Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 17, 2014.

1. The sequence a_{n} = 1 + \frac{1}{n^{2}} converges to its limit of order α. Find α. Then give an example of a quadratically convergent sequence.

2. True or False? Prove it if true, give a counter example if false.

If x = g(x) has exactly one root x^{∗} in [a, b], g([a, b]) ⊆ [a, b] and \underline{|g′(x)| ≤ 1} on [a, b], then the standard fixed point iteration converges to x^{∗} for any x_{0} ∈ [a, b]. Note that the underlined condition is not standard.

3. Find a root of x = 2 cos x with 10 correct decimal digits using Newton’s method. Put (1): the iteration formula, (2): x_{0} and (3): the answer x^{∗}, on the answer sheet, but do not hand in the code. You may use the code in the package or write the code yourself.

4. Show that, if x^{∗} is the unique root of f, f ∈ C^{2}[x^{∗} − δ, x^{∗} + δ], f′ ≠ 0 on [x^{∗} − δ, x^{∗} + δ] and f([x^{∗} − δ, x^{∗} + δ]) ⊆ [x^{∗} − δ, x^{∗} + δ]. Then Newton’s method satisfies |x_{n+1} − x^{∗}| ≤ C|x_{n} − x^{∗}|^{2}.

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 q2p5.txt. Find p_{4}. Also give your formula for finding p_{4} and explain.

Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 17, 2014.

1. The sequence a_{n} = 1 + \frac{1}{n^{2}} converges to its limit of order α. Find α. Then give an example of a quadratically convergent sequence.

2. True or False? Prove it if true, give a counter example if false.

If x = g(x) has exactly one root x^{∗} in [a, b], g([a, b]) ⊆ [a, b] and \underline{|g′(x)| ≤ 1} on [a, b], then the standard fixed point iteration converges to x^{∗} for any x_{0} ∈ [a, b]. Note that the underlined condition is not standard.

3. Find a root of x = 2 cos x with 10 correct decimal digits using Newton’s method. Put (1): the iteration formula, (2): x_{0} and (3): the answer x^{∗}, on the answer sheet, but do not hand in the code. You may use the code in the package or write the code yourself.

4. Show that, if x^{∗} is the unique root of f, f ∈ C^{2}[x^{∗} − δ, x^{∗} + δ], f′ ≠ 0 on [x^{∗} − δ, x^{∗} + δ] and f([x^{∗} − δ, x^{∗} + δ]) ⊆ [x^{∗} − δ, x^{∗} + δ]. Then Newton’s method satisfies |x_{n+1} − x^{∗}| ≤ C|x_{n} − x^{∗}|^{2}.

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 q2p5.txt. Find p_{4}. Also give your formula for finding p_{4} and explain.