Please provide a MATLAB code that accurately answers this question. Let H denote the n x n Hilbert matrix, whose (i, j) entry is 1/(i + j - 1). UseGauss Elimination to solve Hx = b, where b is the vector of all ones, for n =1000, 2000,, 10000. Record the time cost as t(n). Plot t(n), using proper scaleof the axes to verify O(n³).

Answered: Please provide a MATLAB code that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please provide a MATLAB code that accurately answers this question.

Let H denote the n x n Hilbert matrix, whose (i, j) entry is 1/(i + j - 1). Use
Gauss Elimination to solve Hx = b, where b is the vector of all ones, for n =
1000, 2000,, 10000. Record the time cost as t(n). Plot t(n), using proper scale
of the axes to verify O(n³).

Expert Solution

Step 1: MATLAB CODE

1function t = gaussEliminationHilbert(n)
2    % Generate the Hilbert matrix H
3    H = zeros(n, n);
4    for i = 1:n
5        for j = 1:n
6            H(i, j) = 1 / (i + j - 1);
7        end
8    end
9
10    % Generate the vector b (all ones)
11    b = ones(n, 1);
12
13    % Measure the time taken for Gaussian elimination
14    tic;
15    x = H \ b;
16    t = toc;
17end
18
19% Define values of n
20n_values = [1000, 2000, 10000];
21
22% Initialize an array to store time costs
23t_values = zeros(size(n_values));
24
25% Perform Gaussian elimination for each n
26for i = 1:length(n_values)
27    n = n_values(i);
28    t_values(i) = gaussEliminationHilbert(n);
29end
30
31% Plot the results
32figure;
33loglog(n_values, t_values, '-o');
34xlabel('n (Matrix Size)');
35ylabel('Time (seconds)');
36title('Time Complexity of Gaussian Elimination on Hilbert Matrix');
37grid on;
38