数值分析试卷

期
201- 201 学年第 专业班级（教学班） 学期 课程代码 课程名称 考试日期
1.(20%) Fill in the following blanks. (1) Let
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数值分析
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3.(12%)
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8.(10%) (1)Suppose
y ''' (t ) exists on (a, b) .Show that the following two-step explicit scheme
w0 0 , w1 1 , wi 1 wi y (t0 ) 0 , y (t1 ) 1 , y ' (t ) f (t , y ), t [a, b] ba , ti a ih, i 0,1,2, , n n
' p3 ( xi ) f ( xi ) for i 0,1,2 ,and p3 ( x1 ) f ' ( x1 ) .
p3 ( x) of the degree three such that
Then the error (5) If
0
f ( x) 2 x 6 3 x 3 2 x 1 , then the divided differences f [20 ,21 , ,26 ] =____________, and
7.(10%) Determine constant
a, b, c, d that will produce a quadrature formula:

1
1
f ( x)dx af (1) bf (1) cf ' (1) df ' (1) that has degree of 3.
Test the convergence of the Gauss_Seidel iterative scheme by using the norm of its iterative matrix. 5.(12%) Suppose (a) Determine weight function
w( x) 1, P0 ( x) 1, P 1 ( x ) x B1 , P k ( x ) ( x Bk ) P k 1 Ck P k 2 ( x ) for k 2 .
n 2 gives 5,and Composite Simpson’s rule gives e the fact that f (1) f (1) and
f (0.5) f (0.5) 1 to determine f (1), f (0.5), f (0), f (0.5), f (1) .
x* is the root of the equation f ( x) 0 of
multiplicity
m(m 2) ,then the modified
Newton’s iterative formula for finding (4) Let
x* is______________________________________________.
4.(12%) Use the following data to construct an interpolating polynomial
f ( x) x 4 and the cubic Lagrange interpolating polynomial for f on the nodes 0,1,2,3 is p3 ( x) . f ( x) p3 ( x) =_______________________________, and p3 ( x) =_____________________.
(3) Suppose
f ( x) xe x 1 and x0 0.5 .Use Newton’s method to find x2 . f ( x) xe x 1 , x0 0.5 and x1 0.6 .Use the Secant method to find x3 .
[1,1]
命题教师注意事项:1、主考教师必须于考试一周前将“试卷 A” 、 “试卷 B”经教研室主任审批签字后送教务科印刷。 2、请命题教师用黑色水笔工整地书写题目或用 A4 纸横式打印贴在试卷版芯中。
期
201- 201 学年第 专业班级（教学班） 学期 课程代码 课程名称 考试日期
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(1) If the norm of the iterative matrix
B
is
B q 1 , please poof the following
x =5.3001 be an approximation to x =5.300186, then x retains _______________ significant x is __________________.
are orthogonal on [ 1,1] with respect to the B1 , B2 , C2 such that P0 ( x), P 1 ( x ), P 2 ( x)
w( x) . f ( x) x 4 on the interval
(b) Find the least squares approximating polynomial of degree two for the function
命题教师注意事项:1、主考教师必须于考试一周前将“试卷 A” 、 “试卷 B”经教研室主任审批签字后送教务科印刷。 2、请命题教师用黑色水笔工整地书写题目或用 A4 纸横式打印贴在试卷版芯中。
h 3 f (ti , wi ) f (ti 1 , wi 1 ), i 1,2,3,, n 1. 2
T *
*
*
digits,and the relative error of
formula:
x ( k ) x*
(2) Let (3) Let
q x ( k ) x ( k 1) 1 q
- 1 2 (2) Suppose x ( 2,1,3,4) , A , then x 1 =__________, A 2 =___________. 3 4
is a scheme of order two to solve the initial value problem for the ordinary differential equation:
Where
h
2.(12%) Establish the Jacobi iterative scheme and Gauss-Seidel iterative scheme for the following linear system:
4 x1 2 x2 x3 2, 2 x1 6 x2 3 x3 3, x1 2 x2 4 x3 5.
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数值分析
学分 命题教师
√ 选修□ 限修□ 考试形式：闭卷□ √ 课程性质：必修□ 系（所或教研室）主任审批签名
6.(12%) The Midpoint rule for the approximating with

1
1
f ( x)dx gives the value 12,the Composite Midpoint rule
7
i
0 1 2
1 2 4 3
2 3 12ຫໍສະໝຸດ f [2 , ,2 ] =_____________.
(6) Write an
xi
three-point formula to approximate
O(h 2 )
f ' ( x0 )
by
using
of
f ( xi ) f ' ( xi )
f ( x0 ) , f ( x0 h) , f（x0 2h) :__________________________________________________.