MathPHP is the only library you need to integrate mathematical functions into your applications. It is a self-contained library in pure PHP with no external dependencies.
It is actively under development with development (0.y.z) releases.
use MathPHP\Arithmetic;
$³√x = Arithmetic::cubeRoot(-8); // -2// Sum of digits
$digit_sum = Arithmetic::digitSum(99): // 18
$digital_root = Arithmetic::digitalRoot(99); // 9// Equality of numbers within a tolerance
$x = 0.00000003458;
$y = 0.00000003455;
$ε = 0.0000000001;
$almostEqual = Arithmetic::almostEqual($x, $y, $ε); // true// Copy sign
$magnitude = 5;
$sign = -3;
$signed_magnitude = Arithmetic::copySign($magnitude, $sign); // -5
Finance
use MathPHP\Finance;
// Financial payment for a loan or annuity with compound interest
$rate = 0.035 / 12; // 3.5
$periods = 30 * 12; // 30-year mortgage
$present_value = 265000; // Mortgage note of $265,000.00
$future_value = 0;
$beginning = false; // Adjust the payment to the beginning or end of the period
$pmt = Finance::pmt($rate, $periods, $present_value, $future_value, $beginning);
// Interest on a financial payment for a loan or annuity with compound interest.
$period = 1; // First payment period
$ipmt = Finance::ipmt($rate, $period, $periods, $present_value, $future_value, $beginning);
// Principle on a financial payment for a loan or annuity with compound interest
$ppmt = Finance::ppmt($rate, $period, $periods, $present_value, $future_value = 0, $beginning);
// Number of payment periods of an annuity.
$periods = Finance::periods($rate, $payment, $present_value, $future_value, $beginning);
// Annual Equivalent Rate (AER) of an annual percentage rate (APR)
$nominal = 0.035; // APR 3.5
$periods = 12; // Compounded monthly
$aer = Finance::aer($nominal, $periods);
// Annual nominal rate of an annual effective rate (AER)
$nomial = Finance::nominal($aer, $periods);
// Future value for a loan or annuity with compound interest
$payment = 1189.97;
$fv = Finance::fv($rate, $periods, $payment, $present_value, $beginning)
// Present value for a loan or annuity with compound interest
$pv = Finance::pv($rate, $periods, $payment, $future_value, $beginning)
// Net present value of cash flows
$values = [-1000, 100, 200, 300, 400];
$npv = Finance::npv($rate, $values);
// Interest rate per period of an annuity
$beginning = false; // Adjust the payment to the beginning or end of the period
$rate = rate($periods, $payment, $present_value, $future_value, $beginning);
// Internal rate of return
$values = [-100, 50, 40, 30];
$irr = Finance:irr($values); // Rate of return of an initial investment of $100 with returns of $50, $40, and $30// Modified internal rate of return
$finance_rate = 0.05; // 5
$reinvestment_rate = 0.10; // reinvested at 10
$mirr = Finance:mirr($values, $finance_rate); // rate of return of an initial investment of $100 at 5
// Discounted payback of an investment
$values = [-1000, 100, 200, 300, 400, 500];
$rate = 0.1;
$payback = Finance::payback($values, $rate); // The payback period of an investment with a $1,000 investment and future returns of $100, $200, $300, $400, $500 and a discount rate of 0.10// Profitability index
$values = [-100, 50, 50, 50];
$profitability_index = profitabilityIndex($values, $rate); // The profitability index of an initial $100 investment with future returns of $50, $50, $50 with a 10
use MathPHP\NumberTheory\Integer;
$n = 225;
// Prime factorization
$factors = Integer::primeFactorization($n);
// Perfect powers
$bool = Integer::isPerfectPower($n);
list($m, $k) = Integer::perfectPower($n);
// Coprime
$bool = Integer::coprime(4, 35);
// Even and odd
$bool = Integer::isEven($n);
$bool = Integer::isOdd($n);
Numerical Analysis – Interpolation
use MathPHP\NumericalAnalysis\Interpolation;
// Interpolation is a method of constructing new data points with the range// of a discrete set of known data points.// Each integration method can take input in two ways:// 1) As a set of points (inputs and outputs of a function)// 2) As a callback function, and the number of function evaluations to// perform on an interval between a start and end point.// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
// Input as a callback function
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
// Lagrange Polynomial// Returns a function p(x) of x
$p = Interpolation\LagrangePolynomial::interpolate($points); // input as a set of points
$p = Interpolation\LagrangePolynomial::interpolate($f⟮x⟯, $start, $end, $n); // input as a callback function
$p(0) // 1
$p(3) // 16// Nevilles Method// More accurate than Lagrange Polynomial Interpolation given the same input// Returns the evaluation of the interpolating polynomial at the $target point
$target = 2;
$result = Interpolation\NevillesMethod::interpolate($target, $points); // input as a set of points
$result = Interpolation\NevillesMethod::interpolate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function// Newton Polynomial (Forward)// Returns a function p(x) of x
$p = Interpolation\NewtonPolynomialForward::interpolate($points); // input as a set of points
$p = Interpolation\NewtonPolynomialForward::interpolate($f⟮x⟯, $start, $end, $n); // input as a callback function
$p(0) // 1
$p(3) // 16// Natural Cubic Spline// Returns a piecewise polynomial p(x)
$p = Interpolation\NaturalCubicSpline::interpolate($points); // input as a set of points
$p = Interpolation\NaturalCubicSpline::interpolate($f⟮x⟯, $start, $end, $n); // input as a callback function
$p(0) // 1
$p(3) // 16// Clamped Cubic Spline// Returns a piecewise polynomial p(x)// Input as a set of points
$points = [[0, 1, 0], [1, 4, -1], [2, 9, 4], [3, 16, 0]];
// Input as a callback function
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
$f’⟮x⟯ = function ($x) {
return 2*$x + 2;
};
list($start, $end, $n) = [0, 3, 4];
$p = Interpolation\ClampedCubicSpline::interpolate($points); // input as a set of points
$p = Interpolation\ClampedCubicSpline::interpolate($f⟮x⟯, $f’⟮x⟯, $start, $end, $n); // input as a callback function
$p(0) // 1
$p(3) // 16
Numerical Analysis – Numerical Differentiation
use MathPHP\NumericalAnalysis\NumericalDifferentiation;
// Numerical Differentiation approximates the derivative of a function.// Each Differentiation method can take input in two ways:// 1) As a set of points (inputs and outputs of a function)// 2) As a callback function, and the number of function evaluations to// perform on an interval between a start and end point.// Input as a callback function
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
// Three Point Formula// Returns an approximation for the derivative of our input at our target// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9]];
$target = 0;
list($start, $end, $n) = [0, 2, 3];
$derivative = NumericalDifferentiation\ThreePointFormula::differentiate($target, $points); // input as a set of points
$derivative = NumericalDifferentiation\ThreePointFormula::differentiate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function// Five Point Formula// Returns an approximation for the derivative of our input at our target// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9], [3, 16], [4, 25]];
$target = 0;
list($start, $end, $n) = [0, 4, 5];
$derivative = NumericalDifferentiation\FivePointFormula::differentiate($target, $points); // input as a set of points
$derivative = NumericalDifferentiation\FivePointFormula::differentiate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function// Second Derivative Midpoint Formula// Returns an approximation for the second derivative of our input at our target// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9];
$target = 1;
list($start, $end, $n) = [0, 2, 3];
$derivative = NumericalDifferentiation\SecondDerivativeMidpointFormula::differentiate($target, $points); // input as a set of points
$derivative = NumericalDifferentiation\SecondDerivativeMidpointFormula::differentiate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function
Numerical Analysis – Numerical Integration
use MathPHP\NumericalAnalysis\NumericalIntegration;
// Numerical integration approximates the definite integral of a function.// Each integration method can take input in two ways:// 1) As a set of points (inputs and outputs of a function)// 2) As a callback function, and the number of function evaluations to// perform on an interval between a start and end point.// Trapezoidal Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\TrapezoidalRule::approximate($points); // input as a set of points
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
$∫f⟮x⟯dx = NumericalIntegration\TrapezoidalRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function// Simpsons Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16], [4,3]];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsRule::approximate($points); // input as a set of points
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 5];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function// Simpsons 3/8 Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsThreeEighthsRule::approximate($points); // input as a set of points
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 5];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsThreeEighthsRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function// Booles Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16], [4, 25]];
$∫f⟮x⟯dx = NumericalIntegration\BoolesRule::approximate($points); // input as a set of points
$f⟮x⟯ = function ($x) {
return $x**3 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 4, 5];
$∫f⟮x⟯dx = NumericalIntegration\BoolesRuleRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function// Rectangle Method (open Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\RectangleMethod::approximate($points); // input as a set of points
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
$∫f⟮x⟯dx = NumericalIntegration\RectangleMethod::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function// Midpoint Rule (open Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\MidpointRule::approximate($points); // input as a set of points
$f⟮x⟯ = function ($x) {
return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
$∫f⟮x⟯dx = NumericalIntegration\MidpointRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function
Numerical Analysis – Root Finding
use MathPHP\NumericalAnalysis\RootFinding;
// Root-finding methods solve for a root of a polynomial.// f(x) = x⁴ + 8x³ -13x² -92x + 96
$f⟮x⟯ = function($x) {
return $x**4 + 8 * $x**3 - 13 * $x**2 - 92 * $x + 96;
};
// Newton's Method
$args = [-4.1]; // Parameters to pass to callback function (initial guess, other parameters)
$target = 0; // Value of f(x) we a trying to solve for
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$position = 0; // Which element in the $args array will be changed; also serves as initial guess. Defaults to 0.
$x = RootFinding\NewtonsMethod::solve($f⟮x⟯, $args, $target, $tol, $position); // Solve for x where f(x) = $target// Secant Method
$p₀ = -1; // First initial approximation
$p₁ = 2; // Second initial approximation
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$x = RootFinding\SecantMethod::solve($f⟮x⟯, $p₀, $p₁, $tol); // Solve for x where f(x) = 0// Bisection Method
$a = 2; // The start of the interval which contains a root
$b = 5; // The end of the interval which contains a root
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$x = RootFinding\BisectionMethod::solve($f⟮x⟯, $a, $b, $tol); // Solve for x where f(x) = 0// Fixed-Point Iteration// f(x) = x⁴ + 8x³ -13x² -92x + 96// Rewrite f(x) = 0 as (x⁴ + 8x³ -13x² + 96)/92 = x// Thus, g(x) = (x⁴ + 8x³ -13x² + 96)/92
$g⟮x⟯ = function($x) {
return ($x**4 + 8 * $x**3 - 13 * $x**2 + 96)/92;
};
$a = 0; // The start of the interval which contains a root
$b = 2; // The end of the interval which contains a root
$p = 0; // The initial guess for our root
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$x = RootFinding\FixedPointIteration::solve($g⟮x⟯, $a, $b, $p, $tol); // Solve for x where f(x) = 0
Probability – Combinatorics
use MathPHP\Probability\Combinatorics;
list($n, $x, $k) = [10, 3, 4];
// Factorials
$n！ = Combinatorics::factorial($n);
$n‼︎ = Combinatorics::doubleFactorial($n);
$x⁽ⁿ⁾ = Combinatorics::risingFactorial($x, $n);
$x₍ᵢ₎ = Combinatorics::fallingFactorial($x, $n);
$！n = Combinatorics::subfactorial($n);
// Permutations
$nPn = Combinatorics::permutations($n); // Permutations of n things, taken n at a time (same as factorial)
$nPk = Combinatorics::permutations($n, $k); // Permutations of n things, taking only k of them// Combinations
$nCk = Combinatorics::combinations($n, $k); // n choose k without repetition
$nC′k = Combinatorics::combinations($n, $k, Combinatorics::REPETITION); // n choose k with repetition (REPETITION const = true)// Central binomial coefficient
$cbc = Combinatorics::centralBinomialCoefficient($n);
// Catalan number
$Cn = Combinatorics::catalanNumber($n);
// Lah number
$L⟮n、k⟯ = Combinatorics::lahNumber($n, $k)
// Multinomial coefficient
$groups = [5, 2, 3];
$divisions = Combinatorics::multinomial($groups);
Probability – Continuous Distributions
use MathPHP\Probability\Distribution\Continuous;
// Beta distribution
$α = 1; // shape parameter
$β = 1; // shape parameter
$x = 2;
$beta = new Continuous\Beta($α, $β);
$pdf = $beta->pdf($x);
$cdf = $beta->cdf($x);
$μ = $beta->mean();
// Cauchy distribution
$x₀ = 2; // location parameter
$γ = 3; // scale parameter
$x = 1;
$cauchy = new Continuous\Cauchy(x₀, γ);
$pdf = $cauchy->pdf(x);
$cdf = $cauchy->cdf(x);
// χ²-distribution (Chi-Squared)
$k = 2; // degrees of freedom
$x = 1;
$χ² = new Continuous\ChiSquared($k);
$pdf = $χ²->pdf($x);
$cdf = $χ²->cdf($x);
// Dirac delta distribution
$x = 1;
$dirac = new Continuous\DiracDelta();
$pdf = $dirac->pdf($x);
$cdf = $dirac->cdf($x);
// Exponential distribution
$λ = 1; // rate parameter
$x = 2;
$exponential = new Continuous\Exponential($λ);
$pdf = $exponential->pdf($x);
$cdf = $exponential->cdf($x);
$μ = $exponential->mean();
// F-distribution
$d₁ = 3; // degree of freedom v1
$d₂ = 4; // degree of freedom v2
$x = 2;
$f = new Continuous\F($d₁, $d₂);
$pdf = $f->pdf($x);
$cdf = $f->cdf($x);
$μ = $f->mean();
// Gamma distribution
$k = 2; // shape parameter
$θ = 3; // scale parameter
$x = 4;
$gamma = new Continuous\Gamma($k, $θ);
$pdf = $gamma->pdf($x);
$cdf = $gamma->cdf($x);
$μ = $gamma->mean();
// Laplace distribution
$μ = 1; // location parameter
$b = 1.5; // scale parameter (diversity)
$x = 1;
$laplace = new Continuous\Laplace($μ, $b);
$pdf = $laplace->pdf($x);
$cdf = $laplace->cdf($x);
// Logistic distribution
$μ = 2; // location parameter
$s = 1.5; // scale parameter
$x = 3;
$logistic = new Continuous\Logistic($μ, $s);
$pdf = $logistic->pdf($x);
$cdf = $logistic->cdf($x);
// Log-logistic distribution (Fisk distribution)
$α = 1; // scale parameter
$β = 1; // shape parameter
$x = 2;
$logLogistic = new Continuous\LogLogistic($α, $β);
$pdf = $logLogistic->pdf($x);
$cdf = $logLogistic->cdf($x);
$μ = $logLogistic->mean();
// Log-normal distribution
$μ = 6; // scale parameter
$σ = 2; // location parameter
$x = 4.3;
$logNormal = new Continuous\LogNormal($μ, $σ);
$pdf = $logNormal->pdf($x);
$cdf = $logNormal->cdf($x);
$mean = $logNormal->mean();
// Noncentral T distribution
$ν = 50; // degrees of freedom
$μ = 10; // noncentrality parameter
$x = 8;
$noncenetralT = new Continuous\NoncentralT($ν, $μ);
$pdf = $noncenetralT->pdf($x);
$cdf = $noncenetralT->cdf($x);
$mean = $noncenetralT->mean();
// Normal distribution
$σ = 1;
$μ = 0;
$x = 2;
$normal = new Continuous\Normal($μ, $σ);
$pdf = $normal->pdf($x);
$cdf = $normal->cdf($x);
// Pareto distribution
$a = 1; // shape parameter
$b = 1; // scale parameter
$x = 2;
$pareto = new Continuous\Pareto($a, $b);
$pdf = $pareto->pdf($x);
$cdf = $pareto->cdf($x);
$μ = $pareto->mean();
// Standard normal distribution
$z = 2;
$standardNormal = new Continuous\StandardNormal();
$pdf = $standardNormal->pdf($z);
$cdf = $standardNormal->cdf($z);
// Student's t-distribution
$ν = 3; // degrees of freedom
$p = 0.4; // proportion of area
$x = 2;
$studentT = new Continuous\StudentT::pdf($ν);
$pdf = $studentT->pdf($x);
$cdf = $studentT->cdf($x);
$t = $studentT->inverse2Tails($p); // t such that the area greater than t and the area beneath -t is p// Uniform distribution
$a = 1; // lower boundary of the distribution
$b = 4; // upper boundary of the distribution
$x = 2;
$uniform = new Continuous\Uniform($a, $b);
$pdf = $uniform->pdf($x);
$cdf = $uniform->cdf($x);
$μ = $uniform->mean(b);
// Weibull distribution
$k = 1; // shape parameter
$λ = 2; // scale parameter
$x = 2;
$weibull = new Continuous\Weibull($k, $λ);
$pdf = $weibull->pdf($x);
$cdf = $weibull->cdf($x);
$μ = $weibull->mean();
// Other CDFs - All continuous distributions// Replace '$distribution' with desired distribution.
$inv_cdf = $distribution->inverse($target); // Inverse CDF of the distribution
$between = $distribution->between($x₁, $x₂); // Probability of being between two points, x₁ and x₂
$outside = $distribution->outside($x₁, $x); // Probability of being between below x₁ and above x₂
$above = $distribution->above($x); // Probability of being above x to ∞// Random Number Generator
$random = $distribution->rand(); // A random number with a given distribution
Probability – Discrete Distributions
use MathPHP\Probability\Distribution\Discrete;
// Bernoulli distribution (special case of binomial where n = 1)
$p = 0.3;
$k = 0;
$bernoulli = new Discrete\Bernoulli($p);
$pmf = $bernoulli->pmf($k);
$cdf = $bernoulli->cdf($k);
// Binomial distribution
$n = 2; // number of events
$p = 0.5; // probability of success
$r = 1; // number of successful events
$binomial = new Discrete\Binomial($n, $p);
$pmf = $binomial->pmf($r);
$cdf = $binomial->cdf($r);
// Categorical distribution
$k = 3; // number of categories
$probabilities = ['a' => 0.3, 'b' => 0.2, 'c' => 0.5]; // probabilities for categorices a, b, and c
$categorical = new Discrete\Categorical($k, $probabilities);
$pmf_a = $categorical->pmf('a');
$mode = $categorical->mode();
// Geometric distribution (failures before the first success)
$p = 0.5; // success probability
$k = 2; // number of trials
$geometric = new Discrete\Geometric($p);
$pmf = $geometric->pmf($k);
$cdf = $geometric->cdf($k);
// Hypergeometric distribution
$N = 50; // population size
$K = 5; // number of success states in the population
$n = 10; // number of draws
$k = 4; // number of observed successes
$hypergeo = new Discrete\Hypergeometric($N, $K, $n);
$pmf = $hypergeo->pmf($k);
$cdf = $hypergeo->cdf($k);
$μ = $hypergeo->mean();
// Multinomial distribution
$frequencies = [7, 2, 3];
$probabilities = [0.40, 0.35, 0.25];
$multinomial = new Discrete\Multinomial($probabilities);
$pmf = $multinomial->pmf($frequencies);
// Negative binomial distribution (Pascal)
$r = 1; // number of successful events
$P = 0.5; // probability of success on an individual trial
$x = 2; // number of trials required to produce r successes
$negativeBinomial = new Discrete\NegativeBinomial($r, $p);
$pmf = $negativeBinomial->pmf($x);
// Pascal distribution (Negative binomial)
$r = 1; // number of successful events
$P = 0.5; // probability of success on an individual trial
$x = 2; // number of trials required to produce r successes
$pascal = new Discrete\Pascal($r, $p);
$pmf = $pascal->pmf($x);
// Poisson distribution
$λ = 2; // average number of successful events per interval
$k = 3; // events in the interval
$poisson = new Discrete\Poisson($λ);
$pmf = $poisson->pmf($k);
$cdf = $poisson->cdf($k);
// Shifted geometric distribution (probability to get one success)
$p = 0.5; // success probability
$k = 2; // number of trials
$shiftedGeometric = new Discrete\ShiftedGeometric($p);
$pmf = $shiftedGeometric->pmf($k);
$cdf = $shiftedGeometric->cdf($k);
// Uniform distribution
$a = 1; // lower boundary of the distribution
$b = 4; // upper boundary of the distribution
$k = 2; // percentile
$uniform = new Discrete\Uniform($a, $b);
$pmf = $uniform->pmf();
$cdf = $uniform->cdf($k);
$μ = $uniform->mean();
Probability – Multivariate Distributions
use MathPHP\Probability\Distribution\Multivariate;
// Dirichlet distribution
$αs = [1, 2, 3];
$xs = [0.07255081, 0.27811903, 0.64933016];
$dirichlet = new Multivariate\Dirichlet($αs);
$pdf = $dirichlet->pdf($xs);
// Normal distribution
$μ = [1, 1.1];
$∑ = MatrixFactory::create([
[1, 0],
[0, 1],
]);
$X = [0.7, 1.4];
$normal = new Multivariate\Normal($μ, $∑);
$pdf = $normal->pdf($X);
Probability – Distribution Tables
use MathPHP\Probability\Distribution\Table;
// Provided solely for completeness' sake.
// It is statistics tradition to provide these tables.
// MathPHP has dynamic distribution CDF functions you can use instead.
// Standard Normal Table (Z Table)
$table = Table\StandardNormal::Z_SCORES;
$probability = $table[1.5][0]; // Value for Z of 1.50
// t Distribution Tables
$table = Table\TDistribution::ONE_SIDED_CONFIDENCE_LEVEL;
$table = Table\TDistribution::TWO_SIDED_CONFIDENCE_LEVEL;
$ν = 5; // degrees of freedom
$cl = 99; // confidence level
$t = $table[$ν][$cl];
// t Distribution Tables
$table = Table\TDistribution::ONE_SIDED_ALPHA;
$table = Table\TDistribution::TWO_SIDED_ALPHA;
$ν = 5; // degrees of freedom
$α = 0.001; // alpha value
$t = $table[$ν][$α];
// χ² Distribution Table
$table = Table\ChiSquared::CHI_SQUARED_SCORES;
$df = 2; // degrees of freedom
$p = 0.05; // P value
$χ² = $table[$df][$p];
Sequences – Basic
use MathPHP\Sequence\Basic;
$n = 5; // Number of elements in the sequence// Arithmetic progression
$d = 2; // Difference between the elements of the sequence
$a₁ = 1; // Starting number for the sequence
$progression = Basic::arithmeticProgression($n, $d, $a₁);
// [1, 3, 5, 7, 9] - Indexed from 1// Geometric progression (arⁿ⁻¹)
$a = 2; // Scalar value
$r = 3; // Common ratio
$progression = Basic::geometricProgression($n, $a, $r);
// [2(3)⁰, 2(3)¹, 2(3)², 2(3)³] = [2, 6, 18, 54] - Indexed from 1// Square numbers (n²)
$squares = Basic::squareNumber($n);
// [0², 1², 2², 3², 4²] = [0, 1, 4, 9, 16] - Indexed from 0// Cubic numbers (n³)
$cubes = Basic::cubicNumber($n);
// [0³, 1³, 2³, 3³, 4³] = [0, 1, 8, 27, 64] - Indexed from 0// Powers of 2 (2ⁿ)
$po2 = Basic::powersOfTwo($n);
// [2⁰, 2¹, 2², 2³, 2⁴] = [1, 2, 4, 8, 16] - Indexed from 0// Powers of 10 (10ⁿ)
$po10 = Basic::powersOfTen($n);
// [10⁰, 10¹, 10², 10³, 10⁴] = [1, 10, 100, 1000, 10000] - Indexed from 0// Factorial (n!)
$fact = Basic::factorial($n);
// [0!, 1!, 2!, 3!, 4!] = [1, 1, 2, 6, 24] - Indexed from 0// Digit sum
$digit_sum = Basic::digitSum($n);
// [0, 1, 2, 3, 4] - Indexed from 0// Digital root
$digit_root = Basic::digitalRoot($n);
// [0, 1, 2, 3, 4] - Indexed from 0
Sequences – Advanced
use MathPHP\Sequence\Advanced;
$n = 6; // Number of elements in the sequence// Fibonacci (Fᵢ = Fᵢ₋₁ + Fᵢ₋₂)
$fib = Advanced::fibonacci($n);
// [0, 1, 1, 2, 3, 5] - Indexed from 0// Lucas numbers
$lucas = Advanced::lucasNumber($n);
// [2, 1, 3, 4, 7, 11] - Indexed from 0// Pell numbers
$pell = Advanced::pellNumber($n);
// [0, 1, 2, 5, 12, 29] - Indexed from 0// Triangular numbers (figurate number)
$triangles = Advanced::triangularNumber($n);
// [1, 3, 6, 10, 15, 21] - Indexed from 1// Pentagonal numbers (figurate number)
$pentagons = Advanced::pentagonalNumber($n);
// [1, 5, 12, 22, 35, 51] - Indexed from 1// Hexagonal numbers (figurate number)
$hexagons = Advanced::hexagonalNumber($n);
// [1, 6, 15, 28, 45, 66] - Indexed from 1// Heptagonal numbers (figurate number)
$hexagons = Advanced::heptagonalNumber($n);
// [1, 4, 7, 13, 18, 27] - Indexed from 1// Look-and-say sequence (describe the previous term!)
$look_and_say = Advanced::lookAndSay($n);
// ['1', '11', '21', '1211', '111221', '312211'] - Indexed from 1// Lazy caterer's sequence (central polygonal numbers)
$lazy_caterer = Advanced::lazyCaterers($n);
// [1, 2, 4, 7, 11, 16] - Indexed from 0// Magic squares series (magic constants; magic sums)
$magic_squares = Advanced::magicSquares($n);
// [0, 1, 5, 15, 34, 65] - Indexed from 0// Perfect powers sequence
$perfect_powers = Advanced::perfectPowers($n);
// [4, 8, 9, 16, 25, 27] - Indexed from 0// Not perfect powers sequence
$not_perfect_powers = Advanced::notPerfectPowers($n);
// [2, 3, 5, 6, 7, 10] - Indexed from 0// Prime numbers up to n (n is not the number of elements in the sequence)
$primes = Advanced::primesUpTo(30);
// [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] - Indexed from 0
Set Theory
use MathPHP\SetTheory\Set;
use MathPHP\SetTheory\ImmutableSet;
// Sets and immutable sets
$A = new Set([1, 2, 3]); // Can add and remove members
$B = new ImmutableSet([3, 4, 5]); // Cannot modify set once created// Basic set data
$set = $A->asArray();
$cardinality = $A->length();
$bool = $A->isEmpty();
// Set membership
$true = $A->isMember(2);
$true = $A->isNotMember(8);
// Add and remove members
$A->add(4);
$A->add(new Set(['a', 'b']));
$A->addMulti([5, 6, 7]);
$A->remove(7);
$A->removeMulti([5, 6]);
$A->clear();
// Set properties against other sets - return boolean
$bool = $A->isDisjoint($B);
$bool = $A->isSubset($B); // A ⊆ B
$bool = $A->isProperSubset($B); // A ⊆ B & A ≠ B
$bool = $A->isSuperset($B); // A ⊇ B
$bool = $A->isProperSuperset($B); // A ⊇ B & A ≠ B// Set operations with other sets - return a new Set
$A∪B = $A->union($B);
$A∩B = $A->intersect($B);
$A＼B = $A->difference($B); // relative complement
$AΔB = $A->symmetricDifference($B);
$A×B = $A->cartesianProduct($B);
// Other set operations
$P⟮A⟯ = $A->powerSet();
$C = $A->copy();
// Print a set
print($A); // Set{1, 2, 3, 4, Set{a, b}}// PHP Interfaces
$n = count($A); // Countable
foreach ($A as $member) { ... } // Iterator// Fluent interface
$A->add(5)->add(6)->remove(4)->addMulti([7, 8, 9]);
Statistics – ANOVA
use MathPHP\Statistics\ANOVA;
// One-way ANOVA
$sample1 = [1, 2, 3];
$sample2 = [3, 4, 5];
$sample3 = [5, 6, 7];
⋮ ⋮
$anova = ANOVA::oneWay($sample1, $sample2, $sample3);
print_r($anova);
/* Array ( [ANOVA] => Array ( // ANOVA hypothesis test summary data [treatment] => Array ( [SS] => 24 // Sum of squares (between) [df] => 2 // Degrees of freedom [MS] => 12 // Mean squares [F] => 12 // Test statistic [P] => 0.008 // P value ) [error] => Array ( [SS] => 6 // Sum of squares (within) [df] => 6 // Degrees of freedom [MS] => 1 // Mean squares ) [total] => Array ( [SS] => 30 // Sum of squares (total) [df] => 8 // Degrees of freedom ) ) [total_summary] => Array ( // Total summary data [n] => 9 [sum] => 36 [mean] => 4 [SS] => 174 [variance] => 3.75 [sd] => 1.9364916731037 [sem] => 0.6454972243679 ) [data_summary] => Array ( // Data summary (each input sample) [0] => Array ([n] => 3 [sum] => 6 [mean] => 2 [SS] => 14 [variance] => 1 [sd] => 1 [sem] => 0.57735026918963) [1] => Array ([n] => 3 [sum] => 12 [mean] => 4 [SS] => 50 [variance] => 1 [sd] => 1 [sem] => 0.57735026918963) [2] => Array ([n] => 3 [sum] => 18 [mean] => 6 [SS] => 110 [variance] => 1 [sd] => 1 [sem] => 0.57735026918963) )) */// Two-way ANOVA/* | Factor B₁ | Factor B₂ | Factor B₃ | ⋯Factor A₁ | 4, 6, 8 | 6, 6, 9 | 8, 9, 13 | ⋯Factor A₂ | 4, 8, 9 | 7, 10, 13 | 12, 14, 16| ⋯ ⋮ ⋮ ⋮ ⋮ */
$factorA₁ = [
[4, 6, 8], // Factor B₁
[6, 6, 9], // Factor B₂
[8, 9, 13], // Factor B₃
];
$factorA₂ = [
[4, 8, 9], // Factor B₁
[7, 10, 13], // Factor B₂
[12, 14, 16], // Factor B₃
];
⋮
$anova = ANOVA::twoWay($factorA₁, $factorA₂);
print_r($anova);
/* Array ( [ANOVA] => Array ( // ANOVA hypothesis test summary data [factorA] => Array ( [SS] => 32 // Sum of squares [df] => 1 // Degrees of freedom [MS] => 32 // Mean squares [F] => 5.6470588235294 // Test statistic [P] => 0.034994350619895 // P value ) [factorB] => Array ( [SS] => 93 // Sum of squares [df] => 2 // Degrees of freedom [MS] => 46.5 // Mean squares [F] => 8.2058823529412 // Test statistic [P] => 0.0056767297582031 // P value ) [interaction] => Array ( [SS] => 7 // Sum of squares [df] => 2 // Degrees of freedom [MS] => 3.5 // Mean squares [F] => 0.61764705882353 // Test statistic [P] => 0.5555023440712 // P value ) [error] => Array ( [SS] => 68 // Sum of squares (within) [df] => 12 // Degrees of freedom [MS] => 5.6666666666667 // Mean squares ) [total] => Array ( [SS] => 200 // Sum of squares (total) [df] => 17 // Degrees of freedom ) ) [total_summary] => Array ( // Total summary data [n] => 18 [sum] => 162 [mean] => 9 [SS] => 1658 [variance] => 11.764705882353 [sd] => 3.4299717028502 [sem] => 0.80845208345444 ) [summary_factorA] => Array ( ... ) // Summary data of factor A [summary_factorB] => Array ( ... ) // Summary data of factor B [summary_interaction] => Array ( ... ) // Summary data of interactions of factors A and B) */
Statistics – Averages
use MathPHP\Statistics\Average;
$numbers = [13, 18, 13, 14, 13, 16, 14, 21, 13];
// Mean, median, mode
$mean = Average::mean($numbers);
$median = Average::median($numbers);
$mode = Average::mode($numbers); // Returns an array — may be multimodal// Weighted mean
$weights = [12, 1, 23, 6, 12, 26, 21, 12, 1];
$weighted_mean = Average::weightedMean($numbers, $weights)
// Other means of a list of numbers
$geometric_mean = Average::geometricMean($numbers);
$harmonic_mean = Average::harmonicMean($numbers);
$contraharmonic_mean = Average::contraharmonicMean($numbers);
$quadratic_mean = Average::quadraticMean($numbers); // same as rootMeanSquare
$root_mean_square = Average::rootMeanSquare($numbers); // same as quadraticMean
$trimean = Average::trimean($numbers);
$interquartile_mean = Average::interquartileMean($numbers); // same as iqm
$interquartile_mean = Average::iqm($numbers); // same as interquartileMean
$cubic_mean = Average::cubicMean($numbers);
// Truncated mean (trimmed mean)
$trim_percent = 25;
$truncated_mean = Average::truncatedMean($numbers, $trim_percent);
// Generalized mean (power mean)
$p = 2;
$generalized_mean = Average::generalizedMean($numbers, $p); // same as powerMean
$power_mean = Average::powerMean($numbers, $p); // same as generalizedMean// Lehmer mean
$p = 3;
$lehmer_mean = Average::lehmerMean($numbers, $p);
// Moving averages
$n = 3;
$weights = [3, 2, 1];
$SMA = Average::simpleMovingAverage($numbers, $n); // 3 n-point moving average
$CMA = Average::cumulativeMovingAverage($numbers);
$WMA = Average::weightedMovingAverage($numbers, $n, $weights);
$EPA = Average::exponentialMovingAverage($numbers, $n);
// Means of two numbers
list($x, $y) = [24, 6];
$agm = Average::arithmeticGeometricMean($x, $y); // same as agm
$agm = Average::agm($x, $y); // same as arithmeticGeometricMean
$log_mean = Average::logarithmicMean($x, $y);
$heronian_mean = Average::heronianMean($x, $y);
$identric_mean = Average::identricMean($x, $y);
// Averages report
$averages = Average::describe($numbers);
print_r($averages);
/* Array ( [mean] => 15 [median] => 14 [mode] => Array ( [0] => 13 ) [geometric_mean] => 14.789726414533 [harmonic_mean] => 14.605077399381 [contraharmonic_mean] => 15.474074074074 [quadratic_mean] => 15.235193176035 [trimean] => 14.5 [iqm] => 14 [cubic_mean] => 15.492307432707) */
use MathPHP\Statistics\KernelDensityEstimation
$data = [-2.76, -1.09, -0.5, -0.15, 0.22, 0.69, 1.34, 1.75];
$x = 0.5;
// Density estimator with default bandwidth (normal distribution approximation) and kernel function (standard normal)
$kde = new KernelDensityEstimation($data);
$density = $kde->evaluate($x)
// Custom bandwidth
$h = 0.1;
$kde->setBandwidth($h);
// Library of built-in kernel functions
$kde->setKernelFunction(KernelDensityEstimation::STANDARD_NORMAL);
$kde->setKernelFunction(KernelDensityEstimation::NORMAL);
$kde->setKernelFunction(KernelDensityEstimation::UNIFORM);
$kde->setKernelFunction(KernelDensityEstimation::TRIANGULAR);
$kde->setKernelFunction(KernelDensityEstimation::EPANECHNIKOV);
$kde->setKernelFunction(KernelDensityEstimation::TRICUBE);
// Set custom kernel function (user-provided callable)
$kernel = function ($x) {
if (abs($x) > 1) {
return 0;
} else {
return 70 / 81 * ((1 - abs($x) ** 3) ** 3);
}
};
$kde->setKernelFunction($kernel);
// All customization optionally can be done in the constructor
$kde = new KernelDesnsityEstimation($data, $h, $kernel);
Statistics – Random Variables
use MathPHP\Statistics\RandomVariable;
$X = [1, 2, 3, 4];
$Y = [2, 3, 4, 5];
// Central moment (nth moment)
$second_central_moment = RandomVariable::centralMoment($X, 2);
$third_central_moment = RandomVariable::centralMoment($X, 3);
// Skewness (population and sample)
$skewness = RandomVariable::skewness($X); // general method of calculating skewness
$skewness = RandomVariable::populationSkewness($X); // similar to Excel's SKEW.P
$skewness = RandomVariable::sampleSkewness($X); // similar to Excel's SKEW
$SES = RandomVariable::ses(count($X)); // standard error of skewness// Kurtosis (excess)
$kurtosis = RandomVariable::kurtosis($X);
$platykurtic = RandomVariable::isPlatykurtic($X); // true if kurtosis is less than zero
$leptokurtic = RandomVariable::isLeptokurtic($X); // true if kurtosis is greater than zero
$mesokurtic = RandomVariable::isMesokurtic($X); // true if kurtosis is zero
$SEK = RandomVariable::sek(count($X)); // standard error of kurtosis// Standard error of the mean (SEM)
$sem = RandomVariable::standardErrorOfTheMean($X); // same as sem
$sem = RandomVariable::sem($X); // same as standardErrorOfTheMean// Confidence interval
$μ = 90; // sample mean
$n = 9; // sample size
$σ = 36; // standard deviation
$cl = 99; // confidence level
$ci = RandomVariable::confidenceInterval($μ, $n, $σ, $cl); // Array( [ci] => 30.91, [lower_bound] => 59.09, [upper_bound] => 120.91 )
Statistics – Regressions
use MathPHP\Statistics\Regression;
$points = [[1,2], [2,3], [4,5], [5,7], [6,8]];
// Simple linear regression (least squares method)
$regression = new Regression\Linear($points);
$parameters = $regression->getParameters(); // [m => 1.2209302325581, b => 0.6046511627907]
$equation = $regression->getEquation(); // y = 1.2209302325581x + 0.6046511627907
$y = $regression->evaluate(5); // Evaluate for y at x = 5 using regression equation
$ci = $regression->ci(5, 0.5); // Confidence interval for x = 5 with p-value of 0.5
$pi = $regression->pi(5, 0.5); // Prediction interval for x = 5 with p-value of 0.5; Optional number of trials parameter.
$Ŷ = $regression->yHat();
$r = $regression->r(); // same as correlationCoefficient
$r² = $regression->r2(); // same as coefficientOfDetermination
$se = $regression->standardErrors(); // [m => se(m), b => se(b)]
$t = $regression->tValues(); // [m => t, b => t]
$p = $regression->tProbability(); // [m => p, b => p]
$F = $regression->fStatistic();
$p = $regression->fProbability();
$h = $regression->leverages();
$e = $regression->residuals();
$D = $regression->cooksD();
$DFFITS = $regression->dffits();
$SStot = $regression->sumOfSquaresTotal();
$SSreg = $regression->sumOfSquaresRegression();
$SSres = $regression->sumOfSquaresResidual();
$MSR = $regression->meanSquareRegression();
$MSE = $regression->meanSquareResidual();
$MSTO = $regression->meanSquareTotal();
$error = $regression->errorSd(); // Standard error of the residuals
$V = $regression->regressionVariance();
$n = $regression->getSampleSize(); // 5
$points = $regression->getPoints(); // [[1,2], [2,3], [4,5], [5,7], [6,8]]
$xs = $regression->getXs(); // [1, 2, 4, 5, 6]
$ys = $regression->getYs(); // [2, 3, 5, 7, 8]
$ν = $regression->degreesOfFreedom();
// Linear regression through a fixed point (least squares method)
$force_point = [0,0];
$regression = new Regression\LinearThroughPoint($points, $force_point);
$parameters = $regression->getParameters();
$equation = $regression->getEquation();
$y = $regression->evaluate(5);
$Ŷ = $regression->yHat();
$r = $regression->r();
$r² = $regression->r2();
⋮ ⋮
// Theil–Sen estimator (Sen's slope estimator, Kendall–Theil robust line)
$regression = new Regression\TheilSen($points);
$parameters = $regression->getParameters();
$equation = $regression->getEquation();
$y = $regression->evaluate(5);
⋮ ⋮
// Use Lineweaver-Burk linearization to fit data to the Michaelis–Menten model: y = (V * x) / (K + x)
$regression = new Regression\LineweaverBurk($points);
$parameters = $regression->getParameters(); // [V, K]
$equation = $regression->getEquation(); // y = Vx / (K + x)
$y = $regression->evaluate(5);
⋮ ⋮
// Use Hanes-Woolf linearization to fit data to the Michaelis–Menten model: y = (V * x) / (K + x)
$regression = new Regression\HanesWoolf($points);
$parameters = $regression->getParameters(); // [V, K]
$equation = $regression->getEquation(); // y = Vx / (K + x)
$y = $regression->evaluate(5);
⋮ ⋮
// Power law regression - power curve (least squares fitting)
$regression = new Regression\PowerLaw($points);
$parameters = $regression->getParameters(); // [a => 56.483375436574, b => 0.26415375648621]
$equation = $regression->getEquation(); // y = 56.483375436574x^0.26415375648621
$y = $regression->evaluate(5);
⋮ ⋮
// LOESS - Locally Weighted Scatterplot Smoothing (Local regression)
$α = 1/3; // Smoothness parameter
$λ = 1; // Order of the polynomial fit
$regression = new Regression\LOESS($points, $α, $λ);
$y = $regression->evaluate(5);
$Ŷ = $regression->yHat();
⋮ ⋮
Statistics – Significance Testing
use MathPHP\Statistics\Significance;
// Z test - One sample (z and p values)
$Hₐ = 20; // Alternate hypothesis (M Sample mean)
$n = 200; // Sample size
$H₀ = 19.2; // Null hypothesis (μ Population mean)
$σ = 6; // SD of population (Standard error of the mean)
$z = Significance:zTest($Hₐ, $n, $H₀, $σ); // Same as zTestOneSample
$z = Significance:zTestOneSample($Hₐ, $n, $H₀, $σ); // Same as zTest/* [ 'z' => 1.88562, // Z score 'p1' => 0.02938, // one-tailed p value 'p2' => 0.0593, // two-tailed p value] */// Z test - Two samples (z and p values)
$μ₁ = 27; // Sample mean of population 1
$μ₂ = 33; // Sample mean of population 2
$n₁ = 75; // Sample size of population 1
$n₂ = 50; // Sample size of population 2
$σ₁ = 14.1; // Standard deviation of sample mean 1
$σ₂ = 9.5; // Standard deviation of sample mean 2
$z = Significance::zTestTwoSample($μ₁, $μ₂, $n₁, $n₂, $σ₁, $σ₂);
/* [ 'z' => -2.36868418147285, // z score 'p1' => 0.00893, // one-tailed p value 'p2' => 0.0179, // two-tailed p value] */// Z score
$M = 8; // Sample mean
$μ = 7; // Population mean
$σ = 1; // Population SD
$z = Significance::zScore($M, $μ, $σ);
// T test - One sample (from sample data)
$a = [3, 4, 4, 5, 5, 5, 6, 6, 7, 8]; // Data set
$H₀ = 300; // Null hypothesis (μ₀ Population mean)
$tTest = Significance::tTest($a, $H₀)
print_r($tTest);
/* Array ( [t] => 0.42320736951516 // t score [df] => 9 // degrees of freedom [p1] => 0.34103867713806 // one-tailed p value [p2] => 0.68207735427613 // two-tailed p value [mean] => 5.3 // sample mean [sd] => 1.4944341180973 // standard deviation) */// T test - One sample (from summary data)
$Hₐ = 280; // Alternate hypothesis (M Sample mean)
$s = 50; // Standard deviation of sample
$n = 15; // Sample size
$H₀ = 300; // Null hypothesis (μ₀ Population mean)
$tTest = Significance::tTestOneSampleFromSummaryData($Hₐ, $s, $n, $H₀);
print_r($tTest);
/* Array ( [t] => -1.549193338483 // t score [df] => 14 // degreees of freedom [p1] => 0.071820000122611 // one-tailed p value [p2] => 0.14364000024522 // two-tailed p value [mean] => 280 // sample mean [sd] => 50 // standard deviation) */// T test - Two samples (from sample data)
$x₁ = [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4];
$x₂ = [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4];
$tTest = Significance::tTest($x₁, $x₂);
print_r($tTest);
/* Array ( [t] => -2.4553600286929 // t score [df] => 24.988527070145 // degrees of freedom [p1] => 0.010688914613979 // one-tailed p value [p2] => 0.021377829227958 // two-tailed p value [mean1] => 20.82 // mean of sample x₁ [mean2] => 22.98667 // mean of sample x₂ [sd1] => 2.804894 // standard deviation of x₁ [sd2] => 1.952605 // standard deviation of x₂) */// T test - Two samples (from summary data)
$μ₁ = 42.14; // Sample mean of population 1
$μ₂ = 43.23; // Sample mean of population 2
$n₁ = 10; // Sample size of population 1
$n₂ = 10; // Sample size of population 2
$σ₁ = 0.683; // Standard deviation of sample mean 1
$σ₂ = 0.750; // Standard deviation of sample mean 2
$tTest = Significance::tTestTwoSampleFromSummaryData($μ₁, $μ₂, $n₁, $n₂, $σ₁, $σ₂);
print_r($tTest);
/* Array ( [t] => -3.3972305988708 // t score [df] => 17.847298548027 // degrees of freedom [p1] => 0.0016211251126198 // one-tailed p value [p2] => 0.0032422502252396 // two-tailed p value [mean1] => 42.14 [mean2] => 43.23 [sd1] => 0.6834553 [sd2] => 0.7498889] */// T score
$Hₐ = 280; // Alternate hypothesis (M Sample mean)
$s = 50; // SD of sample
$n = 15; // Sample size
$H₀ = 300; // Null hypothesis (μ₀ Population mean)
$t = Significance::tScore($Hₐ, $s, $n, $H);
// χ² test (chi-squared goodness of fit test)
$observed = [4, 6, 17, 16, 8, 9];
$expected = [10, 10, 10, 10, 10, 10];
$χ² = Significance::chiSquaredTest($observed, $expected);
// ['chi-square' => 14.2, 'p' => 0.014388]
Trigonometry
use MathPHP\Trigonometry;
$n = 9;
$points = Trigonometry::unitCircle($n); // Produce n number of points along the unit circle
Unit Tests
Beyond 100
MathPHP has thousands of unit tests testing individual functions directly with numerous data inputs to achieve 100
