2. Show that the approximation 2 1 h) - 5(x- h)- S(x + 2h) - f(x- 2h) (x) can be obtained by a Richardson extrapola...
Show that the approximation -L(mA) rean can be obtained by a Richardson extrapolation. |Hint: Consider p(h)
1. Let f(x) 3e cos z. Determine the analytical forms of f'(a) and f"(x). For h 0.01, a. Determine f'(1.3) using the Richardson extrapolation method b. Use the second derivative midpoint formula to approximate f"(1.3). Compare your results obtained in part (a) and (b) to the results of f(1.3) and f"(1.3) when using analytical formulas, respectively. 1. Let f(x) 3e cos z. Determine the analytical forms of f'(a) and f"(x). For h 0.01, a. Determine f'(1.3) using the Richardson extrapolation...
please clear handwriting (Richardson Extrapolation Applied to Differentiation). (a) Suppose that N(h) is an approximation to M for every h > 0 and that M = N(h) + Kih? + Kh? + K3h3 +... ), and N ) for some constants K1, K2, K3, .... Use the values N(h), N to produce an O(h) approximation to M. (b) Recall that f(xo+h)-f(20) df (20) dc f(x) Use the formula you constructed in part (a) to construct an imation to df 20)...
h 5. In Homework 8 we looked at the finite difference approximation of f'(x). f(x + 1) = f(x) + f'(z)h + }}" (-x)+2 +0(h2). Rearranging terms, and dividing by h leads to D(h) f(x+h)-f(x) != f'(x) +}s"(x)h + O(n?). Find constants a and 8 such that A(h) = aD(h/2) + BD(h) = f'(x) + (h) (similar to convergence acceleration in Chapter 2, we are using our knowledge of the be- haviour of the error to get a better approximation!)...
Suppose that T (A) is the trapezoidal approximation of J f(x) dx·It is known that for every h > 0, f(x)dx=X,(h)-K1 h 2-K2h4-K3 h6+ Use extrapolation (as in Romberg) to derive an integration formula (namely N2 (h)) of order 4 from the trapezoidal approximation N (h)- (f(a)+f(b)) The answer is a familiar formula, what is it warning: do not change the definition of h, use h to denote b-a throughout your solution. Suppose that T (A) is the trapezoidal approximation...
Derive the following numerical approximation to the second derivative of f(x) using Taylor's series. Show all of your steps and derive also the order of accuracy of this approximation in terms of h. - f(x + 2h) + 16f(x + h) – 30f(x) + 16 f(x – h) – f(x – 2h) 12h2 1 (C)
numerical analysis 1 f (x + 2h) f"(x) = 2f (x + h) + f (x) 12 Forward difference II f(x - 2h) f"() = 25(x - hr) 12 method Backward method difference f'(x) = -f(+ 2) + 4(x +h)- 36) 2h Forward difference method Which ones are correct? a) I, II b) Only 11 a d) Only 1 e> I, II, III
Please help me solve this. thanks 5. Using Taylor series, derive the error term for the approximation f' (x) ~ -3 f(x) + 4 f(x + b)-f(x + 2h)]. 2h 5. Using Taylor series, derive the error term for the approximation f' (x) ~ -3 f(x) + 4 f(x + b)-f(x + 2h)]. 2h
5. (10 points) Derive the following formula for f(x) f(4(x+)-3f(x)-f(x+2h)) and show that error is OCh 5. (10 points) Derive the following formula for f(x) f(4(x+)-3f(x)-f(x+2h)) and show that error is OCh
Added the formulas, thank you! Approximating derivatives f(z +h) - f(z) f(x)-f( -h) f(x + h) - f(x - h) Forward difference Backward difference Centered difference for 1st derivative s(a) (3) 2h t)-2e-bCentered diference for 2nd derivative (4) 2 2. Write a short program that uses formulas (1), (3) and (4) to approximate f(1) and f"(1) for f(x)e with h 1, 2-1, 2-2,.., 2-60. Format your output in columns as follows: h (6+f)() error (öf(1 error f error Indicate the...