課程名稱︰ 數值方法 Numerical Methods 課程性質︰資工系選修 課程教師︰林智仁 開課學院：電機資訊學院 開課系所︰資工所 考試日期（年月日）︰2016/6/23 考試時限（分鐘）：150 試題 : -Please give details of your calculation. A direct answer without explanation is not counted. -Your answers must be in English. -Please carefully read problem statements. -During the exam you are not allowed to borrow others' class notes. -Try to work on easier questions first. 1. (15%) Consider the steepest descent method. Does it satisfy r_j* r_(j-1) = 0 If yes, prove the result. Otherwises, give a counter example. 2. (35%) Consider a twice continuously differentiable f(x), x⊆R. Assume f(x) has at least one root, f'(x) > 0 and f"(x) > 0, ∀x, and f(x_0) >= 0, where x_0 is the initial point of Newton methods. (a) Will {x_n} generated by Newton updates satisfy f(x_n) >= 0, ∀n (b) Will the sequence {x_n} converge to a root of f(x)? Theorems proved in our lectures can be considered as known results (though you may not need them). You need to show details of the proof. 3. (30%) Given three points (0,1), (1,0) and (2,2). Find the spline approximation. Draw a figure to show how s_j(x) looks like. (a) Consider the following boundary condition: s_0"(x_0) = 0 and s_(n-1)"(x_n) = 0 (b) Consider the following boundary condition: s_0'(x_0) = -1 and s_(n-1)'(x_n) = 1 4. (20%) In regression we consider a*x +b as the approximate function. Instead we can use only a*x so that the funtion pass through the origin. Assume x_1 = (1,1,0), y_1 = -2 x_2 = (0,0,1), y_2 = 2 x_3 = (0,2,0), y_3 = 2 x_4 = (1,1,1), y_4 = 0 Find the function a*x. 註:V* 為V的transpose --

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