Numerical Analysis I, Fall 2011 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 18, 2011.

1. Suppose that M = N_{1}(h) + K_{1}h^{2} + K_{2}h^{4} + +K_{3}h^{6} + ⋯ and N_{1}(h) = 1.570796, N_{1}(h/2) = 1.896119, N_{1}(h/4) = 1.974232. Construct an extrapolation table and determine N_{3}(h).

2. What is the rate of convergence of N(h) if N(h) = 0.486748, N(h/3) = 0.45788 and N(h/9) = 0.452324?

3. Estimate h or n such that the composite Trapezoidal rule for ∫_{1}^{2} cos(x^{2})dx has absolute error less than 10^{−5}. Then give your numerical value I_{h} with the n you obtained (program it and write down your answer).

4. Write down Simpson’s formula for ∫_{−h}^{h} f(x)dx. Need not derive or prove anything. If you don’t remember and decide to derive, put on details.

5. A quadrature rule takes the form ∫_{−h}^{h} f(x)dx ≈ 2h (α_{−}f(\frac{−h}{2}) + α_{+}f(\frac{h}{2})) . Find α_{−}

and α_{+} that gives the largest degree of precision. Then derive (prove) an error bound for the resulting scheme. (Need not give error identity)

Numerical Analysis I, Fall 2011 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 02

Oct 18, 2011.

1. Suppose that M = N_{1}(h) + K_{1}h^{2} + K_{2}h^{4} + +K_{3}h^{6} + ⋯ and N_{1}(h) = 1.570796, N_{1}(h/2) = 1.896119, N_{1}(h/4) = 1.974232. Construct an extrapolation table and determine N_{3}(h).

2. What is the rate of convergence of N(h) if N(h) = 0.486748, N(h/3) = 0.45788 and N(h/9) = 0.452324?

3. Estimate h or n such that the composite Trapezoidal rule for ∫_{1}^{2} cos(x^{2})dx has absolute error less than 10^{−5}. Then give your numerical value I_{h} with the n you obtained (program it and write down your answer).

4. Write down Simpson’s formula for ∫_{−h}^{h} f(x)dx. Need not derive or prove anything. If you don’t remember and decide to derive, put on details.

5. A quadrature rule takes the form ∫_{−h}^{h} f(x)dx ≈ 2h (α_{−}f(\frac{−h}{2}) + α_{+}f(\frac{h}{2})) . Find α_{−}

and α_{+} that gives the largest degree of precision. Then derive (prove) an error bound for the resulting scheme. (Need not give error identity)