CHOLESKY

Overview

The CHOLESKY function computes the Cholesky decomposition of a Hermitian, positive-definite matrix. The Cholesky decomposition factorizes a matrix $A$ into the product $A = LL^*$ (if lower-triangular) or $A = U^*U$ (if upper-triangular), where $L$ is a lower-triangular matrix and $U$ is an upper-triangular matrix. This decomposition is widely used in numerical linear algebra for solving systems of equations, Monte Carlo simulations, and optimization problems. For more details, see the scipy.linalg.cholesky documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=CHOLESKY(matrix, [lower])

matrix (2D list, required): Hermitian (symmetric if real) positive-definite matrix to decompose. Must be square (n x n).
lower (bool, optional, default=FALSE): If TRUE, returns the lower-triangular Cholesky factor $L$ such that $A = LL^*$ . If FALSE, returns the upper-triangular factor $U$ such that $A = U^*U$ .

The function returns a 2D array (matrix) containing the Cholesky factor, or a 2D array of strings with an error message if the input is invalid.

Examples

Example 1: Lower-triangular Cholesky of a real symmetric matrix

In Excel:


=CHOLESKY({4,12;-2,37}, TRUE)

Expected output:


2.0	0.0
-1.0	6.0

Example 2: Upper-triangular Cholesky of a real symmetric matrix

In Excel:


=CHOLESKY({4,12;-2,37}, FALSE)

Expected output:


2.0	6.0
0.0	1.0

Example 3: Lower-triangular Cholesky of a 3x3 matrix

In Excel:


=CHOLESKY({25,15,-5;15,18,0;-5,0,11}, TRUE)

Expected output:


5.0	0.0	0.0
3.0	3.0	0.0
-1.0	1.0	3.0

Example 4: Lower-triangular Cholesky of a positive-definite matrix with negative off-diagonal

In Excel:


=CHOLESKY({9,-6,3;-6,8,-2;3,-2,3}, TRUE)

Expected output:


3.0	0.0	0.0
-2.0	2.0	0.0
1.0	0.0	1.414214

Python Code


import numpy as np
from scipy.linalg import cholesky as scipy_cholesky
 
def cholesky(matrix, lower=False):
    """
    Compute the Cholesky decomposition of a Hermitian, positive-definite matrix.
 
    Args:
        matrix: 2D list (square), Hermitian (symmetric if real) positive-definite matrix.
        lower: bool, optional (default: False). If True, return lower-triangular factor.
 
    Returns:
        2D list (matrix) of Cholesky factor, or 2D list of str with error message if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input
    if not isinstance(matrix, list) or not matrix or not isinstance(matrix[0], list):
        return [["Error: Input must be a 2D list (matrix)"]]
    n = len(matrix)
    if any(len(row) != n for row in matrix):
        return [["Error: Matrix must be square (n x n)"]]
    try:
        arr = np.array(matrix, dtype=np.complex128)
    except Exception:
        return [["Error: Matrix must contain only numbers"]]
    try:
        result = scipy_cholesky(arr, lower=bool(lower))
    except Exception as e:
        return [[f"Error: {str(e)}"]]
    # Convert result to 2D list of floats (rounded for Excel)
    result_list = result.tolist()
    for i in range(len(result_list)):
        for j in range(len(result_list[i])):
            if abs(result_list[i][j].imag) < 1e-12:
                result_list[i][j] = round(result_list[i][j].real, 6)
            else:
                result_list[i][j] = str(result_list[i][j])
    return result_list

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CHOLESKY

Overview

Usage

Examples

Python Code

Live Notebook

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