NumPy: Get the lower-triangular L in the Cholesky decomposition of a given array
Write a NumPy program to get the lower-triangular L in the Cholesky decomposition of a given array.
Sample Solution:
Python Code :
# Importing the NumPy library
import numpy as np
# Create a 3x3 NumPy array 'a' with specified values and data type as int32
a = np.array([[4, 12, -16], [12, 37, -53], [-16, -53, 98]], dtype=np.int32)
# Display the original array 'a'
print("Original array:")
print(a)
# Compute the Cholesky decomposition of the array 'a' using np.linalg.cholesky()
# Cholesky decomposition calculates the lower-triangular matrix 'L'
L = np.linalg.cholesky(a)
# Display the lower-triangular matrix 'L' obtained from Cholesky decomposition
print("Lower-triangular L in the Cholesky decomposition of the said array:")
print(L)
Sample Output:
Original array: [[ 4 12 -16] [ 12 37 -53] [-16 -53 98]] Lower-triangular L in the Cholesky decomposition of the said array: [[ 2. 0. 0.] [ 6. 1. 0.] [-8. -5. 3.]]
Explanation:
In the above code –
a = np.array([[4, 12, -16], [12, 37, -53], [-16, -53, 98]], dtype=np.int32): This line creates a 3x3 NumPy array a with the specified elements and int32 data type.
L = np.linalg.cholesky(a): This line computes the Cholesky decomposition of the matrix a. The Cholesky decomposition is a technique used to decompose a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. In this case, since a is a real, symmetric, and positive-definite matrix, the Cholesky decomposition can be applied. The result is a lower triangular matrix L such that a = L * L.T, where L.T is the transpose of L.
Python-Numpy Code Editor: