1.
2.
3.
4.
Solve the system of nonlinear equations by an iteration method
 x12  10 x1  x2 2  8  0
 x1( 0)  0
, with  ( 0)     (accurate to within 10-2).

2
x
x

x

10
x

8

0
1
2
 x2  0
 1 2
 1 2  1
Compute the inverse matrix of A=  2 1 0  by some elementary operations.


 1 1 2 
 2 0 1 2
1 1 0 2 
 by the Gaussian elimination.
Obtain the determinant of A= 
2  1 3 1 


 3  1 4 3
 1 1 0 
Find the third iteration obtained by the power method applied to 2 4 2  ,
 0 1 2 
1 
with x = 0  .
 
0 
5. Find a Lagrange interpolating polynomial to fit (2,0.5), (2.5,0.4), and (4,0.25).
Let p(x) denote it, and then compute p(3)=?
2s  3
6. Find the inverse Laplace transform of F(s)= 2
by (a) Heaviside‘s
s  3s  2
formula, (b) Liou’s method, and (c) Wang‘s method.
1 0 0
7. Write down the QR factorization for A= 1 1 0 , where R is an upper


1 1 1
triangular matrix and Q is an orthonormal matrix.
2
0 1
1
 1  1 3 2
 , where R is an upper
8. Write down the QR factorization for A= 
 1  1 3 2


 1 1  3 1 
triangular matrix and Q is an orthonormal matrix. (台大生醫電資所)
1 2
9. Write down the QR factorization for A= 0 1 . (交大電信所)


1 4
10. Check
the
stability
for
an
explicit
multistep
method:
4h
yi+1=yi-1+ [2 f (t i , y i )  f (t i 1 , y i 1 )  2 f (t i  2 , y i  2 )] . Is it strongly or weakly
3
stable?
(0)
11. Use composite Simpson‘s rule with h=0.5 to compute

4
0
e x dx  ?
12. The Romberg integration is used. If R11=0.25 and R22=0.2315, what is R21?
dy y 2  y

13. Use the Runge-Kutta method to solve
, 1  t  2, y(1)=-2 with h=0.5,
dt
t
and then compute y(1.5)=?
14. Use the finite-difference method to solve y”+4y=cos(x), 0  x  π/4,
y(0)=y(π/4)=0 with h=π/12.
15. Find out the approximate eigenvalues of y”+k2y=0, y(0)=y(1)=0 with n=4.
3 dx
?
16. Use the Gaussian quadrature method with n=2 (or n=3) to compute 
1 x
Table 1:
roots
coefficients
0.5773502692
1.0
-0.5773502692
1.0
Table 2:
roots
0.7745966692
0.0000000000
-0.7745966692
coefficients
0.5555555556
1.0000000000
0.5555555556
17. Why is the forward difference method of solving a parabolic differential equation
conditionally stable?
18. How do we combine both the Adams-Bashforth and Adams-Moulton methods
into a predictor-corrector method?
19. Find the minimum of f(x,y,z)=x2+y2+z2 by the steepest descent method. Initial
 x ( 0)  1


guess is given by  y ( 0)   1 .
 z ( 0)  1


20. Can we use the simplex method to solve an equation like x2+x+1=1? Give your
reasons.
x
x
0
0
21. Show that y(x)=1-  (u  x)du  4 (u  x) y(u)du is equivalent to y”+4y=1,
y(0)=1, and y’(0)=0.
22. Let z(n) and w(n) denote the convolution and correlation of x(n) and y(n),
respectively. Calculate z(1) and w(2) for x(0)=1, x(1)=2, x(2)=3, x(3)=4 and
y(0)=0.5, y(1)=1, y(2)=2, y(3)=3.
23. Why can FFT save much more CPU time in executing it than DFT?
24. What is the cubic spline interpolation method? What types of boundary
conditions are imposed upon?
25. How do you compute



f ( x)dx of which upper and lower limits approach
positive and negative infinities?