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