Problem Statement

Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose:

L

{\displaystyle L}

is called the Cholesky factor of

A

{\displaystyle A}

, and can be interpreted as a generalized square root of

A

{\displaystyle A}

, as described in Cholesky decomposition. In a 3x3 example, we have to solve the following system of equations: We can see that for the diagonal elements (

l

k k

{\displaystyle l_{kk}}

) of

L

{\displaystyle L}

there is a calculation pattern: or in general: For the elements below the diagonal (

l

i k

{\displaystyle l_{ik}}

, where

i

k

{\displaystyle i>k}

) there is also a calculation pattern: which can also be expressed in a general formula: Task description The task is to implement a routine which will return a lower Cholesky factor

L

{\displaystyle L}

for every given symmetric, positive definite nxn matrix

A

{\displaystyle A}

. You should then test it on the following two examples and include your output. Example 1: Example 2:

Let's start with the solution:

import "./matrix" for Matrix
import "./fmt" for Fmt

var arrays = [
    [ [25, 15, -5],
      [15, 18,  0],
      [-5,  0, 11] ],

    [ [18, 22,  54,  42],
      [22, 70,  86,  62],
      [54, 86, 174, 134],
      [42, 62, 134, 106] ]
]

for (array in arrays) {
    System.print("Original:")
    Fmt.mprint(array, 3, 0)
    System.print("\nLower Cholesky factor:")
    Fmt.mprint(Matrix.new(array).cholesky(), 8, 5)
    System.print()
}