Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 22, 2010.

1. Find a root of x = cos x with 10 correct decimal digits. You can use any method except bisection. Describe your algorithm, but do not hand in the code. Verify your answer numerically if you are not sure about your answer.

2. Give a convergent fixed point iteration (and not other methods) for solving f(x) = x − 2 cos x = 0. Explain why your formulae converges. The solution x^{∗} ≈ 1.03, f′(x^{∗}) ≈ 1 + 2 · sin(1.03) ≈ 2.71 (it can be obtained from ‘plot(x,x-2*cos(x),’o’)’ in matlab).

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

4. Write down the formula for Newton’s method and secant method for solving f(x) = 0. Give details.

5. Show that, if x^{∗} is a root of f with f′(x^{∗}) ≠ 0, f ∈ C^{2}[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} if x_{0} ∈ [x^{∗} − δ, x^{∗} + δ].

Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 22, 2010.

1. Find a root of x = cos x with 10 correct decimal digits. You can use any method except bisection. Describe your algorithm, but do not hand in the code. Verify your answer numerically if you are not sure about your answer.

2. Give a convergent fixed point iteration (and not other methods) for solving f(x) = x − 2 cos x = 0. Explain why your formulae converges. The solution x^{∗} ≈ 1.03, f′(x^{∗}) ≈ 1 + 2 · sin(1.03) ≈ 2.71 (it can be obtained from ‘plot(x,x-2*cos(x),’o’)’ in matlab).

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

4. Write down the formula for Newton’s method and secant method for solving f(x) = 0. Give details.

5. Show that, if x^{∗} is a root of f with f′(x^{∗}) ≠ 0, f ∈ C^{2}[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} if x_{0} ∈ [x^{∗} − δ, x^{∗} + δ].