|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 |
SUBROUTINE SLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B,
$ LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO ) * * -- LAPACK auxiliary routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. LOGICAL NOINIT, RIGHTV INTEGER INFO, LDB, LDH, N REAL BIGNUM, EPS3, SMLNUM, WI, WR * .. * .. Array Arguments .. REAL B( LDB, * ), H( LDH, * ), VI( * ), VR( * ), $ WORK( * ) * .. * * Purpose * ======= * * SLAEIN uses inverse iteration to find a right or left eigenvector * corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg * matrix H. * * Arguments * ========= * * RIGHTV (input) LOGICAL * = .TRUE. : compute right eigenvector; * = .FALSE.: compute left eigenvector. * * NOINIT (input) LOGICAL * = .TRUE. : no initial vector supplied in (VR,VI). * = .FALSE.: initial vector supplied in (VR,VI). * * N (input) INTEGER * The order of the matrix H. N >= 0. * * H (input) REAL array, dimension (LDH,N) * The upper Hessenberg matrix H. * * LDH (input) INTEGER * The leading dimension of the array H. LDH >= max(1,N). * * WR (input) REAL * WI (input) REAL * The real and imaginary parts of the eigenvalue of H whose * corresponding right or left eigenvector is to be computed. * * VR (input/output) REAL array, dimension (N) * VI (input/output) REAL array, dimension (N) * On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain * a real starting vector for inverse iteration using the real * eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI * must contain the real and imaginary parts of a complex * starting vector for inverse iteration using the complex * eigenvalue (WR,WI); otherwise VR and VI need not be set. * On exit, if WI = 0.0 (real eigenvalue), VR contains the * computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), * VR and VI contain the real and imaginary parts of the * computed complex eigenvector. The eigenvector is normalized * so that the component of largest magnitude has magnitude 1; * here the magnitude of a complex number (x,y) is taken to be * |x| + |y|. * VI is not referenced if WI = 0.0. * * B (workspace) REAL array, dimension (LDB,N) * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= N+1. * * WORK (workspace) REAL array, dimension (N) * * EPS3 (input) REAL * A small machine-dependent value which is used to perturb * close eigenvalues, and to replace zero pivots. * * SMLNUM (input) REAL * A machine-dependent value close to the underflow threshold. * * BIGNUM (input) REAL * A machine-dependent value close to the overflow threshold. * * INFO (output) INTEGER * = 0: successful exit * = 1: inverse iteration did not converge; VR is set to the * last iterate, and so is VI if WI.ne.0.0. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TENTH PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TENTH = 1.0E-1 ) * .. * .. Local Scalars .. CHARACTER NORMIN, TRANS INTEGER I, I1, I2, I3, IERR, ITS, J REAL ABSBII, ABSBJJ, EI, EJ, GROWTO, NORM, NRMSML, $ REC, ROOTN, SCALE, TEMP, VCRIT, VMAX, VNORM, W, $ W1, X, XI, XR, Y * .. * .. External Functions .. INTEGER ISAMAX REAL SASUM, SLAPY2, SNRM2 EXTERNAL ISAMAX, SASUM, SLAPY2, SNRM2 * .. * .. External Subroutines .. EXTERNAL SLADIV, SLATRS, SSCAL * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, REAL, SQRT * .. * .. Executable Statements .. * INFO = 0 * * GROWTO is the threshold used in the acceptance test for an * eigenvector. * ROOTN = SQRT( REAL( N ) ) GROWTO = TENTH / ROOTN NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM * * Form B = H - (WR,WI)*I (except that the subdiagonal elements and * the imaginary parts of the diagonal elements are not stored). * DO 20 J = 1, N DO 10 I = 1, J - 1 B( I, J ) = H( I, J ) 10 CONTINUE B( J, J ) = H( J, J ) - WR 20 CONTINUE * IF( WI.EQ.ZERO ) THEN * * Real eigenvalue. * IF( NOINIT ) THEN * * Set initial vector. * DO 30 I = 1, N VR( I ) = EPS3 30 CONTINUE ELSE * * Scale supplied initial vector. * VNORM = SNRM2( N, VR, 1 ) CALL SSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), VR, $ 1 ) END IF * IF( RIGHTV ) THEN * * LU decomposition with partial pivoting of B, replacing zero * pivots by EPS3. * DO 60 I = 1, N - 1 EI = H( I+1, I ) IF( ABS( B( I, I ) ).LT.ABS( EI ) ) THEN * * Interchange rows and eliminate. * X = B( I, I ) / EI B( I, I ) = EI DO 40 J = I + 1, N TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - X*TEMP B( I, J ) = TEMP 40 CONTINUE ELSE * * Eliminate without interchange. * IF( B( I, I ).EQ.ZERO ) $ B( I, I ) = EPS3 X = EI / B( I, I ) IF( X.NE.ZERO ) THEN DO 50 J = I + 1, N B( I+1, J ) = B( I+1, J ) - X*B( I, J ) 50 CONTINUE END IF END IF 60 CONTINUE IF( B( N, N ).EQ.ZERO ) $ B( N, N ) = EPS3 * TRANS = 'N' * ELSE * * UL decomposition with partial pivoting of B, replacing zero * pivots by EPS3. * DO 90 J = N, 2, -1 EJ = H( J, J-1 ) IF( ABS( B( J, J ) ).LT.ABS( EJ ) ) THEN * * Interchange columns and eliminate. * X = B( J, J ) / EJ B( J, J ) = EJ DO 70 I = 1, J - 1 TEMP = B( I, J-1 ) B( I, J-1 ) = B( I, J ) - X*TEMP B( I, J ) = TEMP 70 CONTINUE ELSE * * Eliminate without interchange. * IF( B( J, J ).EQ.ZERO ) $ B( J, J ) = EPS3 X = EJ / B( J, J ) IF( X.NE.ZERO ) THEN DO 80 I = 1, J - 1 B( I, J-1 ) = B( I, J-1 ) - X*B( I, J ) 80 CONTINUE END IF END IF 90 CONTINUE IF( B( 1, 1 ).EQ.ZERO ) $ B( 1, 1 ) = EPS3 * TRANS = 'T' * END IF * NORMIN = 'N' DO 110 ITS = 1, N * * Solve U*x = scale*v for a right eigenvector * or U**T*x = scale*v for a left eigenvector, * overwriting x on v. * CALL SLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, $ VR, SCALE, WORK, IERR ) NORMIN = 'Y' * * Test for sufficient growth in the norm of v. * VNORM = SASUM( N, VR, 1 ) IF( VNORM.GE.GROWTO*SCALE ) $ GO TO 120 * * Choose new orthogonal starting vector and try again. * TEMP = EPS3 / ( ROOTN+ONE ) VR( 1 ) = EPS3 DO 100 I = 2, N VR( I ) = TEMP 100 CONTINUE VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN 110 CONTINUE * * Failure to find eigenvector in N iterations. * INFO = 1 * 120 CONTINUE * * Normalize eigenvector. * I = ISAMAX( N, VR, 1 ) CALL SSCAL( N, ONE / ABS( VR( I ) ), VR, 1 ) ELSE * * Complex eigenvalue. * IF( NOINIT ) THEN * * Set initial vector. * DO 130 I = 1, N VR( I ) = EPS3 VI( I ) = ZERO 130 CONTINUE ELSE * * Scale supplied initial vector. * NORM = SLAPY2( SNRM2( N, VR, 1 ), SNRM2( N, VI, 1 ) ) REC = ( EPS3*ROOTN ) / MAX( NORM, NRMSML ) CALL SSCAL( N, REC, VR, 1 ) CALL SSCAL( N, REC, VI, 1 ) END IF * IF( RIGHTV ) THEN * * LU decomposition with partial pivoting of B, replacing zero * pivots by EPS3. * * The imaginary part of the (i,j)-th element of U is stored in * B(j+1,i). * B( 2, 1 ) = -WI DO 140 I = 2, N B( I+1, 1 ) = ZERO 140 CONTINUE * DO 170 I = 1, N - 1 ABSBII = SLAPY2( B( I, I ), B( I+1, I ) ) EI = H( I+1, I ) IF( ABSBII.LT.ABS( EI ) ) THEN * * Interchange rows and eliminate. * XR = B( I, I ) / EI XI = B( I+1, I ) / EI B( I, I ) = EI B( I+1, I ) = ZERO DO 150 J = I + 1, N TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - XR*TEMP B( J+1, I+1 ) = B( J+1, I ) - XI*TEMP B( I, J ) = TEMP B( J+1, I ) = ZERO 150 CONTINUE B( I+2, I ) = -WI B( I+1, I+1 ) = B( I+1, I+1 ) - XI*WI B( I+2, I+1 ) = B( I+2, I+1 ) + XR*WI ELSE * * Eliminate without interchanging rows. * IF( ABSBII.EQ.ZERO ) THEN B( I, I ) = EPS3 B( I+1, I ) = ZERO ABSBII = EPS3 END IF EI = ( EI / ABSBII ) / ABSBII XR = B( I, I )*EI XI = -B( I+1, I )*EI DO 160 J = I + 1, N B( I+1, J ) = B( I+1, J ) - XR*B( I, J ) + $ XI*B( J+1, I ) B( J+1, I+1 ) = -XR*B( J+1, I ) - XI*B( I, J ) 160 CONTINUE B( I+2, I+1 ) = B( I+2, I+1 ) - WI END IF * * Compute 1-norm of offdiagonal elements of i-th row. * WORK( I ) = SASUM( N-I, B( I, I+1 ), LDB ) + $ SASUM( N-I, B( I+2, I ), 1 ) 170 CONTINUE IF( B( N, N ).EQ.ZERO .AND. B( N+1, N ).EQ.ZERO ) $ B( N, N ) = EPS3 WORK( N ) = ZERO * I1 = N I2 = 1 I3 = -1 ELSE * * UL decomposition with partial pivoting of conjg(B), * replacing zero pivots by EPS3. * * The imaginary part of the (i,j)-th element of U is stored in * B(j+1,i). * B( N+1, N ) = WI DO 180 J = 1, N - 1 B( N+1, J ) = ZERO 180 CONTINUE * DO 210 J = N, 2, -1 EJ = H( J, J-1 ) ABSBJJ = SLAPY2( B( J, J ), B( J+1, J ) ) IF( ABSBJJ.LT.ABS( EJ ) ) THEN * * Interchange columns and eliminate * XR = B( J, J ) / EJ XI = B( J+1, J ) / EJ B( J, J ) = EJ B( J+1, J ) = ZERO DO 190 I = 1, J - 1 TEMP = B( I, J-1 ) B( I, J-1 ) = B( I, J ) - XR*TEMP B( J, I ) = B( J+1, I ) - XI*TEMP B( I, J ) = TEMP B( J+1, I ) = ZERO 190 CONTINUE B( J+1, J-1 ) = WI B( J-1, J-1 ) = B( J-1, J-1 ) + XI*WI B( J, J-1 ) = B( J, J-1 ) - XR*WI ELSE * * Eliminate without interchange. * IF( ABSBJJ.EQ.ZERO ) THEN B( J, J ) = EPS3 B( J+1, J ) = ZERO ABSBJJ = EPS3 END IF EJ = ( EJ / ABSBJJ ) / ABSBJJ XR = B( J, J )*EJ XI = -B( J+1, J )*EJ DO 200 I = 1, J - 1 B( I, J-1 ) = B( I, J-1 ) - XR*B( I, J ) + $ XI*B( J+1, I ) B( J, I ) = -XR*B( J+1, I ) - XI*B( I, J ) 200 CONTINUE B( J, J-1 ) = B( J, J-1 ) + WI END IF * * Compute 1-norm of offdiagonal elements of j-th column. * WORK( J ) = SASUM( J-1, B( 1, J ), 1 ) + $ SASUM( J-1, B( J+1, 1 ), LDB ) 210 CONTINUE IF( B( 1, 1 ).EQ.ZERO .AND. B( 2, 1 ).EQ.ZERO ) $ B( 1, 1 ) = EPS3 WORK( 1 ) = ZERO * I1 = 1 I2 = N I3 = 1 END IF * DO 270 ITS = 1, N SCALE = ONE VMAX = ONE VCRIT = BIGNUM * * Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector, * or U**T*(xr,xi) = scale*(vr,vi) for a left eigenvector, * overwriting (xr,xi) on (vr,vi). * DO 250 I = I1, I2, I3 * IF( WORK( I ).GT.VCRIT ) THEN REC = ONE / VMAX CALL SSCAL( N, REC, VR, 1 ) CALL SSCAL( N, REC, VI, 1 ) SCALE = SCALE*REC VMAX = ONE VCRIT = BIGNUM END IF * XR = VR( I ) XI = VI( I ) IF( RIGHTV ) THEN DO 220 J = I + 1, N XR = XR - B( I, J )*VR( J ) + B( J+1, I )*VI( J ) XI = XI - B( I, J )*VI( J ) - B( J+1, I )*VR( J ) 220 CONTINUE ELSE DO 230 J = 1, I - 1 XR = XR - B( J, I )*VR( J ) + B( I+1, J )*VI( J ) XI = XI - B( J, I )*VI( J ) - B( I+1, J )*VR( J ) 230 CONTINUE END IF * W = ABS( B( I, I ) ) + ABS( B( I+1, I ) ) IF( W.GT.SMLNUM ) THEN IF( W.LT.ONE ) THEN W1 = ABS( XR ) + ABS( XI ) IF( W1.GT.W*BIGNUM ) THEN REC = ONE / W1 CALL SSCAL( N, REC, VR, 1 ) CALL SSCAL( N, REC, VI, 1 ) XR = VR( I ) XI = VI( I ) SCALE = SCALE*REC VMAX = VMAX*REC END IF END IF * * Divide by diagonal element of B. * CALL SLADIV( XR, XI, B( I, I ), B( I+1, I ), VR( I ), $ VI( I ) ) VMAX = MAX( ABS( VR( I ) )+ABS( VI( I ) ), VMAX ) VCRIT = BIGNUM / VMAX ELSE DO 240 J = 1, N VR( J ) = ZERO VI( J ) = ZERO 240 CONTINUE VR( I ) = ONE VI( I ) = ONE SCALE = ZERO VMAX = ONE VCRIT = BIGNUM END IF 250 CONTINUE * * Test for sufficient growth in the norm of (VR,VI). * VNORM = SASUM( N, VR, 1 ) + SASUM( N, VI, 1 ) IF( VNORM.GE.GROWTO*SCALE ) $ GO TO 280 * * Choose a new orthogonal starting vector and try again. * Y = EPS3 / ( ROOTN+ONE ) VR( 1 ) = EPS3 VI( 1 ) = ZERO * DO 260 I = 2, N VR( I ) = Y VI( I ) = ZERO 260 CONTINUE VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN 270 CONTINUE * * Failure to find eigenvector in N iterations * INFO = 1 * 280 CONTINUE * * Normalize eigenvector. * VNORM = ZERO DO 290 I = 1, N VNORM = MAX( VNORM, ABS( VR( I ) )+ABS( VI( I ) ) ) 290 CONTINUE CALL SSCAL( N, ONE / VNORM, VR, 1 ) CALL SSCAL( N, ONE / VNORM, VI, 1 ) * END IF * RETURN * * End of SLAEIN * END |