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public static interpolate ( $source, $args ) : |
||
| $source | The source of our approximation. Should be either a callback function or a set of arrays. Each array (point) contains precisely three numbers: x, y, and y' Example array: [[1,2,1], [2,3,0], [3,4,2]]. Example callback: function($x) {return $x**2;} | |
| Результат | The interpolting (piecewise) polynomial, as an instance of Piecewise. | |
public static function interpolate($source, ...$args)
{
// Get an array of points from our $source argument
$points = self::getSplinePoints($source, $args);
// Validate input and sort points
self::validateSpline($points, $degree = 1);
$sorted = self::sort($points);
// Descriptive constants
$x = self::X;
$y = self::Y;
$y’ = self::Y’;
// Initialize
$n = count($sorted);
$k = $n - 1;
$x₀ = $sorted[0][$x];
$x₁ = $sorted[1][$x];
$f⟮x₀⟯ = $sorted[0][$y];
// y₀
$f⟮x₁⟯ = $sorted[1][$y];
// y₁
$y’₀ = $sorted[0][$y’];
// y₀-prime
$h = [$x₁ - $x₀];
$a = [3 / $h[0] * ($f⟮x₁⟯ - $f⟮x₀⟯) - 3 * $y’₀];
$μ = [0.5];
$z = [$a[0] / (2 * $h[0])];
$c[$k] = 0;
$poly = [];
for ($i = 0; $i < $k; $i++) {
$xᵢ = $sorted[$i][$x];
$xᵢ₊₁ = $sorted[$i + 1][$x];
$a[$i] = $sorted[$i][$y];
$h[$i] = $xᵢ₊₁ - $xᵢ;
if ($i == 0) {
continue;
}
$xᵢ₋₁ = $sorted[$i - 1][$x];
$f⟮xᵢ⟯ = $sorted[$i][$y];
// yᵢ
$f⟮xᵢ₊₁⟯ = $sorted[$i + 1][$y];
// yᵢ₊₁
$f⟮xᵢ₋₁⟯ = $sorted[$i - 1][$y];
// yᵢ₋₁
$α = 3 / $h[$i] * ($f⟮xᵢ₊₁⟯ - $f⟮xᵢ⟯) - 3 / $h[$i - 1] * ($f⟮xᵢ⟯ - $f⟮xᵢ₋₁⟯);
$l = 2 * ($xᵢ₊₁ - $xᵢ₋₁) - $h[$i - 1] * $μ[$i - 1];
$μ[$i] = $h[$i] / $l;
$z[$i] = ($α - $h[$i - 1] * $z[$i - 1]) / $l;
}
$f⟮xₙ⟯ = $sorted[$k][$y];
// yₙ
$f⟮xₙ₋₁⟯ = $sorted[$k - 1][$y];
// yₙ₋₁
$y’ₙ = $sorted[$k][$y’];
// yₙ-prime
$a[$k] = 3 * $y’ₙ - 3 * ($f⟮xₙ⟯ - $f⟮xₙ₋₁⟯) / $h[$k - 1];
$l = $h[$k - 1] * (2 - $μ[$k - 1]);
$z[$k] = ($a[$k] - $h[$k - 1] * $z[$k - 1]) / $l;
$c[$n] = $z[$k];
for ($i = $k - 1; $i >= 0; $i--) {
$xᵢ = $sorted[$i][$x];
$xᵢ₊₁ = $sorted[$i + 1][$x];
$f⟮xᵢ⟯ = $sorted[$i][$y];
// yᵢ
$f⟮xᵢ₊₁⟯ = $sorted[$i + 1][$y];
// yᵢ₊₁
$c[$i] = $z[$i] - $μ[$i] * $c[$i + 1];
$b[$i] = ($f⟮xᵢ₊₁⟯ - $f⟮xᵢ⟯) / $h[$i] - $h[$i] * ($c[$i + 1] + 2 * $c[$i]) / 3;
$d[$i] = ($c[$i + 1] - $c[$i]) / (3 * $h[$i]);
$poly[$i] = new Polynomial([$d[$i], $c[$i] - 3 * $d[$i] * $xᵢ, $b[$i] - 2 * $c[$i] * $xᵢ + 3 * $d[$i] * $xᵢ ** 2, $a[$i] - $b[$i] * $xᵢ + $c[$i] * $xᵢ ** 2 - $d[$i] * $xᵢ ** 3]);
if ($i == 0) {
$int[$i] = [$xᵢ, $xᵢ₊₁];
} else {
$int[$i] = [$xᵢ, $xᵢ₊₁, true, false];
}
}
$piecewise = new Piecewise($int, $poly);
return $piecewise;
} public function testSolveNonzeroError()
{
// f(x) = x⁴ + 8x³ -13x² -92x + 96
$f = new Polynomial([1, 8, -13, -92, 96]);
$f’ = $f->differentiate();
$f⁽⁴⁾ = $f’->differentiate()->differentiate()->differentiate();
// The error is bounded by:
// |f(x)-p(x)| = tol <= (5/384) * h⁴ * max f⁽⁴⁾(x)
// where h = max hᵢ
// and max f⁽⁴⁾(x) = f⁽⁴⁾(x) for all x given a 4th-degree polynomial f(x)
$a = 0;
$b = 10;
$n = 51;
// So, tol <= (1/24) * (1/5)⁴ * 24 = (1/5)⁴
$h = ($b - $a) / ($n - 1);
$tol = 5 / 384 * $h ** 4 * $f⁽⁴⁾(0);
$roundoff = 1.0E-6;
// round off error
$p = ClampedCubicSpline::interpolate($f, $f’, $a, $b, $n);
// Check that p(x) agrees with f(x) at x = 0
$target = 0;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 2
$target = 2;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 4
$target = 4;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 6
$target = 6;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 8
$target = 8;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 10
$target = 10;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 7.32 (not a node)
$target = 7.32;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
}
