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:

t(chol(matrix(c(25, 15, -5, 15, 18, 0, -5, 0, 11), nrow=3, ncol=3)))
#      [,1] [,2] [,3]
# [1,]    5    0    0
# [2,]    3    3    0
# [3,]   -1    1    3

t(chol(matrix(c(18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106), nrow=4, ncol=4)))
#           [,1]     [,2]     [,3]     [,4]
# [1,]  4.242641 0.000000 0.000000 0.000000
# [2,]  5.185450 6.565905 0.000000 0.000000
# [3,] 12.727922 3.046038 1.649742 0.000000
# [4,]  9.899495 1.624554 1.849711 1.392621