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）︰105/3/31
考試時限（分鐘）：150分鐘
是否需發放獎勵金： 是
（如未明確表示，則不予發放）
試題 :
●Please give details fo your calculation. A direct answer without explanation
is not counted.
●Your answers must be in English.
●You can bring notes and the textbook. Other books or electronic devices are
not allowed.
Problem 1 (15%)
Give the double binary representation of the number +17.3
Problem 2 (20%)
Let
β=10, p=4, x=1.000, y=-0.5555
Consider
x_0 = x
x_1 = (x_0ⓜy)⊕y *註一
...
x_n = (x_(n-1)ⓜy)⊕y
Assume exactly rounded operations using the rounding up scheme. What are the
values of the sequence {x_0, x_1, ...}?
Problem 3 (25%)
Conduct LU factorization with pivoting for the following matrix
A = [4 -10 -3 7; 6 -5 -4 7; 6 27 3 -3; 12 6 -6 6]
(a) What are M_3, P_3, M_2, P_2, M_1, P_1 such that
M_3 P_3 M_2 P_2 M_1 P_1 = U ?
(b) What is
P_1^-1 M_1^-1 P_2^-1 P_3^-1 ?
(c) What is
P_1^-1 M_1^-1 P_2^-1 M_2^-1 P_3^-1 M_3^-1 ?
(d) What is
P_3 P_2 P_1 P_1^-1 M_1^-1 P_2^-1 P_3^-1 ?
(e) What is the P such that
P A = L U
Is it equal to P_3 P_2 P_1 ?
Problem 4 (20%)
Consider a binary number in a floating-point system of β=2 and p=6
1.11111
(1) Transform it to a decimal value with β=10 and p=2. That is,
x.x
Then transform this decimal value to a binary representation with
p=7.
Is the resulting binary the same as (1)?
(2) Redo (1) but a decimal value with p=3.
Note that we assume rounding even is used.
Problem 5 (20%)
In our lecture we derive a version of Cholesky factorization by
L_jj L_:j = A_:j - Σ(k=[1, j-1]) L_jk L_:k
By a similar setting please derive a version that obtains
A = U' U. *註二
You need to give the program.
*註一 :ⓜ 為內有減號的圈圈,即rounded substraction.
*註二 : U' 為 transpose of upper triangular matrix U.