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Start implementing eigenvaluedecomposition

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This commit is contained in:
Dennis Eichhorn 2018-06-05 18:56:17 +02:00
parent 9bf5ed087d
commit 36f63aa97f
1 changed files with 361 additions and 11 deletions
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372
Math/Matrix/EigenvalueDecomposition.php
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@ -93,6 +93,22 @@ final class EigenvalueDecomposition
*/ */
private $ort = []; private $ort = [];
/**
* Complex scalar division
*
* @var float
* @since 1.0.0
*/
private $cdivr = 0.0;
/**
* Complex scalar division
*
* @var float
* @since 1.0.0
*/
private $cdivi = 0.0;
/** /**
* Constructor. * Constructor.
* *
@ -104,6 +120,32 @@ final class EigenvalueDecomposition
{ {
$this->m = $M->getM(); $this->m = $M->getM();
$this->A = $M->toArray(); $this->A = $M->toArray();
for ($j = 0; ($j < $this->m) & $this->isSymmetric; ++$j) {
for ($i = 0; ($i < $this->m) & $this->isSymmetric; ++$i) {
$this->isSymmetric = ($this->A[$i][$j] === $this->A[$j][$i]);
}
}
if ($this->isSymmetric) {
for ($i = 0; $i < $this->m; ++$i) {
for ($j = 0; $j < $this->m; ++$j) {
$this->V[$i][$j] = $this->A[$i][$j];
}
}
$this->tred2();
$this->tql2();
} else {
for ($j = 0; $j < $this->m; ++$j) {
for ($i = 0; $i < $this->m; ++$i) {
$this->H[$i][$j] = $this->A[$i][$j];
}
}
$this->orthes();
$this->hqr2();
}
} }
/** /**
@ -141,11 +183,7 @@ final class EigenvalueDecomposition
} }
$f = $this->D[$i - 1]; $f = $this->D[$i - 1];
$g = \sqrt($h); $g = $f > 0 ? -\sqrt($h) : \sqrt($h);
if ($f > 0) {
$g = -$g;
}
$this->E[$i] = $scale * $g; $this->E[$i] = $scale * $g;
$h -= $f * $g; $h -= $f * $g;
@ -270,11 +308,7 @@ final class EigenvalueDecomposition
$g = $this->D[$l]; $g = $this->D[$l];
$p = ($this->D[$l + 1] - $g) / (2.0 * $this->E[$l]); $p = ($this->D[$l + 1] - $g) / (2.0 * $this->E[$l]);
$r = Triangle::getHypot($p, 1); $r = $p > 0 ? -Triangle::getHypot($p, 1) : Triangle::getHypot($p, 1);
if ($p > 0) {
$r = -$r;
}
$this->D[$l] = $this->E[$l] / ($p + $r); $this->D[$l] = $this->E[$l] / ($p + $r);
$this->D[$l + 1] = $this->E[$l] * ($p + $r); $this->D[$l + 1] = $this->E[$l] * ($p + $r);
@ -308,7 +342,7 @@ final class EigenvalueDecomposition
$this->D[$i + 1] = $h + $s * ($c * $g + $s * $this->D[$i]); $this->D[$i + 1] = $h + $s * ($c * $g + $s * $this->D[$i]);
for ($k = 0; $k < $this->m; ++$k) { for ($k = 0; $k < $this->m; ++$k) {
$h = $this->V[$k][$i + 1]; $h = $this->V[$k][$i + 1];
$this->V[$k][$i + 1] = $s * $this->V[$k][$i] + $c * $h; $this->V[$k][$i + 1] = $s * $this->V[$k][$i] + $c * $h;
$this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h; $this->V[$k][$i] = $c * $this->V[$k][$i] - $s * $h;
} }
@ -350,11 +384,327 @@ final class EigenvalueDecomposition
private function orthes() : void private function orthes() : void
{ {
$low = 0;
$high = $this->m - 1;
for ($m = $low + 1; $m < $high - 1; ++$m) {
$scale = 0.0;
for ($i = $m; $i <= $high; ++$i) {
$scale += \abs($this->H[$i][$m - 1]);
}
if ($scale !== 0.0) {
$h = 0.0;
for ($i = $high; $i >= $m; --$i) {
$this->ort[$i] = $this->H[$i][$m - 1] / $scale;
$h += $this->ort[$i] * $this->ort[$i];
}
$g = $this->ort[$m] > 0 ? -sqrt($h) : sqrt($h);
$h -= $this->ort[$m] * $g;
$this->ort[$m] -= $g;
for ($j = $m; $j < $this->m; ++$j) {
$f = 0.0;
for ($i = $high; $i >= $m; --$i) {
$f += $this->ort[$i] * $this->H[$i][$j];
}
$f /= $h;
for ($i = $m; $i <= $high; ++$i) {
$this->H[$i][$j] -= $f * $this->ort[$i];
}
}
for ($i = 0; $i <= $high; ++$i) {
$f = 0.0;
for ($j = $high; $j >= $m; --$j) {
$f += $this->ort[$j] * $this->H[$i][$j];
}
$f /= $h;
for ($j = $m; $j <= $high; ++$j) {
$this->H[$i][$j] -= $f * $this->ort[$j];
}
}
$this->ort[$m] *= $scale;
$this->H[$m][$m - 1] = $scale * $g;
}
}
for ($i = 0; $i < $this->m; ++$i) {
for ($j = 0; $j < $this->m; ++$j) {
$this->V[$i][$j] = ($i === $j) ? 1.0 : 0.0;
}
}
for ($m = $high - 1; $m >= $low + 1; --$m) {
if ($this->H[$m][$m - 1] !== 0.0) {
for ($i = $m + 1; $i <= $high; ++$i) {
$this->ort[$i] = $this->H[$i][$m - 1];
}
for ($j = $m; $j <= $high; ++$j) {
$g = 0.0;
for ($i = $m; $i <= $high; ++$i) {
$g += $this->ort[$i] * $this->V[$i][$j];
}
$g = ($g / $this->ort[$m]) / $this->H[$m][$m - 1];
for ($i = $m; $i <= $high; ++$i) {
$this->V[$i][$j] += $g * $this->ort[$i];
}
}
}
}
}
private function cdiv(float $xr, float $xi, float $yr, float $yi) : void
{
$r = 0.0;
$d = 0.0;
if (abs($yr) > \abs($yi)) {
$r = $yi / $yr;
$d = $yr + $r * $yi;
$this->cdivr = ($xr + $r * $xi) / $d;
$this->cdivi = ($xi + $r * $xr) / $d;
} else {
$r = $yr / $yi;
$d = $yi + $r * $yr;
$this->cdivr = ($r * $xr + $xi) / $d;
$this->cdivi = ($r * $xi - $xr) / $d;
}
}
private function hqr2() : void
{
$nn = $this->m;
$n = $nn - 1;
$low = 0;
$high = $nn - 1;
$eps = 0.0001;
$exshift = 0.0;
$p = 0;
$q = 0;
$r = 0;
$s = 0;
$z = 0;
$t = 0;
$w = 0;
$x = 0;
$y = 0;
$norm = 0.0;
for ($i = 0; $i < $nn; ++$i) {
if ($i < $low | $i > $high) {
$this->D[$i] = $this->H[$i][$i];
$this->E[$i] = 0.0;
}
for ($j = max($i - 1, 0); $j < $nn; ++$j) {
$norm += \abs($this->H[$i][$j]);
}
}
$iter = 0;
while ($n >= $low) {
$l = $n;
while ($l > $low) {
$s = \abs($this->H[$l - 1][$l - 1]) + \abs($this->H[$l][$l]);
if ($s === 0.0) {
$s = $norm;
}
if (abs($this->H[$l][$l - 1]) < $eps * $s) {
break;
}
--$l;
}
if ($l === $n) {
$this->H[$n][$n] = $this->H[$n][$n] + $exshift;
$this->D[$n] = $this->H[$n][$n];
$this->E[$n] = 0.0;
$iter = 0;
--$n;
} elseif ($l === $n - 1) {
$w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n];
$p = ($this->H[$n - 1][$n - 1] - $this->H[$n][$n]) / 2.0;
$q = $p * $p + $w;
$z = sqrt(abs($q));
$this->H[$n][$n] += $exshift;
$this->H[$n - 1][$n - 1] += $exshift;
$x = $this->H[$n][$n];
if ($q >= 0) {
$z = $p >= 0 ? $p + $z : $p - $z;
$this->D[$n - 1] = $x + $z;
$this->D[$n] = $z !== 0.0 ? $x - $w / $z : $this->D[$n - 1];
$this->E[$n - 1] = 0.0;
$this->E[$n] = 0.0;
$x = $this->H[$n][$n - 1];
$s = \abs($x) + \abs($z);
$p = $x / $s;
$q = $z / $s;
$r = sqrt($p * $p + $q * $q);
$p = $p / $r;
$q = $q / $r;
for ($j = $n - 1; $j < $nn; ++$j) {
$z = $this->H[$n - 1][$j];
$this->H[$n - 1][$j] = $q * $z + $p * $this->H[$n][$j];
$this->H[$n][$j] = $q * $this->H[$n][$j] - $p * $z;
}
for ($i = 0; $i <= $n; ++$i) {
$z = $this->H[$i][$n - 1];
$this->H[$i][$n - 1] = $q * $z + $p * $this->H[$i][$n];
$this->H[$i][$n] = $q * $this->H[$i][$n] - $p * $z;
}
for ($i = $low; $i <= $high; ++$i) {
$z = $this->V[$i][$n - 1];
$this->V[$i][$n - 1] = $q * $z * $this->V[$i][$n];
$this->V[$i][$n] = $q * $this->V[$i][$n] - $p * $z;
}
} else {
$this->D[$n - 1] = $x + $p;
$this->D[$n] = $x + $p;
$this->E[$n - 1] = $z;
$this->E[$n] = -$z;
}
$n -= 2;
$iter = 0;
} else {
$x = $this->H[$n][$n];
$y = 0.0;
$w = 0.0;
if ($l < $n) {
$y = $this->H[$n - 1][$n - 1];
$w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n];
}
if ($iter === 10) {
$exshift += $x;
for ($i = $low; $i <= $n; ++$i) {
$this->H[$i][$i] -= $x;
}
$s = \abs($this->H[$n][$n - 1]) + \abs($this->H[$n - 1][$n - 2]);
$x = 0.75 * $s;
$y = $x;
$w = -0.4375 * $s * $s;
}
if ($iter === 30) {
$s = ($y - $x) / 2.0;
$s = $s * $s + $w;
if ($s > 0) {
$s = $y < $x ? -\sqrt($s) : \sqrt($s);
$s = $x - $w / (($y - $x) / 2.0 + $s);
for ($i = $low; $i <= $n; ++$i) {
$this->H[$i][$i] -= $s;
}
$exshift += $s;
$x = $y = $w = 0.964;
}
}
++$iter;
$m = $n - 2;
while ($m >= 1) {
$z = $this->H[$m][$m];
$r = $x - $z;
$s = $y - $z;
$p = ($r * $s - $w) / $this->H[$m + 1][$m] + $this->H[$m][$m + 1];
$q = $this->H[$m + 1][$m + 1] - $z - $r - $s;
$r = $this->H[$m + 2][$m + 1];
$s = \abs($p) + \abs($q) + \abs($r);
$p = $p / $s;
$q = $q / $s;
$r = $r / $s;
if ($m === $l
|| \abs($this->H[$m][$m - 1]) * (\abs($q) + \abs($r)) < $eps * (\abs($p) * (\abs($this->H[$m - 1][$m - 1] + \abs($z) + \abs($this->H[$m + 1][$m + 1]))))
) {
break;
}
--$m;
}
for ($i = $m + 2; $i <= $n; ++$i) {
$this->H[$i][$i - 2] = 0.0;
if ($i > $m + 2) {
$this->H[$i][$i - 3] = 0.0;
}
}
}
}
} }
public function isSymmetric() : bool public function isSymmetric() : bool
{ {
return $this->isSymmetric; return $this->isSymmetric;
} }
public function getV() : Matrix
{
$matrix = new Matrix();
$matrix->setMatrix($this->V);
return $matrix;
}
public function getRealEigenvalues() : array
{
return $this->D;
}
public function getImagEigenvalues() : array
{
return $this->E;
}
public function getD() : Matrix
{
$matrix = new Matrix();
$D = [[]];
for ($i = 0; $i < $this->m; ++$i) {
for ($j = 0; $j < $this->m; ++$j) {
$D[$i][$j] = 0.0;
}
$D[$i][$i] = $this->D[$i];
if ($this->E[$i] > 0) {
$D[$i][$i + 1] = $this->E[$i];
} elseif ($this->E[$i] < 0) {
$D[$i][$i - 1] = $this->E[$i];
}
}
$matrix->setMatrix($D);
return $matrix;
}
} }