Question

#7. [Extra Credit] is calculus wrong?! Consider f(x) = ex (a) Calculate the derivative of fx) atx 0 using O(h) finite difference (forward and backward) and O(h2) centered finite difference. Vary h in the following manner: 1, 101,102... 1015. (Write a MATLAB script for this purpose and call it pset5_prob7) (b) Modify your script to plot (log-log) the the true percent error in all three cases as a function of h in one plot. (c) In calculus we learned that as h approaches zero, the finite difference should approach the true value of the derivative. Do your results contradict this? Explain.

Answer 1

SMatlab code for forward backward and central difference method clear all close all %function for which derivative have to find f-e (x) exp (x)i texact solution for f (x) exp (x) at x0 ext_val-exp (0) %x value at which derivative have to find x0-0 %loop for forward backward and central difference fprint f ( . \th, \terr-frd, \tbkd-err , \tcnt err\n' ) for i-l:16 formulae tall step size h-10-(i-1); %forward diff fd frwd-(f (x0+h)-f (x0))/h; %backward diff bk_frwd-(f (x0)-f (x0-h))/h; %central diff cn_frwd-(f (x0+h/2)-f (x0-h/2) )/h; %true percent error forward diff er_fd(i)-(abs ( (ext_val-fd frwd)/ext_val))*100; true percent error backward diff er-bk ( İ,-(abs ( (ext-val-bk_frwd ) /ext-val)) * 100; true percent error central diff ercn(i)- (abs((ext_val-cn frwd)/ext_val))*100; %step size fprint f ('\t3.2e, 8.2e, \t %.2e, \t %.2e. n',h,erfd(i),er bk(i), er_cn(i)) ELoglog plot of true error vs h loglog (hh, er_Ed, 'linewidth',2) hold on loglog (hh, erbk, 'linewidth',2) loglog (hh, er_cn, 'linewidth',2) xlabel( 'step size (h)') ylabel( True percent error') title( 'True percent error vs. step size plot') legend Forward difference', 'Backward difference', 'Central difference)

fprint f ("AnUpto h=10"-5 for central diff and h=10^-8 for forward and backward diff error decreases with h.\n') fprintf 'Beyond that by further decreasing h error tends to increases which contradict our assumpsion n') 8888888888움움8888888888 End of Code 움8888888888888888888888움 h, err frd, bkd err, cnt err I.00e+00, 7.18et01, 1.00e-01, 5.17e+00, I.00e-02, 5.02e-01, 1.00e-03, 5.00e-02, I.00e-04, 5.00e-03, 1.00e-05, 5.00e-04 I.00e-06, 5.00e-05, 1.00e-07, 4.94e-06, I.00e-08, 6.0 1.00e-09, 8.27e-06, 2.83e-06, 8.27e-06. I.00e-10, 8.27e-06, 1.00e-11, 8.27e-06, 8.27e-06, 8.27e-06. I.00e-12, 8.89e-03, 1.00e-13, 7.99e-02, 3.11e-02, 7.99e-02. I.00e-14, 7.99e-02, 1.00e-15, 1.10e+01, 7.99e-02, 7.99e-02. 3.68e+01, 4.22e+ 4.84e+00 4.98e-01, 4.17e-04. 5.0 5.00e-03, 4.17e 00. ,4.17e-02. 0e-02, 4.17e-06 -08 4,5.00e-04, 6.55e-10. 2.88e-09 5e-06, 5.26e-08 5.00e-05, ,5.0 8e-07, 6.08e-07, 6.08e-07. 6, 8.27e-06, 8.27e-06. 2.21e-03, 8.89e-03. 7.99e-02, 1.03e+00. Upto h=10^-5 for central diff and h=10^-8 for forward and backward diff error decreases with h Beyond that by further decreasing h error tends to increases which contradict our assumpsion.

True percent err or vs. step size plot -Forward differenc Central difference 100 102 10 10 10 10-10 15 10 10 10° 10 step size (h) Published with MATLAB R2018a

%%Matlab code for forward backward and central difference method
clear all
close all

%function for which derivative have to find
f=@(x) exp(x);

%exact solution for f'(x)=exp(x) at x=0
ext_val=exp(0);

%x value at which derivative have to find
x0=0;

%loop for forward backward and central difference formulae
fprintf('\th,\terr_frd,\tbkd_err,\tcnt_err\n')
for i=1:16
  
    %all step size
    h=10^-(i-1);
    %forward diff
    fd_frwd=(f(x0+h)-f(x0))/h;
    %backward diff
    bk_frwd=(f(x0)-f(x0-h))/h;
    %central diff
    cn_frwd=(f(x0+h/2)-f(x0-h/2))/h;
  
    %true percent error forward diff
    er_fd(i)=(abs((ext_val-fd_frwd)/ext_val))*100;
  
    %true percent error backward diff
    er_bk(i)=(abs((ext_val-bk_frwd)/ext_val))*100;
  
    %true percent error central diff
    er_cn(i)=(abs((ext_val-cn_frwd)/ext_val))*100;
    %step size
    hh(i)=h;
  
    fprintf('\t%.2e, %.2e, \t %.2e, \t %.2e.\n',h,er_fd(i),er_bk(i),er_cn(i))
  
end
  
%Loglog plot of true error vs h
loglog(hh,er_fd,'linewidth',2)
hold on
loglog(hh,er_bk,'linewidth',2)
loglog(hh,er_cn,'linewidth',2)
xlabel('step size (h)')
ylabel('True percent error')
title('True percent error vs. step size plot')
legend('Forward difference','Backward difference','Central difference')

fprintf('\nUpto h=10^-5 for central diff and h=10^-8 for forward and backward diff error decreases with h.\n')
fprintf('Beyond that by further decreasing h error tends to increases which contradict our assumpsion .\n ')

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