def fixed_point_iteration():

    x = 1.0

    for i in range(2):  # (1) 

        x = (x + 3)**0.5  # (2) 

    return x  # (3)

Using the Neumann series, how can the true inverse A⁻¹ be expressed in terms of B and E?

For f(x)=x3−x−2, which initial interval is valid for the Bisection method?

What will the following code snippet output for the LU factorization of matrix A?

import numpy as np

A = np.array([[2, 1], [4, 3]])

n = A.shape[0]

X = np.eye(n)

Y = A.copy()

for i in range(n):

    for j in range(i + 1, n):

        X[j, i] = Y[j, i] / Y[i, i]

        Y[j] = Y[j] - X[j, i] * Y[i]

print(X)

print(Y)

The following code implements Gaussian Elimination to solve Ax=b. What is the error?

import numpy as np

def gaussian_elimination(A, b):

    n = len(b)

    A = A.astype(float)

    b = b.astype(float)

    for k in range(n):

        # Partial pivoting

        max_row = np.argmax(np.abs(A[k:, k])) + k

        A[[k, max_row]] = A[[max_row, k]]

        b[[k, max_row]] = b[[max_row, k]]

        # Elimination

        for i in range(k+1, n):

            factor = A[i, k] / A[k, k]

            A[i, k:] -= factor * A[k, k:]

            b[i] -= factor * b[k]

    # Back substitution

    x = np.zeros(n)

    for k in range(n-1, -1, -1):

        x[k] = (b[k] - np.dot(A[k, k+1:], x[k+1:])) / A[k, k]

    return x

# Test case

A = np.array([[2, 1, -1], [4, 1, 0], [1, -1, 1]])

b = np.array([1, -2, 3])

print(gaussian_elimination(A, b))

What is the Newton-Raphson iterative formula for finding roots of f(x) = 0?

What is the main advantage of the Secant method over Newton-Raphson method?

What does this code output for x_data = [0, 1, 2], y_data = [1, 3, 7], x = 1.5?

def newton_backward(x_data, y_data, x):

    h = x_data[1] - x_data[0]

    s = (x - x_data[-1]) / h

    result = y_data[-1] + s*(y_data[-1] - y_data[-2])

    return result

What is the primary goal of Jacobi’s method for the eigenvalue problem?