This class is used to encompass typical methods and features that you can extend
to polynomials. For example, polynomial differentiation follows a specific rule,
and thus we can build a differentiation method that returns the exact derivative
for polynomials.
Input arguments: simply pass in an array of coefficients in decreasing powers.
Make sure to put a 0 coefficient in place of powers that are not used.
Current features:
o Print a human readable string of a polynomial of any variable (default of x)
o Evaluate a polynomial at any real number
o Return the degree of a polynomial
o Return the coefficients of a polynomial
o Return the variable of a polynomial
o Set the variable of an instantiated polynomial
o Polynomial differentiation (exact)
o Polynomial integration (indefinite integral)
o Polynomial addition
o Polynomial multiplication
Examples:
$polynomial = new Polynomial([1, -8, 12, 3]);
echo $polynomial; // prints 'x³ - 8x² + 12x + 3'
echo $polynomial(4); // prints -31
echo $polynomial->getDegree(); // prints 3
print_r($polynomial->getCoefficients()); // prints [1, -8, 12, 3]
echo $polynomial->differentiate(); // prints '3x² - 16x + 12'
echo $polynomial->integrate(); // prints '0.25x⁴ - 2.6666666666667x³ + 6x² + 3x'
echo $polynomial->add($polynomial); // prints '2x³ - 16x² + 24x + 6'
echo $polynomial->multiply($polynomial); // prints 'x⁶ - 16x⁵ + 88x⁴ - 186x³ + 96x² + 72x + 9'
echo $polynomial->getVariable(); // prints 'x'
$polynomial->setVariable("r");
echo $polynomial; // prints 'r³ - 8r² + 12r + 3'
https://en.wikipedia.org/wiki/Polynomial
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);
}
