Ensures that the length of each subinterval is equal, or equivalently,
that the spacing between each point is equal
| public static isSpacingConstant ( array $sorted ) | ||
| $sorted | array | Points sorted by (increasing) x-component |
/**
* Use Boole's Rule to aproximate the definite integral of a
* function f(x). Our input can support either a set of arrays, or a callback
* function with arguments (to produce a set of arrays). Each array in our
* input contains two numbers which correspond to coordinates (x, y) or
* equivalently, (x, f(x)), of the function f(x) whose definite integral we
* are approximating.
*
* Note: Boole's rule requires that our number of subintervals is a factor
* of four (we must supply an n points such that n-1 is a multiple of four)
* and also that the size of each subinterval is equal (spacing between each
* point is equal).
*
* The bounds of the definite integral to which we are approximating is
* determined by the our inputs.
*
* Example: approximate([0, 10], [2, 5], [4, 7], [6,3]) will approximate the
* definite integral of the function that produces these coordinates with a
* lower bound of 0, and an upper bound of 6.
*
* Example: approximate(function($x) {return $x**2;}, [0, 3 ,5]) will produce
* a set of arrays by evaluating the callback at 5 evenly spaced points
* between 0 and 3. Then, this array will be used in our approximation.
*
* Boole's Rule:
*
* xn ⁿ⁻¹ xᵢ₊₁
* ∫ f(x)dx = ∑ ∫ f(x)dx
* x₁ ⁱ⁼¹ xᵢ
*
* ⁽ⁿ⁻¹⁾/⁴ 2h
* = ∑ -- [7f⟮x₄ᵢ₋₃⟯ + 32f⟮x₄ᵢ₋₂⟯ + 12f⟮x₄ᵢ₋₁⟯ + 32f⟮x₄ᵢ⟯ + 7f⟮x₄ᵢ₊₁⟯] + O(h⁷f⁽⁶⁾(x))
* ⁱ⁼¹ 45
* where h = (xn - x₁) / (n - 1)
*
* @param $source The source of our approximation. Should be either
* a callback function or a set of arrays. Each array
* (point) contains precisely two numbers, an x and y.
* Example array: [[1,2], [2,3], [3,4], [4,5], [5,6]].
* Example callback: function($x) {return $x**2;}
* @param numbers ... $args The arguments of our callback function: start,
* end, and n. Example: approximate($source, 0, 8, 4).
* If $source is a set of points, do not input any
* $args. Example: approximate($source).
*
* @return number The approximation to the integral of f(x)
*/
public static function approximate($source, ...$args)
{
// get an array of points from our $source argument
$points = self::getPoints($source, $args);
// Validate input and sort points
self::validate($points, $degree = 5);
Validation::isSubintervalsMultiple($points, $m = 4);
$sorted = self::sort($points);
Validation::isSpacingConstant($sorted);
// Descriptive constants
$x = self::X;
$y = self::Y;
// Initialize
$n = count($sorted);
$subintervals = $n - 1;
$a = $sorted[0][$x];
$b = $sorted[$n - 1][$x];
$h = ($b - $a) / $subintervals;
$approximation = 0;
/*
* ⁽ⁿ⁻¹⁾/⁴ 2h
* = ∑ -- [7f⟮x₄ᵢ₋₃⟯ + 32f⟮x₄ᵢ₋₂⟯ + 12f⟮x₄ᵢ₋₁⟯ + 32f⟮x₄ᵢ⟯ + 7f⟮x₄ᵢ₊₁⟯] + O(h⁷f⁽⁶⁾(x))
* ⁱ⁼¹ 45
*/
for ($i = 1; $i < $subintervals / 4 + 1; $i++) {
$x₄ᵢ₋₃ = $sorted[4 * $i - 4][$x];
$x₄ᵢ₋₂ = $sorted[4 * $i - 3][$x];
$x₄ᵢ₋₁ = $sorted[4 * $i - 2][$x];
$x₄ᵢ = $sorted[4 * $i - 1][$x];
$x₄ᵢ₊₁ = $sorted[4 * $i][$x];
$f⟮x₄ᵢ₋₃⟯ = $sorted[4 * $i - 4][$y];
// y₄ᵢ₋₃
$f⟮x₄ᵢ₋₂⟯ = $sorted[4 * $i - 3][$y];
// y₄ᵢ₋₂
$f⟮x₄ᵢ₋₁⟯ = $sorted[4 * $i - 2][$y];
// y₄ᵢ₋₁
$f⟮x₄ᵢ⟯ = $sorted[4 * $i - 1][$y];
// y₄ᵢ
$f⟮x₄ᵢ₊₁⟯ = $sorted[4 * $i][$y];
// y₄ᵢ₊₁
$lagrange = LagrangePolynomial::interpolate([[$x₄ᵢ₋₃, $f⟮x₄ᵢ₋₃⟯], [$x₄ᵢ₋₂, $f⟮x₄ᵢ₋₂⟯], [$x₄ᵢ₋₁, $f⟮x₄ᵢ₋₁⟯], [$x₄ᵢ, $f⟮x₄ᵢ⟯], [$x₄ᵢ₊₁, $f⟮x₄ᵢ₊₁⟯]]);
$integral = $lagrange->integrate();
$approximation += $integral($x₄ᵢ₊₁) - $integral($x₄ᵢ₋₃);
// definite integral of lagrange polynomial
}
return $approximation;
}
/**
* Use Simpson's 3/8 Rule to aproximate the definite integral of a
* function f(x). Our input can support either a set of arrays, or a callback
* function with arguments (to produce a set of arrays). Each array in our
* input contains two numbers which correspond to coordinates (x, y) or
* equivalently, (x, f(x)), of the function f(x) whose definite integral we
* are approximating.
*
* Note: Simpson's 3/8 rule requires that our number of subintervals is a factor
* of three (we must supply an n points such that n-1 is a multiple of three)
* and also that the size of each subinterval is equal (spacing between each
* point is equal).
*
* The bounds of the definite integral to which we are approximating is
* determined by the our inputs.
*
* Example: approximate([0, 10], [2, 5], [4, 7], [6,3]) will approximate the
* definite integral of the function that produces these coordinates with a
* lower bound of 0, and an upper bound of 6.
*
* Example: approximate(function($x) {return $x**2;}, [0, 3 ,4]) will produce
* a set of arrays by evaluating the callback at 4 evenly spaced points
* between 0 and 3. Then, this array will be used in our approximation.
*
* Simpson's 3/8 Rule:
*
* xn ⁿ⁻¹ xᵢ₊₁
* ∫ f(x)dx = ∑ ∫ f(x)dx
* x₁ ⁱ⁼¹ xᵢ
*
* ⁽ⁿ⁻¹⁾/³ 3h
* = ∑ - [f⟮x₂ᵢ₋₁⟯ + 3f⟮x₂ᵢ⟯ + 3f⟮x₂ᵢ₊₁⟯ + f⟮x₂ᵢ₊₂⟯] + O(h⁵f⁗(x))
* ⁱ⁼¹ 8
* where h = (xn - x₁) / (n - 1)
*
* @param $source The source of our approximation. Should be either
* a callback function or a set of arrays. Each array
* (point) contains precisely two numbers, an x and y.
* Example array: [[1,2], [2,3], [3,4], [4,5]].
* Example callback: function($x) {return $x**2;}
* @param numbers ... $args The arguments of our callback function: start,
* end, and n. Example: approximate($source, 0, 8, 4).
* If $source is a set of points, do not input any
* $args. Example: approximate($source).
*
* @return number The approximation to the integral of f(x)
*/
public static function approximate($source, ...$args)
{
// get an array of points from our $source argument
$points = self::getPoints($source, $args);
// Validate input and sort points
self::validate($points, $degree = 4);
Validation::isSubintervalsMultiple($points, $m = 3);
$sorted = self::sort($points);
Validation::isSpacingConstant($sorted);
// Descriptive constants
$x = self::X;
$y = self::Y;
// Initialize
$n = count($sorted);
$subintervals = $n - 1;
$a = $sorted[0][$x];
$b = $sorted[$n - 1][$x];
$h = ($b - $a) / $subintervals;
$approximation = 0;
/*
* ⁽ⁿ⁻¹⁾/³ 3h
* ∑ -- [f⟮x₂ᵢ₋₁⟯ + 3f⟮x₂ᵢ⟯ + 3f⟮x₂ᵢ₊₁⟯ + f⟮x₂ᵢ₊₂⟯] + O(h⁵f⁗(x))
* ⁱ⁼¹ 8
* where h = (xn - x₁) / (n - 1)
*/
for ($i = 1; $i < $subintervals / 3 + 1; $i++) {
$x₂ᵢ₋₁ = $sorted[2 * $i - 2][$x];
$x₂ᵢ = $sorted[2 * $i - 1][$x];
$x₂ᵢ₊₁ = $sorted[2 * $i][$x];
$x₂ᵢ₊₂ = $sorted[2 * $i + 1][$x];
$f⟮x₂ᵢ₋₁⟯ = $sorted[2 * $i - 2][$y];
// y₂₋₁
$f⟮x₂ᵢ⟯ = $sorted[2 * $i - 1][$y];
// y₂ᵢ
$f⟮x₂ᵢ₊₁⟯ = $sorted[2 * $i][$y];
// y₂ᵢ₊₁
$f⟮x₂ᵢ₊₂⟯ = $sorted[2 * $i + 1][$y];
// y₂ᵢ₊₂
$lagrange = LagrangePolynomial::interpolate([[$x₂ᵢ₋₁, $f⟮x₂ᵢ₋₁⟯], [$x₂ᵢ, $f⟮x₂ᵢ⟯], [$x₂ᵢ₊₁, $f⟮x₂ᵢ₊₁⟯], [$x₂ᵢ₊₂, $f⟮x₂ᵢ₊₂⟯]]);
$integral = $lagrange->integrate();
$approximation += $integral($x₂ᵢ₊₂) - $integral($x₂ᵢ₋₁);
// definite integral of lagrange polynomial
}
return $approximation;
}