Matlab question, please use matlab to solve the following questions and provide
ID: 3210315 • Letter: M
Question
Matlab question, please use matlab to solve the following questions and provide the codes.
Three formulae for approximating f' are h) ( h) 2h CD,(h) = 12h Consider f(x) je2z at Z 1. -2, . . . , 10-16} to approximate f,(z). a) Use each formula and h - 1, 10,10 b) Compute the absolute error for each formula and value of h. c) Does the numerical rate of convergence support the predicted rate of convergence in each case? (explain your answer) d) When do numerical errors become evident for each formula?Explanation / Answer
clc;
clear all;
format long
f=@(x)(1/2)*exp(2*x);
df=@(x) exp(2*x);
N=16;
x=1;
for i=1:N+1
h(i)=10^(-(i-1));
end
for i=1:N+1
FD(i)=(f(x+h(i))-f(x))/h(i);
CD2(i)=(f(x+h(i))-f(x-h(i)))/(2*h(i));
CD4(i)=(-f(x+2*h(i))+8*f(x+h(i))-8*f(x-h(i))+f(x-2*h(i)))/(12*h(i));
err_FD(i)=abs(df(x)-FD(i));
err_CD2(i)=abs(df(x)-CD2(i));
err_CD4(i)=abs(df(x)-CD4(i));
end
disp(' h Forward(DD) error ')
disp('__________________________________________________________________________________')
for i=1:N+1
fprintf('%e %15e %15e ',h(i),FD(i),err_FD(i))
end
disp('_________________________________________________________________________________')
disp(' h CD2 CD2_err ')
disp('________________________________________________________________________')
for i=1:N+1
fprintf('%e %15e %15e ',h(i),CD2(i),err_CD2(i))
end
disp('_________________________________________________________________________________')
disp(' h CD4 CD4_error ')
disp('_________________________________________________________________________________')
for i=1:N+1
fprintf('%e %15e %15e ',h(i),CD4(i),err_CD4(i))
end
%%% Solution
h Forward(DD) error
__________________________________________________________________________________
1.000000e+00 2.360455e+01 1.621549e+01
1.000000e-01 8.179787e+00 7.907309e-01
1.000000e-02 7.463442e+00 7.438564e-02
1.000000e-03 7.396450e+00 7.393985e-03
1.000000e-04 7.389795e+00 7.389549e-04
1.000000e-05 7.389130e+00 7.389108e-05
1.000000e-06 7.389063e+00 7.388365e-06
1.000000e-07 7.389057e+00 7.430137e-07
1.000000e-08 7.389056e+00 2.358918e-08
1.000000e-09 7.389056e+00 3.788605e-07
1.000000e-10 7.389058e+00 2.155217e-06
1.000000e-11 7.389023e+00 3.337192e-05
1.000000e-12 7.389644e+00 5.883530e-04
1.000000e-13 7.380763e+00 8.293431e-03
1.000000e-14 7.371881e+00 1.717522e-02
1.000000e-15 7.993606e+00 6.045497e-01
1.000000e-16 0.000000e+00 7.389056e+00
_________________________________________________________________________________
h CD2 CD2_err
________________________________________________________________________
1.000000e+00 1.339954e+01 6.010481e+00
1.000000e-01 7.438415e+00 4.935899e-02
1.000000e-02 7.389549e+00 4.926136e-04
1.000000e-03 7.389061e+00 4.926038e-06
1.000000e-04 7.389056e+00 4.925976e-08
1.000000e-05 7.389056e+00 4.965441e-10
1.000000e-06 7.389056e+00 1.695906e-10
1.000000e-07 7.389056e+00 1.384722e-09
1.000000e-08 7.389056e+00 1.384722e-09
1.000000e-09 7.389056e+00 1.568159e-07
1.000000e-10 7.389058e+00 2.155217e-06
1.000000e-11 7.389045e+00 1.116746e-05
1.000000e-12 7.389200e+00 1.442638e-04
1.000000e-13 7.387424e+00 1.632093e-03
1.000000e-14 7.371881e+00 1.717522e-02
1.000000e-15 7.771561e+00 3.825051e-01
1.000000e-16 4.440892e+00 2.948164e+00
_________________________________________________________________________________
h CD4 CD4_error
_________________________________________________________________________________
1.000000e+00 1.062156e+00 6.326900e+00
1.000000e-01 7.388660e+00 3.959640e-04
1.000000e-02 7.389056e+00 3.941016e-08
1.000000e-03 7.389056e+00 4.648726e-12
1.000000e-04 7.389056e+00 4.671818e-13
1.000000e-05 7.389056e+00 3.056222e-12
1.000000e-06 7.389056e+00 2.436060e-10
1.000000e-07 7.389056e+00 3.605168e-09
1.000000e-08 7.389056e+00 1.341825e-08
1.000000e-09 7.389056e+00 3.418531e-07
1.000000e-10 7.389058e+00 1.415069e-06
1.000000e-11 7.389052e+00 3.765972e-06
1.000000e-12 7.389385e+00 3.293009e-04
1.000000e-13 7.386314e+00 2.742316e-03
1.000000e-14 7.375582e+00 1.347447e-02
1.000000e-15 7.845576e+00 4.565199e-01
1.000000e-16 3.700743e+00 3.688313e+00
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