Rank-polymorphic convolution using fast fourier transform

Arithmetic mean

Geometric mean

Harmonic mean

Quadratic mean

Median TODO: O(n) median algorithm?

Variance

Standard deviation

Matrix Determinant

Perform Gauss-Jordan elimination

Matrix Inverse

Matrix Cofactors

Pervasive factorial

Gamma function Γ(z)

Divisors of N (including 1 and N) The values are in increasing order and distinct.

Error function

Powerset of a list

Rank-polymorphic fast fourier transform

Dot product

Cross product

Matrix product

Computes AB

Matrix-vector product

Computes MV where V is represented as a 1D list of numbers

Matrix power

Quaternion product

Computes PQ for quaternion arrays P and Q.

Semi-pervasive; the last axis of the inputs must be of length 4, and will be interpreted as the input quaternions' real, i, j, and k.

# Computes (1+2i+3j+4k)(1+3i+5j+7k) and (2+4i+6j+8k)(5+6i+7j+8k)
Qqp [1_2_3_4 2_4_6_8] [1_3_5_7 5_6_7_8]
## ╭─
## ╷  ¯48  6  6 12
##   ¯120 24 60 48
##                 ╯

Create 3D rotation quaternions.

Expects an angle array θ and a unit vector axis array U.

Semi-pervasive; the unit vector array must have one extra trailing axis compared to the angle array, and that axis must be length 3.

For each angle-vector pair, computes cos(θ/2) + Usin(θ/2).

# Creates two rotation quaternions:
# - η radians about 0_0_1
# - π radians about 3/5_4/5_0
RQuat η_π [0_0_1 3/5_4/5_0]

To use rotation quaternions created by RQuat to rotate 3D vectors, see QRot.

Rotate a 3D vector array using a quaternion array.

To create rotation quaternions to use with QRot, see RQuat.

Expects a quaternion array Q and a 3D vector array V.

Semi-pervasive; the input arrays should have matching shapes aside from the final axis, which should be of lengths 4 and 3 respectively.

For each quaternion-vector pair, computes Q * V * Q'.

RQuat η 0_0_1        # Create a rotation by η radians about 0_0_1
QRot ¤:[3_2_0 1_3_2] # Use the quaternion to rotate 3_2_0 and 1_3_2

QRot can be used with ⍜ to undo the rotations afterward.

Greatest common divisor

Ignores sign

Least common multiple

Ignores sign

Encode an array of numbers into digits of a given base

Analogous to ⋯ for base 2; the least significant digit is first.

Base allows multiple bases as input.

# Encodes 6 in base 2, 7 in base 3, 8 in base 4, and 9 in base 5.
Base [2_3 4_5] [6_7 8_9]
## ╭─
## ╷ 0 1 1
## ╷ 1 2 0
##
##   0 2 0
##   4 1 0
##         ╯

You can use ⌝Base to decode from a base/bases.

# Decodes 6 in base 2, 7 in base 3, 8 in base 4, and 9 in base 5.
⌝Base [2_3 4_5] [[0_1_1 1_2_0] [0_2_0 4_1_0]]
## ╭─
## ╷ 6 7
##   8 9
##       ╯

Inverse of N modulo M (M if nonexistent)

Multiplicative order of N modulo M

Binomial coefficient aka N choose K

Convert a number X to a continued fraction to N terms.

Use °CFrac to convert to a number from a continued fraction.

Box-Muller transform

Converts uniformly distributed pairs of input values to standard normally distributed pairs of output values

Generate a 2 dimensional square Gaussian kernel

Geometric distribution

Probability of X failures before the first success of repeated trials with success probability p

Cumulative geometric distribution

Probability of X or fewer failures before the first success of repeated trials with success probability p

Poisson distribution

Probability of X occurrences of an unlikely event over many trials with expected value λ

Cumulative Poisson distribution

Probability of X or fewer occurrences of an unlikely event over many trials with expected value λ

Seeded random indices from a distribution array

Given a seed, a shape, and a probability distribution array of any rank containing positive numbers, DistRand normalizes the distribution if necessary so it sums to 1, then returns an array of deep indices into the distribution arrays, chosen randomly weighted by the distribution. The generated random indices will fill the shape given to the function. If the distribution array is rank 1, the output will have the shape given. If the distribution array is rank >1, the output will have a shape equal to the shape given suffixed with the rank of the distribution array.

# Generate random indices in the shape 2_3 with a seed of 8
DistRand [0.1 0.4 0.3 0.2] 2_3 8
## ╭─
## ╷ 1 2 3
##   2 1 2
##         ╯

Binomial distribution

Probability that n Bernoulli trials each with success probability p will amount to X successes

Cumulative binomial distribution

Probability that n Bernoulli trials each with success probability p will amount to X or fewer successes

NOTE: This function only accepts scalars for X. Use each or rows if multiple X values are desired

Normal distribution

Probability density of a gaussian/bell curve with mean μ and standard deviation σ

Cumulative normal distribution

Probability that a normally distributed random variable with mean μ and standard deviation σ is less than X

Po-Shen quadratic solver

Returns roots as a list

Outputs complex-typed numbers only if the roots are complex