Passed
03:28
created

#### Upgrade to new PHP Analysis Engine

These results are based on our legacy PHP analysis, consider migrating to our new PHP analysis engine instead. Learn more

1
`<?php`
2
3
`declare(strict_types=1);`
4
5
`/**`
6
` *	Class to obtain eigenvalues and eigenvectors of a real matrix.`
7
` *`
8
` *	If A is symmetric, then A = V*D*V' where the eigenvalue matrix D`
9
` *	is diagonal and the eigenvector matrix V is orthogonal (i.e.`
10
` *	A = V.times(D.times(V.transpose())) and V.times(V.transpose())`
11
` *	equals the identity matrix).`
12
` *`
13
` *	If A is not symmetric, then the eigenvalue matrix D is block diagonal`
14
` *	with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,`
15
` *	lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The`
16
` *	columns of V represent the eigenvectors in the sense that A*V = V*D,`
17
` *	i.e. A.times(V) equals V.times(D).  The matrix V may be badly`
18
` *	conditioned, or even singular, so the validity of the equation`
19
` *	A = V*D*inverse(V) depends upon V.cond().`
20
` *`
21
` *	@author  Paul Meagher`
22
` *	@license PHP v3.0`
23
` *`
24
` *	@version 1.1`
25
` *`
26
` *  Slightly changed to adapt the original code to PHP-ML library`
27
` *  @date 2017/04/11`
28
` *`
29
` *  @author Mustafa Karabulut`
30
` */`
31
32
`namespace Phpml\Math\LinearAlgebra;`
33
34
`use Phpml\Math\Matrix;`
35
36
`class EigenvalueDecomposition`
37
`{`
38
`    /**`
39
`     *	Row and column dimension (square matrix).`
40
`     *`
41
`     *	@var int`
42
`     */`
43
`    private \$n;`
44
45
`    /**`
46
`     *	Internal symmetry flag.`
47
`     *`
48
`     *	@var bool`
49
`     */`
50
`    private \$symmetric;`
51
52
`    /**`
53
`     *	Arrays for internal storage of eigenvalues.`
54
`     *`
55
`     *	@var array`
56
`     */`
57
`    private \$d = [];`
58
59
`    private \$e = [];`
60
61
`    /**`
62
`     *	Array for internal storage of eigenvectors.`
63
`     *`
64
`     *	@var array`
65
`     */`
66
`    private \$V = [];`
67
68
`    /**`
69
`     *	Array for internal storage of nonsymmetric Hessenberg form.`
70
`     *`
71
`     *	@var array`
72
`     */`
73
`    private \$H = [];`
74
75
`    /**`
76
`     *	Working storage for nonsymmetric algorithm.`
77
`     *`
78
`     *	@var array`
79
`     */`
80
`    private \$ort = [];`
81
82
`    /**`
83
`     *	Used for complex scalar division.`
84
`     *`
85
`     *	@var float`
86
`     */`
87
`    private \$cdivr;`
88
89
`    private \$cdivi;`
90
91
`    private \$A;`
92
93
`    /**`
94
`     *	Constructor: Check for symmetry, then construct the eigenvalue decomposition`
95
`     */`
96
`    public function __construct(array \$Arg)`
97
`    {`
98
`        \$this->A = \$Arg;`
99
`        \$this->n = count(\$Arg);`
100
`        \$this->symmetric = true;`
101
102
`        for (\$j = 0; (\$j < \$this->n) && \$this->symmetric; ++\$j) {`
103
`            for (\$i = 0; (\$i < \$this->n) & \$this->symmetric; ++\$i) {`
104
`                \$this->symmetric = (\$this->A[\$i][\$j] == \$this->A[\$j][\$i]);`
105
`            }`
106
`        }`
107
108
`        if (\$this->symmetric) {`
109
`            \$this->V = \$this->A;`
110
`            // Tridiagonalize.`
111
`            \$this->tred2();`
112
`            // Diagonalize.`
113
`            \$this->tql2();`
114
`        } else {`
115
`            \$this->H = \$this->A;`
116
`            \$this->ort = [];`
117
`            // Reduce to Hessenberg form.`
118
`            \$this->orthes();`
119
`            // Reduce Hessenberg to real Schur form.`
120
`            \$this->hqr2();`
121
`        }`
122
`    }`
123
124
`    /**`
125
`     * Return the eigenvector matrix`
126
`     */`
127
`    public function getEigenvectors(): array`
128
`    {`
129
`        \$vectors = \$this->V;`
130
131
`        // Always return the eigenvectors of length 1.0`
132
`        \$vectors = new Matrix(\$vectors);`
133
`        \$vectors = array_map(function (\$vect) {`
134
`            \$sum = 0;`
135 View Code Duplication
`            for (\$i = 0; \$i < count(\$vect); ++\$i) {`
0 ignored issues
show
Performance Best Practice introduced by It seems like you are calling the size function `count()` as part of the test condition. You might want to compute the size beforehand, and not on each iteration.

If the size of the collection does not change during the iteration, it is generally a good practice to compute it beforehand, and not on each iteration:

```for (\$i=0; \$i<count(\$array); \$i++) { // calls count() on each iteration
}

// Better
for (\$i=0, \$c=count(\$array); \$i<\$c; \$i++) { // calls count() just once
}
``` Loading history...
136
`                \$sum += \$vect[\$i] ** 2;`
137
`            }`
138
139
`            \$sum = sqrt(\$sum);`
140 View Code Duplication
`            for (\$i = 0; \$i < count(\$vect); ++\$i) {`
0 ignored issues
show
Performance Best Practice introduced by It seems like you are calling the size function `count()` as part of the test condition. You might want to compute the size beforehand, and not on each iteration.

If the size of the collection does not change during the iteration, it is generally a good practice to compute it beforehand, and not on each iteration:

```for (\$i=0; \$i<count(\$array); \$i++) { // calls count() on each iteration
}

// Better
for (\$i=0, \$c=count(\$array); \$i<\$c; \$i++) { // calls count() just once
}
``` Loading history...
141
`                \$vect[\$i] /= \$sum;`
142
`            }`
143
144
`            return \$vect;`
145
`        }, \$vectors->transpose()->toArray());`
146
147
`        return \$vectors;`
148
`    }`
149
150
`    /**`
151
`     *	Return the real parts of the eigenvalues<br>`
152
`     *  d = real(diag(D));`
153
`     */`
154
`    public function getRealEigenvalues(): array`
155
`    {`
156
`        return \$this->d;`
157
`    }`
158
159
`    /**`
160
`     *	Return the imaginary parts of the eigenvalues <br>`
161
`     *  d = imag(diag(D))`
162
`     */`
163
`    public function getImagEigenvalues(): array`
164
`    {`
165
`        return \$this->e;`
166
`    }`
167
168
`    /**`
169
`     *	Return the block diagonal eigenvalue matrix`
170
`     */`
171
`    public function getDiagonalEigenvalues(): array`
172
`    {`
173
`        \$D = [];`
174
175
`        for (\$i = 0; \$i < \$this->n; ++\$i) {`
176
`            \$D[\$i] = array_fill(0, \$this->n, 0.0);`
177
`            \$D[\$i][\$i] = \$this->d[\$i];`
178
`            if (\$this->e[\$i] == 0) {`
179
`                continue;`
180
`            }`
181
182
`            \$o = (\$this->e[\$i] > 0) ? \$i + 1 : \$i - 1;`
183
`            \$D[\$i][\$o] = \$this->e[\$i];`
184
`        }`
185
186
`        return \$D;`
187
`    }`
188
189
`    /**`
190
`     *	Symmetric Householder reduction to tridiagonal form.`
191
`     */`
192
`    private function tred2(): void`
193
`    {`
194
`        //  This is derived from the Algol procedures tred2 by`
195
`        //  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for`
196
`        //  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding`
197
`        //  Fortran subroutine in EISPACK.`
198
`        \$this->d = \$this->V[\$this->n - 1];`
199
`        // Householder reduction to tridiagonal form.`
200
`        for (\$i = \$this->n - 1; \$i > 0; --\$i) {`
201
`            \$i_ = \$i - 1;`
202
`            // Scale to avoid under/overflow.`
203
`            \$h = \$scale = 0.0;`
204
`            \$scale += array_sum(array_map('abs', \$this->d));`
205
`            if (\$scale == 0.0) {`
206
`                \$this->e[\$i] = \$this->d[\$i_];`
207
`                \$this->d = array_slice(\$this->V[\$i_], 0, \$i_);`
208
`                for (\$j = 0; \$j < \$i; ++\$j) {`
209
`                    \$this->V[\$j][\$i] = \$this->V[\$i][\$j] = 0.0;`
210
`                }`
211
`            } else {`
212
`                // Generate Householder vector.`
213
`                for (\$k = 0; \$k < \$i; ++\$k) {`
214
`                    \$this->d[\$k] /= \$scale;`
215
`                    \$h += pow(\$this->d[\$k], 2);`
216
`                }`
217
218
`                \$f = \$this->d[\$i_];`
219
`                \$g = sqrt(\$h);`
220
`                if (\$f > 0) {`
221
`                    \$g = -\$g;`
222
`                }`
223
224
`                \$this->e[\$i] = \$scale * \$g;`
225
`                \$h = \$h - \$f * \$g;`
226
`                \$this->d[\$i_] = \$f - \$g;`
227
228
`                for (\$j = 0; \$j < \$i; ++\$j) {`
229
`                    \$this->e[\$j] = 0.0;`
230
`                }`
231
232
`                // Apply similarity transformation to remaining columns.`
233
`                for (\$j = 0; \$j < \$i; ++\$j) {`
234
`                    \$f = \$this->d[\$j];`
235
`                    \$this->V[\$j][\$i] = \$f;`
236
`                    \$g = \$this->e[\$j] + \$this->V[\$j][\$j] * \$f;`
237
238
`                    for (\$k = \$j + 1; \$k <= \$i_; ++\$k) {`
239
`                        \$g += \$this->V[\$k][\$j] * \$this->d[\$k];`
240
`                        \$this->e[\$k] += \$this->V[\$k][\$j] * \$f;`
241
`                    }`
242
243
`                    \$this->e[\$j] = \$g;`
244
`                }`
245
246
`                \$f = 0.0;`
247
`                if (\$h === 0 || \$h < 1e-32) {`
248
`                    \$h = 1e-32;`
249
`                }`
250
251
`                for (\$j = 0; \$j < \$i; ++\$j) {`
252
`                    \$this->e[\$j] /= \$h;`
253
`                    \$f += \$this->e[\$j] * \$this->d[\$j];`
254
`                }`
255
256
`                \$hh = \$f / (2 * \$h);`
257
`                for (\$j = 0; \$j < \$i; ++\$j) {`
258
`                    \$this->e[\$j] -= \$hh * \$this->d[\$j];`
259
`                }`
260
261
`                for (\$j = 0; \$j < \$i; ++\$j) {`
262
`                    \$f = \$this->d[\$j];`
263
`                    \$g = \$this->e[\$j];`
264
`                    for (\$k = \$j; \$k <= \$i_; ++\$k) {`
265
`                        \$this->V[\$k][\$j] -= (\$f * \$this->e[\$k] + \$g * \$this->d[\$k]);`
266
`                    }`
267
268
`                    \$this->d[\$j] = \$this->V[\$i - 1][\$j];`
269
`                    \$this->V[\$i][\$j] = 0.0;`
270
`                }`
271
`            }`
272
273
`            \$this->d[\$i] = \$h;`
274
`        }`
275
276
`        // Accumulate transformations.`
277
`        for (\$i = 0; \$i < \$this->n - 1; ++\$i) {`
278
`            \$this->V[\$this->n - 1][\$i] = \$this->V[\$i][\$i];`
279
`            \$this->V[\$i][\$i] = 1.0;`
280
`            \$h = \$this->d[\$i + 1];`
281
`            if (\$h != 0.0) {`
282
`                for (\$k = 0; \$k <= \$i; ++\$k) {`
283
`                    \$this->d[\$k] = \$this->V[\$k][\$i + 1] / \$h;`
284
`                }`
285
286
`                for (\$j = 0; \$j <= \$i; ++\$j) {`
287
`                    \$g = 0.0;`
288
`                    for (\$k = 0; \$k <= \$i; ++\$k) {`
289
`                        \$g += \$this->V[\$k][\$i + 1] * \$this->V[\$k][\$j];`
290
`                    }`
291
292
`                    for (\$k = 0; \$k <= \$i; ++\$k) {`
293
`                        \$this->V[\$k][\$j] -= \$g * \$this->d[\$k];`
294
`                    }`
295
`                }`
296
`            }`
297
298
`            for (\$k = 0; \$k <= \$i; ++\$k) {`
299
`                \$this->V[\$k][\$i + 1] = 0.0;`
300
`            }`
301
`        }`
302
303
`        \$this->d = \$this->V[\$this->n - 1];`
304
`        \$this->V[\$this->n - 1] = array_fill(0, \$j, 0.0);`
305
`        \$this->V[\$this->n - 1][\$this->n - 1] = 1.0;`
306
`        \$this->e = 0.0;`
307
`    }`
308
309
`    /**`
310
`     *	Symmetric tridiagonal QL algorithm.`
311
`     *`
312
`     *	This is derived from the Algol procedures tql2, by`
313
`     *	Bowdler, Martin, Reinsch, and Wilkinson, Handbook for`
314
`     *	Auto. Comp., Vol.ii-Linear Algebra, and the corresponding`
315
`     *	Fortran subroutine in EISPACK.`
316
`     */`
317
`    private function tql2(): void`
318
`    {`
319
`        for (\$i = 1; \$i < \$this->n; ++\$i) {`
320
`            \$this->e[\$i - 1] = \$this->e[\$i];`
321
`        }`
322
323
`        \$this->e[\$this->n - 1] = 0.0;`
324
`        \$f = 0.0;`
325
`        \$tst1 = 0.0;`
326
`        \$eps = pow(2.0, -52.0);`
327
328
`        for (\$l = 0; \$l < \$this->n; ++\$l) {`
329
`            // Find small subdiagonal element`
330
`            \$tst1 = max(\$tst1, abs(\$this->d[\$l]) + abs(\$this->e[\$l]));`
331
`            \$m = \$l;`
332
`            while (\$m < \$this->n) {`
333
`                if (abs(\$this->e[\$m]) <= \$eps * \$tst1) {`
334
`                    break;`
335
`                }`
336
337
`                ++\$m;`
338
`            }`
339
340
`            // If m == l, \$this->d[l] is an eigenvalue,`
341
`            // otherwise, iterate.`
342
`            if (\$m > \$l) {`
343
`                \$iter = 0;`
344
`                do {`
345
`                    // Could check iteration count here.`
346
`                    \$iter += 1;`
347
`                    // Compute implicit shift`
348
`                    \$g = \$this->d[\$l];`
349
`                    \$p = (\$this->d[\$l + 1] - \$g) / (2.0 * \$this->e[\$l]);`
350
`                    \$r = hypot(\$p, 1.0);`
351
`                    if (\$p < 0) {`
352
`                        \$r *= -1;`
353
`                    }`
354
355
`                    \$this->d[\$l] = \$this->e[\$l] / (\$p + \$r);`
356
`                    \$this->d[\$l + 1] = \$this->e[\$l] * (\$p + \$r);`
357
`                    \$dl1 = \$this->d[\$l + 1];`
358
`                    \$h = \$g - \$this->d[\$l];`
359
`                    for (\$i = \$l + 2; \$i < \$this->n; ++\$i) {`
360
`                        \$this->d[\$i] -= \$h;`
361
`                    }`
362
363
`                    \$f += \$h;`
364
`                    // Implicit QL transformation.`
365
`                    \$p = \$this->d[\$m];`
366
`                    \$c = 1.0;`
367
`                    \$c2 = \$c3 = \$c;`
368
`                    \$el1 = \$this->e[\$l + 1];`
369
`                    \$s = \$s2 = 0.0;`
370
`                    for (\$i = \$m - 1; \$i >= \$l; --\$i) {`
371
`                        \$c3 = \$c2;`
372
`                        \$c2 = \$c;`
373
`                        \$s2 = \$s;`
374
`                        \$g = \$c * \$this->e[\$i];`
375
`                        \$h = \$c * \$p;`
376
`                        \$r = hypot(\$p, \$this->e[\$i]);`
377
`                        \$this->e[\$i + 1] = \$s * \$r;`
378
`                        \$s = \$this->e[\$i] / \$r;`
379
`                        \$c = \$p / \$r;`
380
`                        \$p = \$c * \$this->d[\$i] - \$s * \$g;`
381
`                        \$this->d[\$i + 1] = \$h + \$s * (\$c * \$g + \$s * \$this->d[\$i]);`
382
`                        // Accumulate transformation.`
383
`                        for (\$k = 0; \$k < \$this->n; ++\$k) {`
384
`                            \$h = \$this->V[\$k][\$i + 1];`
385
`                            \$this->V[\$k][\$i + 1] = \$s * \$this->V[\$k][\$i] + \$c * \$h;`
386
`                            \$this->V[\$k][\$i] = \$c * \$this->V[\$k][\$i] - \$s * \$h;`
387
`                        }`
388
`                    }`
389
390
`                    \$p = -\$s * \$s2 * \$c3 * \$el1 * \$this->e[\$l] / \$dl1;`
391
`                    \$this->e[\$l] = \$s * \$p;`
392
`                    \$this->d[\$l] = \$c * \$p;`
393
`                    // Check for convergence.`
394
`                } while (abs(\$this->e[\$l]) > \$eps * \$tst1);`
395
`            }`
396
397
`            \$this->d[\$l] = \$this->d[\$l] + \$f;`
398
`            \$this->e[\$l] = 0.0;`
399
`        }`
400
401
`        // Sort eigenvalues and corresponding vectors.`
402
`        for (\$i = 0; \$i < \$this->n - 1; ++\$i) {`
403
`            \$k = \$i;`
404
`            \$p = \$this->d[\$i];`
405
`            for (\$j = \$i + 1; \$j < \$this->n; ++\$j) {`
406
`                if (\$this->d[\$j] < \$p) {`
407
`                    \$k = \$j;`
408
`                    \$p = \$this->d[\$j];`
409
`                }`
410
`            }`
411
412
`            if (\$k != \$i) {`
413
`                \$this->d[\$k] = \$this->d[\$i];`
414
`                \$this->d[\$i] = \$p;`
415 View Code Duplication
`                for (\$j = 0; \$j < \$this->n; ++\$j) {`
416
`                    \$p = \$this->V[\$j][\$i];`
417
`                    \$this->V[\$j][\$i] = \$this->V[\$j][\$k];`
418
`                    \$this->V[\$j][\$k] = \$p;`
419
`                }`
420
`            }`
421
`        }`
422
`    }`
423
424
`    /**`
425
`     *	Nonsymmetric reduction to Hessenberg form.`
426
`     *`
427
`     *	This is derived from the Algol procedures orthes and ortran,`
428
`     *	by Martin and Wilkinson, Handbook for Auto. Comp.,`
429
`     *	Vol.ii-Linear Algebra, and the corresponding`
430
`     *	Fortran subroutines in EISPACK.`
431
`     */`
432
`    private function orthes(): void`
433
`    {`
434
`        \$low = 0;`
435
`        \$high = \$this->n - 1;`
436
437
`        for (\$m = \$low + 1; \$m <= \$high - 1; ++\$m) {`
438
`            // Scale column.`
439
`            \$scale = 0.0;`
440 View Code Duplication
`            for (\$i = \$m; \$i <= \$high; ++\$i) {`
441
`                \$scale = \$scale + abs(\$this->H[\$i][\$m - 1]);`
442
`            }`
443
444
`            if (\$scale != 0.0) {`
445
`                // Compute Householder transformation.`
446
`                \$h = 0.0;`
447
`                for (\$i = \$high; \$i >= \$m; --\$i) {`
448
`                    \$this->ort[\$i] = \$this->H[\$i][\$m - 1] / \$scale;`
449
`                    \$h += \$this->ort[\$i] * \$this->ort[\$i];`
450
`                }`
451
452
`                \$g = sqrt(\$h);`
453
`                if (\$this->ort[\$m] > 0) {`
454
`                    \$g *= -1;`
455
`                }`
456
457
`                \$h -= \$this->ort[\$m] * \$g;`
458
`                \$this->ort[\$m] -= \$g;`
459
`                // Apply Householder similarity transformation`
460
`                // H = (I -u * u' / h) * H * (I -u * u') / h)`
461 View Code Duplication
`                for (\$j = \$m; \$j < \$this->n; ++\$j) {`
462
`                    \$f = 0.0;`
463
`                    for (\$i = \$high; \$i >= \$m; --\$i) {`
464
`                        \$f += \$this->ort[\$i] * \$this->H[\$i][\$j];`
465
`                    }`
466
467
`                    \$f /= \$h;`
468
`                    for (\$i = \$m; \$i <= \$high; ++\$i) {`
469
`                        \$this->H[\$i][\$j] -= \$f * \$this->ort[\$i];`
470
`                    }`
471
`                }`
472
473 View Code Duplication
`                for (\$i = 0; \$i <= \$high; ++\$i) {`
474
`                    \$f = 0.0;`
475
`                    for (\$j = \$high; \$j >= \$m; --\$j) {`
476
`                        \$f += \$this->ort[\$j] * \$this->H[\$i][\$j];`
477
`                    }`
478
479
`                    \$f = \$f / \$h;`
480
`                    for (\$j = \$m; \$j <= \$high; ++\$j) {`
481
`                        \$this->H[\$i][\$j] -= \$f * \$this->ort[\$j];`
482
`                    }`
483
`                }`
484
485
`                \$this->ort[\$m] = \$scale * \$this->ort[\$m];`
486
`                \$this->H[\$m][\$m - 1] = \$scale * \$g;`
487
`            }`
488
`        }`
489
490
`        // Accumulate transformations (Algol's ortran).`
491
`        for (\$i = 0; \$i < \$this->n; ++\$i) {`
492
`            for (\$j = 0; \$j < \$this->n; ++\$j) {`
493
`                \$this->V[\$i][\$j] = (\$i == \$j ? 1.0 : 0.0);`
494
`            }`
495
`        }`
496
497
`        for (\$m = \$high - 1; \$m >= \$low + 1; --\$m) {`
498
`            if (\$this->H[\$m][\$m - 1] != 0.0) {`
499 View Code Duplication
`                for (\$i = \$m + 1; \$i <= \$high; ++\$i) {`
500
`                    \$this->ort[\$i] = \$this->H[\$i][\$m - 1];`
501
`                }`
502
503
`                for (\$j = \$m; \$j <= \$high; ++\$j) {`
504
`                    \$g = 0.0;`
505
`                    for (\$i = \$m; \$i <= \$high; ++\$i) {`
506
`                        \$g += \$this->ort[\$i] * \$this->V[\$i][\$j];`
507
`                    }`
508
509
`                    // Double division avoids possible underflow`
510
`                    \$g = (\$g / \$this->ort[\$m]) / \$this->H[\$m][\$m - 1];`
511
`                    for (\$i = \$m; \$i <= \$high; ++\$i) {`
512
`                        \$this->V[\$i][\$j] += \$g * \$this->ort[\$i];`
513
`                    }`
514
`                }`
515
`            }`
516
`        }`
517
`    }`
518
519
`    /**`
520
`     * Performs complex division.`
521
`     *`
522
`     * @param int|float \$xr`
523
`     * @param int|float \$xi`
524
`     * @param int|float \$yr`
525
`     * @param int|float \$yi`
526
`     */`
527
`    private function cdiv(\$xr, \$xi, \$yr, \$yi): void`
528
`    {`
529
`        if (abs(\$yr) > abs(\$yi)) {`
530
`            \$r = \$yi / \$yr;`
531
`            \$d = \$yr + \$r * \$yi;`
532
`            \$this->cdivr = (\$xr + \$r * \$xi) / \$d;`
533
`            \$this->cdivi = (\$xi - \$r * \$xr) / \$d;`
534
`        } else {`
535
`            \$r = \$yr / \$yi;`
536
`            \$d = \$yi + \$r * \$yr;`
537
`            \$this->cdivr = (\$r * \$xr + \$xi) / \$d;`
538
`            \$this->cdivi = (\$r * \$xi - \$xr) / \$d;`
539
`        }`
540
`    }`
541
542
`    /**`
543
`     *	Nonsymmetric reduction from Hessenberg to real Schur form.`
544
`     *`
545
`     *	Code is derived from the Algol procedure hqr2,`
546
`     *	by Martin and Wilkinson, Handbook for Auto. Comp.,`
547
`     *	Vol.ii-Linear Algebra, and the corresponding`
548
`     *	Fortran subroutine in EISPACK.`
549
`     */`
550
`    private function hqr2(): void`
551
`    {`
552
`        //  Initialize`
553
`        \$nn = \$this->n;`
554
`        \$n = \$nn - 1;`
555
`        \$low = 0;`
556
`        \$high = \$nn - 1;`
557
`        \$eps = pow(2.0, -52.0);`
558
`        \$exshift = 0.0;`
559
`        \$p = \$q = \$r = \$s = \$z = 0;`
560
`        // Store roots isolated by balanc and compute matrix norm`
561
`        \$norm = 0.0;`
562
563
`        for (\$i = 0; \$i < \$nn; ++\$i) {`
564
`            if ((\$i < \$low) or (\$i > \$high)) {`
0 ignored issues
show
Comprehensibility Best Practice introduced by Using logical operators such as `or` instead of `||` is generally not recommended.

PHP has two types of connecting operators (logical operators, and boolean operators):

Logical Operators Boolean Operator
AND - meaning `and` `&&`
OR - meaning `or` `||`

The difference between these is the order in which they are executed. In most cases, you would want to use a boolean operator like `&&`, or `||`.

Let’s take a look at a few examples:

```// Logical operators have lower precedence:
\$f = false or true;

// is executed like this:
(\$f = false) or true;

// Boolean operators have higher precedence:
\$f = false || true;

// is executed like this:
\$f = (false || true);
```

#### Logical Operators are used for Control-Flow

One case where you explicitly want to use logical operators is for control-flow such as this:

```\$x === 5
or die('\$x must be 5.');

if (\$x !== 5) {
die('\$x must be 5.');
}
```

Since `die` introduces problems of its own, f.e. it makes our code hardly testable, and prevents any kind of more sophisticated error handling; you probably do not want to use this in real-world code. Unfortunately, logical operators cannot be combined with `throw` at this point:

```// The following is currently a parse error.
\$x === 5
or throw new RuntimeException('\$x must be 5.');
```

These limitations lead to logical operators rarely being of use in current PHP code. Loading history...
565
`                \$this->d[\$i] = \$this->H[\$i][\$i];`
566
`                \$this->e[\$i] = 0.0;`
567
`            }`
568
569
`            for (\$j = max(\$i - 1, 0); \$j < \$nn; ++\$j) {`
570
`                \$norm = \$norm + abs(\$this->H[\$i][\$j]);`
571
`            }`
572
`        }`
573
574
`        // Outer loop over eigenvalue index`
575
`        \$iter = 0;`
576
`        while (\$n >= \$low) {`
577
`            // Look for single small sub-diagonal element`
578
`            \$l = \$n;`
579
`            while (\$l > \$low) {`
580
`                \$s = abs(\$this->H[\$l - 1][\$l - 1]) + abs(\$this->H[\$l][\$l]);`
581
`                if (\$s == 0.0) {`
582
`                    \$s = \$norm;`
583
`                }`
584
585
`                if (abs(\$this->H[\$l][\$l - 1]) < \$eps * \$s) {`
586
`                    break;`
587
`                }`
588
589
`                --\$l;`
590
`            }`
591
592
`            // Check for convergence`
593
`            // One root found`
594
`            if (\$l == \$n) {`
595
`                \$this->H[\$n][\$n] = \$this->H[\$n][\$n] + \$exshift;`
596
`                \$this->d[\$n] = \$this->H[\$n][\$n];`
597
`                \$this->e[\$n] = 0.0;`
598
`                --\$n;`
599
`                \$iter = 0;`
600
`                // Two roots found`
601
`            } elseif (\$l == \$n - 1) {`
602
`                \$w = \$this->H[\$n][\$n - 1] * \$this->H[\$n - 1][\$n];`
603
`                \$p = (\$this->H[\$n - 1][\$n - 1] - \$this->H[\$n][\$n]) / 2.0;`
604
`                \$q = \$p * \$p + \$w;`
605
`                \$z = sqrt(abs(\$q));`
606
`                \$this->H[\$n][\$n] = \$this->H[\$n][\$n] + \$exshift;`
607
`                \$this->H[\$n - 1][\$n - 1] = \$this->H[\$n - 1][\$n - 1] + \$exshift;`
608
`                \$x = \$this->H[\$n][\$n];`
609
`                // Real pair`
610
`                if (\$q >= 0) {`
611
`                    if (\$p >= 0) {`
612
`                        \$z = \$p + \$z;`
613
`                    } else {`
614
`                        \$z = \$p - \$z;`
615
`                    }`
616
617
`                    \$this->d[\$n - 1] = \$x + \$z;`
618
`                    \$this->d[\$n] = \$this->d[\$n - 1];`
619
`                    if (\$z != 0.0) {`
620
`                        \$this->d[\$n] = \$x - \$w / \$z;`
621
`                    }`
622
623
`                    \$this->e[\$n - 1] = 0.0;`
624
`                    \$this->e[\$n] = 0.0;`
625
`                    \$x = \$this->H[\$n][\$n - 1];`
626
`                    \$s = abs(\$x) + abs(\$z);`
627
`                    \$p = \$x / \$s;`
628
`                    \$q = \$z / \$s;`
629
`                    \$r = sqrt(\$p * \$p + \$q * \$q);`
630
`                    \$p = \$p / \$r;`
631
`                    \$q = \$q / \$r;`
632
`                    // Row modification`
633 View Code Duplication
`                    for (\$j = \$n - 1; \$j < \$nn; ++\$j) {`
634
`                        \$z = \$this->H[\$n - 1][\$j];`
635
`                        \$this->H[\$n - 1][\$j] = \$q * \$z + \$p * \$this->H[\$n][\$j];`
636
`                        \$this->H[\$n][\$j] = \$q * \$this->H[\$n][\$j] - \$p * \$z;`
637
`                    }`
638
639
`                    // Column modification`
640 View Code Duplication
`                    for (\$i = 0; \$i <= \$n; ++\$i) {`
641
`                        \$z = \$this->H[\$i][\$n - 1];`
642
`                        \$this->H[\$i][\$n - 1] = \$q * \$z + \$p * \$this->H[\$i][\$n];`
643
`                        \$this->H[\$i][\$n] = \$q * \$this->H[\$i][\$n] - \$p * \$z;`
644
`                    }`
645
646
`                    // Accumulate transformations`
647 View Code Duplication
`                    for (\$i = \$low; \$i <= \$high; ++\$i) {`
648
`                        \$z = \$this->V[\$i][\$n - 1];`
649
`                        \$this->V[\$i][\$n - 1] = \$q * \$z + \$p * \$this->V[\$i][\$n];`
650
`                        \$this->V[\$i][\$n] = \$q * \$this->V[\$i][\$n] - \$p * \$z;`
651
`                    }`
652
653
`                    // Complex pair`
654
`                } else {`
655
`                    \$this->d[\$n - 1] = \$x + \$p;`
656
`                    \$this->d[\$n] = \$x + \$p;`
657
`                    \$this->e[\$n - 1] = \$z;`
658
`                    \$this->e[\$n] = -\$z;`
659
`                }`
660
661
`                \$n = \$n - 2;`
662
`                \$iter = 0;`
663
`                // No convergence yet`
664
`            } else {`
665
`                // Form shift`
666
`                \$x = \$this->H[\$n][\$n];`
667
`                \$y = 0.0;`
668
`                \$w = 0.0;`
669
`                if (\$l < \$n) {`
670
`                    \$y = \$this->H[\$n - 1][\$n - 1];`
671
`                    \$w = \$this->H[\$n][\$n - 1] * \$this->H[\$n - 1][\$n];`
672
`                }`
673
674
`                // Wilkinson's original ad hoc shift`
675
`                if (\$iter == 10) {`
676
`                    \$exshift += \$x;`
677
`                    for (\$i = \$low; \$i <= \$n; ++\$i) {`
678
`                        \$this->H[\$i][\$i] -= \$x;`
679
`                    }`
680
681
`                    \$s = abs(\$this->H[\$n][\$n - 1]) + abs(\$this->H[\$n - 1][\$n - 2]);`
682
`                    \$x = \$y = 0.75 * \$s;`
683
`                    \$w = -0.4375 * \$s * \$s;`
684
`                }`
685
686
`                // MATLAB's new ad hoc shift`
687
`                if (\$iter == 30) {`
688
`                    \$s = (\$y - \$x) / 2.0;`
689
`                    \$s = \$s * \$s + \$w;`
690
`                    if (\$s > 0) {`
691
`                        \$s = sqrt(\$s);`
692
`                        if (\$y < \$x) {`
693
`                            \$s = -\$s;`
694
`                        }`
695
696
`                        \$s = \$x - \$w / ((\$y - \$x) / 2.0 + \$s);`
697
`                        for (\$i = \$low; \$i <= \$n; ++\$i) {`
698
`                            \$this->H[\$i][\$i] -= \$s;`
699
`                        }`
700
701
`                        \$exshift += \$s;`
702
`                        \$x = \$y = \$w = 0.964;`
703
`                    }`
704
`                }`
705
706
`                // Could check iteration count here.`
707
`                \$iter = \$iter + 1;`
708
`                // Look for two consecutive small sub-diagonal elements`
709
`                \$m = \$n - 2;`
710
`                while (\$m >= \$l) {`
711
`                    \$z = \$this->H[\$m][\$m];`
712
`                    \$r = \$x - \$z;`
713
`                    \$s = \$y - \$z;`
714
`                    \$p = (\$r * \$s - \$w) / \$this->H[\$m + 1][\$m] + \$this->H[\$m][\$m + 1];`
715
`                    \$q = \$this->H[\$m + 1][\$m + 1] - \$z - \$r - \$s;`
716
`                    \$r = \$this->H[\$m + 2][\$m + 1];`
717
`                    \$s = abs(\$p) + abs(\$q) + abs(\$r);`
718
`                    \$p = \$p / \$s;`
719
`                    \$q = \$q / \$s;`
720
`                    \$r = \$r / \$s;`
721
`                    if (\$m == \$l) {`
722
`                        break;`
723
`                    }`
724
725
`                    if (abs(\$this->H[\$m][\$m - 1]) * (abs(\$q) + abs(\$r)) <`
726
`                        \$eps * (abs(\$p) * (abs(\$this->H[\$m - 1][\$m - 1]) + abs(\$z) + abs(\$this->H[\$m + 1][\$m + 1])))) {`
727
`                        break;`
728
`                    }`
729
730
`                    --\$m;`
731
`                }`
732
733
`                for (\$i = \$m + 2; \$i <= \$n; ++\$i) {`
734
`                    \$this->H[\$i][\$i - 2] = 0.0;`
735
`                    if (\$i > \$m + 2) {`
736
`                        \$this->H[\$i][\$i - 3] = 0.0;`
737
`                    }`
738
`                }`
739
740
`                // Double QR step involving rows l:n and columns m:n`
741
`                for (\$k = \$m; \$k <= \$n - 1; ++\$k) {`
742
`                    \$notlast = (\$k != \$n - 1);`
743
`                    if (\$k != \$m) {`
744
`                        \$p = \$this->H[\$k][\$k - 1];`
745
`                        \$q = \$this->H[\$k + 1][\$k - 1];`
746
`                        \$r = (\$notlast ? \$this->H[\$k + 2][\$k - 1] : 0.0);`
747
`                        \$x = abs(\$p) + abs(\$q) + abs(\$r);`
748
`                        if (\$x != 0.0) {`
749
`                            \$p = \$p / \$x;`
750
`                            \$q = \$q / \$x;`
751
`                            \$r = \$r / \$x;`
752
`                        }`
753
`                    }`
754
755
`                    if (\$x == 0.0) {`
756
`                        break;`
757
`                    }`
758
759
`                    \$s = sqrt(\$p * \$p + \$q * \$q + \$r * \$r);`
760
`                    if (\$p < 0) {`
761
`                        \$s = -\$s;`
762
`                    }`
763
764
`                    if (\$s != 0) {`
765
`                        if (\$k != \$m) {`
766
`                            \$this->H[\$k][\$k - 1] = -\$s * \$x;`
767
`                        } elseif (\$l != \$m) {`
768
`                            \$this->H[\$k][\$k - 1] = -\$this->H[\$k][\$k - 1];`
769
`                        }`
770
771
`                        \$p = \$p + \$s;`
772
`                        \$x = \$p / \$s;`
773
`                        \$y = \$q / \$s;`
774
`                        \$z = \$r / \$s;`
775
`                        \$q = \$q / \$p;`
776
`                        \$r = \$r / \$p;`
777
`                        // Row modification`
778 View Code Duplication
`                        for (\$j = \$k; \$j < \$nn; ++\$j) {`
779
`                            \$p = \$this->H[\$k][\$j] + \$q * \$this->H[\$k + 1][\$j];`
780
`                            if (\$notlast) {`
781
`                                \$p = \$p + \$r * \$this->H[\$k + 2][\$j];`
782
`                                \$this->H[\$k + 2][\$j] = \$this->H[\$k + 2][\$j] - \$p * \$z;`
783
`                            }`
784
785
`                            \$this->H[\$k][\$j] = \$this->H[\$k][\$j] - \$p * \$x;`
786
`                            \$this->H[\$k + 1][\$j] = \$this->H[\$k + 1][\$j] - \$p * \$y;`
787
`                        }`
788
789
`                        // Column modification`
790 View Code Duplication
`                        for (\$i = 0; \$i <= min(\$n, \$k + 3); ++\$i) {`
791
`                            \$p = \$x * \$this->H[\$i][\$k] + \$y * \$this->H[\$i][\$k + 1];`
792
`                            if (\$notlast) {`
793
`                                \$p = \$p + \$z * \$this->H[\$i][\$k + 2];`
794
`                                \$this->H[\$i][\$k + 2] = \$this->H[\$i][\$k + 2] - \$p * \$r;`
795
`                            }`
796
797
`                            \$this->H[\$i][\$k] = \$this->H[\$i][\$k] - \$p;`
798
`                            \$this->H[\$i][\$k + 1] = \$this->H[\$i][\$k + 1] - \$p * \$q;`
799
`                        }`
800
801
`                        // Accumulate transformations`
802 View Code Duplication
`                        for (\$i = \$low; \$i <= \$high; ++\$i) {`
803
`                            \$p = \$x * \$this->V[\$i][\$k] + \$y * \$this->V[\$i][\$k + 1];`
804
`                            if (\$notlast) {`
805
`                                \$p = \$p + \$z * \$this->V[\$i][\$k + 2];`
806
`                                \$this->V[\$i][\$k + 2] = \$this->V[\$i][\$k + 2] - \$p * \$r;`
807
`                            }`
808
809
`                            \$this->V[\$i][\$k] = \$this->V[\$i][\$k] - \$p;`
810
`                            \$this->V[\$i][\$k + 1] = \$this->V[\$i][\$k + 1] - \$p * \$q;`
811
`                        }`
812
`                    }  // (\$s != 0)`
0 ignored issues
show
Unused Code Comprehensibility introduced by `56%` of this comment could be valid code. Did you maybe forget this after debugging?

Sometimes obsolete code just ends up commented out instead of removed. In this case it is better to remove the code once you have checked you do not need it.

The code might also have been commented out for debugging purposes. In this case it is vital that someone uncomments it again or your project may behave in very unexpected ways in production.

This check looks for comments that seem to be mostly valid code and reports them. Loading history...
813
`                }  // k loop`
814
`            }  // check convergence`
815
`        }  // while (\$n >= \$low)`
0 ignored issues
show
Unused Code Comprehensibility introduced by `46%` of this comment could be valid code. Did you maybe forget this after debugging?

Sometimes obsolete code just ends up commented out instead of removed. In this case it is better to remove the code once you have checked you do not need it.

The code might also have been commented out for debugging purposes. In this case it is vital that someone uncomments it again or your project may behave in very unexpected ways in production.

This check looks for comments that seem to be mostly valid code and reports them. Loading history...
816
817
`        // Backsubstitute to find vectors of upper triangular form`
818
`        if (\$norm == 0.0) {`
819
`            return;`
820
`        }`
821
822
`        for (\$n = \$nn - 1; \$n >= 0; --\$n) {`
823
`            \$p = \$this->d[\$n];`
824
`            \$q = \$this->e[\$n];`
825
`            // Real vector`
826
`            if (\$q == 0) {`
827
`                \$l = \$n;`
828
`                \$this->H[\$n][\$n] = 1.0;`
829
`                for (\$i = \$n - 1; \$i >= 0; --\$i) {`
830
`                    \$w = \$this->H[\$i][\$i] - \$p;`
831
`                    \$r = 0.0;`
832 View Code Duplication
`                    for (\$j = \$l; \$j <= \$n; ++\$j) {`
833
`                        \$r = \$r + \$this->H[\$i][\$j] * \$this->H[\$j][\$n];`
834
`                    }`
835
836
`                    if (\$this->e[\$i] < 0.0) {`
837
`                        \$z = \$w;`
838
`                        \$s = \$r;`
839
`                    } else {`
840
`                        \$l = \$i;`
841
`                        if (\$this->e[\$i] == 0.0) {`
842
`                            if (\$w != 0.0) {`
843
`                                \$this->H[\$i][\$n] = -\$r / \$w;`
844
`                            } else {`
845
`                                \$this->H[\$i][\$n] = -\$r / (\$eps * \$norm);`
846
`                            }`
847
848
`                            // Solve real equations`
849
`                        } else {`
850
`                            \$x = \$this->H[\$i][\$i + 1];`
851
`                            \$y = \$this->H[\$i + 1][\$i];`
852
`                            \$q = (\$this->d[\$i] - \$p) * (\$this->d[\$i] - \$p) + \$this->e[\$i] * \$this->e[\$i];`
853
`                            \$t = (\$x * \$s - \$z * \$r) / \$q;`
854
`                            \$this->H[\$i][\$n] = \$t;`
855
`                            if (abs(\$x) > abs(\$z)) {`
856
`                                \$this->H[\$i + 1][\$n] = (-\$r - \$w * \$t) / \$x;`
857
`                            } else {`
858
`                                \$this->H[\$i + 1][\$n] = (-\$s - \$y * \$t) / \$z;`
859
`                            }`
860
`                        }`
861
862
`                        // Overflow control`
863
`                        \$t = abs(\$this->H[\$i][\$n]);`
864
`                        if ((\$eps * \$t) * \$t > 1) {`
865 View Code Duplication
`                            for (\$j = \$i; \$j <= \$n; ++\$j) {`
866
`                                \$this->H[\$j][\$n] = \$this->H[\$j][\$n] / \$t;`
867
`                            }`
868
`                        }`
869
`                    }`
870
`                }`
871
872
`                // Complex vector`
873
`            } elseif (\$q < 0) {`
874
`                \$l = \$n - 1;`
875
`                // Last vector component imaginary so matrix is triangular`
876
`                if (abs(\$this->H[\$n][\$n - 1]) > abs(\$this->H[\$n - 1][\$n])) {`
877
`                    \$this->H[\$n - 1][\$n - 1] = \$q / \$this->H[\$n][\$n - 1];`
878
`                    \$this->H[\$n - 1][\$n] = -(\$this->H[\$n][\$n] - \$p) / \$this->H[\$n][\$n - 1];`
879
`                } else {`
880
`                    \$this->cdiv(0.0, -\$this->H[\$n - 1][\$n], \$this->H[\$n - 1][\$n - 1] - \$p, \$q);`
881
`                    \$this->H[\$n - 1][\$n - 1] = \$this->cdivr;`
882
`                    \$this->H[\$n - 1][\$n] = \$this->cdivi;`
883
`                }`
884
885
`                \$this->H[\$n][\$n - 1] = 0.0;`
886
`                \$this->H[\$n][\$n] = 1.0;`
887
`                for (\$i = \$n - 2; \$i >= 0; --\$i) {`
888
`                    // double ra,sa,vr,vi;`
0 ignored issues
show
Unused Code Comprehensibility introduced by `37%` of this comment could be valid code. Did you maybe forget this after debugging?

Sometimes obsolete code just ends up commented out instead of removed. In this case it is better to remove the code once you have checked you do not need it.

The code might also have been commented out for debugging purposes. In this case it is vital that someone uncomments it again or your project may behave in very unexpected ways in production.

This check looks for comments that seem to be mostly valid code and reports them. Loading history...
889
`                    \$ra = 0.0;`
890
`                    \$sa = 0.0;`
891
`                    for (\$j = \$l; \$j <= \$n; ++\$j) {`
892
`                        \$ra = \$ra + \$this->H[\$i][\$j] * \$this->H[\$j][\$n - 1];`
893
`                        \$sa = \$sa + \$this->H[\$i][\$j] * \$this->H[\$j][\$n];`
894
`                    }`
895
896
`                    \$w = \$this->H[\$i][\$i] - \$p;`
897
`                    if (\$this->e[\$i] < 0.0) {`
898
`                        \$z = \$w;`
899
`                        \$r = \$ra;`
900
`                        \$s = \$sa;`
901
`                    } else {`
902
`                        \$l = \$i;`
903
`                        if (\$this->e[\$i] == 0) {`
904
`                            \$this->cdiv(-\$ra, -\$sa, \$w, \$q);`
905
`                            \$this->H[\$i][\$n - 1] = \$this->cdivr;`
906
`                            \$this->H[\$i][\$n] = \$this->cdivi;`
907
`                        } else {`
908
`                            // Solve complex equations`
909
`                            \$x = \$this->H[\$i][\$i + 1];`
910
`                            \$y = \$this->H[\$i + 1][\$i];`
911
`                            \$vr = (\$this->d[\$i] - \$p) * (\$this->d[\$i] - \$p) + \$this->e[\$i] * \$this->e[\$i] - \$q * \$q;`
912
`                            \$vi = (\$this->d[\$i] - \$p) * 2.0 * \$q;`
913
`                            if (\$vr == 0.0 & \$vi == 0.0) {`
0 ignored issues
show
Comprehensibility introduced by Consider adding parentheses for clarity. Current Interpretation: `(\$vr == 0.0) & \$vi == 0.0`, Probably Intended Meaning: `\$vr == (0.0 & \$vi == 0.0)`

When comparing the result of a bit operation, we suggest to add explicit parenthesis and not to rely on PHP’s built-in operator precedence to ensure the code behaves as intended and to make it more readable.

Let’s take a look at these examples:

```// Returns always int(0).
return 0 === \$foo & 4;
return (0 === \$foo) & 4;

// More likely intended return: true/false
return 0 === (\$foo & 4);
``` Loading history...
914
`                                \$vr = \$eps * \$norm * (abs(\$w) + abs(\$q) + abs(\$x) + abs(\$y) + abs(\$z));`
915
`                            }`
916
917
`                            \$this->cdiv(\$x * \$r - \$z * \$ra + \$q * \$sa, \$x * \$s - \$z * \$sa - \$q * \$ra, \$vr, \$vi);`
918
`                            \$this->H[\$i][\$n - 1] = \$this->cdivr;`
919
`                            \$this->H[\$i][\$n] = \$this->cdivi;`
920
`                            if (abs(\$x) > (abs(\$z) + abs(\$q))) {`
921
`                                \$this->H[\$i + 1][\$n - 1] = (-\$ra - \$w * \$this->H[\$i][\$n - 1] + \$q * \$this->H[\$i][\$n]) / \$x;`
922
`                                \$this->H[\$i + 1][\$n] = (-\$sa - \$w * \$this->H[\$i][\$n] - \$q * \$this->H[\$i][\$n - 1]) / \$x;`
923
`                            } else {`
924
`                                \$this->cdiv(-\$r - \$y * \$this->H[\$i][\$n - 1], -\$s - \$y * \$this->H[\$i][\$n], \$z, \$q);`
925
`                                \$this->H[\$i + 1][\$n - 1] = \$this->cdivr;`
926
`                                \$this->H[\$i + 1][\$n] = \$this->cdivi;`
927
`                            }`
928
`                        }`
929
930
`                        // Overflow control`
931
`                        \$t = max(abs(\$this->H[\$i][\$n - 1]), abs(\$this->H[\$i][\$n]));`
932
`                        if ((\$eps * \$t) * \$t > 1) {`
933
`                            for (\$j = \$i; \$j <= \$n; ++\$j) {`
934
`                                \$this->H[\$j][\$n - 1] = \$this->H[\$j][\$n - 1] / \$t;`
935
`                                \$this->H[\$j][\$n] = \$this->H[\$j][\$n] / \$t;`
936
`                            }`
937
`                        }`
938
`                    } // end else`
939
`                } // end for`
940
`            } // end else for complex case`
941
`        } // end for`
942
943
`        // Vectors of isolated roots`
944
`        for (\$i = 0; \$i < \$nn; ++\$i) {`
945
`            if (\$i < \$low | \$i > \$high) {`
946
`                for (\$j = \$i; \$j < \$nn; ++\$j) {`
947
`                    \$this->V[\$i][\$j] = \$this->H[\$i][\$j];`
948
`                }`
949
`            }`
950
`        }`
951
952
`        // Back transformation to get eigenvectors of original matrix`
953
`        for (\$j = \$nn - 1; \$j >= \$low; --\$j) {`
954
`            for (\$i = \$low; \$i <= \$high; ++\$i) {`
955
`                \$z = 0.0;`
956
`                for (\$k = \$low; \$k <= min(\$j, \$high); ++\$k) {`
957
`                    \$z = \$z + \$this->V[\$i][\$k] * \$this->H[\$k][\$j];`
958
`                }`
959
960
`                \$this->V[\$i][\$j] = \$z;`
961
`            }`
962
`        }`
963
`    }`
964
`}`
965