課程名稱︰數值方法 課程性質︰計算機運算處理 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）︰100/05/10 考試時限（分鐘）：150 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Problem 1(15%): Give an n by n matrix A, what is |A| = max |Ax| ? ∞ |x| = 1 ∞ That is, write |A| as an expression of A's elements. ∞ Problem 2(25%): Assume A is symmetric and invertible. Formally prove that cond(A) >= 1 where cond(.) is the condition number. Assume 2-norm is used. Hint: no need to check eigenvalues of matrices Problem 3(20%): Assume L is a lower triangular square matrix stored in the compressed row format. That is, assume arrays val, col_ind, and row_ptr are already available. No need to worry about declaring variables and memory allocation. Write the MATLAB code to solve Lx = b We further make the following assumptions and requirements: (1) L ≠ 0, for all i ii (2) In col_ind, indices of each row are in ascending order. (3) Assume simple loops are used (It is like that you are writing Fortran/C/Java code though you use MATLAB syntax here and assume arrays start with index 1. Thus, you cannot use thinds such as x(i:j), find, etc.). (4) The complexity must be no more than O(nnz). Problem 4(30%): Assume a square matrix A is stroed in the compressed column format and we would like to find T B = A . We stroe B in compressed column format as well. In the beginning, we try to find the number of non-zero elements in each column of B: for i=1:n+1 bcol_ptr(i) = 0; nnz = acol_ptr(n+1)-1; for i=1:nnz bcol_ptr(arow_ind(i)+1) = bcol_ptr(arow_ind(i)+1) +1; bcol_ptr(1) = 1; for i=2:n+1 bcol_ptr(i) = bcol_ptr(i) + bcol_ptr(i-1); Explain this segment of code and continue to finish the MATLAB code for obtaining B. We further make the following assumptions and requirements: (1) Assume simple loops are used (It is like that your are writing Fortran/C/Java code though you use MATLAB syntax and assume arrays start with index 1. Thus, you cannot use thins such as x(i:j), find, etc.). (2) The complexity must be no more than O(nnz). (3) You can use only arrays aval, arow_ind, acol_ptr, bval, bcol_ind, brow_ptr. That is, you cannot allocate other arrays. Problem 5(10%): Given a linear system [ 1 2 1] [-1] [ 3 1 -2] x = [ 1] [ 1 2 1] [ 2] [0] Use [0] as the initial point. [0] (a) Do three itreations of the Jacobi method (b) Do three iterations of the Gauss-seidal method --

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