Equis

A library for solving systems of linear equations, using Idris.

Overview

Sometimes we need to solve systems of linear equations like the following:

x + y + z = 6
2y + 5z = −4
2x + 5y − z = 27

Using this library, we can do so in Idris:

> :exec solve [[1.0, 1.0, 1.0], [0.0, 2.0, 5.0], [2.0, 5.0, -1.0]] [6.0, -4.0, 27.0]
[5, 3, -2]

This is the same system of linear equations used over on Maths is Fun so you can check the solution over there if you don't believe me. The library uses the same well-known approach detailed in that tutorial.

If we attempt to solve a system for which there is no solution, we will get nan back:

> :exec solve [[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]] [6.0, 9.0]
[nan, nan, nan]

Polynomial Curve Fitting

We can also perform least-squares polynomial curve fitting using this library. If we would like the result as a vector of coefficients, we can use leastSquares:

> :let coeffs = leastSquares 3 [1.0, 2.0, 3.0, 4.0] [4.0, 8.0, 4.0, 8.0]
> coeffs
[-23.999999999999318, 45.3333333333336, -19.999999999999403, 2.6666666666667425] : Vect 4 Double

To get a function of the form f(x) = y we can use leastSquaresFn:

> :let f = leastSquaresFn 3 [1.0, 2.0, 3.0, 4.0] [4.0, 8.0, 4.0, 8.0]
> f 3
4.000000000008896 : Double
> f 4
8.000000000016144 : Double

Note: The above example using LeastSquaresFn should work in compiled code, but doesn't work in the REPL because we're hitting a piece of missing functionality in Erasure.hs for some reason. Instead, try the following:

> :let coeffs = leastSquares 3 [1.0, 2.0, 3.0, 4.0] [4.0, 8.0, 4.0, 8.0]
> sum (zipWith (*) (powers 3 3.0) coeffs)
4.000000000008896 : Double
> sum (zipWith (*) (powers 3 4.0) coeffs)
8.000000000016144 : Double

References

• Pierce, Rod, 2018, 'Solving Systems of Linear Equations Using Matrices', Math Is Fun, Available at: http://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html. [Accessed 8 Feb 2019]