1- Write a C++ program to implement the Example on Chapter 6 - Lecture 1 Slide 5. You need to print a table like Table 6.2 and to find optimum step size from the table (minimum error). 2- Complete your program by implementing the Example on Chapter 6 - Lecture 1 Slide 8. And compare the result with optimum step size found on part 1 above. Central-difference formula - Example Let f(x) = cos(x). Use central-difference formula of order 0(ha) with step size h = 0.1 , 0.01 , 0.001 , 0.0001 . To approximate f'(0.8) Note that the exact value of f'(0.8) = sin(0.8) = -0.717356090899 . If h = 0.01 Table 6.2 Numerical Differentiation Using For Step size f'(0.8) = f(0.81)-f(0.79) 2x0.01 Approximation by formula (3) -0.716161095 -0.717344150 -0.717356000 -0.717360000 0.1 0.0! 0.001 0.0001 Error using formula (3) -0.001194996 -0.00001941 -0.000000091 -0.000003909 = -0.717344150 11 Optimum step-size . f(xo h) = y-1 +e-1 f(xo + h) = y + ei y-1,9 approximate values for f(xo h) and f(xo + h) e-1,61 round-off error O . Central-difference formula of order (h?) f'(xo) = 15Y-1 + E(f,h) 2h Ef,h) total error = E(f, h)round-off + E(f,h), truncated Numerical Analysis - prepared by: Eng Shatha Al-Hasan Optimum step-size (cont'd) Ef,h) = E(f, h)round-off + Ef,h) total error truncated ha f(3)(c) e1-4-1 + 2h 3! Assume: \e_11 se,leil SE, M = max{]f(3)(x)]} 2 Mh2 Then, error bound: [E(f, hs 2h 6 To find optimum value of h: a(h) = 0 h = Numerical Analysis - prepared by: Eng Shatha Al-Hasan Optimum step-size - Example Let f(x) = cos(x) and = 0.5 x 10-9. Find optimum step size for Central-difference formula of order 0(ha) |F3)(x) sin(x)= 1 M = 1 (3x0.5X10-9 .h= = = 0.001144714 M 1 Note that from previous example, optimum step size h = 0.001