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Berkeley MATH 128A - Numerical Analysis Sample Final

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Prof. Ming Gu, 861 Evans, tel: 2-3145Email: [email protected]://www.math.berkeley.edu/∼mgu/MA128A2008SMath128A: Numerical Analysis Sample FinalThis is a closed book, closed notes exam. You need to justify every one of youranswers. Completely correct answers given without justification will receive littlecredit. Do as much as you can. Partial solutions will get partial credit. Look overthe whole exam to find problems that you can do quickly. You need not simplify youranswers unless you are specifically asked to do so.1. (4 Points)Your Name:Your SID:Math128A: Numerical A nalysis Sample Final 22. (12 Points)(a) Describe a method to evaluateZ∞1e−xx2. No actual calculation is required.(b) EvaluateZ1−1Z2−2x2+ y2+ xydxdy.Math128A: Numerical A nalysis Sample Final 33. (12 Points)(a) From the Taylor expansion of a function f(x), derive a first-order approximation tof0(x).(b) Use Richardson’s extrapolation method to find a 3 point 2nd order formula.Math128A: Numerical A nalysis Sample Final 44. (12 Points)(a) Let A and B be n × n matrices. Prove or find a counter example: If AB = 0 thenA = 0 or B = 0.(b) Let A and B be n × n matrices. Prove or find a counter example: If AB = 0 thendet(A) = 0 or det(B) = 0.Math128A: Numerical A nalysis Sample Final 55. Consider the iterationxk+1= 2xk− αx2k, k = 0, 1, · · · ,where α > 0 is given. Show that the iteration converges quadratically to 1/α for any initialguess x0satisfying 0 < x0< 2/α.Math128A: Numerical A nalysis Sample Final 66. (12 Points)(a) For a function f and distinct points α, β, and γ, define what is meant by f[α, β, γ].(b) Find the Lagrange form of the polynomial P (x) which interpolates f(x) = 4x/(x + 1)at 0, 1, and 3.Math128A: Numerical A nalysis Sample Final 77. For the following linear systemx − αy = 1,αx − y = 1,describe for which values of α the system has an infinite number of solutions, no solutions,and exactly one solution, and find the solution when it is unique.Math128A: Numerical A nalysis Sample Final 88. Determine the free cubic spline that apprixmates the data f (−1) = 1, f(0) = 0 and f(1) = 1.Math128A: Numerical A nalysis Sample Final 99. (a) Define what is meant by the local truncation error, and the local order, for a single-stepmethod for solving the ODE’s.(b) Derive a specific Runge-Kutta method of local order 2. Show your work.Math128A: Numerical A nalysis Sample Final 1010. Given matrix A =1 αα 1.(a) Find the ∞ norm and the spectral radius of A.(b) Consider the iterative methodxk+1= Axk+ c, k = 0, 1, · · · ,Find the conditions on α so that this iteration converges for any initial


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