1. You wish to find a numerical approximation of the number ln 2. There are a number of ways to do this.  The first way is to define ln 2 = ₀∫¹ 1 / (1 + t) dt. We can also use the power series for

ln(1 + x) = x - (x²/2) + (x³/3) - ··· + (-1)^n-1 (xⁿ/n) ····

which is conditionally convergent when x = 1.  We could use a Pad´e approximant for ln(1 + x) centered at x = 0. Finally, for our purposes, we could consider the fact that ln(1 + x) is the solution of the initial value problem x' = 1 / (1 + t), x(0) = 0 and use a Runge-Kutta method on this equation to approximate x(1) = ln 2.

(a) Compare and contrast the strengths and weaknesses of each of these methods paying attention to sources and magnitudes of errors and relative efficiencies.

(b) Compute the Romberg R(3, 3) approximation for the integral representation. Give your answer to 6 decimal places.

(c) Use the Alternating Series Theorem to determine how many terms are needed for 5 decimal place accuracy in the  representation

ln(2) = 1 - (1/2) + (1/3) - (1/4) ··· + (-1)^n-1 (xⁿ/n) ····

(d) Compute the (2, 2)-Pade approximant for ln(1 + x) and use it to evaluate ln 2.  Give your answer to 6 decimal places.

(e) Use the classical Runge-Kutta method of order 2 for first order equations with step size h = 0.1 from t = 0 to t = 1 to approximate ln 2. Give your answer to 6 decimal places. (f) Compare your answers and comment on your results. How could each method be improved?

2. Consider the system of equations

(a) Solve this system exactly using Naive Gaussian Elimination ( i.e., forward elimination and  back substitution).

(b) Let x₀^→  = (1, 0, 0, 0)^T. Use Jacobi iteration to find x₄^→.

(c) Let x₀^→ = (1, 0, 0, 0)^T. Use Gauss-Seidel iteration to find x₄^→.

(d) Given what you have calculated above, which iterative method gives a better l₂ approx- imation of the actual solution?

(e) For the method that gives the better approximation, calculate x₆^→.

3. Find the LU factorization of the matrix A given below. Use this factorization to efficiently solve the systems of equations Ax^→ = L(U x^→) = b_i^→ for the b_i^→'s given below.

Use these solutions to find A^-1.

4. Calculate the l₁, l₂ and l_∞ condition numbers of the block diagonal matrix

What is the significance of the condition number of a matrix and what do the values obtained above mean to the matrix A?