ad_producer/vendor/markrogoyski/math-php/tests/Algebra/AlgebraTest.php

<?php

namespace MathPHP\Tests\Algebra;

use MathPHP\Algebra;
use MathPHP\Number;

class AlgebraTest extends \PHPUnit\Framework\TestCase
{
    /**
     * @test         gcd returns the greatest common divisor of two integers.
     * @dataProvider dataProviderForGcd
     * @param        int $a
     * @param        int $b
     * @param        int $expected_gcd
     * @param        int $_
     * @param        int $__
     */
    public function testGCD(int $a, int $b, int $expected_gcd, int $_, int $__)
    {
        // When
        $gcd = Algebra::gcd($a, $b);

        // Then
        $this->assertEquals($expected_gcd, $gcd);
    }

    /**
     * @test         extendedGCD returns the extended greatest common divisor of two integers.
     * @dataProvider dataProviderForGcd
     * @param        int $a
     * @param        int $b
     * @param        int $expected_gcd
     * @param        int $expected_alpha
     * @param        int $expected_beta
     */
    public function testExtendedGcd(int $a, int $b, int $expected_gcd, int $expected_alpha, int $expected_beta)
    {
        // When
        [$gcd, $alpha, $beta] = Algebra::extendedGcd($a, $b);

        // Then
        $this->assertEquals($expected_gcd, $gcd);
        $this->assertEquals($expected_alpha, $alpha);
        $this->assertEquals($expected_beta, $beta);
    }

    public function dataProviderForGcd(): array
    {
        return [
            [0, 0, 0, 0, 1],
            [8, 0, 8, 1, 0],
            [0, 8, 8, 0, 1],
            [8, 12, 4, -1, 1],
            [12, 8, 4, 1, -1],
            [54, 24, 6, 1, -2],
            [24, 54, 6, -2, 1],
            [18, 84, 6, 5, -1],
            [84, 18, 6, -1, 5],
            [244, 343, 1, 97, -69],
            [343, 244, 1, -69, 97],
            [97, 577, 1, 232, -39],
            [577, 97, 1, -39, 232],
            [40902, 24140, 34, 337, -571],
            [24140, 40902, 34, -571, 337],
            [1234, 54, 2, -7, 160],
            [54, 1234, 2, 160, -7],
        ];
    }

    /**
     * @test         lcm returns the least-common multiple of two integers.
     * @dataProvider dataProviderForLcm
     * @param        int $a
     * @param        int $b
     * @param        int $expected_lcm
     */
    public function testLCM(int $a, int $b, int $expected_lcm)
    {
        // When
        $lcm = Algebra::lcm($a, $b);

        // Then
        $this->assertEquals($expected_lcm, $lcm);
    }

    /**
     * @return array [a, b, lcm]
     */
    public function dataProviderForLcm(): array
    {
        return [
            [0, 0, 0],
            [8, 0, 0],
            [0, 8, 0],
            [5, 2, 10],
            [2, 5, 10],
            [4, 6, 12],
            [6, 4, 12],
            [21, 6, 42],
            [6, 21, 42],
            [598, 352, 105248],
            [352, 598, 105248],
        ];
    }

    /**
     * @test         factors returns the expected factors of an integer.
     * @dataProvider dataProviderForFactors
     * @param        int $x
     * @param        array $expected_factors
     */
    public function testFactors(int $x, array $expected_factors)
    {
        // When
        $factors = Algebra::factors($x);

        // Then
        $this->assertEquals($expected_factors, $factors);
    }

    /**
     * @return array [x, factors]
     */
    public function dataProviderForFactors(): array
    {
        return [
            [ 0, [\INF] ],
            [ 1, [1] ],
            [ 4, [1, 2, 4] ],
            [ 12, [1, 2, 3, 4, 6, 12] ],
            [ 14, [1, 2, 7, 14] ],
            [ 30, [1, 2, 3, 5, 6, 10, 15, 30] ],
            [ 2248, [1, 2, 4, 8, 281, 562, 1124, 2248] ],
            [ 983928, [1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264, 3727, 7454, 11181, 14908, 22362, 29816, 40997, 44724, 81994, 89448, 122991, 163988, 245982, 327976, 491964, 983928] ],
            [ 9938938492, [1, 2, 4, 7, 14, 28, 79, 158, 283, 316, 553, 566, 1106, 1132, 1981, 2212, 3962, 7924, 15877, 22357, 31754, 44714, 63508, 89428, 111139, 156499, 222278, 312998, 444556, 625996, 1254283, 2508566, 4493191, 5017132, 8779981, 8986382, 17559962, 17972764, 31452337, 35119924, 62904674, 125809348, 354962089, 709924178, 1419848356, 2484734623, 4969469246, 9938938492] ],
            [ 9938938492873, [ 1, 13, 22637, 294281, 33773633, 439057229, 764533730221, 9938938492873]],
        ];
    }

    /**
     * @test         linear returns the expected root
     * @dataProvider dataProviderForLinear
     * @param        float $a
     * @param        float $b
     * @param        float $expected
     */
    public function testLinear(float $a, float $b, float $expected)
    {
        // When
        $root = Algebra::linear($a, $b);

        // Then
        $this->assertEqualsWithDelta($expected, $root, 0.00001);
    }

    /**
     * Test data created with Python numpy.roots([a, b])
     * @return array (a, b, root)
     */
    public function dataProviderForLinear(): array
    {
        return [
            [-1, 0, 0],
            [1, 0, 0],
            [5, 0, 0],
            [1, 1, -1],
            [-1, 1, 1],
            [1, -1, 1],
            [-1, -1, -1],
            [1, 2, -2],
            [-1, 2, 2],
            [1, -2, 2],
            [-1, -2, -2],
            [2, 4, -2],
            [-2, 4, 2],
            [2, -4, 2],
            [-2, -4, -2],
            [0.5, 1, -2],
            [-0.5, 1, 2],
            [0.5, -1, 2],
            [-0.5, -1, -2],
            [1, 0.5, -0.5],
            [-1, 0.5, 0.5],
            [1, -0.5, 0.5],
            [-1, -0.5, -0.5],
            [1, -3, 3],
            [35, 8, -0.22857143],
            [8, 35, -4.375],
            [82376, 984398, -11.95005827],
            [3.2, 2, -0.625],
            [6.2, 8.7, -1.40322581],
            [8.7, 6.2, -0.71264368],
            [0.001, 3, -3000],
        ];
    }

    /**
     * @test         linear returns the no root when a = 0
     * @dataProvider dataProviderForLinearNoRoot
     * @param        float $a
     * @param        float $b
     */
    public function testLinearNoRoot(float $a, float $b)
    {
        // When
        $root = Algebra::linear($a, $b);

        // Then
        $this->assertNull($root);
    }

    /**
     * @return array (a, b)
     */
    public function dataProviderForLinearNoRoot(): array
    {
        return [
            [0, 0],
            [0, 1],
            [0, 5],
            [0, -1],
            [0, -5],
            [-0, 1],
            [0.0, 1],
            [-0.0, 1],
        ];
    }

    /**
     * @test         quadratic returns the expected roots.
     * @dataProvider dataProviderForQuadratic
     * @param        float $a
     * @param        float $b
     * @param        float $c
     * @param        array $expected_quadratic
     * @throws       \MathPHP\Exception\IncorrectTypeException
     */
    public function testQuadratic(float $a, float $b, float $c, array $expected_quadratic)
    {
        // Given
        $expected_r1 = $expected_quadratic[0];
        $expected_r2 = $expected_quadratic[1];

        // When
        [$r1, $r2] = Algebra::quadratic($a, $b, $c);

        // Then
        $this->assertEqualsWithDelta($expected_r1, $r1, 0.00000001);
        $this->assertEqualsWithDelta($expected_r2, $r2, 0.00000001);
    }

    /**
     * Many data examples from: http://www.themathpage.com/alg/quadratic-equations.htm
     */
    public function dataProviderForQuadratic(): array
    {
        return [
            [2, 4, -4, [-1 - \sqrt(3), -1 + \sqrt(3)]],
            [1, -3, -4, [-1, 4]],
            [1, 1, -4, [-2.56155281280883, 1.56155281280883]],
            [1, 0, -4, [-2, 2]],
            [6, 11, -35, [-7 / 2, 5 / 3]],
            [1, 0, -48, [-4 * \sqrt(3), 4 * \sqrt(3)]],
            [1, -7, 0, [0, 7]],
            [5, 6, 1, [-1, -0.2]],
            [1, 2, -8, [-4, 2]],
            [1, 2, -3, [-3, 1]],
            [1, -12, 36, [6, 6]],
            [2, 9, -5, [-5, 1 / 2]],
            [1, -3, 2, [1, 2]],
            [1, 7, 12, [-4, -3]],
            [1, 3, -10, [-5, 2]],
            [1, -1, -30, [-5, 6]],
            [2, 7, 3, [-3, -1 / 2]],
            [3, 1, -2, [-1, 2 / 3]],
            [1, 12, 36, [-6, -6]],
            [1, -2, 1, [1, 1]],
            [1, -5, 0, [0, 5]],
            [1, 1, 0, [-1, 0]],
            [3, 4, 0, [-4 / 3, 0]],
            [2, -1, 0, [0, 1 / 2]],
            [1, 0, -3, [-sqrt(3), \sqrt(3)]],
            [1, 0, -25, [-5, 5]],
            [1, 0, -10, [-sqrt(10), \sqrt(10)]],
            [1, -5, 6, [2, 3]],
            [1, -8, 12, [2, 6]],
            [3, 1, -10, [-2, 5 / 3]],
            [2, -1, 0, [0, 1 / 2]],
            [3, 5 / 2, -3, [-3 / 2, 2 / 3]],
            [5, 11 / 2, -3, [-3 / 2, 2 / 5]],
            [5, -11 / 3, -4, [-3 / 5, 4 / 3]],
            [1, 1, -20, [-5, 4]],
            [1, -3, -18, [-3, 6]],
            [2, -5, -3, [-1 / 2, 3]],
        ];
    }

    /**
     * @test         quadratic returns the expected root for edge case where a = 0 and formula is not quadratic.
     * @dataProvider dataProviderForQuadraticAIsZero
     * @param        float $a
     * @param        float $b
     * @param        float $c
     * @param        array $expected_quadratic
     * @throws      \MathPHP\Exception\IncorrectTypeException
     */
    public function testQuadraticAIsZero($a, $b, $c, array $expected_quadratic)
    {
        // When
        $quadratic = Algebra::quadratic($a, $b, $c);

        // Then
        $this->assertEqualsWithDelta($expected_quadratic, $quadratic, 0.00000001);
    }

    /**
     * @return array [a, b, c, quadratic]
     */
    public function dataProviderForQuadraticAIsZero(): array
    {
        return [
            [0, -5, -3, [-3 / 5]],
            [0, 5, -3, [3 / 5]],
            [0, 12, 6, [-1 / 2]],
            [0, 3, 7, [-7 / 3]],
        ];
    }

    /**
     * @test         quadratic returns array of [NAN, NAN] if the discriminant is negative.
     * @dataProvider dataProviderForQuadraticNegativeDiscriminant
     * @param        float $a
     * @param        float $b
     * @param        float $c
     * @throws       \MathPHP\Exception\IncorrectTypeException
     */
    public function testQuadraticNegativeDiscriminant(float $a, float $b, float $c)
    {
        // When
        $roots = Algebra::quadratic($a, $b, $c);

        // Then
        $this->assertIsArray($roots);
        $this->assertNotEmpty($roots);
        $this->assertEquals(2, count($roots));
        foreach ($roots as $root) {
            $this->assertTrue(\is_nan($root));
        }
    }

    /**
     * @return array [a, b, c]
     */
    public function dataProviderForQuadraticNegativeDiscriminant(): array
    {
        return [
            [10, 1, 1, [\NAN, \NAN]],
            [3, 4, 20, [\NAN, \NAN]],
        ];
    }

    /**
     * @test         quadratic returns array of Complex Number objects if the discriminant is negative.
     * @dataProvider dataProviderForQuadraticNegativeDiscriminantComplex
     * @param        float $a
     * @param        float $b
     * @param        float $c
     * @param        array $expected
     * @throws       \Exception
     */
    public function testQuadraticNegativeDiscriminantComplex(float $a, float $b, float $c, array $expected)
    {
        // Given
        $complex0 = new Number\Complex($expected[0][0], $expected[0][1]);
        $complex1 = new Number\Complex($expected[1][0], $expected[1][1]);

        // When
        $roots = Algebra::quadratic($a, $b, $c, true);

        // Then
        $this->assertIsArray($roots);
        $this->assertInstanceOf(Number\Complex::class, $roots[0]);
        $this->assertInstanceOf(Number\Complex::class, $roots[1]);
        $this->assertNotEmpty($roots);
        $this->assertEquals(2, count($roots));
        $this->assertTrue($roots[0]->equals($complex0));
        $this->assertTrue($roots[1]->equals($complex1));
    }

    /**
     * @return array [a, b, c, quadratic]
     */
    public function dataProviderForQuadraticNegativeDiscriminantComplex(): array
    {
        return [
            [10, 1, 1, [[-.05, -1 * \sqrt(39) / 20], [-.05, \sqrt(39) / 20]]],
            [3, 4, 20, [[-2 / 3, -1 * \sqrt(14) * 2 / 3], [-2 / 3, \sqrt(14) * 2 / 3]]],
        ];
    }

    /**
     * @test         discriminant returns the expected value.
     * @dataProvider dataProviderForDiscriminant
     * @param        float $a
     * @param        float $b
     * @param        float $c
     * @param        float $expected_discriminant
     */
    public function testDiscriminant(float $a, float $b, float $c, float $expected_discriminant)
    {
        // When
        $discriminant = Algebra::discriminant($a, $b, $c);

        // Then
        $this->assertEqualsWithDelta($expected_discriminant, $discriminant, 0.00000001);
    }

    /**
     * @return array [a, b, c, discriminant]
     */
    public function dataProviderForDiscriminant(): array
    {
        return [
            [2, 3, 4, -23],
            [2, 4, -4, 48],
            [1, -3, -4, 25],
            [1, 1, -4, 17],
            [1, 0, -4, 16],
            [6, 11, -35, 961],
            [1, 0, -48, 192],
            [1, -7, 0, 49],
            [10, 1, 1, -39],
            [3, 4, 20, -224],
        ];
    }

    /**
     * @test         cubic returns the expected three real roots when D < 0 or D = 0.
     * @dataProvider dataProviderForCubic
     * @param        int $a
     * @param        int $b
     * @param        int $c
     * @param        int $d
     * @param        array $expected_cubic expected roots
     * @throws       \Exception
     */
    public function testCubic(int $a, int $b, int $c, int $d, array $expected_cubic)
    {
        // When
        $cubic = Algebra::cubic($a, $b, $c, $d);

        // Then
        $this->assertEqualsWithDelta($expected_cubic, $cubic, 0.00000001);
    }

    /**
     * Calculator used to generate and validate examples: http://www.1728.org/cubic.htm
     * Some examples from: http://www.mash.dept.shef.ac.uk/Resources/web-cubicequations-john.pdf
     * Some examples from: https://trans4mind.com/personal_development/mathematics/polynomials/cardanoMethodExamples.htm
     * @return array
     */
    public function dataProviderForCubic(): array
    {
        return [
            // D < 0: Three real roots. Nice even numbers.
            [1, 0, 0, 0, [0, 0, 0]],
            [1, -6, 11, -6, [3, 1, 2]],
            [1, -5, -2, 24, [4, -2, 3]],
            [1, 0, -7, -6, [3, -2, -1]],
            [1, -4, -9, 36, [4, -3, 3]],
            [1, 3, -6, -8, [2, -4, -1]],
            [1, 2, -21, 18, [3, -6, 1]],
            [1, -7, 4, 12, [6, -1, 2]],
            [1, 9, 26, 24, [-2, -4, -3]],
            [1, 0, -19, -30, [5, -3, -2]],
            [1, 2, -25, -50, [5, -5, -2]],
            [1, 6, 11, 6, [-1, -3, -2]],
            [1, 4, 1, -6, [1, -3, -2]],
            [2, 9, 3, -4, [0.5, -4, -1]],
            [2, -4, -22, 24, [4, -3, 1]],
            [2, 3, -11, -6, [2, -3, -1 / 2]],
            [2, -9, 1, 12, [4, -1, 1.5]],
            [2, -3, -5, 6, [2, -1.5, 1]],
            [3, -1, -10, 8, [4 / 3, -2, 1]],
            [6, -5, -17, 6, [2, -1.5, 1 / 3]],
            [45, 24, -7, -2, [1 / 3, -2 / 3, -0.2]],
            [-1, -1, 22, 40, [5, -4, -2]],
            [-1, 0, 19, -30, [3, -5, 2]],
            [-1, 6, -5, -12, [4, -1, 3]],

            // D < 0: Three real roots. Floats.
            [1, 6, 3, -5, [0.66966384064222, -5.24655136455856, -1.42311247608366]],
            [1, 4, 1, -5, [0.9122291784844, -3.198691243516, -1.7135379349684]],
            [1, -4, -6, 5, [5, -1.61803398874989, 0.61803398874989]],
            [1, -3, -1, 1, [3.21431974337754, -0.67513087056665, 0.46081112718911]],
            [1, -2, -6, 4, [3.41421356237309, -2, 0.58578643762691]],
            [1, 1, -16, 0, [3.53112887414927, -4.53112887414927, 0]],
            [2, -3, -22, 24, [3.62221312679243, -3.16796177749228,  1.04574865069985]],
            [2, -2, -22, 24, [3.2488979294409, -3.35109344639606,  1.10219551695516]],
            [1000, -1254, -496, 191, [1.49979930548345, -0.50033136443491, 0.25453205895145]],

            // D = 0: All real roots--at least two are equal. Nice even numbers.
            [1, -5, 8, -4, [2, 1, 2]],
            [1, -3, 3, -1, [1, 1, 1]],
            [1, 3, 3, 1, [-1, -1, -1]],
            [1, 2, -20, 24, [2, -6, 2]],
            [64, -48, 12, -1, [0.25, 0.25, 0.25]],
        ];
    }

    /**
     * @test         cubic returns the expected roots when D > 0: one root is real, 2 are complex conjugates.
     * @dataProvider dataProviderForCubicOneRealRoot
     * @param        float $a
     * @param        float $b
     * @param        float $c
     * @param        float $d
     * @param        float $real_root
     * @throws       \Exception
     */
    public function testCubicOneRealRoot(float $a, float $b, float $c, float $d, float $real_root)
    {
        // When
        [$z₁, $z₂, $z₃] = Algebra::cubic($a, $b, $c, $d);

        // Then
        $this->assertEqualsWithDelta($real_root, $z₁, 0.00000001);
        $this->assertNan($z₂);
        $this->assertNan($z₃);
    }

    /**
     * Calculator used to generate and validate examples: http://www.1728.org/cubic.htm
     * Some examples from: http://www.mash.dept.shef.ac.uk/Resources/web-cubicequations-john.pdf
     * @return array
     */
    public function dataProviderForCubicOneRealRoot(): array
    {
        return [
            // D > 0: one root is real, 2 are complex conjugates.
            [1, 1, 1, -3, 0.9999999999999984],
            [1, -6, -6, -7, 7],
            [1, 1, 4, -8, 1.202981258316938],
            [1, 2, 3, -4, 0.7760454350285383],
            [1, -2.7, 4.5, -6, 1.9641774065933375],
            [1, 3, 3, -2, 0.4422495703074083],
            [1, 2, 10, -20, 1.3688081078213727],
            [1, 1, 10, -3, 0.28921621924406943],
            [2, -3, -4, -35, 3.5000000000000027],
            [2, -5, 23, -10, 0.4744277602198689],
            [2, -6, 7, -1, 0.1648776515186341],
            [2, 0, 4, 1, -0.24283973258548086],
        ];
    }

    /**
     * @test         cubic returns the expected roots when D > 0: one root is real, 2 are complex conjugates.
     * @dataProvider dataProviderForCubicOneRealRootWithComplex
     * @param        int   $a
     * @param        int   $b
     * @param        int   $c
     * @param        int   $d
     * @param        array $roots
     * @throws       \Exception
     */
    public function testCubicOneRealRootWithComplex(int $a, int $b, int $c, int $d, array $roots)
    {
        // Given
        $real_root = $roots[0];
        $complex0 = new Number\Complex($roots[1]['r'], $roots[1]['i']);
        $complex1 = new Number\Complex($roots[2]['r'], $roots[2]['i']);

        // When
        [$z₁, $z₂, $z₃] = Algebra::cubic($a, $b, $c, $d, true);

        // Then
        $this->assertEqualsWithDelta($real_root, $z₁, 0.00000001);
        $this->assertInstanceOf(Number\Complex::class, $z₂);
        $this->assertInstanceOf(Number\Complex::class, $z₃);
        $this->assertTrue($z₂->equals($complex0), "Expecting $complex0 but saw $z₂");
        $this->assertTrue($z₃->equals($complex1), "Expecting $complex1 but saw $z₃");
    }

    /**
     * @return array [a, b, c, d, roots]
     */
    public function dataProviderForCubicOneRealRootWithComplex(): array
    {
        return [
            // D > 0: one root is real, 2 are complex conjugates.
            [1, 0, 1, 0, [0, ['r' => 0, 'i' => -1], ['r' => 0, 'i' => 1]]],
            [1, -1, 1, -1, [1, ['r' => 0, 'i' => -1], ['r' => 0, 'i' => 1]]],
        ];
    }

    /**
     * @test         cubic with a₃ coefficient of z³ of 0 is the same as quadratic.
     * @dataProvider dataProviderForQuadratic
     * @param        float $b
     * @param        float $c
     * @param        float $d
     * @param        array $quadratic
     * @throws       \Exception
     */
    public function testCubicCubeCoefficientZeroSameAsQuadratic(float $b, float $c, float $d, array $quadratic)
    {
        $a = 0;
        $this->assertEqualsWithDelta($quadratic, Algebra::cubic($a, $b, $c, $d), 0.00000001);
    }

    /**
     * @test         cubic returns array of Complex Number objects if the quadradic discriminant is negative.
     * @dataProvider dataProviderForQuadraticNegativeDiscriminantComplex
     * @param        float $a₂
     * @param        float $a₁
     * @param        float $a₀
     * @param        array $expected
     * @throws       \Exception
     */
    public function testCubicNegativeDiscriminantComplex(float $a₂, float $a₁, float $a₀, array $expected)
    {
        // Given
        $a₃       = 0;
        $complex0 = new Number\Complex($expected[0][0], $expected[0][1]);
        $complex1 = new Number\Complex($expected[1][0], $expected[1][1]);

        // When
        $roots = Algebra::cubic($a₃, $a₂, $a₁, $a₀, true);

        // Then
        $this->assertIsArray($roots);
        $this->assertInstanceOf(Number\Complex::class, $roots[0]);
        $this->assertInstanceOf(Number\Complex::class, $roots[1]);
        $this->assertNotEmpty($roots);
        $this->assertEquals(2, count($roots));
        $this->assertTrue($roots[0]->equals($complex0));
        $this->assertTrue($roots[1]->equals($complex1));
    }

    /**
     * @test         quartic
     * @dataProvider dataProviderForQuartic
     * @param        int   $a
     * @param        int   $b
     * @param        int   $c
     * @param        int   $d
     * @param        int   $e
     * @param        array $expected_quartic
     * @throws       \Exception
     */
    public function testQuartic(int $a, int $b, int $c, int $d, int $e, array $expected_quartic)
    {
        // When
        $quartic = Algebra::quartic($a, $b, $c, $d, $e);

        // Then
        $this->assertEqualsWithDelta($expected_quartic, $quartic, 0.00000001);
    }

    /**
     * @return array [a, b, c, d, e, quartic]
     */
    public function dataProviderForQuartic(): array
    {
        return [
            [3, 6, -123, -126, 1080, [5, -6, 3, -4]],
            [1, -10, 35, -50, 24, [4, 1, 3, 2]],
            [1, -4, 6, -4, 1, [1, 1, 1, 1]],

            // Actually a cubic
            [0, 1, -6, 11, -6, [3, 1, 2]],

            // Zero Root
            [1, -6, 11, -6, 0, [0, 3, 1, 2]],

            // Biquadratic
            [1, 0, -5, 0, 4, [2, -2, 1, -1]],

            // Depressed quartic is biquadratic
            [1, 12, 49, 78, 40, [-1, -5, -2, -4]],

            // Depressed quartic and has 4 real roots.
            [1, 0, -25, 60, -36, [-6, 1, 2, 3]],
         ];
    }

    /**
     * @test         quartic with two complex roots - not set to return complex
     * @dataProvider dataProviderForQuarticTwoComplex
     * @param        int $a
     * @param        int $b
     * @param        int $c
     * @param        int $d
     * @param        int $e
     * @param        array $quartic expected roots
     * @throws       \Exception
     */
    public function testQuarticTwoComplexNotSetToReturnComplex($a, $b, $c, $d, $e, $quartic)
    {
        // When
        [$z₁, $z₂, $z₃, $z₄] = Algebra::quartic($a, $b, $c, $d, $e);

        // Then
        $this->assertEqualsWithDelta($quartic[0], \floatval($z₁), 0.00000001);
        $this->assertEqualsWithDelta($quartic[1], \floatval($z₂), 0.00000001);
        $this->assertNan($z₃, '');
        $this->assertNan($z₄, '');
    }

    /**
     * @test         quartic with two complex roots - set to return complex
     * @dataProvider dataProviderForQuarticTwoComplex
     * @param        int $a
     * @param        int $b
     * @param        int $c
     * @param        int $d
     * @param        int $e
     * @param        array $quartic expected roots
     * @throws       \Exception
     */
    public function testQuarticTwoComplex($a, $b, $c, $d, $e, $quartic)
    {
        // Given
        $complex0 = new Number\Complex($quartic[2]['r'], $quartic[2]['i']);
        $complex1 = new Number\Complex($quartic[3]['r'], $quartic[3]['i']);

        // When
        [$z₁, $z₂, $z₃, $z₄] = Algebra::quartic($a, $b, $c, $d, $e, true);

        // Then
        $this->assertEqualsWithDelta($quartic[0], \floatval($z₁), 0.00000001);
        $this->assertEqualsWithDelta($quartic[1], \floatval($z₂), 0.00000001);
        $this->assertTrue($z₃->equals($complex0), "Expecting $complex0 but saw $z₃, complex conjugate is $z₄");
        $this->assertTrue($z₄->equals($complex1), "Expecting $complex1 but saw $z₄, complex conjugate is $z₃");
    }

    /**
     * @return array
     */
    public function dataProviderForQuarticTwoComplex(): array
    {
        return [
            // Two Complex Roots
            [1, -5, 10, -10, 4, [1, 2, ['r' => 1, 'i' => -1], ['r' => 1, 'i' => 1]]],

            // And is a depressed quartic. (sum of roots=0)
            [1, 0, 5 / 8, -5 / 8, -51 / 256, [-.25, .75, ['r' => -.25, 'i' => -1], ['r' => -.25, 'i' => 1]]],
            [1, 0, -5, 10, -6, [-3, 1, ['r' => 1, 'i' => -1], ['r' => 1, 'i' => 1]]],

            // Biquadratic with two complex roots
            [1, 0, -5, 0, -36, [3, -3, ['r' => 0, 'i' => 2], ['r' => 0, 'i' => -2]]],
         ];
    }

    /**
     * @test         quartic with four complex roots - not set to return complex
     * @dataProvider dataProviderForQuarticFourComplex
     * @param        int   $a
     * @param        int   $b
     * @param        int   $c
     * @param        int   $d
     * @param        int   $e
     * @param        array $quartic expected roots
     * @throws       \Exception
     */
    public function testQuarticFourComplexReturnsNansIfNotSetToReturnComplex(int $a, int $b, int $c, int $d, int $e, array $quartic)
    {
        // When
        [$z₁, $z₂, $z₃, $z₄] = Algebra::quartic($a, $b, $c, $d, $e);

        // Then
        $this->assertNan($z₁);
        $this->assertNan($z₂);
        $this->assertNan($z₃);
        $this->assertNan($z₄);
    }

    /**
     * @test         quartic with four complex roots - set to return complex
     * @dataProvider dataProviderForQuarticFourComplex
     * @param        int   $a
     * @param        int   $b
     * @param        int   $c
     * @param        int   $d
     * @param        int   $e
     * @param        array $quartic expected roots
     * @throws       \Exception
     */
    public function testQuarticFourComplex(int $a, int $b, int $c, int $d, int $e, array $quartic)
    {
        // Given
        $complex0 = new Number\Complex($quartic[0]['r'], $quartic[0]['i']);
        $complex1 = new Number\Complex($quartic[1]['r'], $quartic[1]['i']);
        $complex2 = new Number\Complex($quartic[2]['r'], $quartic[2]['i']);
        $complex3 = new Number\Complex($quartic[3]['r'], $quartic[3]['i']);

        // When
        [$z₁, $z₂, $z₃, $z₄] = Algebra::quartic($a, $b, $c, $d, $e, true);

        // Then
        $this->assertTrue($z₁->equals($complex0), "Expecting $complex0 but saw $z₁");
        $this->assertTrue($z₂->equals($complex1), "Expecting $complex1 but saw $z₂");
        $this->assertTrue($z₃->equals($complex2), "Expecting $complex2 but saw $z₃");
        $this->assertTrue($z₄->equals($complex3), "Expecting $complex3 but saw $z₄");
    }

    /**
     * @return array
     */
    public function dataProviderForQuarticFourComplex(): array
    {
        return [
            // Four Complex Roots
            [1, -6, 18, -24, 16, [['r' => 1, 'i' => -1], ['r' => 1, 'i' => 1], ['r' => 2, 'i' => -2], ['r' => 2, 'i' => 2]]],
            [1, 0, -3, 12, 40, [['r' => -2, 'i' => -1], ['r' => -2, 'i' => 1], ['r' => 2, 'i' => -2], ['r' => 2, 'i' => 2]]],

            // Biquadratic with four complex roots
            [1, 0, 13, 0, 36, [['r' => 0, 'i' => 2], ['r' => 0, 'i' => -2], ['r' => 0, 'i' => 3], ['r' => 0, 'i' => -3]]],
         ];
    }
}