRHS solution vectors of the ratios: 00172 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00173 *> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 00174 *> \endverbatim 00175 * 00176 * Authors: 00177 * ======== 00178 * 00179 *> \author Univ. of Tennessee 00180 *> \author Univ. of California Berkeley 00181 *> \author Univ. of Colorado Denver 00182 *> \author NAG Ltd. 00183 * 00184 *> \date November 2011 00185 * 00186 *> \ingroup complex16_lin 00187 * 00188 * ===================================================================== 00189 SUBROUTINE ZTBT05( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, 00190 $ LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00191 * 00192 * -- LAPACK test routine (version 3.4.0) -- 00193 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00195 * November 2011 00196 * 00197 * .. Scalar Arguments .. 00198 CHARACTER DIAG, TRANS, UPLO 00199 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 00200 * .. 00201 * .. Array Arguments .. 00202 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) 00203 COMPLEX*16 AB( LDAB, * ), B( LDB, * ), X( LDX, * ), 00204 $ XACT( LDXACT, * ) 00205 * .. 00206 * 00207 * ===================================================================== 00208 * 00209 * .. Parameters .. 00210 DOUBLE PRECISION ZERO, ONE 00211 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00212 * .. 00213 * .. Local Scalars .. 00214 LOGICAL NOTRAN, UNIT, UPPER 00215 INTEGER I, IFU, IMAX, J, K, NZ 00216 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00217 COMPLEX*16 ZDUM 00218 * .. 00219 * .. External Functions .. 00220 LOGICAL LSAME 00221 INTEGER IZAMAX 00222 DOUBLE PRECISION DLAMCH 00223 EXTERNAL LSAME, IZAMAX, DLAMCH 00224 * .. 00225 * .. Intrinsic Functions .. 00226 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 00227 * .. 00228 * .. Statement Functions .. 00229 DOUBLE PRECISION CABS1 00230 * .. 00231 * .. Statement Function definitions .. 00232 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00233 * .. 00234 * .. Executable Statements .. 00235 * 00236 * Quick exit if N = 0 or NRHS = 0. 00237 * 00238 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00239 RESLTS( 1 ) = ZERO 00240 RESLTS( 2 ) = ZERO 00241 RETURN 00242 END IF 00243 * 00244 EPS = DLAMCH( 'Epsilon' ) 00245 UNFL = DLAMCH( 'Safe minimum' ) 00246 OVFL = ONE / UNFL 00247 UPPER = LSAME( UPLO, 'U' ) 00248 NOTRAN = LSAME( TRANS, 'N' ) 00249 UNIT = LSAME( DIAG, 'U' ) 00250 NZ = MIN( KD, N-1 ) + 1 00251 * 00252 * Test 1: Compute the maximum of 00253 * norm(X - XACT) / ( norm(X) * FERR ) 00254 * over all the vectors X and XACT using the infinity-norm. 00255 * 00256 ERRBND = ZERO 00257 DO 30 J = 1, NRHS 00258 IMAX = IZAMAX( N, X( 1, J ), 1 ) 00259 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00260 DIFF = ZERO 00261 DO 10 I = 1, N 00262 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00263 10 CONTINUE 00264 * 00265 IF( XNORM.GT.ONE ) THEN 00266 GO TO 20 00267 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00268 GO TO 20 00269 ELSE 00270 ERRBND = ONE / EPS 00271 GO TO 30 00272 END IF 00273 * 00274 20 CONTINUE 00275 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00276 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00277 ELSE 00278 ERRBND = ONE / EPS 00279 END IF 00280 30 CONTINUE 00281 RESLTS( 1 ) = ERRBND 00282 * 00283 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 00284 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00285 * 00286 IFU = 0 00287 IF( UNIT ) 00288 $ IFU = 1 00289 DO 90 K = 1, NRHS 00290 DO 80 I = 1, N 00291 TMP = CABS1( B( I, K ) ) 00292 IF( UPPER ) THEN 00293 IF( .NOT.NOTRAN ) THEN 00294 DO 40 J = MAX( I-KD, 1 ), I - IFU 00295 TMP = TMP + CABS1( AB( KD+1-I+J, I ) )* 00296 $ CABS1( X( J, K ) ) 00297 40 CONTINUE 00298 IF( UNIT ) 00299 $ TMP = TMP + CABS1( X( I, K ) ) 00300 ELSE 00301 IF( UNIT ) 00302 $ TMP = TMP + CABS1( X( I, K ) ) 00303 DO 50 J = I + IFU, MIN( I+KD, N ) 00304 TMP = TMP + CABS1( AB( KD+1+I-J, J ) )* 00305 $ CABS1( X( J, K ) ) 00306 50 CONTINUE 00307 END IF 00308 ELSE 00309 IF( NOTRAN ) THEN 00310 DO 60 J = MAX( I-KD, 1 ), I - IFU 00311 TMP = TMP + CABS1( AB( 1+I-J, J ) )* 00312 $ CABS1( X( J, K ) ) 00313 60 CONTINUE 00314 IF( UNIT ) 00315 $ TMP = TMP + CABS1( X( I, K ) ) 00316 ELSE 00317 IF( UNIT ) 00318 $ TMP = TMP + CABS1( X( I, K ) ) 00319 DO 70 J = I + IFU, MIN( I+KD, N ) 00320 TMP = TMP + CABS1( AB( 1+J-I, I ) )* 00321 $ CABS1( X( J, K ) ) 00322 70 CONTINUE 00323 END IF 00324 END IF 00325 IF( I.EQ.1 ) THEN 00326 AXBI = TMP 00327 ELSE 00328 AXBI = MIN( AXBI, TMP ) 00329 END IF 00330 80 CONTINUE 00331 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 00332 IF( K.EQ.1 ) THEN 00333 RESLTS( 2 ) = TMP 00334 ELSE 00335 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00336 END IF 00337 90 CONTINUE 00338 * 00339 RETURN 00340 * 00341 * End of ZTBT05 00342 * 00343 END