Reference no: EM132519580

30604 Numerical Analysis Assignment - University of Salford, UK

Question 1 -

(a) Show that the equation e^x + 2^-x + 2 cos x - 6 = 0 must have at least one solution in [1, 2].

(b) For 3-digit floating point form compute the following using both rounding and chopping

12.3 ⊕ 0.0234,  -0.0321 ⊕ 0.000136

(c) Find an approximation to √2 correct to within 10^-4 using the Bisection method, and comment on your result.

Question 2 -

(a) Consider the nonlinear equation g(x) = ½e^0.5x defined on the interval [0, 1]. Then

(i) Show that there exists a unique fixed-point for g in [0, 1].

(ii) Use the fixed-point iteration method to compute p₃, set P₀ = 0.

(iii) Compute an error bound for your approximation in part (ii), using |p₃ - p| ≤ (kⁿ/(1-k))|p₀ - p=|.

(b) Let f (x) = e^x + 3x²,

i. Find the Newton's formula g(p_n-1),

ii. Start with P₀ = 4 and compute p₄.

iii. Start with P₀ = 0.5 and compute p₄.

iv. From parts i and ii which is the better choice of P₀ and why?

(c) Use the error term of a Taylor polynomial to estimate the degree of the Taylor polynomial which approximates cos x for |x| ≤ π/4, with an error of no greater than 10^-5.

Question 3 -

(a) Determine a polynomial p(x) of degree at most 2 such that p(-1) = 1, p(0) = 0, and p(1) = 1.

(b) Use the following date to approximate f'(1.005) using the three-point formula

x	1.00	1.01	1.02
f(x)	1.27	1.32	1.38

(c) If p^~ approximates p to four significant digits. Find the interval in which p^~ must lie if p = 9990.

Question 4 -

(a) Show that the sequence p_n = (1 + p_n-1)^1/5 converges to a unique root of x⁵ - x -1 = 0 in [0, 2] using the fixed-point method.

(b) Show that the Simpson's composite rule is more efficient to use than the trapezoidal composite rule to approximate the integral ₀∫^π cos 2x dx with an error at most 0.00002.

(c) Compute the condition number of the following matrix relative to ||.||_∞ using the matrix norm:

||A||_∞ = max_{1≤j≤n j=1}∑ⁿ|a_ij|