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. --

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