Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student ID: . . . . . . . . . . . . . . . . . MATH3820: Numerical Methods - Quiz Instructions There are 6 questions. You are given 55 minutes. You are allowed to use one A4-size two-sided handwritten help sheet. Non-programmable calculators are allowed. id=qn3 rand=2345 id=qn2 rand=2345 For all questions, write your solution carefully in the space provided. Show all your working. 1. The function f (x) satisfies f (1) = 1, f (0) = 2 and f (2) = 1. Find the Lagrange interpolation polynomial which passes through the given points. 2. Use a Taylor series expansion to compute an error estimate in approximating the derivative of the function f : R R using the formula f 0 (x0 ) MATH3820: Quiz Mathematics, University of Newcastle f (x0 2h) 4f (x0 h) + 3f (x0 ) . 2h Page 1 of 6 id=qn4 rand=2345 3. Consider the following tabulated values of a function f (x) and its derivative at three points: x f (x) f 0 (x) 0 1 2 1 0 4 0.5 2 1 id=qn5 rand=2345 Find the correct Hermite interpolation polynomial for interpolating the function values and its derivatives at these points. Hint: It is not necessary to simplify the expression. 4. Write down the Newton's method to approximate a solution to the following nonlinear system in terms of xk and xk1 . Let the initial guess be 1 x0 = 1 . 1 Write down the Jacobian, and find the value x1 using this initial guess. Here x = (x, y, z)T . x3 2y 2 = 0 x3 5z 2 + z = 0 yz 3 1 = 0 id=qn5 rand=2345 5. Let ( 3x2 + ax3 if 1 x 0 f (x) = 3x2 + bx3 + c if 0 < x 1. id=qn6 rand=2345 be a natural cubic spline on [1, 1] with nodes at {1, 0, 1}. Determine a, b and c. Using the values of a, b and c, verify that f , f 0 and f 00 are continuous at x = 0, and f satisfies the end conditions f 00 (1) = f 00 (1) = 0. 6. Determine a fixed-point function g in the interval [0, 1] that produces an approximation to a positive solution of 3x2 ex = 0. Show that the fixed point function maps [0, 1] into itself and |g 0 (x)| < 1 in the given interval. Starting with a suitable initial guess, compute two iterations of the fixed point sequence. MATH3820: Quiz Mathematics, University of Newcastle Page 2 of 6