HW 4.15
Nasser Abbasi
OUTPUT
This shows that N=17, the abs value of maximum element in D is close to 1. The plot of condition number vs matrix size follows the same pattern as how the maximum element size grows vs matrix size as well. We see that the condition number becomes very large when the limit of double accuracy in matlab is reached. (around 16 decimal points).
» nma_HW_4_15
Program to solve problem 4.15
by Nasser Abbasi
Enter the maximum number of points (N) to try for:16
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
1.158588e-016.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
1.764763e-018.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
2.483244e-020.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
3.246740e-022.
>
In d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
3.964100e-024.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
»
» nma_HW_4_15
Program to solve problem 4.15
by Nasser Abbasi
Enter the maximum number of points (N) to try for:20
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
1.158588e-016.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
1.764763e-018.
>
In d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
2.483244e-020.
>
In d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
3.246740e-022.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
3.964100e-024.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
4.539855e-026.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
4.892271e-028.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
5.228577e-030.
>
In d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
Warning:
Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND =
5.996228e-032.
>
In
d:\nabbasi\data\nabbasi_web_page\academic\my_courses\phys_240\HW10\nma_HW_4_15.m
at line 17
»
CODE
function nma_HW_4_15()
%
%Program to solve problem 4.15
%by Nasser Abbasi
clear all; help nma_HW_4_15;
data = struct('matrixSize',0,...
'DmaxElement',0,...
'VconditionNumber',0);
N= input('Enter the
maximum number of points (N) to try for:');
for(i=2:N)
V =
generate_V(i);
D = (
V*inv(V) ) - eye(i);
data.matrixSize(i) = i;
data.DmaxElement(i) = abs(max(max(D)));
data.VconditionNumber(i) =
cond(V);
end
figure;
plot(data.matrixSize, data.DmaxElement,'-');
title('Vandermonde
matrix size vs maximum element size in D');
xlabel('Matrix size N');
ylabel('abs value of
maximum element in V*inv(V) - I');
set(gca,'Xtick',[1:N]);
grid on;
figure;
semilogy(data.matrixSize, data.DmaxElement,'-');
set(gca,'Xtick',[1:N]);
title('Vandermonde
matrix size vs maximum element size in D');
xlabel('Matrix size N');
ylabel('Log of abs
value of maximum element in V*inv(V) - I');
grid on;
figure;
plot(data.matrixSize, data.VconditionNumber,'-');
title('condition
number of Vandermonde matrix vs matrix size');
xlabel('matrix size');
ylabel('condition
number');
set(gca,'Xtick',[1:N]);
grid on;
figure;
semilogy(data.matrixSize, data.VconditionNumber,'-');
title('condition number
of Vandermonde matrix vs matrix size');
xlabel('matrix size');
ylabel('Log of
condition number');
set(gca,'Xtick',[1:N]);
grid on;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function to generate Vendermond
matrix gievn N.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function V=generate_V(N)
for(i=1:N)
for(j=1:N)
V(i,j) = i^(j-1);
end
end