数值分析上机题目

数值分析上机题目
1、 分别用不动点迭代与Newton 法求解方程250x x e -+=的正根与负根。

2、 Use each of the following methods to find a solution in [0.1,1] accurate to within 10^-4 for
4326005502002010x x x x -+--=
a. Bisection method
b. Newton’s method
c. Secant method
d. Method of False Position
e. Muller’s method
3、 应用Newton 法求f （x ）的零点，e=10^-6,这里f （x ）=x-sin （x ）。

再用求重根的两种方法求f （x ）的零点。

4、 应用Newton 法求f （x ）的零点，e=10^-6,f(x)=x-sin(x) 再用Steffensen’s method 加速其收敛。

5、 用Neville’s 迭代差值算法，对于函数2
1
(),11125f x x x =
-≤≤+进行lagrange
插值。

取不同的等分数n=5,10，将区间[-1,1]n 等分，取等距节点。

把f(x)和插值多项式的曲线画在同一张图上进行比较。

6、 画狗的轮廓图
7、 Use Romberg integration to compute the following approximations to
⎰
a 、
Determine R1,1,R2,1,R3,1,R4,1and R5,1,and use these approximations to predict the value of the integral.
b 、 Determine R2,2 ,R3,3 ,R4,4 ,and R5,5,and modify your prediction.
c 、
Determine R6,1 ,R6,2 ,R6,3 ,R6,4 ,R6,5 and R6,6,and modify your prediction.
d 、 Determin
e R7,7 ,R8,8 ,R9,9 ,and R10,10,and make a final prediction.
e 、
Explain why this integral causes difficulty with Romberg integration and how it can be reformulated to more easily determine an accurate approximation.
8、 分别用1）Euler method 2）Modified Euler Method 3）Runge-Kutta Order Four
求解P264 ：7a 并计算其误差； Given the initial-value problem :
=-1y y t '++05t ≤≤(0)1y =
With exact solution
-t
()y t e t =+； A: Approximate y(5) with h=0.2 , h=0.1, h=0.05.
9、 课本P279 Example 4:
For the problem y’=y -t^2+1 , 0<=t<=2 , y(0)=0.5, Euler’s Method with h=0.025, the
Modified Euler Method with h=0.05,and the Runge-Kutta fourth-order method with h=0.1 are compared at the common mesh points of these methods 0.1, 0.2, 0.3, 0.4 and 0.5 .Each of these techniques requires 20 functional evaluations to determine the values listed in Table 5.8 to approximate y(0.5). In this example , the fourth-order method is clearly superior. 10、 P 322 Exercise 5.9
1 Use the Runge-Kutta method for systems to approximate the solutions of the following systems of first-order differential equations, and compare the results to the actual solutions. A ：
221
12222
1232(21)4(24)t t u u u t e u u u t t e '=+-+'=+++- 01t ≤≤1(0)1u =2(0)1u =0.2h =
actual solutions
521()(1/3)(1/3)t t t u t e e e -=-+ 5222()(1/3)(2/3)t t t u t e e t e -=++
11、 P 322 Exercise 5.9
1 Use the Runge-Kutta method for Systems Algorithm to approximate the solutions of the following higher-order differential equations, and compare the results to the actual solutions. A:
2t y y y te t ''-+=-01t ≤≤(0)(0)0y y '==0.1h =
actual solutions 3()(1/6)22t t t
y t t e te e t =-+--
end
12、 P 368Exercise 6.2
第一问:
Use Gaussian elimination and three-digit chopping arithmetic to solve the following linear systems, and compare the approximations to the actual solution.
E:
1234123423412341.19 2.11100 1.12
14.20.12212.2 3.4410099.9 2.1515.30.1113.1 4.16
x x x x x x x x x x x x x x x +-+=-+-=-+=+--=
Actual solution x1=0.17682530 x2=0.01269269 x3=-0.02065405 x4=-1.18260870
第二问:
Repeat Exercise 6 using Gaussian elimination with partial pivoting. 第三问:
Repeat Exericise 5 using Gaussian elimination with scaled partial pivoting.
13、 P411 7：
Let A be the 10*10 tridiagonal matrix given by a(1,1)=2, a(i,i+1)=a(i,i-1)=-1,for each i=2,……9,anda(1,1)=a(10,10)=2,
a(1,2)=a(10,9)=-1,let b be the ten-dimensional column vector given by
b1=b10=1,and bi=0,for each i=2,3,……，9 Solve Ax=b,using the Crout factorization for tridiagonal systems.
14、
P 453 Exericse 7.3 15
Use all the applicable methods in this section to solve the linear system Ax=b to within 10^(-5) in the
l ∞ norm ,where the entries of A are
2i, when j=i and i=1,2,……，80
j=i+2 and i=1,2,……,78
0.5i, when j=i-2 and i=1,2,……, 80
a (i,j) = 0.25i when j=i+4 and i=1,2,……,76
j=1-4 and i=1,2,……,80
0, otherwise
And those of bi=π ,for each i=1,2, (80)。