/maths

Maths includes mathematical functions not defined in the standard Go math package.

Primary LanguageGoBSD 3-Clause "New" or "Revised" LicenseBSD-3-Clause

Maths

Go Reference Go Go Report Card

maths includes mathematical functions not defined in the standard Go math package. Most functions support any primitive integer or float type through generics.

Installation

go get github.com/theriault/maths

What's Included

Combinatorics

import "github.com/theriault/maths/combinatorics"

Factorial

Source | Tests | Wikipedia | OEIS

$\displaystyle n! \; = \; \prod_{i=1}^{n} i$

combinatorics.Factorial(10) // will return uint64(3628800)

Falling Factorial

Source | Tests | Wikipedia

$\displaystyle x^{\underline{n}} \; = \; \prod _{k=1}^{n}(x-k+1)$

combinatorics.FallingFactorial(8, 3) // will return uint64(336)
combinatorics.PartialPermutations(8, 3) // will return uint64(336)

Rising Factorial

Source | Tests | Wikipedia

$\displaystyle x^{\overline{n}} \; = \; \prod _{k=1}^{n}(x+k-1)$

combinatorics.RisingFactorial(2, 3) // will return uint64(24)

Number Theory

import "github.com/theriault/maths/numbertheory"

Aliquot Sum

Source | Tests | Wikipedia | OEIS

$\displaystyle s(n) = \sigma_1(n) - n = \sum_{\substack{i = 1 \\ i | n}}^{n-1} i$

numbertheory.AliquotSum(60) // will return uint64(108)

Coprime

Source | Tests | Wikipedia

$\displaystyle f(a,b) = \begin{cases}\text{true} &\text{if}\ \gcd(a,b) = 1 \\ \text{false} &\text{else} \end{cases}$

numbertheory.Coprime(3*5*7, 11*13*17) // will return true

Digit Sum

Source | Tests | Wikipedia

$\displaystyle f_b(n) = \sum_{i=0}^{\lfloor \log_b{n} \rfloor} \frac{n \bmod b^{i+1} - n \bmod b^i}{b^i}$

numbertheory.DigitSum(9045, 10) // will return int(18)

Digital Root

Source | Tests | Wikipedia

$\displaystyle f_{b}(n)={\begin{cases} 0 &\text{if}\ n=0\\ n\ \bmod (b-1)&{\text{if}}\ n\not \equiv 0{\pmod {b-1}} \\ b-1 &\text{else} \end{cases}}$

numbertheory.DigitalRoot(9045, 10) // will return int(9)

Divisors function

$\displaystyle \sigma_z(n) = \sum_{\substack{i = 1 \\ i | n}}^{n} i^{z}$

Number-of-divisors (z = 0)

Source | Tests | Wikipedia | OEIS

numbertheory.NumberOfDivisors(48) // will return uint64(10)
Sum-of-divisors (z = 1)

Source | Tests | Wikipedia | OEIS

numbertheory.SumOfDivisors(48) // will return uint64(124)

Greatest Common Divisor

Source | Tests | Wikipedia

numbertheory.GCD(48,18) // will return int(6)

Least Common Multiple

Source | Tests | Wikipedia

numbertheory.LCM(48,18) // will return int(144)

Möbius function

Source | Tests | Wikipedia | OEIS

$\displaystyle \mu(n) = \begin{cases} +1 & n \text{ is square-free with even number of prime factors} \\ -1 & n \text{ is square-free with odd number of prime factors} \\ 0 & n \text{ is not square-free} \end{cases}$

numbertheory.Mobius(70) // will return int8(-1)

Politeness

Source | Tests | Wikipedia | OEIS

$\displaystyle p(n) = \left( \prod_{\substack{p |n \\ p \neq 2}}^{n} v_p(n) + 1\right)-1$

where $v_p(n)$ is the $p$-adic order

numbertheory.Politeness(32) // will return uint64(0)

Polygonal Numbers

Source | Tests | Wikipedia

$\displaystyle p(s, n) = \frac{(s-2)n^2-(s-4)n}{2}$

numbertheory.PolygonalNumber(3, 4) // will return uint64(10)

Finding $n$:

$\displaystyle p(s, x) = \frac{\sqrt{8(s-2)x+(s-4)^2}+(s-4)}{2(s-2)}$

numbertheory.PolygonalRoot(3, 10) // will return float64(4)

Finding $s$:

$\displaystyle p(n, x) = 2+\frac{2}{n} \cdot \frac{x-n}{n-1}$

numbertheory.PolygonalSides(4, 10) // will return float64(3)

Prime Factorization

Source | Tests | Wikipedia

numbertheory.PrimeFactorization(184756) // will return []uint64{2, 2, 11, 13, 17, 19}

Primorial

Source | Tests | Wikipedia | OEIS

$\displaystyle n\# = \prod_{\substack{i=2 \\ i \in \mathbb{P}}}^{n} i$

numbertheory.Primorial(30) // will return uint64(6469693230)

Radical

Source | Tests | Wikipedia | OEIS

$\displaystyle rad(n) = \prod_{p | n}p$

numbertheory.Radical(60) // will return uint64(30)

Totient

Euler's Totient

Source | Tests | Wikipedia | OEIS

$\displaystyle \varphi(n) = n \prod_{p | n} \left(1 - \frac{1}{p}\right) $

numbertheory.Totient(68) // will return uint64(32)
Jordan's Totient

Source | Tests | Wikipedia

$\displaystyle J_k(n) = n^k \prod_{p | n} \left(1 - \frac{1}{p^k}\right) $

numbertheory.TotientK(60, 2) // will return uint64(2304)

Statistics

import "github.com/theriault/maths/statistics"

Average/Mean

Generalized Mean

Source | Tests | Wikipedia

statistics.GeneralizedMean([]float64{1, 1000}, 2) // will return float64(707.1071347398497)
statistics.RootMeanSquare(1, 1000)  // will return float64(707.1071347398497)
Arithmetic Mean

Source | Tests | Wikipedia

$\displaystyle \bar{x} = \frac{1}{n}\left (\sum_{i=1}^n{x_i}\right ) = \frac{x_1+x_2+\cdots +x_n}{n}$

statistics.Mean(1, 1000) // will return float64(500.5)
Geometric Mean

Source | Tests | Wikipedia

$\displaystyle \bar{x} = \left( \prod_{i=1}^n{x_i} \right )^\frac{1}{n} = \exp{\left( {\frac{1}{n}\sum\limits_{i=1}^{n}\ln x_i} \right)} = \left(x_1 x_2 \cdots x_n \right)^\frac{1}{n}$

statistics.GeometricMean(1, 1000) // will return float64(31.62...)
Harmonic Mean

Source | Tests | Wikipedia

$\displaystyle \bar{x} = n \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}$

statistics.HarmonicMean(1, 1000) // will return float64(1.99...)

Central Moment

Source | Tests | Wikipedia

statistics.CentralMoment([]uint8{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2) // returns float64(8.25)

Interquartile Range (IQR)

Source | Tests | Wikipedia

statistics.InterquartileRange(3, 6, 7, 8, 8, 10, 13, 15, 16, 20) // returns float64(7.25)

Kurtosis

Population

Source | Tests | Wikipedia

statistics.Kurtosis(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // returns float64(1.5133)
Sample

Source | Tests | Wikipedia

statistics.SampleKurtosis(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // returns float64(2.522167)
Excess Sample Kurtosis

Source | Tests | Wikipedia

statistics.ExcessSampleKurtosis([]uint8{8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8}) // returns float64(-1.6445)

Logistic Function

Source | Tests | Wikipedia

$\displaystyle f(x) = \frac{L}{1+e^{-k(x-x_0)}}$

  • $L$ - curve's max value
  • $x_0$ - sigmoid's midpoint
  • $k$ - logistic growth rate
maxValue := 1.0
midpoint := 0.0
growthRate := 1.0
fx := statistics.LogisticFunction(maxValue, midpoint, growthRate) // will return func (x float64) float64

Mode

Source | Tests | Wikipedia

statistics.Mode(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}

Moving Averages

Simple Moving Average

Source | Tests | Wikipedia

$\displaystyle {\begin{aligned}{\textit {SMA}}{k}&={\frac{p{n-k+1}+p_{n-k+2}\cdots +p_{n}}{k}}\&={\frac{1}{k}}\sum_{i=n-k+1}^np_i\end{aligned}}$

statistics.SimpleMovingAverage(3, 1, 2, 3, 4, 5, 6, 7, 8, 9) // will return []float64{2, 3, 4, 5, 6, 7, 8}

Power Sum

Source | Tests | Wikipedia

$\displaystyle S_p(x_1, x_2, ..., x_n) = \sum_{i=1}^{n} x_i^p$

statistics.PowerSum([]float64{2, 3, 4}, 2) // will return float64(29)

Power Sum Around

Source | Wikipedia

$\displaystyle S_{p,y}(x_1, x_2, ..., x_n) = \sum_{i=1}^{n} (x_i - y)^p$

statistics.PowerSumAround([]float64{2, 3, 4}, 3, 2) // will return float64(29)

Quantiles (Median/Tertile/Quartile/.../Percentile)

Source | Tests | Wikipedia

n := []float64{3, 6, 7, 8, 8, 10, 13, 15, 16, 20}
statistics.Quantile(n, 2) // median: will return []float64{9}
statistics.Quantile(n, 3) // tertiles: will return []float64{8, 13}
statistics.Quantile(n, 4) // quartiles: will return []float64{7.25, 9, 14.5}
statistics.Quantile(n, 100) // percentile: will return []float64{3.27, 3.54, 3.81, 4.08, ...95 other values...}

// aliases
statistics.Tertile(n) // will return []float64{8, 13}
statistics.Quartile(n) // will return []float64{7.25, 9, 14.5}
statistics.Percentile(n) // will return []float64{3.27, 3.54, 3.81, 4.08, ...95 other values...}

Median (Source | Tests | Wikipedia)

n := []float64{3, 6, 7, 8, 8, 10, 13, 15, 16, 20}
statistics.Median(n) // will return float64(9)

Sample Extrema (Max/Min/Range)

Sample Maximum / Largest Observation

Source | Tests | Wikipedia

n := []uint8{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
statistics.Max(n...) // will return uint8(10)
Sample Minimum / Smallest Observation

Source | Tests | Wikipedia

n := []uint8{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
statistics.Min(n...) // will return uint8(1)
Range

Source | Tests | Wikipedia

n := []uint8{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
statistics.Range(n...) // will return uint8(9)

Skewness

Population

Source | Tests | Wikipedia

statistics.Skewness(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // returns float64(-0.274241)
Sample

Source | Tests | Wikipedia

statistics.SampleSkewness(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // returns float64(-0.319584)

Standard Deviation

Population

Source | Tests | Wikipedia

$\displaystyle \sigma = \sqrt{\left(\frac{1}{N} \sum_{i=1}^{N} x_{i}^{2} \right) - \mu^2}$

statistics.StandardDeviation(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}
Sample

Source | Tests | Wikipedia

$\displaystyle s = \sqrt{\sigma^2 \frac{N}{N-1}}$

statistics.SampleStandardDeviation(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}

Standard Error

Population

Source | Tests | Wikipedia

$\displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

statistics.StandardError(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}
Sample

Source | Tests | Wikipedia

$\displaystyle s_{\bar{x}} = \frac{s}{\sqrt{n}}$

statistics.SampleStandardError(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}

Softmax

Source | Tests | Wikipedia

X := []float64{1, 2, 3, 4, 1, 2, 3}
statistics.Softmax(X) // will return []float64{0.0236405, 0.0642617, 0.1746813, 0.4748330, 0.0236405, 0.0642617, 0.1746813}
statistics.MutableSoftmax(X) // same as above, but will modify X and return X

X = []float64{1, 2, 3, 4, 1, 2, 3}
statistics.GeneralizedSoftmax(X, 0.5) // will return []float64{0.0018889, 0.0139573, 0.1031315, 0.7620445, 0.0018889, 0.0139573, 0.1031315}

Sum/Summation

Source | Tests | Wikipedia

$\displaystyle S(x_1, x_2, ..., x_n) = \sum_{i=1}^{n} x_i $

statistics.Sum(1.1, 1.2, 1.3) // will return float64(3.6)

Variance

Population

Source | Tests | Wikipedia

$\displaystyle \sigma^2 = \left(\frac{1}{N} \sum_{i=1}^{N} x_{i}^{2} \right) - \mu^2$

statistics.Variance(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}
Sample

Source | Tests | Wikipedia

$\displaystyle s^2 = \sigma^2 \frac{N}{N-1}$

statistics.SampleVariance(8, 3, 6, 2, 7, 1, 8, 3, 7, 4, 8) // will return []float64{8}

Weighted Average/Mean

Weighted Generalized Mean

Source | Tests | Wikipedia

X := []uint8{8, 7, 3, 2, 6, 11, 6, 7, 2, 1, 7}
W := []uint8{1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2}
mean, err := statistics.WeightedGeneralizedMean(X, Y, 1) // will return float64(5.5)
Weighted Arithmetic Mean

Source | Tests | Wikipedia

X := []uint8{8, 7, 3, 2, 6, 11, 6, 7, 2, 1, 7}
W := []uint8{1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2}
mean, err := statistics.WeightedMean(X, Y) // will return float64(5.5)
Weighted Geometric Mean

Source | Tests | Wikipedia

X := []uint8{8, 7, 3, 2, 6, 11, 6, 7, 2, 1, 7}
W := []uint8{1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2}
mean, err := statistics.WeightedGeometricMean(X, Y) // will return float64(4.6)
Weighted Harmonic Mean

Source | Tests | Wikipedia

X := []uint8{8, 7, 3, 2, 6, 11, 6, 7, 2, 1, 7}
W := []uint8{1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2}
mean, err := statistics.WeightedHarmonicMean(X, Y) // will return float64(5.8)

Complexity

See docs/complexity.md for information on time complexity and space complexity.