Let fe C[a, b] and ro € [a, b]. In this problem, we derive a difference formula to approxi-mate f"(ro).(a) Let f(x) = P3(x) + R(x), where P3(x) is the fourth Taylor Polynomial of f about zo.Find out the expressions of P3(1) and R(æ).(b) Assume that for small h > 0, zo ±h e [a, b]. Compute f(xo + h) + f(zo – h) in termsof the Taylor Polynomial P3 obtained in part (a).(c) Using the result in part (b), derive a difference formula to approximate f"(ro). Findout the leading order of h in the truncation error.(d) Assume that M := max f() (z) is given. Consider the effect of round-off error in thecomputation. Assume that the round-off error in each term can be bounded by a givennumber e > 0. Show that the total error of approximating f"(x0) using the derivedPe[a,b]formula satisfies:Mh?|Round off plus Truncation errors| <12(e) Find the optimal step size h, (in terms of M and e) such that this minimizes the errorbound obtained in part (d).

Answered: Image /qna-images/answer/f5008158-f37a-414d-bb98-0f585b4c6818.jpg

Let fe C[a, b] and ro E [a, b]. In this problem, we derive a difference formula to approxi- mate f"(ro). (a) Let f(z) = P3(x) + R(x), where P3(x) is the fourth Taylor Polynomial of f about zo. Find out the expressions of P3(r) and R(x). (b) Assume that for small h > 0, xo + h e [a, b]. Compute f(xo + h) + f(xo – h) in terms of the Taylor Polynomial P3 obtained in part (a). (c) Using the result in part (b), derive a difference formula to approximate f"(ro). Find out the leading order of h in the truncation error.

Let fe C[a, b] and ro E [a, b]. In this problem, we derive a difference formula to approxi- mate f"(ro). (a) Let f(z) = P3(x) + R(x), where P3(x) is the fourth Taylor Polynomial of f about zo. Find out the expressions of P3(r) and R(x). (b) Assume that for small h > 0, xo + h e [a, b]. Compute f(xo + h) + f(xo – h) in terms of the Taylor Polynomial P3 obtained in part (a). (c) Using the result in part (b), derive a difference formula to approximate f"(ro). Find out the leading order of h in the truncation error.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let fe C[a, b] and ro € [a, b]. In this problem, we derive a difference formula to approxi-
mate f"(ro).
(a) Let f(x) = P3(x) + R(x), where P3(x) is the fourth Taylor Polynomial of f about zo.
Find out the expressions of P3(1) and R(æ).
(b) Assume that for small h > 0, zo ±h e [a, b]. Compute f(xo + h) + f(zo – h) in terms
of the Taylor Polynomial P3 obtained in part (a).
(c) Using the result in part (b), derive a difference formula to approximate f"(ro). Find
out the leading order of h in the truncation error.
(d) Assume that M := max f() (z) is given. Consider the effect of round-off error in the
computation. Assume that the round-off error in each term can be bounded by a given
number e > 0. Show that the total error of approximating f"(x0) using the derived
Pe[a,b]
formula satisfies:
Mh?
|Round off plus Truncation errors| <
12
(e) Find the optimal step size h, (in terms of M and e) such that this minimizes the error
bound obtained in part (d).

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