numerical analysis with python

n Linear least squares problem Ax≅b, we seek a solution x such that the residual

‖b−Ax‖22

is minimized. A regularization term can be included to give preference to a solution x with a desired property. This regularization is minimizing,

‖b−Ax‖22+α‖x‖22.

This is equivalent to the augmented least squares problem [Aα‾‾√I]x≅[b0]

You are given a matrix A, vector b and scalar α. Calculate x such that it minimizes the residual for Ax≅b and x2 such that it minimizes the regularization condition. Also calculate the residual and norm for both.

code below

import numpy as np

import scipy.sparse

INPUT:

• A: numpy array of shape (m, n)
• b: numpy array of shape (m,)
• alpha: Python float

OUTPUT:

• x: numpy array of shape (n,) that minimizes the residual.
• x2: numpy array of shape (n,) that minimizes the L2-regularization condition.
• norm_x: Norm of x.
• norm_x2: Norm of x2.
• residual_x: Norm of the residual of x.
• residual_x2: Norm of the residual of x2.

n = np.random.randint(100, 200)

A = scipy.sparse.diags([-1,2,-1],[-1,0,1],shape=(n, n//4)).A

b = np.ones(n)

alpha = 1.5