Checking for diagonal dominate matrix in julia

When working with matrices in Julia, it is often necessary to check if a matrix is diagonally dominant. A matrix is said to be diagonally dominant if the absolute value of each diagonal element is greater than or equal to the sum of the absolute values of the other elements in the same row.

Option 1: Using the isdiagdom() function

Julia provides a built-in function called isdiagdom() that can be used to check if a matrix is diagonally dominant. This function returns a boolean value indicating whether the matrix is diagonally dominant or not.

# Define a matrix
A = [2 1 1; 1 3 2; 1 2 4]

# Check if the matrix is diagonally dominant
is_diag_dom = isdiagdom(A)

# Print the result
println("Is the matrix diagonally dominant? ", is_diag_dom)

This code defines a matrix A and uses the isdiagdom() function to check if it is diagonally dominant. The result is then printed to the console.

Option 2: Manual calculation

If you prefer to manually calculate the diagonal dominance of a matrix, you can use a loop to iterate over each row and compare the diagonal element with the sum of the absolute values of the other elements in the same row.

# Define a matrix
A = [2 1 1; 1 3 2; 1 2 4]

# Initialize a variable to track diagonal dominance
is_diag_dom = true

# Iterate over each row
for i in 1:size(A, 1)
    # Calculate the sum of absolute values of other elements in the row
    row_sum = sum(abs.(A[i, [1:i-1, i+1:end]]))
    
    # Check if the diagonal element is less than the row sum
    if abs(A[i, i]) < row_sum
        is_diag_dom = false
        break
    end
end

# Print the result
println("Is the matrix diagonally dominant? ", is_diag_dom)

This code manually calculates the diagonal dominance of a matrix A by iterating over each row and comparing the diagonal element with the sum of the absolute values of the other elements in the same row. The result is then printed to the console.

Option 3: Using matrix properties

Another way to check for diagonal dominance is by utilizing the properties of a diagonally dominant matrix. In a diagonally dominant matrix, the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in the same row. We can use this property to check for diagonal dominance.

# Define a matrix
A = [2 1 1; 1 3 2; 1 2 4]

# Calculate the sum of absolute values of other elements in each row
row_sums = sum(abs.(A), dims=2) .- abs.(diag(A))

# Check if all diagonal elements are greater than row sums
is_diag_dom = all(diag(A) .> row_sums)

# Print the result
println("Is the matrix diagonally dominant? ", is_diag_dom)

This code calculates the sum of absolute values of other elements in each row of matrix A using the sum() function and the abs() function. It then subtracts the absolute values of the diagonal elements using the diag() function. Finally, it checks if all diagonal elements are greater than the row sums using the all() function. The result is printed to the console.

Among the three options, using the isdiagdom() function (Option 1) is the most straightforward and concise way to check for diagonal dominance in Julia. It provides a built-in function specifically designed for this purpose, making the code more readable and easier to understand. Therefore, Option 1 is the recommended approach.