Numerical Analysis I, Fall 2017 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 13, 2017.

1. Consider the following recursive equation p_{0} = 1, p_{1} = a_{1}, p_{n} = \frac{10}{3}p_{n−1} − p_{n−2}. For what values of a_{1} is it stable in relative error? Explain.

2. The file an.txt contains a sequence stored as ’n, a_{n}’ at nth line. Find its rate of convergence. Express your answer as O(β_{n}) and find β_{n} explicitly. Show details.

3. Find a root of x = 2 cos x with 10 correct decimal digits using any numerical method of your choice. Put (1): the detail formula, (2): x_{0}, x_{1} and (3): the answer x^{∗}, on the answer sheet, but need not hand in the code.

4. Show that the nonlinear equation x = 1 + cos(x)/2 has a solution in [1, 1.5]. Let x_{0} = 1.25, give an estimate on N (need not be optimal) such that |x_{n} − x^{∗}| < 10^{−5} for all n ≥ N.

5. The first few iteration (p_{i}, f(p_{i})), i = 0, 1, 2, 3 of method of false position for some equation f(x) = 0 is given in q2p4.txt. Find p_{4} (4 digits will do). Also give your formula for finding p_{4} and explain.

Need not hand in your code this time. Double check your answers carefully.

Numerical Analysis I, Fall 2017 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 13, 2017.

1. Consider the following recursive equation p_{0} = 1, p_{1} = a_{1}, p_{n} = \frac{10}{3}p_{n−1} − p_{n−2}. For what values of a_{1} is it stable in relative error? Explain.

2. The file an.txt contains a sequence stored as ’n, a_{n}’ at nth line. Find its rate of convergence. Express your answer as O(β_{n}) and find β_{n} explicitly. Show details.

3. Find a root of x = 2 cos x with 10 correct decimal digits using any numerical method of your choice. Put (1): the detail formula, (2): x_{0}, x_{1} and (3): the answer x^{∗}, on the answer sheet, but need not hand in the code.

4. Show that the nonlinear equation x = 1 + cos(x)/2 has a solution in [1, 1.5]. Let x_{0} = 1.25, give an estimate on N (need not be optimal) such that |x_{n} − x^{∗}| < 10^{−5} for all n ≥ N.

5. The first few iteration (p_{i}, f(p_{i})), i = 0, 1, 2, 3 of method of false position for some equation f(x) = 0 is given in q2p4.txt. Find p_{4} (4 digits will do). Also give your formula for finding p_{4} and explain.

Need not hand in your code this time. Double check your answers carefully.