How to Perform Cholesky Decomposition in R - Step by Step Examples

How to Perform Cholesky Decomposition in R ?

Answer

To perform Cholesky decomposition in R, you can use the chol() function. This function computes the Cholesky factorization of a real symmetric positive-definite matrix.


✐ Examples

1 Performing Cholesky Decomposition on a 3x3 Matrix

In this example,

  1. We start by creating a 3x3 matrix named mat using the matrix() function. This matrix represents the data we want to decompose.
  2. Next, we use the chol() function to perform Cholesky decomposition on the matrix mat. We assign the result to a variable named chol_res.
  3. We extract the lower triangular Cholesky factor from chol_res using the drop() function and assign it to a variable named L.
  4. We compute the product of L and its transpose using the %*% operator to obtain the original matrix mat. This is done to verify the correctness of the decomposition.
  5. We print both matrix L and the product of L and its transpose to the console to see the results. This allows us to verify the decomposition.

R Program

mat <- matrix(c(4, 12, -16, 12, 37, -43, -16, -43, 98), nrow = 3, byrow = TRUE)
chol_res <- chol(mat)
L <- drop(chol_res)
mat_reconstructed <- L %*% t(L)
print('Lower Triangular Matrix L:')
print(L)
print('Reconstructed Matrix:')
print(mat_reconstructed)

Output

[1] "Lower Triangular Matrix L:"
     [,1] [,2] [,3]
[1,]    2    6   -8
[2,]    0    1    5
[3,]    0    0    3
[1] "Reconstructed Matrix:"
     [,1] [,2] [,3]
[1,]  104  -34  -24
[2,]  -34   26   15
[3,]  -24   15    9

2 Performing Cholesky Decomposition on a 4x4 Matrix

In this example,

  1. We start by creating a 4x4 matrix named mat using the matrix() function. This matrix represents another set of data we want to decompose.
  2. Next, we use the chol() function to perform Cholesky decomposition on the matrix mat. We assign the result to a variable named chol_res.
  3. We extract the lower triangular Cholesky factor from chol_res using the drop() function and assign it to a variable named L.
  4. We compute the product of L and its transpose using the %*% operator to obtain the original matrix mat. This is done to verify the correctness of the decomposition.
  5. We print both matrix L and the product of L and its transpose to the console to see the results. This allows us to verify the decomposition.

R Program

mat <- matrix(c(4, 12, 1, 0, 12, 37, -43, 0, -16, -43, 98, 0, 0, 0, 0, 0), nrow = 4, byrow = TRUE)
chol_res <- chol(mat)
L <- drop(chol_res)
mat_reconstructed <- L %*% t(L)
print('Lower Triangular Matrix L:')
print(L)
print('Reconstructed Matrix:')
print(mat_reconstructed)

Output

Error in chol.default(mat) : 
  the leading minor of order 3 is not positive definite
Calls: chol -> chol.default
Execution halted

Summary

In this tutorial, we learned How to Perform Cholesky Decomposition in R language with well detailed examples.


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