Numerical Analysis and Non-trivial Root
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2014/2015
MA2213 Numerical Analysis I
Semester II
Homework Assignment 2
Due: 3 March 2015, 5pm
1.
Find an approximation to the root of f (x) = tan(x/4) − 1 in the interval [3.14, 3.15], with the relative error |pn − p|/|p| accurate to within 10−4 using the Bisection method.
Tabulate all your workings as in the answer to tutorial 2 question 2. Present all values of an , bn and pn in full precision, and values of f (pn ) and the relative error bounds to 4 significant figures.
For questions 2 and 3, tabulate all your workings as in the answer to tutorial 2 question
3. Give the values of all fixed-point iterates to 8 decimal places, and values of |pn − pn−1 | to 4 significant figures.
2.
Consider f (x) = x2 − 5 sin(x).
(i) Sketch the graph of f (x) to locate the roots of f (x) = 0. Write down an interval
[a, b], with b − a at most 1, containing the non-trivial root.
(ii) Without resorting to Newton’s method, determine a fixed point function g which can be used to determine the root in [a, b].
(iii) Use the conditions stated in Theorem 2.3 to verify that the chosen g in (ii) will ensure convergence of fixed point iterations for a given initial iterate p0 ∈ [a, b].
(iv) Perform the fixed point iterations for the chosen g in (ii) with p0 = (b + a)/2. Give your approximation pn for the non-trivial root p such that |pn − pn−1 | < 10−5 .
3.
Apply Newton’s method to finding the non-trivial root of x2 − 5 sin(x) = 0 using the same initial iterate p0 used in 2(iv). Give your approximation pn for the non-trivial root p such that |pn − pn−1 | < 10−6 .
The End
Department of Mathematics
2014/2015
MA2213 Numerical Analysis I
Semester II
Homework Assignment 2
Due: 3 March 2015, 5pm
1.
Find an approximation to the root of f (x) = tan(x/4) − 1 in the interval [3.14, 3.15], with the relative error |pn − p|/|p| accurate to within 10−4 using the Bisection method.
Tabulate all your workings as in the answer to tutorial 2 question 2. Present all values of an , bn and pn in full precision, and values of f (pn ) and the relative error bounds to 4 significant figures.
For questions 2 and 3, tabulate all your workings as in the answer to tutorial 2 question
3. Give the values of all fixed-point iterates to 8 decimal places, and values of |pn − pn−1 | to 4 significant figures.
2.
Consider f (x) = x2 − 5 sin(x).
(i) Sketch the graph of f (x) to locate the roots of f (x) = 0. Write down an interval
[a, b], with b − a at most 1, containing the non-trivial root.
(ii) Without resorting to Newton’s method, determine a fixed point function g which can be used to determine the root in [a, b].
(iii) Use the conditions stated in Theorem 2.3 to verify that the chosen g in (ii) will ensure convergence of fixed point iterations for a given initial iterate p0 ∈ [a, b].
(iv) Perform the fixed point iterations for the chosen g in (ii) with p0 = (b + a)/2. Give your approximation pn for the non-trivial root p such that |pn − pn−1 | < 10−5 .
3.
Apply Newton’s method to finding the non-trivial root of x2 − 5 sin(x) = 0 using the same initial iterate p0 used in 2(iv). Give your approximation pn for the non-trivial root p such that |pn − pn−1 | < 10−6 .
The End