1*> \brief \b CTPT05 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8* Definition: 9* =========== 10* 11* SUBROUTINE CTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 12* XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER DIAG, TRANS, UPLO 16* INTEGER LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* REAL BERR( * ), FERR( * ), RESLTS( * ) 20* COMPLEX AP( * ), B( LDB, * ), X( LDX, * ), 21* $ XACT( LDXACT, * ) 22* .. 23* 24* 25*> \par Purpose: 26* ============= 27*> 28*> \verbatim 29*> 30*> CTPT05 tests the error bounds from iterative refinement for the 31*> computed solution to a system of equations A*X = B, where A is a 32*> triangular matrix in packed storage format. 33*> 34*> RESLTS(1) = test of the error bound 35*> = norm(X - XACT) / ( norm(X) * FERR ) 36*> 37*> A large value is returned if this ratio is not less than one. 38*> 39*> RESLTS(2) = residual from the iterative refinement routine 40*> = the maximum of BERR / ( (n+1)*EPS + (*) ), where 41*> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 42*> \endverbatim 43* 44* Arguments: 45* ========== 46* 47*> \param[in] UPLO 48*> \verbatim 49*> UPLO is CHARACTER*1 50*> Specifies whether the matrix A is upper or lower triangular. 51*> = 'U': Upper triangular 52*> = 'L': Lower triangular 53*> \endverbatim 54*> 55*> \param[in] TRANS 56*> \verbatim 57*> TRANS is CHARACTER*1 58*> Specifies the form of the system of equations. 59*> = 'N': A * X = B (No transpose) 60*> = 'T': A'* X = B (Transpose) 61*> = 'C': A'* X = B (Conjugate transpose = Transpose) 62*> \endverbatim 63*> 64*> \param[in] DIAG 65*> \verbatim 66*> DIAG is CHARACTER*1 67*> Specifies whether or not the matrix A is unit triangular. 68*> = 'N': Non-unit triangular 69*> = 'U': Unit triangular 70*> \endverbatim 71*> 72*> \param[in] N 73*> \verbatim 74*> N is INTEGER 75*> The number of rows of the matrices X, B, and XACT, and the 76*> order of the matrix A. N >= 0. 77*> \endverbatim 78*> 79*> \param[in] NRHS 80*> \verbatim 81*> NRHS is INTEGER 82*> The number of columns of the matrices X, B, and XACT. 83*> NRHS >= 0. 84*> \endverbatim 85*> 86*> \param[in] AP 87*> \verbatim 88*> AP is COMPLEX array, dimension (N*(N+1)/2) 89*> The upper or lower triangular matrix A, packed columnwise in 90*> a linear array. The j-th column of A is stored in the array 91*> AP as follows: 92*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 93*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 94*> If DIAG = 'U', the diagonal elements of A are not referenced 95*> and are assumed to be 1. 96*> \endverbatim 97*> 98*> \param[in] B 99*> \verbatim 100*> B is COMPLEX array, dimension (LDB,NRHS) 101*> The right hand side vectors for the system of linear 102*> equations. 103*> \endverbatim 104*> 105*> \param[in] LDB 106*> \verbatim 107*> LDB is INTEGER 108*> The leading dimension of the array B. LDB >= max(1,N). 109*> \endverbatim 110*> 111*> \param[in] X 112*> \verbatim 113*> X is COMPLEX array, dimension (LDX,NRHS) 114*> The computed solution vectors. Each vector is stored as a 115*> column of the matrix X. 116*> \endverbatim 117*> 118*> \param[in] LDX 119*> \verbatim 120*> LDX is INTEGER 121*> The leading dimension of the array X. LDX >= max(1,N). 122*> \endverbatim 123*> 124*> \param[in] XACT 125*> \verbatim 126*> XACT is COMPLEX array, dimension (LDX,NRHS) 127*> The exact solution vectors. Each vector is stored as a 128*> column of the matrix XACT. 129*> \endverbatim 130*> 131*> \param[in] LDXACT 132*> \verbatim 133*> LDXACT is INTEGER 134*> The leading dimension of the array XACT. LDXACT >= max(1,N). 135*> \endverbatim 136*> 137*> \param[in] FERR 138*> \verbatim 139*> FERR is REAL array, dimension (NRHS) 140*> The estimated forward error bounds for each solution vector 141*> X. If XTRUE is the true solution, FERR bounds the magnitude 142*> of the largest entry in (X - XTRUE) divided by the magnitude 143*> of the largest entry in X. 144*> \endverbatim 145*> 146*> \param[in] BERR 147*> \verbatim 148*> BERR is REAL array, dimension (NRHS) 149*> The componentwise relative backward error of each solution 150*> vector (i.e., the smallest relative change in any entry of A 151*> or B that makes X an exact solution). 152*> \endverbatim 153*> 154*> \param[out] RESLTS 155*> \verbatim 156*> RESLTS is REAL array, dimension (2) 157*> The maximum over the NRHS solution vectors of the ratios: 158*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 159*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 160*> \endverbatim 161* 162* Authors: 163* ======== 164* 165*> \author Univ. of Tennessee 166*> \author Univ. of California Berkeley 167*> \author Univ. of Colorado Denver 168*> \author NAG Ltd. 169* 170*> \date December 2016 171* 172*> \ingroup complex_lin 173* 174* ===================================================================== 175 SUBROUTINE CTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 176 $ XACT, LDXACT, FERR, BERR, RESLTS ) 177* 178* -- LAPACK test routine (version 3.7.0) -- 179* -- LAPACK is a software package provided by Univ. of Tennessee, -- 180* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 181* December 2016 182* 183* .. Scalar Arguments .. 184 CHARACTER DIAG, TRANS, UPLO 185 INTEGER LDB, LDX, LDXACT, N, NRHS 186* .. 187* .. Array Arguments .. 188 REAL BERR( * ), FERR( * ), RESLTS( * ) 189 COMPLEX AP( * ), B( LDB, * ), X( LDX, * ), 190 $ XACT( LDXACT, * ) 191* .. 192* 193* ===================================================================== 194* 195* .. Parameters .. 196 REAL ZERO, ONE 197 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 198* .. 199* .. Local Scalars .. 200 LOGICAL NOTRAN, UNIT, UPPER 201 INTEGER I, IFU, IMAX, J, JC, K 202 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 203 COMPLEX ZDUM 204* .. 205* .. External Functions .. 206 LOGICAL LSAME 207 INTEGER ICAMAX 208 REAL SLAMCH 209 EXTERNAL LSAME, ICAMAX, SLAMCH 210* .. 211* .. Intrinsic Functions .. 212 INTRINSIC ABS, AIMAG, MAX, MIN, REAL 213* .. 214* .. Statement Functions .. 215 REAL CABS1 216* .. 217* .. Statement Function definitions .. 218 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) ) 219* .. 220* .. Executable Statements .. 221* 222* Quick exit if N = 0 or NRHS = 0. 223* 224 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 225 RESLTS( 1 ) = ZERO 226 RESLTS( 2 ) = ZERO 227 RETURN 228 END IF 229* 230 EPS = SLAMCH( 'Epsilon' ) 231 UNFL = SLAMCH( 'Safe minimum' ) 232 OVFL = ONE / UNFL 233 UPPER = LSAME( UPLO, 'U' ) 234 NOTRAN = LSAME( TRANS, 'N' ) 235 UNIT = LSAME( DIAG, 'U' ) 236* 237* Test 1: Compute the maximum of 238* norm(X - XACT) / ( norm(X) * FERR ) 239* over all the vectors X and XACT using the infinity-norm. 240* 241 ERRBND = ZERO 242 DO 30 J = 1, NRHS 243 IMAX = ICAMAX( N, X( 1, J ), 1 ) 244 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 245 DIFF = ZERO 246 DO 10 I = 1, N 247 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 248 10 CONTINUE 249* 250 IF( XNORM.GT.ONE ) THEN 251 GO TO 20 252 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 253 GO TO 20 254 ELSE 255 ERRBND = ONE / EPS 256 GO TO 30 257 END IF 258* 259 20 CONTINUE 260 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 261 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 262 ELSE 263 ERRBND = ONE / EPS 264 END IF 265 30 CONTINUE 266 RESLTS( 1 ) = ERRBND 267* 268* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 269* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 270* 271 IFU = 0 272 IF( UNIT ) 273 $ IFU = 1 274 DO 90 K = 1, NRHS 275 DO 80 I = 1, N 276 TMP = CABS1( B( I, K ) ) 277 IF( UPPER ) THEN 278 JC = ( ( I-1 )*I ) / 2 279 IF( .NOT.NOTRAN ) THEN 280 DO 40 J = 1, I - IFU 281 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) ) 282 40 CONTINUE 283 IF( UNIT ) 284 $ TMP = TMP + CABS1( X( I, K ) ) 285 ELSE 286 JC = JC + I 287 IF( UNIT ) THEN 288 TMP = TMP + CABS1( X( I, K ) ) 289 JC = JC + I 290 END IF 291 DO 50 J = I + IFU, N 292 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) 293 JC = JC + J 294 50 CONTINUE 295 END IF 296 ELSE 297 IF( NOTRAN ) THEN 298 JC = I 299 DO 60 J = 1, I - IFU 300 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) 301 JC = JC + N - J 302 60 CONTINUE 303 IF( UNIT ) 304 $ TMP = TMP + CABS1( X( I, K ) ) 305 ELSE 306 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 307 IF( UNIT ) 308 $ TMP = TMP + CABS1( X( I, K ) ) 309 DO 70 J = I + IFU, N 310 TMP = TMP + CABS1( AP( JC+J-I ) )* 311 $ CABS1( X( J, K ) ) 312 70 CONTINUE 313 END IF 314 END IF 315 IF( I.EQ.1 ) THEN 316 AXBI = TMP 317 ELSE 318 AXBI = MIN( AXBI, TMP ) 319 END IF 320 80 CONTINUE 321 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 322 $ MAX( AXBI, ( N+1 )*UNFL ) ) 323 IF( K.EQ.1 ) THEN 324 RESLTS( 2 ) = TMP 325 ELSE 326 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 327 END IF 328 90 CONTINUE 329* 330 RETURN 331* 332* End of CTPT05 333* 334 END 335