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| 1 | <?php |
| 2 | /** |
| 3 | * Jingga |
| 4 | * |
| 5 | * PHP Version 8.1 |
| 6 | * |
| 7 | * @package phpOMS\Math\Matrix |
| 8 | * @copyright Dennis Eichhorn |
| 9 | * @copyright JAMA - https://math.nist.gov/javanumerics/jama/ |
| 10 | * @license OMS License 2.0 |
| 11 | * @version 1.0.0 |
| 12 | * @link https://jingga.app |
| 13 | */ |
| 14 | declare(strict_types=1); |
| 15 | |
| 16 | namespace phpOMS\Math\Matrix; |
| 17 | |
| 18 | use phpOMS\Math\Geometry\Shape\D2\Triangle; |
| 19 | |
| 20 | /** |
| 21 | * Eigenvalue decomposition |
| 22 | * |
| 23 | * A symmetric then A = V*D*V' |
| 24 | * A not symmetric then (potentially) A = V*D*inverse(V) |
| 25 | * |
| 26 | * @package phpOMS\Math\Matrix |
| 27 | * @license OMS License 2.0 |
| 28 | * @link https://jingga.app |
| 29 | * @since 1.0.0 |
| 30 | */ |
| 31 | final class EigenvalueDecomposition |
| 32 | { |
| 33 | /** |
| 34 | * Epsilon for float comparison. |
| 35 | * |
| 36 | * @var float |
| 37 | * @since 1.0.0 |
| 38 | */ |
| 39 | public const EPSILON = 4.88e-04; |
| 40 | |
| 41 | /** |
| 42 | * Dimension m |
| 43 | * |
| 44 | * @var int |
| 45 | * @since 1.0.0 |
| 46 | */ |
| 47 | private int $m = 0; |
| 48 | |
| 49 | /** |
| 50 | * Is symmetric |
| 51 | * |
| 52 | * @var bool |
| 53 | * @since 1.0.0 |
| 54 | */ |
| 55 | private bool $isSymmetric = true; |
| 56 | |
| 57 | /** |
| 58 | * A square matrix. |
| 59 | * |
| 60 | * @var array |
| 61 | * @since 1.0.0 |
| 62 | */ |
| 63 | private array $A = []; |
| 64 | |
| 65 | /** |
| 66 | * Eigenvectors |
| 67 | * |
| 68 | * @var array |
| 69 | * @since 1.0.0 |
| 70 | */ |
| 71 | private array $V = []; |
| 72 | |
| 73 | /** |
| 74 | * Eigenvalues |
| 75 | * |
| 76 | * @var array |
| 77 | * @since 1.0.0 |
| 78 | */ |
| 79 | private array $D = []; |
| 80 | |
| 81 | /** |
| 82 | * Eigenvalues |
| 83 | * |
| 84 | * @var array |
| 85 | * @since 1.0.0 |
| 86 | */ |
| 87 | private array $E = []; |
| 88 | |
| 89 | /** |
| 90 | * Hessenberg form |
| 91 | * |
| 92 | * @var array |
| 93 | * @since 1.0.0 |
| 94 | */ |
| 95 | private array $H = []; |
| 96 | |
| 97 | /** |
| 98 | * Non-symmetric storage |
| 99 | * |
| 100 | * @var array |
| 101 | * @since 1.0.0 |
| 102 | */ |
| 103 | private array $ort = []; |
| 104 | |
| 105 | /** |
| 106 | * Complex scalar division |
| 107 | * |
| 108 | * @var float |
| 109 | * @since 1.0.0 |
| 110 | */ |
| 111 | private float $cdivr = 0.0; |
| 112 | |
| 113 | /** |
| 114 | * Complex scalar division |
| 115 | * |
| 116 | * @var float |
| 117 | * @since 1.0.0 |
| 118 | */ |
| 119 | private float $cdivi = 0.0; |
| 120 | |
| 121 | /** |
| 122 | * Constructor. |
| 123 | * |
| 124 | * @param Matrix $M Matrix |
| 125 | * |
| 126 | * @since 1.0.0 |
| 127 | */ |
| 128 | public function __construct(Matrix $M) |
| 129 | { |
| 130 | $this->m = $M->getM(); |
| 131 | $this->A = $M->toArray(); |
| 132 | |
| 133 | for ($j = 0; ($j < $this->m) & $this->isSymmetric; ++$j) { |
| 134 | for ($i = 0; ($i < $this->m) & $this->isSymmetric; ++$i) { |
| 135 | $this->isSymmetric = ($this->A[$i][$j] === $this->A[$j][$i]); |
| 136 | } |
| 137 | } |
| 138 | |
| 139 | if ($this->isSymmetric) { |
| 140 | $this->V = $this->A; |
| 141 | |
| 142 | $this->tred2(); |
| 143 | $this->tql2(); |
| 144 | } else { |
| 145 | $this->H = $this->A; |
| 146 | |
| 147 | $this->orthes(); |
| 148 | $this->hqr2(); |
| 149 | } |
| 150 | } |
| 151 | |
| 152 | /** |
| 153 | * Housholder tridiagonal form reduction. |
| 154 | * |
| 155 | * @return void |
| 156 | * |
| 157 | * @since 1.0.0 |
| 158 | */ |
| 159 | private function tred2() : void |
| 160 | { |
| 161 | for ($j = 0; $j < $this->m; ++$j) { |
| 162 | $this->D[$j] = $this->V[$this->m - 1][$j]; |
| 163 | } |
| 164 | |
| 165 | for ($i = $this->m - 1; $i > 0; --$i) { |
| 166 | $scale = 0.0; |
| 167 | $h = 0.0; |
| 168 | |
| 169 | for ($k = 0; $k < $i; ++$k) { |
| 170 | $scale += \abs($this->D[$k]); |
| 171 | } |
| 172 | |
| 173 | if ($scale == 0) { |
| 174 | $this->E[$i] = $this->D[$i - 1]; |
| 175 | |
| 176 | /* @phpstan-ignore-next-line */ |
| 177 | for ($j = 0; $j > $i; ++$j) { |
| 178 | $this->D[$j] = $this->V[$i - 1][$j]; |
| 179 | $this->V[$i][$j] = 0.0; |
| 180 | $this->V[$j][$i] = 0.0; |
| 181 | } |
| 182 | } else { |
| 183 | for ($k = 0; $k < $i; ++$k) { |
| 184 | $this->D[$k] /= $scale; |
| 185 | $h += $this->D[$k] * $this->D[$k]; |
| 186 | } |
| 187 | |
| 188 | $f = $this->D[$i - 1]; |
| 189 | $g = $f > 0 ? -\sqrt($h) : \sqrt($h); |
| 190 | |
| 191 | $this->E[$i] = $scale * $g; |
| 192 | $h -= $f * $g; |
| 193 | $this->D[$i - 1] = $f - $g; |
| 194 | |
| 195 | for ($j = 0; $j < $i; ++$j) { |
| 196 | $this->E[$j] = 0.0; |
| 197 | } |
| 198 | |
| 199 | for ($j = 0; $j < $i; ++$j) { |
| 200 | $f = $this->D[$j]; |
| 201 | $this->V[$j][$i] = $f; |
| 202 | $g = $this->E[$j] + $this->V[$j][$j] * $f; |
| 203 | |
| 204 | for ($k = $j + 1; $k < $i; ++$k) { |
| 205 | $g += $this->V[$k][$j] * $this->D[$k]; |
| 206 | $this->E[$k] += $this->V[$k][$j] * $f; |
| 207 | } |
| 208 | |
| 209 | $this->E[$j] = $g; |
| 210 | } |
| 211 | |
| 212 | $f = 0.0; |
| 213 | for ($j = 0; $j < $i; ++$j) { |
| 214 | $this->E[$j] /= $h; |
| 215 | $f += $this->E[$j] * $this->D[$j]; |
| 216 | } |
| 217 | |
| 218 | $hh = $f / ($h + $h); |
| 219 | for ($j = 0; $j < $i; ++$j) { |
| 220 | $this->E[$j] -= $hh * $this->D[$j]; |
| 221 | } |
| 222 | |
| 223 | for ($j = 0; $j < $i; ++$j) { |
| 224 | $f = $this->D[$j]; |
| 225 | $g = $this->E[$j]; |
| 226 | |
| 227 | for ($k = $j; $k < $i; ++$k) { |
| 228 | $this->V[$k][$j] -= ($f * $this->E[$k] + $g * $this->D[$k]); |
| 229 | } |
| 230 | |
| 231 | $this->D[$j] = $this->V[$i - 1][$j]; |
| 232 | $this->V[$i][$j] = 0.0; |
| 233 | } |
| 234 | } |
| 235 | |
| 236 | $this->D[$i] = $h; |
| 237 | } |
| 238 | |
| 239 | for ($i = 0; $i < $this->m - 1; ++$i) { |
| 240 | $this->V[$this->m - 1][$i] = $this->V[$i][$i]; |
| 241 | $this->V[$i][$i] = 1.0; |
| 242 | $h = $this->D[$i + 1]; |
| 243 | |
| 244 | if ($h != 0) { |
| 245 | for ($k = 0; $k <= $i; ++$k) { |
| 246 | $this->D[$k] = $this->V[$k][$i + 1] / $h; |
| 247 | } |
| 248 | |
| 249 | for ($j = 0; $j <= $i; ++$j) { |
| 250 | $g = 0.0; |
| 251 | for ($k = 0; $k <= $i; ++$k) { |
| 252 | $g += $this->V[$k][$i + 1] * $this->V[$k][$j]; |
| 253 | } |
| 254 | |
| 255 | for ($k = 0; $k <= $i; ++$k) { |
| 256 | $this->V[$k][$j] -= $g * $this->D[$k]; |
| 257 | } |
| 258 | } |
| 259 | } |
| 260 | |
| 261 | for ($k = 0; $k <= $i; ++$k) { |
| 262 | $this->V[$k][$i + 1] = 0.0; |
| 263 | } |
| 264 | } |
| 265 | |
| 266 | for ($j = 0; $j < $this->m; ++$j) { |
| 267 | $this->D[$j] = $this->V[$this->m - 1][$j]; |
| 268 | $this->V[$this->m - 1][$j] = 0.0; |
| 269 | } |
| 270 | |
| 271 | $this->V[$this->m - 1][$this->m - 1] = 1.0; |
| 272 | $this->E[0] = 0.0; |
| 273 | } |
| 274 | |
| 275 | /** |
| 276 | * Symmetric tridiagonal QL algorithm |
| 277 | * |
| 278 | * @return void |
| 279 | * |
| 280 | * @since 1.0.0 |
| 281 | */ |
| 282 | private function tql2() : void |
| 283 | { |
| 284 | for ($i = 1; $i < $this->m; ++$i) { |
| 285 | $this->E[$i - 1] = $this->E[$i]; |
| 286 | } |
| 287 | |
| 288 | $this->E[$this->m - 1] = 0.0; |
| 289 | |
| 290 | $f = 0.0; |
| 291 | $tst1 = 0.0; |
| 292 | |
| 293 | for ($l = 0; $l < $this->m; ++$l) { |
| 294 | $tst1 = \max($tst1, \abs($this->D[$l]) + \abs($this->E[$l])); |
| 295 | $m = $l; |
| 296 | |
| 297 | while ($m < $this->m) { |
| 298 | if (\abs($this->E[$m]) <= self::EPSILON * $tst1) { |
| 299 | break; |
| 300 | } |
| 301 | |
| 302 | ++$m; |
| 303 | } |
| 304 | |
| 305 | if ($m > $l) { |
| 306 | $iter = 0; |
| 307 | |
| 308 | do { |
| 309 | ++$iter; |
| 310 | |
| 311 | $g = $this->D[$l]; |
| 312 | $p = ($this->D[$l + 1] - $g) / (2.0 * $this->E[$l]); |
| 313 | $r = $p < 0 ? -Triangle::getHypot($p, 1) : Triangle::getHypot($p, 1); |
| 314 | |
| 315 | $this->D[$l] = $this->E[$l] / ($p + $r); |
| 316 | $this->D[$l + 1] = $this->E[$l] * ($p + $r); |
| 317 | $dl1 = $this->D[$l + 1]; |
| 318 | $h = $g - $this->D[$l]; |
| 319 | |
| 320 | for ($i = $l + 2; $i < $this->m; ++$i) { |
| 321 | $this->D[$i] -= $h; |
| 322 | } |
| 323 | |
| 324 | $f += $h; |
| 325 | $p = $this->D[$m]; |
| 326 | $c = 1.0; |
| 327 | $c2 = 1.0; |
| 328 | $c3 = 1.0; |
| 329 | $el1 = $this->E[$l + 1]; |
| 330 | $s = 0.0; |
| 331 | $s2 = 0.0; |
| 332 | |
| 333 | for ($i = $m - 1; $i >= $l; --$i) { |
| 334 | $c3 = $c2; |
| 335 | $c2 = $c; |
| 336 | $s2 = $s; |
| 337 | $g = $c * $this->E[$i]; |
| 338 | $h = $c * $p; |
| 339 | $r = Triangle::getHypot($p, $this->E[$i]); |
| 340 | $this->E[$i + 1] = $s * $r; |
| 341 | $s = $this->E[$i] / $r; |
| 342 | $c = $p / $r; |
| 343 | $p = $c * $this->D[$i] - $s * $g; |
| 344 | $this->D[$i + 1] = $h + $s * ($c * $g + $s * $this->D[$i]); |
| 345 | |
| 346 | for ($k = 0; $k < $this->m; ++$k) { |
| 347 | $h = $this->V[$k][$i + 1]; |
| 348 | $this->V[$k][$i + 1] = $s * $this->V[$k][$i] + $c * $h; |
| 349 | $this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h; |
| 350 | } |
| 351 | } |
| 352 | |
| 353 | $p = -$s * $s2 * $c3 * $el1 * $this->E[$l] / $dl1; |
| 354 | $this->E[$l] = $s * $p; |
| 355 | $this->D[$l] = $c * $p; |
| 356 | } while (\abs($this->E[$l]) > self::EPSILON * $tst1); |
| 357 | } |
| 358 | |
| 359 | $this->D[$l] += $f; |
| 360 | $this->E[$l] = 0.0; |
| 361 | } |
| 362 | |
| 363 | for ($i = 0; $i < $this->m - 1; ++$i) { |
| 364 | $k = $i; |
| 365 | $p = $this->D[$i]; |
| 366 | |
| 367 | for ($j = $i + 1; $j < $this->m; ++$j) { |
| 368 | if ($this->D[$j] < $p) { |
| 369 | $k = $j; |
| 370 | $p = $this->D[$j]; |
| 371 | } |
| 372 | } |
| 373 | |
| 374 | if ($k !== $i) { |
| 375 | $this->D[$k] = $this->D[$i]; |
| 376 | $this->D[$i] = $p; |
| 377 | |
| 378 | for ($j = 0; $j < $this->m; ++$j) { |
| 379 | $p = $this->V[$j][$i]; |
| 380 | $this->V[$j][$i] = $this->V[$j][$k]; |
| 381 | $this->V[$j][$k] = $p; |
| 382 | } |
| 383 | } |
| 384 | } |
| 385 | } |
| 386 | |
| 387 | /** |
| 388 | * Create the orthogonal eigenvectors |
| 389 | * |
| 390 | * @return void |
| 391 | * |
| 392 | * @since 1.0.0 |
| 393 | */ |
| 394 | private function orthes() : void |
| 395 | { |
| 396 | $low = 0; |
| 397 | $high = $this->m - 1; |
| 398 | |
| 399 | for ($m = $low + 1; $m < $high; ++$m) { |
| 400 | $scale = 0.0; |
| 401 | |
| 402 | for ($i = $m; $i <= $high; ++$i) { |
| 403 | $scale += \abs($this->H[$i][$m - 1]); |
| 404 | } |
| 405 | |
| 406 | if ($scale != 0) { |
| 407 | $h = 0.0; |
| 408 | for ($i = $high; $i >= $m; --$i) { |
| 409 | $this->ort[$i] = $this->H[$i][$m - 1] / $scale; |
| 410 | $h += $this->ort[$i] * $this->ort[$i]; |
| 411 | } |
| 412 | |
| 413 | $g = $this->ort[$m] > 0 ? -\sqrt($h) : \sqrt($h); |
| 414 | $h -= $this->ort[$m] * $g; |
| 415 | $this->ort[$m] -= $g; |
| 416 | |
| 417 | for ($j = $m; $j < $this->m; ++$j) { |
| 418 | $f = 0.0; |
| 419 | for ($i = $high; $i >= $m; --$i) { |
| 420 | $f += $this->ort[$i] * $this->H[$i][$j]; |
| 421 | } |
| 422 | |
| 423 | $f /= $h; |
| 424 | for ($i = $m; $i <= $high; ++$i) { |
| 425 | $this->H[$i][$j] -= $f * $this->ort[$i]; |
| 426 | } |
| 427 | } |
| 428 | |
| 429 | for ($i = 0; $i <= $high; ++$i) { |
| 430 | $f = 0.0; |
| 431 | for ($j = $high; $j >= $m; --$j) { |
| 432 | $f += $this->ort[$j] * $this->H[$i][$j]; |
| 433 | } |
| 434 | |
| 435 | $f /= $h; |
| 436 | for ($j = $m; $j <= $high; ++$j) { |
| 437 | $this->H[$i][$j] -= $f * $this->ort[$j]; |
| 438 | } |
| 439 | } |
| 440 | |
| 441 | $this->ort[$m] *= $scale; |
| 442 | $this->H[$m][$m - 1] = $scale * $g; |
| 443 | } |
| 444 | } |
| 445 | |
| 446 | for ($i = 0; $i < $this->m; ++$i) { |
| 447 | for ($j = 0; $j < $this->m; ++$j) { |
| 448 | $this->V[$i][$j] = $i === $j ? 1.0 : 0.0; |
| 449 | } |
| 450 | } |
| 451 | |
| 452 | for ($m = $high - 1; $m > $low; --$m) { |
| 453 | if ($this->H[$m][$m - 1] != 0) { |
| 454 | for ($i = $m + 1; $i <= $high; ++$i) { |
| 455 | $this->ort[$i] = $this->H[$i][$m - 1]; |
| 456 | } |
| 457 | |
| 458 | for ($j = $m; $j <= $high; ++$j) { |
| 459 | $g = 0.0; |
| 460 | for ($i = $m; $i <= $high; ++$i) { |
| 461 | $g += $this->ort[$i] * $this->V[$i][$j]; |
| 462 | } |
| 463 | |
| 464 | $g = ($g / $this->ort[$m]) / $this->H[$m][$m - 1]; |
| 465 | for ($i = $m; $i <= $high; ++$i) { |
| 466 | $this->V[$i][$j] += $g * $this->ort[$i]; |
| 467 | } |
| 468 | } |
| 469 | } |
| 470 | } |
| 471 | } |
| 472 | |
| 473 | /** |
| 474 | * Perform complex division |
| 475 | * |
| 476 | * @param float $xr Real value |
| 477 | * @param float $xi Imaginary value |
| 478 | * @param float $yr Real value |
| 479 | * @param float $yi Imaginary value |
| 480 | * |
| 481 | * @return void |
| 482 | * |
| 483 | * @since 1.0.0 |
| 484 | */ |
| 485 | private function cdiv(float $xr, float $xi, float $yr, float $yi) : void |
| 486 | { |
| 487 | $r = 0.0; |
| 488 | $d = 0.0; |
| 489 | |
| 490 | if (\abs($yr) > \abs($yi)) { |
| 491 | $r = $yi / $yr; |
| 492 | $d = $yr + $r * $yi; |
| 493 | |
| 494 | $this->cdivr = ($xr + $r * $xi) / $d; |
| 495 | $this->cdivi = ($xi - $r * $xr) / $d; |
| 496 | } else { |
| 497 | $r = $yr / $yi; |
| 498 | $d = $yi + $r * $yr; |
| 499 | |
| 500 | $this->cdivr = ($r * $xr + $xi) / $d; |
| 501 | $this->cdivi = ($r * $xi - $xr) / $d; |
| 502 | } |
| 503 | } |
| 504 | |
| 505 | /** |
| 506 | * QR algorithm |
| 507 | * |
| 508 | * @return void |
| 509 | * |
| 510 | * @since 1.0.0 |
| 511 | */ |
| 512 | private function hqr2() : void |
| 513 | { |
| 514 | $nn = $this->m; |
| 515 | $n = $nn - 1; |
| 516 | $low = 0; |
| 517 | $high = $nn - 1; |
| 518 | $exshift = 0.0; |
| 519 | $p = 0; |
| 520 | $q = 0; |
| 521 | $r = 0; |
| 522 | $s = 0; |
| 523 | $z = 0; |
| 524 | $norm = 0.0; |
| 525 | |
| 526 | for ($i = 0; $i < $nn; ++$i) { |
| 527 | /* @phpstan-ignore-next-line */ |
| 528 | if ($i < $low || $i > $high) { |
| 529 | $this->D[$i] = $this->H[$i][$i]; |
| 530 | $this->E[$i] = 0.0; |
| 531 | } |
| 532 | |
| 533 | for ($j = \max($i - 1, 0); $j < $nn; ++$j) { |
| 534 | $norm += \abs($this->H[$i][$j]); |
| 535 | } |
| 536 | } |
| 537 | |
| 538 | $iter = 0; |
| 539 | while ($n >= $low) { |
| 540 | $l = $n; |
| 541 | while ($l > $low) { |
| 542 | $s = \abs($this->H[$l - 1][$l - 1]) + \abs($this->H[$l][$l]); |
| 543 | if ($s == 0) { |
| 544 | $s = $norm; |
| 545 | } |
| 546 | |
| 547 | if (\abs($this->H[$l][$l - 1]) < self::EPSILON * $s) { |
| 548 | break; |
| 549 | } |
| 550 | |
| 551 | --$l; |
| 552 | } |
| 553 | |
| 554 | if ($l === $n) { |
| 555 | $this->H[$n][$n] += $exshift; |
| 556 | $this->D[$n] = $this->H[$n][$n]; |
| 557 | $this->E[$n] = 0.0; |
| 558 | $iter = 0; |
| 559 | |
| 560 | --$n; |
| 561 | } elseif ($l === $n - 1) { |
| 562 | $w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n]; |
| 563 | $p = ($this->H[$n - 1][$n - 1] - $this->H[$n][$n]) / 2.0; |
| 564 | $q = $p * $p + $w; |
| 565 | $z = \sqrt(\abs($q)); |
| 566 | |
| 567 | $this->H[$n][$n] += $exshift; |
| 568 | $this->H[$n - 1][$n - 1] += $exshift; |
| 569 | |
| 570 | $x = $this->H[$n][$n]; |
| 571 | |
| 572 | if ($q >= 0) { |
| 573 | $z = $p >= 0 ? $p + $z : $p - $z; |
| 574 | $this->D[$n - 1] = $x + $z; |
| 575 | $this->D[$n] = $z != 0 ? $x - $w / $z : $this->D[$n - 1]; |
| 576 | $this->E[$n - 1] = 0.0; |
| 577 | $this->E[$n] = 0.0; |
| 578 | |
| 579 | $x = $this->H[$n][$n - 1]; |
| 580 | $s = \abs($x) + \abs($z); |
| 581 | $p = $x / $s; |
| 582 | $q = $z / $s; |
| 583 | $r = \sqrt($p * $p + $q * $q); |
| 584 | $p /= $r; |
| 585 | $q /= $r; |
| 586 | |
| 587 | for ($j = $n - 1; $j < $nn; ++$j) { |
| 588 | $z = $this->H[$n - 1][$j]; |
| 589 | $this->H[$n - 1][$j] = $q * $z + $p * $this->H[$n][$j]; |
| 590 | $this->H[$n][$j] = $q * $this->H[$n][$j] - $p * $z; |
| 591 | } |
| 592 | |
| 593 | for ($i = 0; $i <= $n; ++$i) { |
| 594 | $z = $this->H[$i][$n - 1]; |
| 595 | $this->H[$i][$n - 1] = $q * $z + $p * $this->H[$i][$n]; |
| 596 | $this->H[$i][$n] = $q * $this->H[$i][$n] - $p * $z; |
| 597 | } |
| 598 | |
| 599 | for ($i = $low; $i <= $high; ++$i) { |
| 600 | $z = $this->V[$i][$n - 1]; |
| 601 | $this->V[$i][$n - 1] = $q * $z + $p * $this->V[$i][$n]; |
| 602 | $this->V[$i][$n] = $q * $this->V[$i][$n] - $p * $z; |
| 603 | } |
| 604 | } else { |
| 605 | $this->D[$n - 1] = $x + $p; |
| 606 | $this->D[$n] = $x + $p; |
| 607 | $this->E[$n - 1] = $z; |
| 608 | $this->E[$n] = -$z; |
| 609 | } |
| 610 | |
| 611 | $n -= 2; |
| 612 | $iter = 0; |
| 613 | } else { |
| 614 | $x = $this->H[$n][$n]; |
| 615 | $y = 0.0; |
| 616 | $w = 0.0; |
| 617 | |
| 618 | if ($l < $n) { |
| 619 | $y = $this->H[$n - 1][$n - 1]; |
| 620 | $w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n]; |
| 621 | } |
| 622 | |
| 623 | if ($iter === 10) { |
| 624 | $exshift += $x; |
| 625 | for ($i = $low; $i <= $n; ++$i) { |
| 626 | $this->H[$i][$i] -= $x; |
| 627 | } |
| 628 | |
| 629 | $s = \abs($this->H[$n][$n - 1]) + \abs($this->H[$n - 1][$n - 2]); |
| 630 | $x = 0.75 * $s; |
| 631 | $y = $x; |
| 632 | $w = -0.4375 * $s * $s; |
| 633 | } |
| 634 | |
| 635 | if ($iter === 30) { |
| 636 | $s = ($y - $x) / 2.0; |
| 637 | $s = $s * $s + $w; |
| 638 | |
| 639 | if ($s > 0) { |
| 640 | $s = $y < $x ? -\sqrt($s) : \sqrt($s); |
| 641 | $s = $x - $w / (($y - $x) / 2.0 + $s); |
| 642 | |
| 643 | for ($i = $low; $i <= $n; ++$i) { |
| 644 | $this->H[$i][$i] -= $s; |
| 645 | } |
| 646 | |
| 647 | $exshift += $s; |
| 648 | $x = $y = $w = 0.964; |
| 649 | } |
| 650 | } |
| 651 | |
| 652 | ++$iter; |
| 653 | $m = $n - 2; |
| 654 | |
| 655 | while ($m >= $l) { |
| 656 | $z = $this->H[$m][$m]; |
| 657 | $r = $x - $z; |
| 658 | $s = $y - $z; |
| 659 | $p = ($r * $s - $w) / $this->H[$m + 1][$m] + $this->H[$m][$m + 1]; |
| 660 | $q = $this->H[$m + 1][$m + 1] - $z - $r - $s; |
| 661 | $r = $this->H[$m + 2][$m + 1]; |
| 662 | $s = \abs($p) + \abs($q) + \abs($r); |
| 663 | $p /= $s; |
| 664 | $q /= $s; |
| 665 | $r /= $s; |
| 666 | |
| 667 | if ($m === $l |
| 668 | || \abs($this->H[$m][$m - 1]) * (\abs($q) + \abs($r)) < self::EPSILON * (\abs($p) * (\abs($this->H[$m - 1][$m - 1]) + \abs($z) + \abs($this->H[$m + 1][$m + 1]))) |
| 669 | ) { |
| 670 | break; |
| 671 | } |
| 672 | |
| 673 | --$m; |
| 674 | } |
| 675 | |
| 676 | for ($i = $m + 2; $i <= $n; ++$i) { |
| 677 | $this->H[$i][$i - 2] = 0.0; |
| 678 | |
| 679 | if ($i > $m + 2) { |
| 680 | $this->H[$i][$i - 3] = 0.0; |
| 681 | } |
| 682 | } |
| 683 | |
| 684 | for ($k = $m; $k < $n; ++$k) { |
| 685 | $notlast = ($k !== $n - 1); |
| 686 | |
| 687 | if ($k !== $m) { |
| 688 | $p = $this->H[$k][$k - 1]; |
| 689 | $q = $this->H[$k + 1][$k - 1]; |
| 690 | $r = ($notlast ? $this->H[$k + 2][$k - 1] : 0.0); |
| 691 | $x = \abs($p) + \abs($q) + \abs($r); |
| 692 | |
| 693 | if ($x == 0) { |
| 694 | continue; |
| 695 | } |
| 696 | |
| 697 | $p /= $x; |
| 698 | $q /= $x; |
| 699 | $r /= $x; |
| 700 | } |
| 701 | |
| 702 | $s = $p < 0 ? -\sqrt($p * $p + $q * $q + $r * $r) : \sqrt($p * $p + $q * $q + $r * $r); |
| 703 | |
| 704 | if ($s == 0) { |
| 705 | continue; |
| 706 | } |
| 707 | |
| 708 | if ($k !== $m) { |
| 709 | $this->H[$k][$k - 1] = -$s * $x; |
| 710 | } elseif ($l !== $m) { |
| 711 | $this->H[$k][$k - 1] = -$this->H[$k][$k - 1]; |
| 712 | } |
| 713 | |
| 714 | $p += $s; |
| 715 | $x = $p / $s; |
| 716 | $y = $q / $s; |
| 717 | $z = $r / $s; |
| 718 | $q /= $p; |
| 719 | $r /= $p; |
| 720 | |
| 721 | for ($j = $k; $j < $nn; ++$j) { |
| 722 | $p = $this->H[$k][$j] + $q * $this->H[$k + 1][$j]; |
| 723 | if ($notlast) { |
| 724 | $p += $r * $this->H[$k + 2][$j]; |
| 725 | $this->H[$k + 2][$j] -= $p * $z; |
| 726 | } |
| 727 | |
| 728 | $this->H[$k][$j] -= $p * $x; |
| 729 | $this->H[$k + 1][$j] -= $p * $y; |
| 730 | } |
| 731 | |
| 732 | $min = \min($n, $k + 3); |
| 733 | for ($i = 0; $i <= $min; ++$i) { |
| 734 | $p = $x * $this->H[$i][$k] + $y * $this->H[$i][$k + 1]; |
| 735 | |
| 736 | if ($notlast) { |
| 737 | $p += $z * $this->H[$i][$k + 2]; |
| 738 | $this->H[$i][$k + 2] -= $p * $r; |
| 739 | } |
| 740 | |
| 741 | $this->H[$i][$k] -= $p; |
| 742 | $this->H[$i][$k + 1] -= $p * $q; |
| 743 | } |
| 744 | |
| 745 | for ($i = $low; $i <= $high; ++$i) { |
| 746 | $p = $x * $this->V[$i][$k] + $y * $this->V[$i][$k + 1]; |
| 747 | |
| 748 | if ($notlast) { |
| 749 | $p += $z * $this->V[$i][$k + 2]; |
| 750 | $this->V[$i][$k + 2] -= $p * $r; |
| 751 | } |
| 752 | $this->V[$i][$k] -= $p; |
| 753 | $this->V[$i][$k + 1] -= $p * $q; |
| 754 | } |
| 755 | } |
| 756 | } |
| 757 | } |
| 758 | |
| 759 | if ($norm == 0) { |
| 760 | return; |
| 761 | } |
| 762 | |
| 763 | for ($n = $nn - 1; $n >= 0; --$n) { |
| 764 | $p = $this->D[$n]; |
| 765 | $q = $this->E[$n]; |
| 766 | |
| 767 | if ($q == 0) { |
| 768 | $l = $n; |
| 769 | $this->H[$n][$n] = 1.0; |
| 770 | |
| 771 | for ($i = $n - 1; $i >= 0; --$i) { |
| 772 | $w = $this->H[$i][$i] - $p; |
| 773 | $r = 0.0; |
| 774 | |
| 775 | for ($j = $l; $j <= $n; ++$j) { |
| 776 | $r += $this->H[$i][$j] * $this->H[$j][$n]; |
| 777 | } |
| 778 | |
| 779 | if ($this->E[$i] < 0.0) { |
| 780 | $z = $w; |
| 781 | $s = $r; |
| 782 | } else { |
| 783 | $l = $i; |
| 784 | |
| 785 | if ($this->E[$i] == 0) { |
| 786 | $this->H[$i][$n] = $w != 0 ? -$r / $w : -$r / (self::EPSILON * $norm); |
| 787 | } else { |
| 788 | $x = $this->H[$i][$i + 1]; |
| 789 | $y = $this->H[$i + 1][$i]; |
| 790 | $q = ($this->D[$i] - $p) * ($this->D[$i] - $p) + $this->E[$i] * $this->E[$i]; |
| 791 | $t = ($x * $s - $z * $r) / $q; |
| 792 | $this->H[$i][$n] = $t; |
| 793 | $this->H[$i + 1][$n] = \abs($x) > \abs($z) ? (-$r - $w * $t) / $x : (-$s - $y * $t) / $z; |
| 794 | } |
| 795 | |
| 796 | $t = \abs($this->H[$i][$n]); |
| 797 | if ((self::EPSILON * $t) * $t > 1) { |
| 798 | for ($j = $i; $j <= $n; ++$j) { |
| 799 | $this->H[$j][$n] /= $t; |
| 800 | } |
| 801 | } |
| 802 | } |
| 803 | } |
| 804 | } elseif ($q < 0) { |
| 805 | $l = $n - 1; |
| 806 | |
| 807 | if (\abs($this->H[$n][$n - 1]) > \abs($this->H[$n - 1][$n])) { |
| 808 | $this->H[$n - 1][$n - 1] = $q / $this->H[$n][$n - 1]; |
| 809 | $this->H[$n - 1][$n] = -($this->H[$n][$n] - $p) / $this->H[$n][$n - 1]; |
| 810 | } else { |
| 811 | $this->cdiv(0.0, -$this->H[$n - 1][$n], $this->H[$n - 1][$n - 1] - $p, $q); |
| 812 | $this->H[$n - 1][$n - 1] = $this->cdivr; |
| 813 | $this->H[$n - 1][$n] = $this->cdivi; |
| 814 | } |
| 815 | |
| 816 | $this->H[$n][$n - 1] = 0.0; |
| 817 | $this->H[$n][$n] = 1.0; |
| 818 | |
| 819 | for ($i = $n - 2; $i >= 0; --$i) { |
| 820 | $ra = 0.0; |
| 821 | $sa = 0.0; |
| 822 | |
| 823 | for ($j = $l; $j <= $n; ++$j) { |
| 824 | $ra += $this->H[$i][$j] * $this->H[$j][$n - 1]; |
| 825 | $sa += $this->H[$i][$j] * $this->H[$j][$n]; |
| 826 | } |
| 827 | |
| 828 | $w = $this->H[$i][$i] - $p; |
| 829 | if ($this->E[$i] < 0.0) { |
| 830 | $z = $w; |
| 831 | $r = $ra; |
| 832 | $s = $sa; |
| 833 | } else { |
| 834 | $l = $i; |
| 835 | |
| 836 | if ($this->E[$i] == 0) { |
| 837 | $this->cdiv(-$ra, -$sa, $w, $q); |
| 838 | |
| 839 | $this->H[$i][$n - 1] = $this->cdivr; |
| 840 | $this->H[$i][$n] = $this->cdivi; |
| 841 | } else { |
| 842 | $x = $this->H[$i][$i + 1]; |
| 843 | $y = $this->H[$i + 1][$i]; |
| 844 | $vr = ($this->D[$i] - $p) * ($this->D[$i] - $p) + $this->E[$i] * $this->E[$i] - $q * $q; |
| 845 | $vi = ($this->D[$i] - $p) * 2.0 * $q; |
| 846 | |
| 847 | if (($vr == 0 & $vi == 0) !== 0) { |
| 848 | $vr = self::EPSILON * $norm * (\abs($w) + \abs($q) + \abs($x) + \abs($y) + \abs($z)); |
| 849 | } |
| 850 | |
| 851 | $this->cdiv($x * $r - $z * $ra + $q * $sa, $x * $s - $z * $sa - $q * $ra, $vr, $vi); |
| 852 | |
| 853 | $this->H[$i][$n - 1] = $this->cdivr; |
| 854 | $this->H[$i][$n] = $this->cdivi; |
| 855 | |
| 856 | if (\abs($x) > (\abs($z) + \abs($q))) { |
| 857 | $this->H[$i + 1][$n - 1] = (-$ra - $w * $this->H[$i][$n - 1] + $q * $this->H[$i][$n]) / $x; |
| 858 | $this->H[$i + 1][$n] = (-$sa - $w * $this->H[$i][$n] - $q * $this->H[$i][$n - 1]) / $x; |
| 859 | } else { |
| 860 | $this->cdiv(-$r - $y * $this->H[$i][$n - 1], -$s - $y * $this->H[$i][$n], $z, $q); |
| 861 | $this->H[$i + 1][$n - 1] = $this->cdivr; |
| 862 | $this->H[$i + 1][$n] = $this->cdivi; |
| 863 | } |
| 864 | } |
| 865 | |
| 866 | $t = \max(\abs($this->H[$i][$n - 1]), \abs($this->H[$i][$n])); |
| 867 | if ((self::EPSILON * $t) * $t > 1) { |
| 868 | for ($j = $i; $j <= $n; ++$j) { |
| 869 | $this->H[$j][$n - 1] /= $t; |
| 870 | $this->H[$j][$n] /= $t; |
| 871 | } |
| 872 | } |
| 873 | } |
| 874 | } |
| 875 | } |
| 876 | } |
| 877 | |
| 878 | for ($i = 0; $i < $nn; ++$i) { |
| 879 | /* @phpstan-ignore-next-line */ |
| 880 | if ($i < $low || $i > $high) { |
| 881 | for ($j = $i; $j < $nn; ++$j) { |
| 882 | $this->V[$i][$j] = $this->H[$i][$j]; |
| 883 | } |
| 884 | } |
| 885 | } |
| 886 | |
| 887 | for ($j = $nn - 1; $j >= $low; --$j) { |
| 888 | for ($i = $low; $i <= $high; ++$i) { |
| 889 | $z = 0.0; |
| 890 | |
| 891 | $min = \min($j, $high); |
| 892 | for ($k = $low; $k <= $min; ++$k) { |
| 893 | $z += $this->V[$i][$k] * $this->H[$k][$j]; |
| 894 | } |
| 895 | |
| 896 | $this->V[$i][$j] = $z; |
| 897 | } |
| 898 | } |
| 899 | } |
| 900 | |
| 901 | /** |
| 902 | * Is matrix symmetric? |
| 903 | * |
| 904 | * @return bool |
| 905 | * |
| 906 | * @since 1.0.0 |
| 907 | */ |
| 908 | public function isSymmetric() : bool |
| 909 | { |
| 910 | return $this->isSymmetric; |
| 911 | } |
| 912 | |
| 913 | /** |
| 914 | * Get V matrix |
| 915 | * |
| 916 | * @return Matrix |
| 917 | * |
| 918 | * @since 1.0.0 |
| 919 | */ |
| 920 | public function getV() : Matrix |
| 921 | { |
| 922 | $matrix = new Matrix(); |
| 923 | $matrix->setMatrix($this->V); |
| 924 | |
| 925 | return $matrix; |
| 926 | } |
| 927 | |
| 928 | /** |
| 929 | * Get real eigenvalues |
| 930 | * |
| 931 | * @return Vector |
| 932 | * |
| 933 | * @since 1.0.0 |
| 934 | */ |
| 935 | public function getRealEigenvalues() : Vector |
| 936 | { |
| 937 | $vector = new Vector(); |
| 938 | $vector->setMatrix($this->D); |
| 939 | |
| 940 | return $vector; |
| 941 | } |
| 942 | |
| 943 | /** |
| 944 | * Get imaginary eigenvalues |
| 945 | * |
| 946 | * @return Vector |
| 947 | * |
| 948 | * @since 1.0.0 |
| 949 | */ |
| 950 | public function getImagEigenvalues() : Vector |
| 951 | { |
| 952 | $vector = new Vector(); |
| 953 | $vector->setMatrix($this->E); |
| 954 | |
| 955 | return $vector; |
| 956 | } |
| 957 | |
| 958 | /** |
| 959 | * Get D matrix |
| 960 | * |
| 961 | * @return Matrix |
| 962 | * |
| 963 | * @since 1.0.0 |
| 964 | */ |
| 965 | public function getD() : Matrix |
| 966 | { |
| 967 | $matrix = new Matrix(); |
| 968 | |
| 969 | $D = [[]]; |
| 970 | for ($i = 0; $i < $this->m; ++$i) { |
| 971 | for ($j = 0; $j < $this->m; ++$j) { |
| 972 | $D[$i][$j] = 0.0; |
| 973 | } |
| 974 | |
| 975 | $D[$i][$i] = $this->D[$i]; |
| 976 | if ($this->E[$i] > 0) { |
| 977 | $D[$i][$i + 1] = $this->E[$i]; |
| 978 | } elseif ($this->E[$i] < 0) { |
| 979 | $D[$i][$i - 1] = $this->E[$i]; |
| 980 | } |
| 981 | } |
| 982 | |
| 983 | $matrix->setMatrix($D); |
| 984 | |
| 985 | return $matrix; |
| 986 | } |
| 987 | } |