news_producer/vendor/markrogoyski/math-php/tests/NumericalAnalysis/NumericalDifferentiation/SecondDerivativeMidpointFormulaTest.php

<?php

namespace MathPHP\Tests\NumericalAnalysis\NumericalDifferentiation;

use MathPHP\NumericalAnalysis\NumericalDifferentiation\SecondDerivativeMidpointFormula;
use MathPHP\Exception;

class SecondDerivativeMidpointFormulaTest extends \PHPUnit\Framework\TestCase
{
    /**
     * @test   differentiate zero error - callback
     * @throws \Exception
     *
     * f(x)    = 13x² -92x + 96
     * f'(x)   = 26x - 92
     * f''(x)  = 26
     *
     *                                                        h²
     * Error term for the Second Derivative Midpoint Formula: - f⁽⁴⁾(ζ)
     *                                                        12
     * where ζ lies between x₀ - h and x₀ + h
     *
     * f'(x)   = 26x - 92
     * f''(x)  = 26
     * f⁽³⁾(x) = 0
     * f⁽⁴⁾(x) = 0
     * Thus, our error is zero in our formula.
     */
    public function testZeroErrorCallback()
    {
        // Given f(x) = 13x² -92x + 96
        $f = function ($x) {
            return 13 * $x ** 2 - 92 * $x + 96;
        };

        // And $f’’ - 26
        $f’’ = function ($x) {
            return 26;
        };

        // And
        $n = 3;
        $a = 0;
        $b = 4;

        // And f'(x) at x = 2
        $target   = 2;
        $expected = $f’’($target);

        // When
        $actual = SecondDerivativeMidpointFormula::differentiate($target, $f, $a, $b, $n);

        // Then
        $this->assertEquals($expected, $actual);
    }

    /**
     * @test   differentiate nonzero error - callback
     * @throws \Exception
     *
     * f(x)    = x⁴ - 13x² -92x + 96
     * f'(x)   = 4x³ - 26x - 92
     * f''(x)  = 12x² - 26
     *
     *                                                        h²
     * Error term for the Second Derivative Midpoint Formula: - f⁽⁴⁾(ζ)
     *                                                        12
     * where ζ lies between x₀ - h and x₀ + h
     *
     * f'(x)   = 4x³ - 26x - 92
     * f''(x)  = 12x² - 26
     * f⁽³⁾(x) = 24x
     * f⁽⁴⁾(x) = 24
     * Error in Second Derivative Midpoint Formula on [0,2] (where h=1) < 2
     */
    public function testNonZeroErrorCallback()
    {
        // Given f(x) = x⁴ - 13x² -92x + 96
        $f = function ($x) {
            return $x ** 4 - 13 * $x ** 2 - 92 * $x + 96;
        };

        // And $f’’(x) = 12x² - 26
        $f’’ = function ($x) {
            return 12 * $x ** 2 - 26;
        };

        // And
        $n = 3;
        $a = 0;
        $b = 2;

        // And f’’(x) at x = 1
        $target   = 1;
        $tol      = 2;
        $expected = $f’’($target);

        // When
        $actual = SecondDerivativeMidpointFormula::differentiate($target, $f, $a, $b, $n);

        // Then
        $this->assertEqualsWithDelta($expected, $actual, $tol);
    }

    /**
     * @test   differentiate zero error - using points
     * @throws \Exception
     *
     * f(x)    = 13x² -92x + 96
     * f'(x)   = 26x - 92
     * f''(x)  = 26
     *
     *                                                        h²
     * Error term for the Second Derivative Midpoint Formula: - f⁽⁴⁾(ζ)
     *                                                        12
     * where ζ lies between x₀ - h and x₀ + h
     *
     * f'(x)   = 26x - 92
     * f''(x)  = 26
     * f⁽³⁾(x) = 0
     * f⁽⁴⁾(x) = 0
     * Thus, our error is zero in our formula.
     */
    public function testZeroErrorPoints()
    {
        // Given f(x) = 13x² -92x + 96
        $f = function ($x) {
            return 13 * $x ** 2 - 92 * $x + 96;
        };
        $points = [[0, $f(0)], [2, $f(2)], [4, $f(4)]];

        // And $f’’ - 26
        $f’’ = function ($x) {
            return 26;
        };

        // And f'(x) at x = 2
        $target   = 2;
        $expected = $f’’($target);

        // When
        $actual = SecondDerivativeMidpointFormula::differentiate($target, $points);

        // Then
        $this->assertEquals($expected, $actual);
    }

    /**
     * @test   differentiate target exception
     * @throws \Exception
     */
    public function testTargetException()
    {
        // Given f(x) = 13x² -92x + 96
        $f = function ($x) {
            return 13 * $x ** 2 - 92 * $x + 96;
        };

        $points = [[0, $f(0)], [2, $f(2)], [4, $f(4)]];

        // And
        $f’’ = function ($x) {
            return 26;
        };

        $target = 87348738473;

        // Then
        $this->expectException(Exception\BadDataException::class);

        // When
        $actual = SecondDerivativeMidpointFormula::differentiate($target, $points);
    }
}