2007合肥大学数值分析期末考试参考答案-(2007-06-29).doc

Numerical AnalysisAnswers to Test A (June 29, 2007)1Fill in the following blanks(1) Suppose , then the 2008th divided difference (差商(均差) 0 .(2) Let，then the number approximate with 4 significant digits. , , . (3) Suppose .Then 12 , 9 .(4) The Trapezoidal rule (梯形求积公式) applied to gives the value 4, and Simpsons rule gives the value 2. Then 1/2 .(5) A quadratic spline S for a function on is defined by, Then 2 ， 1 .2. a) Show that the sequence is generated by Newtons method for finding the root of equation . b) The sequence converges to of order 2 whenever . c) Use to compute with 6 significant digits.Proof: a) Define then the sequence generated by Newtons method for finding the root of equation isthat is. b) Since , it is easy to get that , and by induction it follows ,and.Therefore the sequenceconverges to some constant Hence the sequenceconverges to of order 2 which follows fromc) With , from the iterative scheme, it follows 3. Use the following data to construct an interpolating polynomial of degree four so that for and 01201101Solution: Build up the divided-difference table as follows :0000011111110-1210-1-1/21/4 So the polynomial interpolating the given data is 4. Find the constants and, so that the quadrature formula（求积公式） has the highest possible degree of precision (代数精度). Solution: For ,we have by the definition of degree of precision Solving the equation systems for ,we get 5. The forward-difference formula can be expressed asUse Richardsons extrapolation (Richardson 外推) to derive an formula for Solution: Define . By Richadsons extrapolation, substituting by into the forward diffence formula gives (2)From , one gets (3)Similarly, changing by into (3), we have (4)From , we have6. Use Eulers method and the Modified Euler method to approximate the solution for the initial-value problem with Solution Define ,then . By the Modified Euler method, we get the iterative scheme or Therefore7. Establish the convergent (收敛的) Jacobi iterative scheme (迭代格式) and Gauss-Seidel iterative scheme for the following linear system and explain why these schemes are convergent? Solution Rearrange the linear system as follows The corresponding coefficient matrix is a strictly diagonal dominant matrix, so the Jacobi iterative scheme and Gauss-S