数值分析试卷（英文版）
- 格式：pdf
- 大小：34.21 KB
- 文档页数：3
数值分析试卷（英⽂版）
1.(20%) Fill in the following blanks.(1) Let
*x =5.3001 be an approximation to x =5.300186, then *x retains _______________ significant
digits,and the relative error of
*x is __________________.
(2) Suppose T
x )4,3,1,2(-=, ??
=4321-A , then 1x =__________, 2A =___________.
(3) Suppose
*x is the root of the equation 0)(=x f of multiplicity )2(≥m m ,then the modified
Newton ’s iterative formula for finding *x is______________________________________________.
(4) Let
4)(x x f = and the cubic Lagrange interpolating polynomial for f on the nodes 0,1,2,3 is )(3x p .
Then the error )()(3x p x f -=_______________________________, and )(3x p =_____________________.
(5) If
1232)(36+++-=x x x x f , then the divided differences ]2,,2,2[610 f =____________, and
]2,,2[70 f =_____________.
(6)
Write
an
)(2h O three-point
formula to approximate
)
(0'x f by using of
)(0x f ,)(0h x f +,)20h x f +（:__________________________________________________.2.(12%) Establish the Jacobi iterative scheme and Gauss-Seidel iterative scheme for the following linear system:
.
542,3362,
224321321321=++=++=+-x x x x x x x x xTest the convergence of the Gauss_Seidel iterative scheme by using the norm of its iterative matrix.
3.(12%) (1) If the norm of the iterative matrix
B
is
1<=q B ,
please poof the following
formula:
)1()(*)(1---≤
-k k k x x q
q
x x (2) Let 1)(-=x xe x f and 5.00=x .Use Newton ’s method to find 2x .
(3) Let 1)(-=x xe x f ,5.00=x and 6.01=x .Use the Secant method to find 3x .
4.(12%) Use the following data to construct an interpolating polynomial
)(3x p of the degree three such that
)()(3i i x f x p = for 2,1,0=i ,and )()(1'1'
3x f x p =.
5.(12%) Suppose )()()(,)(,1)(,1)(21110x P C P B x x P B x x P x P x w k k k k k ----=-=≡≡ for
2≥k . (a) Determine 221,,C B B such that )(),(),(210x P x P x P are orthogonal on ]1,1
[- with respect to the weight function
)(x w .
(b) Find the least squares approximating polynomial of degree two for the function
4)(x x f = on the interval
]1,1[-
6.(12%) The Midpoint rule for the approximating
-1
1
)(dx x f gives the value 12,the Composite Midpoint rule
with 2=n gives 5,and Composite Simpson ’s rule giveshttp://www.doczj.com/doc/cf6d3e45ab00b52acfc789eb172ded630a1c9874.html e the fact that
)1()1(f f =- and
1)5.0()5.0(-=-f f to determine )1(),5.0(),0(),5.0(),1(f f f f f --.
7.(10%) Determine constant
d c b a ,,,that will produc
e a quadrature formula:
)1()1()1()1()('1
1
'df cf bf af dx x f +-++-=?- that has degree of 3.
8.(10%) (1)Suppose
)('
''t y exists on ),(b a .Show that the following two-step explicit scheme
[].1,,3,2,1,),(),(32
,,
1111100-=-+===--+n i w t f w t f h
w w w w i i i i i i αα
is a scheme of order two to solve the initial value problem for the ordinary differential equation:
]
,[),,()(,
)(,
)('1100b a t y t f t y t y t y ∈===αα
Where
n i ih a t n
a
b h i ,,2,1,0,, =+=-=
