Page 255 - Applied Numerical Methods Using MATLAB

244 NUMERICAL DIFFERENTIATION/ INTEGRATION
PROBLEMS

5.1 Numerical Differentiation of Basic Functions
If we want to find the derivative of a polynomial/trigonometric/exponential
function, it would be more convenient and accurate to use an analytical
computation (by hand) than to use a numerical computation (by computer).
But, in order to test the accuracy of the numerical derivative formulas,
consider the three basic functions as

3
f 1 (x) = x − 2x, f 2 (x) = sin x, f 3 (x) = e x (P5.1.1)
(a) To find the first derivatives of these functions by using the formulas
(5.1.8) and (5.1.9) listed in Table 5.3 (Section 5.3), modify the program
“nm5p01.m”, which uses the MATLAB routine “difapx()” (Section
5.3) for generating the coefficients of the numerical derivative formulas.
Fill in the following table with the error results obtained from running
the program.

f 1 − f −1 −f 2 + 8f 1 − 8f −1 + f −2
First Derivatives h
2h 12h
3
(x − 2x) | x=1 0.1 1.0000e-02
= 1.00000000
0.01 9.1038e-15

(sin x) | x=π/3 0.1 8.3292e-04
= 0.50000000
0.01 8.3333e-06

x
(e ) | x=0 0.1 3.3373e-06
= 1.00000000
0.01 1.6667e-05

%nm5p01
f = inline(’x.*(x.*x-2)’, ’x’);
n = [1 -1]; x0 = 1; h = 0.1; DT = 1;
c = difapx(1,n); i = 1:length(c);
num = c*feval(f,x0 + (n(1)+1- i)*h)’; drv = num/h;
fprintf(’with h = %6.4f, %12.6f %12.4e\n’, h,drv,drv - DT);

(b) Likewise in (a), modify the program “nm5p01.m”insuchawaythat
the formulas (5.3.1) and (5.3.2) in Table 5.3 are generated and used to
find the second numerical derivatives. Fill in the following table with
the error results obtained from running the program.