that uie buttu ercise 2.6.5 Use MATLAB to explore the extent of cancellation whe

Question

that uie buttu ercise 2.6.5 Use MATLAB to explore the extent of cancellation when Gaussian elimination is performed on larger Hilbert matrices. (a) In MATLAB, type A = hilb (7) to get H7. To get the LU decomposition with partial pivoting, type [L,U] = lu (A) . Notice that the matrix L is not itself unit lower triangular, but it can be made unit lower triangular by permuting the rows. This is because MATLAB's lu command incorporates the row interchanges in the L matrix. Our real object of interest is the matrix U, which is the triangular matrix resulting from Gaussian elimination. Notice that the further down in U you go, the smaller the numbers become. The ones at the bottom appear to be zero. To get a more accurate picture, type format long and redisplay U. (b) Generate H12 and its LU decomposition. Observe the U matrix using format long. It suffices to print out the last column of U, as this tells the story.

Explanation / Answer

clc;
clear all;
close all;
format short
%%%%% a)
A=hilb(7);
[l1 u1]=lu(A);
fprintf(' U matrix : ');
disp(u1);
format long
fprintf('after using long format U matrix : ');
disp(u1);

%%%% b)
A=hilb(12);
[l2 u2]=lu(A);
fprintf('last column of U matrix : ');
disp(u2(:,end));

OUTPUT:

U matrix :
1.0000 0.5000 0.3333 0.2500 0.2000 0.1667 0.1429
0 0.0833 0.0889 0.0833 0.0762 0.0694 0.0635
0 0 0.0063 0.0107 0.0134 0.0149 0.0157
0 0 0 0.0010 0.0022 0.0031 0.0038
0 0 0 0 -0.0000 -0.0001 -0.0001
0 0 0 0 0 0.0000 0.0000
0 0 0 0 0 0 -0.0000

after using long format U matrix :
Columns 1 through 6

1.000000000000000 0.500000000000000 0.333333333333333 0.250000000000000 0.200000000000000 0.166666666666667
0 0.083333333333333 0.088888888888889 0.083333333333333 0.076190476190476 0.069444444444444
0 0 0.006349206349206 0.010714285714286 0.013358070500928 0.014880952380952
0 0 0 0.001041666666667 0.002164502164502 0.003100198412698
0 0 0 0 -0.000032982890126 -0.000085034013605
0 0 0 0 0 0.000000885770975
0 0 0 0 0 0

Column 7

0.142857142857143
0.063492063492063
0.015698587127159
0.003815628815629
-0.000142714428429
0.000002675895533
-0.000000030032498

last column of U matrix :
0.083333333333333
0.043650793650794
0.016751949317739
0.005964430306536
0.001003095780172
0.000121017426510
-0.000004610187676
-0.000000362689590
-0.000000009030553
-0.000000000150119
0.000000000001834
-0.000000000000005

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