php-ai /
php-ml
Duplicate code is one of the most pungent code smells. A rule that is often used is to re-structure code once it is duplicated in three or more places.
Common duplication problems, and corresponding solutions are:
Complex classes like EigenvalueDecomposition often do a lot of different things. To break such a class down, we need to identify a cohesive component within that class. A common approach to find such a component is to look for fields/methods that share the same prefixes, or suffixes. You can also have a look at the cohesion graph to spot any un-connected, or weakly-connected components.
Once you have determined the fields that belong together, you can apply the Extract Class refactoring. If the component makes sense as a sub-class, Extract Subclass is also a candidate, and is often faster.
While breaking up the class, it is a good idea to analyze how other classes use EigenvalueDecomposition, and based on these observations, apply Extract Interface, too.
| 1 | <?php |
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| 36 | class EigenvalueDecomposition |
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| 37 | { |
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| 38 | /** |
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| 39 | * Row and column dimension (square matrix). |
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| 40 | * |
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| 41 | * @var int |
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| 42 | */ |
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| 43 | private $n; |
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| 44 | |||
| 45 | /** |
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| 46 | * Internal symmetry flag. |
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| 47 | * |
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| 48 | * @var bool |
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| 49 | */ |
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| 50 | private $symmetric; |
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| 51 | |||
| 52 | /** |
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| 53 | * Arrays for internal storage of eigenvalues. |
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| 54 | * |
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| 55 | * @var array |
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| 56 | */ |
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| 57 | private $d = []; |
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| 58 | |||
| 59 | private $e = []; |
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| 60 | |||
| 61 | /** |
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| 62 | * Array for internal storage of eigenvectors. |
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| 63 | * |
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| 64 | * @var array |
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| 65 | */ |
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| 66 | private $V = []; |
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| 67 | |||
| 68 | /** |
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| 69 | * Array for internal storage of nonsymmetric Hessenberg form. |
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| 70 | * |
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| 71 | * @var array |
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| 72 | */ |
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| 73 | private $H = []; |
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| 74 | |||
| 75 | /** |
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| 76 | * Working storage for nonsymmetric algorithm. |
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| 77 | * |
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| 78 | * @var array |
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| 79 | */ |
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| 80 | private $ort = []; |
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| 81 | |||
| 82 | /** |
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| 83 | * Used for complex scalar division. |
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| 84 | * |
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| 85 | * @var float |
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| 86 | */ |
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| 87 | private $cdivr; |
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| 88 | |||
| 89 | private $cdivi; |
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| 90 | |||
| 91 | private $A; |
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| 92 | |||
| 93 | /** |
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| 94 | * Constructor: Check for symmetry, then construct the eigenvalue decomposition |
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| 95 | */ |
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| 96 | public function __construct(array $Arg) |
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| 97 | { |
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| 98 | $this->A = $Arg; |
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| 99 | $this->n = count($Arg[0]); |
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| 100 | $this->symmetric = true; |
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| 101 | |||
| 102 | for ($j = 0; ($j < $this->n) & $this->symmetric; ++$j) { |
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| 103 | for ($i = 0; ($i < $this->n) & $this->symmetric; ++$i) { |
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| 104 | $this->symmetric = ($this->A[$i][$j] == $this->A[$j][$i]); |
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| 105 | } |
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| 106 | } |
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| 107 | |||
| 108 | if ($this->symmetric) { |
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| 109 | $this->V = $this->A; |
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| 110 | // Tridiagonalize. |
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| 111 | $this->tred2(); |
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| 112 | // Diagonalize. |
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| 113 | $this->tql2(); |
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| 114 | } else { |
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| 115 | $this->H = $this->A; |
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| 116 | $this->ort = []; |
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| 117 | // Reduce to Hessenberg form. |
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| 118 | $this->orthes(); |
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| 119 | // Reduce Hessenberg to real Schur form. |
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| 120 | $this->hqr2(); |
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| 121 | } |
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| 122 | } |
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| 123 | |||
| 124 | /** |
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| 125 | * Return the eigenvector matrix |
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| 126 | */ |
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| 127 | public function getEigenvectors(): array |
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| 128 | { |
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| 129 | $vectors = $this->V; |
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| 130 | |||
| 131 | // Always return the eigenvectors of length 1.0 |
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| 132 | $vectors = new Matrix($vectors); |
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| 133 | $vectors = array_map(function ($vect) { |
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| 134 | $sum = 0; |
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| 135 | for ($i = 0; $i < count($vect); ++$i) { |
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| 136 | $sum += $vect[$i] ** 2; |
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| 137 | } |
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| 138 | |||
| 139 | $sum = sqrt($sum); |
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| 140 | for ($i = 0; $i < count($vect); ++$i) { |
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| 141 | $vect[$i] /= $sum; |
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| 142 | } |
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| 143 | |||
| 144 | return $vect; |
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| 145 | }, $vectors->transpose()->toArray()); |
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| 146 | |||
| 147 | return $vectors; |
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| 148 | } |
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| 149 | |||
| 150 | /** |
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| 151 | * Return the real parts of the eigenvalues<br> |
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| 152 | * d = real(diag(D)); |
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| 153 | */ |
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| 154 | public function getRealEigenvalues(): array |
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| 155 | { |
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| 156 | return $this->d; |
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| 157 | } |
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| 158 | |||
| 159 | /** |
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| 160 | * Return the imaginary parts of the eigenvalues <br> |
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| 161 | * d = imag(diag(D)) |
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| 162 | */ |
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| 163 | public function getImagEigenvalues(): array |
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| 164 | { |
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| 165 | return $this->e; |
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| 166 | } |
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| 167 | |||
| 168 | /** |
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| 169 | * Return the block diagonal eigenvalue matrix |
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| 170 | */ |
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| 171 | public function getDiagonalEigenvalues(): array |
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| 172 | { |
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| 173 | $D = []; |
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| 174 | |||
| 175 | for ($i = 0; $i < $this->n; ++$i) { |
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| 176 | $D[$i] = array_fill(0, $this->n, 0.0); |
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| 177 | $D[$i][$i] = $this->d[$i]; |
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| 178 | if ($this->e[$i] == 0) { |
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| 179 | continue; |
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| 180 | } |
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| 181 | |||
| 182 | $o = ($this->e[$i] > 0) ? $i + 1 : $i - 1; |
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| 183 | $D[$i][$o] = $this->e[$i]; |
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| 184 | } |
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| 185 | |||
| 186 | return $D; |
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| 187 | } |
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| 188 | |||
| 189 | /** |
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| 190 | * Symmetric Householder reduction to tridiagonal form. |
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| 191 | */ |
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| 192 | private function tred2(): void |
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| 193 | { |
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| 194 | // This is derived from the Algol procedures tred2 by |
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| 195 | // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for |
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| 196 | // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding |
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| 197 | // Fortran subroutine in EISPACK. |
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| 198 | $this->d = $this->V[$this->n - 1]; |
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| 199 | // Householder reduction to tridiagonal form. |
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| 200 | for ($i = $this->n - 1; $i > 0; --$i) { |
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| 201 | $i_ = $i - 1; |
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| 202 | // Scale to avoid under/overflow. |
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| 203 | $h = $scale = 0.0; |
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| 204 | $scale += array_sum(array_map('abs', $this->d)); |
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| 205 | if ($scale == 0.0) { |
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| 206 | $this->e[$i] = $this->d[$i_]; |
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| 207 | $this->d = array_slice($this->V[$i_], 0, $this->n - 1); |
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| 208 | for ($j = 0; $j < $i; ++$j) { |
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| 209 | $this->V[$j][$i] = $this->V[$i][$j] = 0.0; |
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| 210 | } |
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| 211 | } else { |
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| 212 | // Generate Householder vector. |
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| 213 | for ($k = 0; $k < $i; ++$k) { |
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| 214 | $this->d[$k] /= $scale; |
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| 215 | $h += pow($this->d[$k], 2); |
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| 216 | } |
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| 217 | |||
| 218 | $f = $this->d[$i_]; |
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| 219 | $g = sqrt($h); |
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| 220 | if ($f > 0) { |
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| 221 | $g = -$g; |
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| 222 | } |
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| 223 | |||
| 224 | $this->e[$i] = $scale * $g; |
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| 225 | $h = $h - $f * $g; |
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| 226 | $this->d[$i_] = $f - $g; |
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| 227 | |||
| 228 | for ($j = 0; $j < $i; ++$j) { |
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| 229 | $this->e[$j] = 0.0; |
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| 230 | } |
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| 231 | |||
| 232 | // Apply similarity transformation to remaining columns. |
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| 233 | for ($j = 0; $j < $i; ++$j) { |
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| 234 | $f = $this->d[$j]; |
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| 235 | $this->V[$j][$i] = $f; |
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| 236 | $g = $this->e[$j] + $this->V[$j][$j] * $f; |
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| 237 | |||
| 238 | for ($k = $j + 1; $k <= $i_; ++$k) { |
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| 239 | $g += $this->V[$k][$j] * $this->d[$k]; |
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| 240 | $this->e[$k] += $this->V[$k][$j] * $f; |
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| 241 | } |
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| 242 | |||
| 243 | $this->e[$j] = $g; |
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| 244 | } |
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| 245 | |||
| 246 | $f = 0.0; |
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| 247 | |||
| 248 | if ($h == 0.0) { |
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| 249 | $h = 1e-32; |
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| 250 | } |
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| 251 | |||
| 252 | for ($j = 0; $j < $i; ++$j) { |
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| 253 | $this->e[$j] /= $h; |
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| 254 | $f += $this->e[$j] * $this->d[$j]; |
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| 255 | } |
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| 256 | |||
| 257 | $hh = $f / (2 * $h); |
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| 258 | for ($j = 0; $j < $i; ++$j) { |
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| 259 | $this->e[$j] -= $hh * $this->d[$j]; |
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| 260 | } |
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| 261 | |||
| 262 | for ($j = 0; $j < $i; ++$j) { |
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| 263 | $f = $this->d[$j]; |
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| 264 | $g = $this->e[$j]; |
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| 265 | for ($k = $j; $k <= $i_; ++$k) { |
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| 266 | $this->V[$k][$j] -= ($f * $this->e[$k] + $g * $this->d[$k]); |
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| 267 | } |
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| 268 | |||
| 269 | $this->d[$j] = $this->V[$i - 1][$j]; |
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| 270 | $this->V[$i][$j] = 0.0; |
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| 271 | } |
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| 272 | } |
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| 273 | |||
| 274 | $this->d[$i] = $h; |
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| 275 | } |
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| 276 | |||
| 277 | // Accumulate transformations. |
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| 278 | for ($i = 0; $i < $this->n - 1; ++$i) { |
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| 279 | $this->V[$this->n - 1][$i] = $this->V[$i][$i]; |
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| 280 | $this->V[$i][$i] = 1.0; |
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| 281 | $h = $this->d[$i + 1]; |
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| 282 | if ($h != 0.0) { |
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| 283 | for ($k = 0; $k <= $i; ++$k) { |
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| 284 | $this->d[$k] = $this->V[$k][$i + 1] / $h; |
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| 285 | } |
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| 286 | |||
| 287 | for ($j = 0; $j <= $i; ++$j) { |
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| 288 | $g = 0.0; |
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| 289 | for ($k = 0; $k <= $i; ++$k) { |
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| 290 | $g += $this->V[$k][$i + 1] * $this->V[$k][$j]; |
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| 291 | } |
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| 292 | |||
| 293 | for ($k = 0; $k <= $i; ++$k) { |
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| 294 | $this->V[$k][$j] -= $g * $this->d[$k]; |
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| 295 | } |
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| 296 | } |
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| 297 | } |
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| 298 | |||
| 299 | for ($k = 0; $k <= $i; ++$k) { |
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| 300 | $this->V[$k][$i + 1] = 0.0; |
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| 301 | } |
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| 302 | } |
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| 303 | |||
| 304 | $this->d = $this->V[$this->n - 1]; |
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| 305 | $this->V[$this->n - 1] = array_fill(0, $this->n, 0.0); |
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| 306 | $this->V[$this->n - 1][$this->n - 1] = 1.0; |
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| 307 | $this->e[0] = 0.0; |
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| 308 | } |
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| 309 | |||
| 310 | /** |
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| 311 | * Symmetric tridiagonal QL algorithm. |
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| 312 | * |
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| 313 | * This is derived from the Algol procedures tql2, by |
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| 314 | * Bowdler, Martin, Reinsch, and Wilkinson, Handbook for |
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| 315 | * Auto. Comp., Vol.ii-Linear Algebra, and the corresponding |
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| 316 | * Fortran subroutine in EISPACK. |
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| 317 | */ |
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| 318 | private function tql2(): void |
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| 319 | { |
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| 320 | for ($i = 1; $i < $this->n; ++$i) { |
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| 321 | $this->e[$i - 1] = $this->e[$i]; |
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| 322 | } |
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| 323 | |||
| 324 | $this->e[$this->n - 1] = 0.0; |
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| 325 | $f = 0.0; |
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| 326 | $tst1 = 0.0; |
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| 327 | $eps = pow(2.0, -52.0); |
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| 328 | |||
| 329 | for ($l = 0; $l < $this->n; ++$l) { |
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| 330 | // Find small subdiagonal element |
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| 331 | $tst1 = max($tst1, abs($this->d[$l]) + abs($this->e[$l])); |
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| 332 | $m = $l; |
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| 333 | while ($m < $this->n) { |
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| 334 | if (abs($this->e[$m]) <= $eps * $tst1) { |
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| 335 | break; |
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| 336 | } |
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| 337 | |||
| 338 | ++$m; |
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| 339 | } |
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| 340 | |||
| 341 | // If m == l, $this->d[l] is an eigenvalue, |
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| 342 | // otherwise, iterate. |
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| 343 | if ($m > $l) { |
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| 344 | $iter = 0; |
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| 345 | do { |
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| 346 | // Could check iteration count here. |
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| 347 | $iter += 1; |
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| 348 | // Compute implicit shift |
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| 349 | $g = $this->d[$l]; |
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| 350 | $p = ($this->d[$l + 1] - $g) / (2.0 * $this->e[$l]); |
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| 351 | $r = hypot($p, 1.0); |
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| 352 | if ($p < 0) { |
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| 353 | $r *= -1; |
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| 354 | } |
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| 355 | |||
| 356 | $this->d[$l] = $this->e[$l] / ($p + $r); |
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| 357 | $this->d[$l + 1] = $this->e[$l] * ($p + $r); |
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| 358 | $dl1 = $this->d[$l + 1]; |
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| 359 | $h = $g - $this->d[$l]; |
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| 360 | for ($i = $l + 2; $i < $this->n; ++$i) { |
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| 361 | $this->d[$i] -= $h; |
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| 362 | } |
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| 363 | |||
| 364 | $f += $h; |
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| 365 | // Implicit QL transformation. |
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| 366 | $p = $this->d[$m]; |
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| 367 | $c = 1.0; |
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| 368 | $c2 = $c3 = $c; |
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| 369 | $el1 = $this->e[$l + 1]; |
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| 370 | $s = $s2 = 0.0; |
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| 371 | for ($i = $m - 1; $i >= $l; --$i) { |
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| 372 | $c3 = $c2; |
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| 373 | $c2 = $c; |
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| 374 | $s2 = $s; |
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| 375 | $g = $c * $this->e[$i]; |
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| 376 | $h = $c * $p; |
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| 377 | $r = hypot($p, $this->e[$i]); |
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| 378 | $this->e[$i + 1] = $s * $r; |
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| 379 | $s = $this->e[$i] / $r; |
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| 380 | $c = $p / $r; |
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| 381 | $p = $c * $this->d[$i] - $s * $g; |
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| 382 | $this->d[$i + 1] = $h + $s * ($c * $g + $s * $this->d[$i]); |
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| 383 | // Accumulate transformation. |
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| 384 | for ($k = 0; $k < $this->n; ++$k) { |
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| 385 | $h = $this->V[$k][$i + 1]; |
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| 386 | $this->V[$k][$i + 1] = $s * $this->V[$k][$i] + $c * $h; |
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| 387 | $this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h; |
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| 388 | } |
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| 389 | } |
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| 390 | |||
| 391 | $p = -$s * $s2 * $c3 * $el1 * $this->e[$l] / $dl1; |
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| 392 | $this->e[$l] = $s * $p; |
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| 393 | $this->d[$l] = $c * $p; |
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| 394 | // Check for convergence. |
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| 395 | } while (abs($this->e[$l]) > $eps * $tst1); |
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| 396 | } |
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| 397 | |||
| 398 | $this->d[$l] = $this->d[$l] + $f; |
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| 399 | $this->e[$l] = 0.0; |
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| 400 | } |
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| 401 | |||
| 402 | // Sort eigenvalues and corresponding vectors. |
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| 403 | for ($i = 0; $i < $this->n - 1; ++$i) { |
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| 404 | $k = $i; |
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| 405 | $p = $this->d[$i]; |
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| 406 | for ($j = $i + 1; $j < $this->n; ++$j) { |
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| 407 | if ($this->d[$j] < $p) { |
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| 408 | $k = $j; |
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| 409 | $p = $this->d[$j]; |
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| 410 | } |
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| 411 | } |
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| 412 | |||
| 413 | if ($k != $i) { |
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| 414 | $this->d[$k] = $this->d[$i]; |
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| 415 | $this->d[$i] = $p; |
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| 416 | for ($j = 0; $j < $this->n; ++$j) { |
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| 417 | $p = $this->V[$j][$i]; |
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| 418 | $this->V[$j][$i] = $this->V[$j][$k]; |
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| 419 | $this->V[$j][$k] = $p; |
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| 420 | } |
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| 421 | } |
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| 422 | } |
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| 423 | } |
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| 424 | |||
| 425 | /** |
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| 426 | * Nonsymmetric reduction to Hessenberg form. |
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| 427 | * |
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| 428 | * This is derived from the Algol procedures orthes and ortran, |
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| 429 | * by Martin and Wilkinson, Handbook for Auto. Comp., |
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| 430 | * Vol.ii-Linear Algebra, and the corresponding |
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| 431 | * Fortran subroutines in EISPACK. |
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| 432 | */ |
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| 433 | private function orthes(): void |
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| 434 | { |
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| 435 | $low = 0; |
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| 436 | $high = $this->n - 1; |
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| 437 | |||
| 438 | for ($m = $low + 1; $m <= $high - 1; ++$m) { |
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| 439 | // Scale column. |
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| 440 | $scale = 0.0; |
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| 441 | for ($i = $m; $i <= $high; ++$i) { |
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| 442 | $scale = $scale + abs($this->H[$i][$m - 1]); |
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| 443 | } |
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| 444 | |||
| 445 | if ($scale != 0.0) { |
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| 446 | // Compute Householder transformation. |
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| 447 | $h = 0.0; |
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| 448 | for ($i = $high; $i >= $m; --$i) { |
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| 449 | $this->ort[$i] = $this->H[$i][$m - 1] / $scale; |
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| 450 | $h += $this->ort[$i] * $this->ort[$i]; |
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| 451 | } |
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| 452 | |||
| 453 | $g = sqrt($h); |
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| 454 | if ($this->ort[$m] > 0) { |
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| 455 | $g *= -1; |
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| 456 | } |
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| 457 | |||
| 458 | $h -= $this->ort[$m] * $g; |
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| 459 | $this->ort[$m] -= $g; |
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| 460 | // Apply Householder similarity transformation |
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| 461 | // H = (I -u * u' / h) * H * (I -u * u') / h) |
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| 462 | for ($j = $m; $j < $this->n; ++$j) { |
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| 463 | $f = 0.0; |
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| 464 | for ($i = $high; $i >= $m; --$i) { |
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| 465 | $f += $this->ort[$i] * $this->H[$i][$j]; |
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| 466 | } |
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| 467 | |||
| 468 | $f /= $h; |
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| 469 | for ($i = $m; $i <= $high; ++$i) { |
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| 470 | $this->H[$i][$j] -= $f * $this->ort[$i]; |
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| 471 | } |
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| 472 | } |
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| 473 | |||
| 474 | for ($i = 0; $i <= $high; ++$i) { |
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| 475 | $f = 0.0; |
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| 476 | for ($j = $high; $j >= $m; --$j) { |
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| 477 | $f += $this->ort[$j] * $this->H[$i][$j]; |
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| 478 | } |
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| 479 | |||
| 480 | $f = $f / $h; |
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| 481 | for ($j = $m; $j <= $high; ++$j) { |
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| 482 | $this->H[$i][$j] -= $f * $this->ort[$j]; |
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| 483 | } |
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| 484 | } |
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| 485 | |||
| 486 | $this->ort[$m] = $scale * $this->ort[$m]; |
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| 487 | $this->H[$m][$m - 1] = $scale * $g; |
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| 488 | } |
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| 489 | } |
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| 490 | |||
| 491 | // Accumulate transformations (Algol's ortran). |
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| 492 | for ($i = 0; $i < $this->n; ++$i) { |
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| 493 | for ($j = 0; $j < $this->n; ++$j) { |
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| 494 | $this->V[$i][$j] = ($i == $j ? 1.0 : 0.0); |
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| 495 | } |
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| 496 | } |
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| 497 | |||
| 498 | for ($m = $high - 1; $m >= $low + 1; --$m) { |
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| 499 | if ($this->H[$m][$m - 1] != 0.0) { |
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| 500 | for ($i = $m + 1; $i <= $high; ++$i) { |
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| 501 | $this->ort[$i] = $this->H[$i][$m - 1]; |
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| 502 | } |
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| 503 | |||
| 504 | for ($j = $m; $j <= $high; ++$j) { |
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| 505 | $g = 0.0; |
||
| 506 | for ($i = $m; $i <= $high; ++$i) { |
||
| 507 | $g += $this->ort[$i] * $this->V[$i][$j]; |
||
| 508 | } |
||
| 509 | |||
| 510 | // Double division avoids possible underflow |
||
| 511 | $g = ($g / $this->ort[$m]) / $this->H[$m][$m - 1]; |
||
| 512 | for ($i = $m; $i <= $high; ++$i) { |
||
| 513 | $this->V[$i][$j] += $g * $this->ort[$i]; |
||
| 514 | } |
||
| 515 | } |
||
| 516 | } |
||
| 517 | } |
||
| 518 | } |
||
| 519 | |||
| 520 | /** |
||
| 521 | * Performs complex division. |
||
| 522 | * |
||
| 523 | * @param int|float $xr |
||
| 524 | * @param int|float $xi |
||
| 525 | * @param int|float $yr |
||
| 526 | * @param int|float $yi |
||
| 527 | */ |
||
| 528 | private function cdiv($xr, $xi, $yr, $yi): void |
||
| 529 | { |
||
| 530 | if (abs($yr) > abs($yi)) { |
||
| 531 | $r = $yi / $yr; |
||
| 532 | $d = $yr + $r * $yi; |
||
| 533 | $this->cdivr = ($xr + $r * $xi) / $d; |
||
| 534 | $this->cdivi = ($xi - $r * $xr) / $d; |
||
| 535 | } else { |
||
| 536 | $r = $yr / $yi; |
||
| 537 | $d = $yi + $r * $yr; |
||
| 538 | $this->cdivr = ($r * $xr + $xi) / $d; |
||
| 539 | $this->cdivi = ($r * $xi - $xr) / $d; |
||
| 540 | } |
||
| 541 | } |
||
| 542 | |||
| 543 | /** |
||
| 544 | * Nonsymmetric reduction from Hessenberg to real Schur form. |
||
| 545 | * |
||
| 546 | * Code is derived from the Algol procedure hqr2, |
||
| 547 | * by Martin and Wilkinson, Handbook for Auto. Comp., |
||
| 548 | * Vol.ii-Linear Algebra, and the corresponding |
||
| 549 | * Fortran subroutine in EISPACK. |
||
| 550 | */ |
||
| 551 | private function hqr2(): void |
||
| 552 | { |
||
| 553 | // Initialize |
||
| 554 | $nn = $this->n; |
||
| 555 | $n = $nn - 1; |
||
| 556 | $low = 0; |
||
| 557 | $high = $nn - 1; |
||
| 558 | $eps = pow(2.0, -52.0); |
||
| 559 | $exshift = 0.0; |
||
| 560 | $p = $q = $r = $s = $z = 0; |
||
| 561 | // Store roots isolated by balanc and compute matrix norm |
||
| 562 | $norm = 0.0; |
||
| 563 | |||
| 564 | for ($i = 0; $i < $nn; ++$i) { |
||
| 565 | if (($i < $low) or ($i > $high)) { |
||
| 566 | $this->d[$i] = $this->H[$i][$i]; |
||
| 567 | $this->e[$i] = 0.0; |
||
| 568 | } |
||
| 569 | |||
| 570 | for ($j = max($i - 1, 0); $j < $nn; ++$j) { |
||
| 571 | $norm = $norm + abs($this->H[$i][$j]); |
||
| 572 | } |
||
| 573 | } |
||
| 574 | |||
| 575 | // Outer loop over eigenvalue index |
||
| 576 | $iter = 0; |
||
| 577 | while ($n >= $low) { |
||
| 578 | // Look for single small sub-diagonal element |
||
| 579 | $l = $n; |
||
| 580 | while ($l > $low) { |
||
| 581 | $s = abs($this->H[$l - 1][$l - 1]) + abs($this->H[$l][$l]); |
||
| 582 | if ($s == 0.0) { |
||
| 583 | $s = $norm; |
||
| 584 | } |
||
| 585 | |||
| 586 | if (abs($this->H[$l][$l - 1]) < $eps * $s) { |
||
| 587 | break; |
||
| 588 | } |
||
| 589 | |||
| 590 | --$l; |
||
| 591 | } |
||
| 592 | |||
| 593 | // Check for convergence |
||
| 594 | // One root found |
||
| 595 | if ($l == $n) { |
||
| 596 | $this->H[$n][$n] = $this->H[$n][$n] + $exshift; |
||
| 597 | $this->d[$n] = $this->H[$n][$n]; |
||
| 598 | $this->e[$n] = 0.0; |
||
| 599 | --$n; |
||
| 600 | $iter = 0; |
||
| 601 | // Two roots found |
||
| 602 | } elseif ($l == $n - 1) { |
||
| 603 | $w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n]; |
||
| 604 | $p = ($this->H[$n - 1][$n - 1] - $this->H[$n][$n]) / 2.0; |
||
| 605 | $q = $p * $p + $w; |
||
| 606 | $z = sqrt(abs($q)); |
||
| 607 | $this->H[$n][$n] = $this->H[$n][$n] + $exshift; |
||
| 608 | $this->H[$n - 1][$n - 1] = $this->H[$n - 1][$n - 1] + $exshift; |
||
| 609 | $x = $this->H[$n][$n]; |
||
| 610 | // Real pair |
||
| 611 | if ($q >= 0) { |
||
| 612 | if ($p >= 0) { |
||
| 613 | $z = $p + $z; |
||
| 614 | } else { |
||
| 615 | $z = $p - $z; |
||
| 616 | } |
||
| 617 | |||
| 618 | $this->d[$n - 1] = $x + $z; |
||
| 619 | $this->d[$n] = $this->d[$n - 1]; |
||
| 620 | if ($z != 0.0) { |
||
| 621 | $this->d[$n] = $x - $w / $z; |
||
| 622 | } |
||
| 623 | |||
| 624 | $this->e[$n - 1] = 0.0; |
||
| 625 | $this->e[$n] = 0.0; |
||
| 626 | $x = $this->H[$n][$n - 1]; |
||
| 627 | $s = abs($x) + abs($z); |
||
| 628 | $p = $x / $s; |
||
| 629 | $q = $z / $s; |
||
| 630 | $r = sqrt($p * $p + $q * $q); |
||
| 631 | $p = $p / $r; |
||
| 632 | $q = $q / $r; |
||
| 633 | // Row modification |
||
| 634 | for ($j = $n - 1; $j < $nn; ++$j) { |
||
| 635 | $z = $this->H[$n - 1][$j]; |
||
| 636 | $this->H[$n - 1][$j] = $q * $z + $p * $this->H[$n][$j]; |
||
| 637 | $this->H[$n][$j] = $q * $this->H[$n][$j] - $p * $z; |
||
| 638 | } |
||
| 639 | |||
| 640 | // Column modification |
||
| 641 | for ($i = 0; $i <= $n; ++$i) { |
||
| 642 | $z = $this->H[$i][$n - 1]; |
||
| 643 | $this->H[$i][$n - 1] = $q * $z + $p * $this->H[$i][$n]; |
||
| 644 | $this->H[$i][$n] = $q * $this->H[$i][$n] - $p * $z; |
||
| 645 | } |
||
| 646 | |||
| 647 | // Accumulate transformations |
||
| 648 | for ($i = $low; $i <= $high; ++$i) { |
||
| 649 | $z = $this->V[$i][$n - 1]; |
||
| 650 | $this->V[$i][$n - 1] = $q * $z + $p * $this->V[$i][$n]; |
||
| 651 | $this->V[$i][$n] = $q * $this->V[$i][$n] - $p * $z; |
||
| 652 | } |
||
| 653 | |||
| 654 | // Complex pair |
||
| 655 | } else { |
||
| 656 | $this->d[$n - 1] = $x + $p; |
||
| 657 | $this->d[$n] = $x + $p; |
||
| 658 | $this->e[$n - 1] = $z; |
||
| 659 | $this->e[$n] = -$z; |
||
| 660 | } |
||
| 661 | |||
| 662 | $n = $n - 2; |
||
| 663 | $iter = 0; |
||
| 664 | // No convergence yet |
||
| 665 | } else { |
||
| 666 | // Form shift |
||
| 667 | $x = $this->H[$n][$n]; |
||
| 668 | $y = 0.0; |
||
| 669 | $w = 0.0; |
||
| 670 | if ($l < $n) { |
||
| 671 | $y = $this->H[$n - 1][$n - 1]; |
||
| 672 | $w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n]; |
||
| 673 | } |
||
| 674 | |||
| 675 | // Wilkinson's original ad hoc shift |
||
| 676 | if ($iter == 10) { |
||
| 677 | $exshift += $x; |
||
| 678 | for ($i = $low; $i <= $n; ++$i) { |
||
| 679 | $this->H[$i][$i] -= $x; |
||
| 680 | } |
||
| 681 | |||
| 682 | $s = abs($this->H[$n][$n - 1]) + abs($this->H[$n - 1][$n - 2]); |
||
| 683 | $x = $y = 0.75 * $s; |
||
| 684 | $w = -0.4375 * $s * $s; |
||
| 685 | } |
||
| 686 | |||
| 687 | // MATLAB's new ad hoc shift |
||
| 688 | if ($iter == 30) { |
||
| 689 | $s = ($y - $x) / 2.0; |
||
| 690 | $s = $s * $s + $w; |
||
| 691 | if ($s > 0) { |
||
| 692 | $s = sqrt($s); |
||
| 693 | if ($y < $x) { |
||
| 694 | $s = -$s; |
||
| 695 | } |
||
| 696 | |||
| 697 | $s = $x - $w / (($y - $x) / 2.0 + $s); |
||
| 698 | for ($i = $low; $i <= $n; ++$i) { |
||
| 699 | $this->H[$i][$i] -= $s; |
||
| 700 | } |
||
| 701 | |||
| 702 | $exshift += $s; |
||
| 703 | $x = $y = $w = 0.964; |
||
| 704 | } |
||
| 705 | } |
||
| 706 | |||
| 707 | // Could check iteration count here. |
||
| 708 | $iter = $iter + 1; |
||
| 709 | // Look for two consecutive small sub-diagonal elements |
||
| 710 | $m = $n - 2; |
||
| 711 | while ($m >= $l) { |
||
| 712 | $z = $this->H[$m][$m]; |
||
| 713 | $r = $x - $z; |
||
| 714 | $s = $y - $z; |
||
| 715 | $p = ($r * $s - $w) / $this->H[$m + 1][$m] + $this->H[$m][$m + 1]; |
||
| 716 | $q = $this->H[$m + 1][$m + 1] - $z - $r - $s; |
||
| 717 | $r = $this->H[$m + 2][$m + 1]; |
||
| 718 | $s = abs($p) + abs($q) + abs($r); |
||
| 719 | $p = $p / $s; |
||
| 720 | $q = $q / $s; |
||
| 721 | $r = $r / $s; |
||
| 722 | if ($m == $l) { |
||
| 723 | break; |
||
| 724 | } |
||
| 725 | |||
| 726 | if (abs($this->H[$m][$m - 1]) * (abs($q) + abs($r)) < |
||
| 727 | $eps * (abs($p) * (abs($this->H[$m - 1][$m - 1]) + abs($z) + abs($this->H[$m + 1][$m + 1])))) { |
||
| 728 | break; |
||
| 729 | } |
||
| 730 | |||
| 731 | --$m; |
||
| 732 | } |
||
| 733 | |||
| 734 | for ($i = $m + 2; $i <= $n; ++$i) { |
||
| 735 | $this->H[$i][$i - 2] = 0.0; |
||
| 736 | if ($i > $m + 2) { |
||
| 737 | $this->H[$i][$i - 3] = 0.0; |
||
| 738 | } |
||
| 739 | } |
||
| 740 | |||
| 741 | // Double QR step involving rows l:n and columns m:n |
||
| 742 | for ($k = $m; $k <= $n - 1; ++$k) { |
||
| 743 | $notlast = ($k != $n - 1); |
||
| 744 | if ($k != $m) { |
||
| 745 | $p = $this->H[$k][$k - 1]; |
||
| 746 | $q = $this->H[$k + 1][$k - 1]; |
||
| 747 | $r = ($notlast ? $this->H[$k + 2][$k - 1] : 0.0); |
||
| 748 | $x = abs($p) + abs($q) + abs($r); |
||
| 749 | if ($x != 0.0) { |
||
| 750 | $p = $p / $x; |
||
| 751 | $q = $q / $x; |
||
| 752 | $r = $r / $x; |
||
| 753 | } |
||
| 754 | } |
||
| 755 | |||
| 756 | if ($x == 0.0) { |
||
| 757 | break; |
||
| 758 | } |
||
| 759 | |||
| 760 | $s = sqrt($p * $p + $q * $q + $r * $r); |
||
| 761 | if ($p < 0) { |
||
| 762 | $s = -$s; |
||
| 763 | } |
||
| 764 | |||
| 765 | if ($s != 0) { |
||
| 766 | if ($k != $m) { |
||
| 767 | $this->H[$k][$k - 1] = -$s * $x; |
||
| 768 | } elseif ($l != $m) { |
||
| 769 | $this->H[$k][$k - 1] = -$this->H[$k][$k - 1]; |
||
| 770 | } |
||
| 771 | |||
| 772 | $p = $p + $s; |
||
| 773 | $x = $p / $s; |
||
| 774 | $y = $q / $s; |
||
| 775 | $z = $r / $s; |
||
| 776 | $q = $q / $p; |
||
| 777 | $r = $r / $p; |
||
| 778 | // Row modification |
||
| 779 | for ($j = $k; $j < $nn; ++$j) { |
||
| 780 | $p = $this->H[$k][$j] + $q * $this->H[$k + 1][$j]; |
||
| 781 | if ($notlast) { |
||
| 782 | $p = $p + $r * $this->H[$k + 2][$j]; |
||
| 783 | $this->H[$k + 2][$j] = $this->H[$k + 2][$j] - $p * $z; |
||
| 784 | } |
||
| 785 | |||
| 786 | $this->H[$k][$j] = $this->H[$k][$j] - $p * $x; |
||
| 787 | $this->H[$k + 1][$j] = $this->H[$k + 1][$j] - $p * $y; |
||
| 788 | } |
||
| 789 | |||
| 790 | // Column modification |
||
| 791 | for ($i = 0; $i <= min($n, $k + 3); ++$i) { |
||
| 792 | $p = $x * $this->H[$i][$k] + $y * $this->H[$i][$k + 1]; |
||
| 793 | if ($notlast) { |
||
| 794 | $p = $p + $z * $this->H[$i][$k + 2]; |
||
| 795 | $this->H[$i][$k + 2] = $this->H[$i][$k + 2] - $p * $r; |
||
| 796 | } |
||
| 797 | |||
| 798 | $this->H[$i][$k] = $this->H[$i][$k] - $p; |
||
| 799 | $this->H[$i][$k + 1] = $this->H[$i][$k + 1] - $p * $q; |
||
| 800 | } |
||
| 801 | |||
| 802 | // Accumulate transformations |
||
| 803 | for ($i = $low; $i <= $high; ++$i) { |
||
| 804 | $p = $x * $this->V[$i][$k] + $y * $this->V[$i][$k + 1]; |
||
| 805 | if ($notlast) { |
||
| 806 | $p = $p + $z * $this->V[$i][$k + 2]; |
||
| 807 | $this->V[$i][$k + 2] = $this->V[$i][$k + 2] - $p * $r; |
||
| 808 | } |
||
| 809 | |||
| 810 | $this->V[$i][$k] = $this->V[$i][$k] - $p; |
||
| 811 | $this->V[$i][$k + 1] = $this->V[$i][$k + 1] - $p * $q; |
||
| 812 | } |
||
| 813 | } // ($s != 0) |
||
| 814 | } // k loop |
||
| 815 | } // check convergence |
||
| 816 | } // while ($n >= $low) |
||
| 817 | |||
| 818 | // Backsubstitute to find vectors of upper triangular form |
||
| 819 | if ($norm == 0.0) { |
||
| 820 | return; |
||
| 821 | } |
||
| 822 | |||
| 823 | for ($n = $nn - 1; $n >= 0; --$n) { |
||
| 824 | $p = $this->d[$n]; |
||
| 825 | $q = $this->e[$n]; |
||
| 826 | // Real vector |
||
| 827 | if ($q == 0) { |
||
| 828 | $l = $n; |
||
| 829 | $this->H[$n][$n] = 1.0; |
||
| 830 | for ($i = $n - 1; $i >= 0; --$i) { |
||
| 831 | $w = $this->H[$i][$i] - $p; |
||
| 832 | $r = 0.0; |
||
| 833 | for ($j = $l; $j <= $n; ++$j) { |
||
| 834 | $r = $r + $this->H[$i][$j] * $this->H[$j][$n]; |
||
| 835 | } |
||
| 836 | |||
| 837 | if ($this->e[$i] < 0.0) { |
||
| 838 | $z = $w; |
||
| 839 | $s = $r; |
||
| 840 | } else { |
||
| 841 | $l = $i; |
||
| 842 | if ($this->e[$i] == 0.0) { |
||
| 843 | if ($w != 0.0) { |
||
| 844 | $this->H[$i][$n] = -$r / $w; |
||
| 845 | } else { |
||
| 846 | $this->H[$i][$n] = -$r / ($eps * $norm); |
||
| 847 | } |
||
| 848 | |||
| 849 | // Solve real equations |
||
| 850 | } else { |
||
| 851 | $x = $this->H[$i][$i + 1]; |
||
| 852 | $y = $this->H[$i + 1][$i]; |
||
| 853 | $q = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i]; |
||
| 854 | $t = ($x * $s - $z * $r) / $q; |
||
| 855 | $this->H[$i][$n] = $t; |
||
| 856 | if (abs($x) > abs($z)) { |
||
| 857 | $this->H[$i + 1][$n] = (-$r - $w * $t) / $x; |
||
| 858 | } else { |
||
| 859 | $this->H[$i + 1][$n] = (-$s - $y * $t) / $z; |
||
| 860 | } |
||
| 861 | } |
||
| 862 | |||
| 863 | // Overflow control |
||
| 864 | $t = abs($this->H[$i][$n]); |
||
| 865 | if (($eps * $t) * $t > 1) { |
||
| 866 | for ($j = $i; $j <= $n; ++$j) { |
||
| 867 | $this->H[$j][$n] = $this->H[$j][$n] / $t; |
||
| 868 | } |
||
| 869 | } |
||
| 870 | } |
||
| 871 | } |
||
| 872 | |||
| 873 | // Complex vector |
||
| 874 | } elseif ($q < 0) { |
||
| 875 | $l = $n - 1; |
||
| 876 | // Last vector component imaginary so matrix is triangular |
||
| 877 | if (abs($this->H[$n][$n - 1]) > abs($this->H[$n - 1][$n])) { |
||
| 878 | $this->H[$n - 1][$n - 1] = $q / $this->H[$n][$n - 1]; |
||
| 879 | $this->H[$n - 1][$n] = -($this->H[$n][$n] - $p) / $this->H[$n][$n - 1]; |
||
| 880 | } else { |
||
| 881 | $this->cdiv(0.0, -$this->H[$n - 1][$n], $this->H[$n - 1][$n - 1] - $p, $q); |
||
| 882 | $this->H[$n - 1][$n - 1] = $this->cdivr; |
||
| 883 | $this->H[$n - 1][$n] = $this->cdivi; |
||
| 884 | } |
||
| 885 | |||
| 886 | $this->H[$n][$n - 1] = 0.0; |
||
| 887 | $this->H[$n][$n] = 1.0; |
||
| 888 | for ($i = $n - 2; $i >= 0; --$i) { |
||
| 889 | // double ra,sa,vr,vi; |
||
| 890 | $ra = 0.0; |
||
| 891 | $sa = 0.0; |
||
| 892 | for ($j = $l; $j <= $n; ++$j) { |
||
| 893 | $ra = $ra + $this->H[$i][$j] * $this->H[$j][$n - 1]; |
||
| 894 | $sa = $sa + $this->H[$i][$j] * $this->H[$j][$n]; |
||
| 895 | } |
||
| 896 | |||
| 897 | $w = $this->H[$i][$i] - $p; |
||
| 898 | if ($this->e[$i] < 0.0) { |
||
| 899 | $z = $w; |
||
| 900 | $r = $ra; |
||
| 901 | $s = $sa; |
||
| 902 | } else { |
||
| 903 | $l = $i; |
||
| 904 | if ($this->e[$i] == 0) { |
||
| 905 | $this->cdiv(-$ra, -$sa, $w, $q); |
||
| 906 | $this->H[$i][$n - 1] = $this->cdivr; |
||
| 907 | $this->H[$i][$n] = $this->cdivi; |
||
| 908 | } else { |
||
| 909 | // Solve complex equations |
||
| 910 | $x = $this->H[$i][$i + 1]; |
||
| 911 | $y = $this->H[$i + 1][$i]; |
||
| 912 | $vr = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i] - $q * $q; |
||
| 913 | $vi = ($this->d[$i] - $p) * 2.0 * $q; |
||
| 914 | if ($vr == 0.0 & $vi == 0.0) { |
||
| 915 | $vr = $eps * $norm * (abs($w) + abs($q) + abs($x) + abs($y) + abs($z)); |
||
| 916 | } |
||
| 917 | |||
| 918 | $this->cdiv($x * $r - $z * $ra + $q * $sa, $x * $s - $z * $sa - $q * $ra, $vr, $vi); |
||
| 919 | $this->H[$i][$n - 1] = $this->cdivr; |
||
| 920 | $this->H[$i][$n] = $this->cdivi; |
||
| 921 | if (abs($x) > (abs($z) + abs($q))) { |
||
| 922 | $this->H[$i + 1][$n - 1] = (-$ra - $w * $this->H[$i][$n - 1] + $q * $this->H[$i][$n]) / $x; |
||
| 923 | $this->H[$i + 1][$n] = (-$sa - $w * $this->H[$i][$n] - $q * $this->H[$i][$n - 1]) / $x; |
||
| 924 | } else { |
||
| 925 | $this->cdiv(-$r - $y * $this->H[$i][$n - 1], -$s - $y * $this->H[$i][$n], $z, $q); |
||
| 926 | $this->H[$i + 1][$n - 1] = $this->cdivr; |
||
| 927 | $this->H[$i + 1][$n] = $this->cdivi; |
||
| 928 | } |
||
| 929 | } |
||
| 930 | |||
| 931 | // Overflow control |
||
| 932 | $t = max(abs($this->H[$i][$n - 1]), abs($this->H[$i][$n])); |
||
| 933 | if (($eps * $t) * $t > 1) { |
||
| 934 | for ($j = $i; $j <= $n; ++$j) { |
||
| 935 | $this->H[$j][$n - 1] = $this->H[$j][$n - 1] / $t; |
||
| 936 | $this->H[$j][$n] = $this->H[$j][$n] / $t; |
||
| 937 | } |
||
| 938 | } |
||
| 939 | } // end else |
||
| 940 | } // end for |
||
| 941 | } // end else for complex case |
||
| 942 | } // end for |
||
| 943 | |||
| 944 | // Vectors of isolated roots |
||
| 945 | for ($i = 0; $i < $nn; ++$i) { |
||
| 946 | if ($i < $low | $i > $high) { |
||
| 947 | for ($j = $i; $j < $nn; ++$j) { |
||
| 948 | $this->V[$i][$j] = $this->H[$i][$j]; |
||
| 949 | } |
||
| 950 | } |
||
| 951 | } |
||
| 952 | |||
| 953 | // Back transformation to get eigenvectors of original matrix |
||
| 954 | for ($j = $nn - 1; $j >= $low; --$j) { |
||
| 955 | for ($i = $low; $i <= $high; ++$i) { |
||
| 956 | $z = 0.0; |
||
| 957 | for ($k = $low; $k <= min($j, $high); ++$k) { |
||
| 958 | $z = $z + $this->V[$i][$k] * $this->H[$k][$j]; |
||
| 959 | } |
||
| 960 | |||
| 961 | $this->V[$i][$j] = $z; |
||
| 962 | } |
||
| 963 | } |
||
| 964 | } |
||
| 965 | } |
||
| 966 |
