i) Consider the floating-point number system that have binary representa- tions of the form (0.b1b2)2 x 2*, where k, b,b = {0,1}. a) Write down all the numbers which can be represented in this floating point system. b) How many different numbers can be represented using this represen- tation? c) Design a new binary representation using three bits , m, n = {0,1} which can represent 8 different numbers. This set of 8 numbers must include all the numbers from part a) except 0. Write down all the numbers which can be represented with your new binary representa- tion. d) Let M be a 10 x 10 matrix whose entries are floating point numbers using the representation (0.b1b) x 2 introduced above. How many bits need to be stored to store the numbers in this matrix? ii) Letz= 10-0. The goal is to compute 1-cos(2) A straightforward Matlab implementation shows that >> x = le-10; >> 1/(1-cos(x)) ans = Inf Show how this computation can be modified such that catastrophic can- cellation can be avoided. iii) Let f [0, 1] R be given by f(x) = e. a) Compute the quadratic spline with the node points 0.1/2.1 with first derivative at the point 1/2 given by f'(1/2). 1/2 -1/2 b) Find an approximation for feda by evaluating f/2s(2)de. iv) Derive the Gaussian quadrature rule of the form [ f(x)xdz ~ af(-a) + bf (0) + cf (a) (1) that is exact for all polynomials of as high a degree as possible; that is, determine a. a, b,and c. v) The formula for the secant method can also be written as ti+kN In Xn+1 = Establish this, and explain why it is inferior to the expression of *n+1 = - (1 (72) f)) (4.) in a computer program. xn-if(n) - Enf(n-1) f(an) - f(n-1) vi) Let T> 0 be a real number, N> 1 be an odd integer and 20, 21, 2N-1 be a set of real numbers. You are given the values = T (i+kN); and y+kN 2, for i= 0,1,2,... N-1 and all k Z. N a) Define a trigonometric polynomial p: R R such that p(te) ye for all l Z. = b) Outline a Matlab program which computes the coefficients of the trigonometric polynomial p.