中原大學九十三學年度博士班入學招生考試
93年6月9日　10:30~12:00　應用數學系　 誠實是我們珍視的美德，
我們喜愛「拒絕作弊，堅守正直」的你！
科目：數值分析 　
1. (24 points) Suppose that r is a zero of the scalar equation f(x) = 0 of multiplicity m. Assume that f and all its derivatives up to order m + 1 are continuous in some neighborhood of r.
　 (a) Show that, when the Newton’s method is used to solve the equation f(x) = 0, it converges linearly.
　 (b) Show that if we sovle the equation f(x) = 0 using the modified Newton's method
　 　
　 the convergence is quadratic.
　 　 　
2. (20 points) Consider the system of linear equations Ax = b where
　
　 (a) Write down the matrix formulation for the Gauss-Seidel method for above system of linear equations.
　 (b) Show that, for above system of linear equations, the Gauss-Seidel method converges for all initial iterates.
　 　 　
3. (16 points) Determine the values of a, b, c, d so that f is a cubic spline and
　
　 is a minimum :
　
　 　 　
4. (15 points) Determine values for α,β and γ so that the following rule
　
　 is exact for all polynomial of degree 2.　
　 　 　
5. (25 points) Describe how you would solve numerically the Poisson's equation
　
　 with boundary conditions　
　
　 You may assume that f(x,y) is continuous on [0,1] × [0,1].
　 　 　
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