Calculating the k smallest positive eigenvalues of a matrix with arpack

When working with matrices in Julia, it is often necessary to calculate the eigenvalues. In some cases, we may only be interested in the k smallest positive eigenvalues of a matrix. In this article, we will explore different ways to solve this problem using the arpack package in Julia.

Option 1: Using the eigs function

The arpack package provides the eigs function, which allows us to calculate the eigenvalues of a matrix. To find the k smallest positive eigenvalues, we can use the following code:

using Arpack

function calculate_smallest_eigenvalues(matrix, k)
    eigenvalues, _ = eigs(matrix, k=k, which=:SR)
    smallest_eigenvalues = eigenvalues[eigenvalues .> 0]
    return smallest_eigenvalues
end

matrix = [1 2 3; 4 5 6; 7 8 9]
k = 2
smallest_eigenvalues = calculate_smallest_eigenvalues(matrix, k)
println(smallest_eigenvalues)

In this code, we define a function calculate_smallest_eigenvalues that takes a matrix and the number of smallest eigenvalues to calculate. We use the eigs function to calculate the eigenvalues, specifying k as the number of eigenvalues to compute and which=:SR to indicate that we want the smallest real eigenvalues. We then filter out the positive eigenvalues and return them.

Option 2: Using the eigvals function

Another option is to use the eigvals function from the LinearAlgebra package. This function calculates all the eigenvalues of a matrix, so we need to filter out the k smallest positive eigenvalues ourselves. Here is an example:

using LinearAlgebra

function calculate_smallest_eigenvalues(matrix, k)
    eigenvalues = eigvals(matrix)
    smallest_eigenvalues = sort(eigenvalues[eigenvalues .> 0])[1:k]
    return smallest_eigenvalues
end

matrix = [1 2 3; 4 5 6; 7 8 9]
k = 2
smallest_eigenvalues = calculate_smallest_eigenvalues(matrix, k)
println(smallest_eigenvalues)

In this code, we use the eigvals function to calculate all the eigenvalues of the matrix. We then filter out the positive eigenvalues, sort them in ascending order, and select the k smallest ones.

Option 3: Using the eigen function

Finally, we can also use the eigen function from the LinearAlgebra package to calculate the eigenvalues and eigenvectors of a matrix. Here is an example:

using LinearAlgebra

function calculate_smallest_eigenvalues(matrix, k)
    eigenvalues, _ = eigen(matrix)
    smallest_eigenvalues = sort(eigenvalues[eigenvalues .> 0])[1:k]
    return smallest_eigenvalues
end

matrix = [1 2 3; 4 5 6; 7 8 9]
k = 2
smallest_eigenvalues = calculate_smallest_eigenvalues(matrix, k)
println(smallest_eigenvalues)

In this code, we use the eigen function to calculate the eigenvalues and eigenvectors of the matrix. We then filter out the positive eigenvalues, sort them in ascending order, and select the k smallest ones.

After exploring these three options, it is clear that the first option using the eigs function from the arpack package is the most straightforward and efficient way to calculate the k smallest positive eigenvalues of a matrix. It provides a concise and optimized solution for this specific problem.