Sample Test 1
20 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
1. Write out the cardinal functions Li (x) appropriate to the problem of interpolating the
following table, and give the Lagrange form of the interpolating polynomial:
x
f (x)
20 points
1
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
Sample Test 2
20 points
February 16, 2012
1. Taylors theorem can be used to show that centered-dierence formula to approximate
f (x0 ) can be expressed with an error formula
f ( x0 ) =
1
h2
h4 (5)
[f (x0 + h) f (
Sample Test 1
20 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
January 31, 2012
1. Write out the cardinal functions Li (x) appropriate to the problem of interpolating the
following table, and give the Lagrange form of the interpolating polynomial:
1
2
Sample Test 3
40 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
March 6, 2012
1. Consider the initial value problem
y = 2y + te3t ,
0 t 1,
y (0) = 0.
(1)
(a) Use Eulers method with h = 0.5 to approximate the solution to equation (1).
(b) The exact solt
Sample Test 3
40 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
March 6, 2012
1. Consider the initial value problem
y = 2y + te3t ,
0 t 1,
y (0) = 0.
(1)
(a) Use Eulers method with h = 0.5 to approximate the solution to equation (1).
Solution: Let f (t
MATH 4365: Numerical Analysis
Spring 2015
Computer Project: due on May 8 by 4PM
(1). Solve the following elliptic problem by a nite dierence method with the Gauss-Seidel
method for solving the resulting linear system
(P)
u = f in = (0, 1) (0, 1),
u = g on
an:
P| 60455f
I of
:cf? apoadg "9% 50V7é: '
I
I w A 73 5-mmef7c 057%: m 7777f:
I I. a {mmc'm :5 5
I
3m: 3% x lie >2<
I A0212 K} Ts am \nec 7n / n .
Xh
I >7=3 I MN
I
I
- . X
3m=zx\,x1x3>x'=
3
V
an 6332 5C3; 3
AItxl +41: Xx+Q3X3
I t \ ()(b X) X2) an) 6 43
MATH 4365: Numerical Analysis
Spring 2015
Computer Project # 1: due by 4 PM, Mar. 27, 2015
1. Let A be a 10 10 tridiagonal matrix and b a 10 1 vector given in the following:
3
4 1 0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2 4 1 0
0
0
0
0
0
1
0 2 4 1 0
0
0 2
Solutions for Homework 5
Zhuo Liu
April 17, 2015
#1 of Section 11.1
By applying the linear shooting method, numerical solution for h = 0.05 is:
xi
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
u1,i
0
0.077618
0.
Solutions for Homework 4
Zhuo Liu
April 10, 2015
1(c) of Section 9.1
1
A = 1
1
1
1
0
1
det(A I) = det 1
1
1
0 ,
1
1
1
1
0
0
1
=(1 )(2 2 1) = 0.
The roots above equation, or the eigenvalues for A are 1 = 1, 2 = 1 2,
for
3 = 1 + 2. The corresponding eigenv
Solutions for Homework 6
Zhuo Liu
April 24, 2015
#1 of Section 12.1
Part (a)
The nite dierence scheme for this equation is:
wi,j1 2wi,j + wi,j+1
wi1,j 2wi,j + wi+1,j
+
= 4,
h2
k2
where wi,j is numerical approximation to u(xi , yj ). If h = k, the above eq
Solutions for Homework 2
Zhuo Liu
March 05, 2015
1(c) of Section 5.4
Applying Modied Eulers formula:
h
(f (ti , wi ) + f (ti+1 , wi + hf (ti , wi )
2
wi + 0.25 (1 + wii )
h
wi
t
= wi + (1 +
+1+
)
2
ti
ti+1
wi+1 = wi +
with ti = 1 + 0.25i, i = 0, 1, 2, 3,
Solutions for Homework 3
Zhuo Liu
March 20, 2015
1(a),3(a) of Section 7.3
The formula for Jacobi method is:
1
(k+1)
xi
=
(bi
aii
(k)
aij xj ),
i = 1, ., n,
j=i
and the formula for Gauss-Seidel method is:
n
i1
(k+1)
xi
=
1
(k)
(k+1)
aij xj ),
aij xj
(bi
Sample Test 3
40 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
March 6, 2012
1. Consider the initial value problem
y = 2y + te3t ,
0 t 1,
y (0) = 0.
(1)
(a) Use Eulers method with h = 0.5 to approximate the solution to equation (1).
Solution: Let f (t
Sample Test 4
25 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
April 10, 2012
1. Apply the Linear Shooting method to solve
y = 4(y x), x (0, 1),
y (0) = 0, y (1) = 2
and write down the detailed algorithm using the 4th order Runge-Kutta method.
25 poin
Sample Test 1
20 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
January 31, 2012
1. Write out the cardinal functions Li (x) appropriate to the problem of interpolating the
following table, and give the Lagrange form of the interpolating polynomial:
1
2
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
Sample Test 2
20 points
February 16, 2012
1. Taylors theorem can be used to show that centered-dierence formula to approximate
f (x0 ) can be expressed with an error formula
f (x0 ) =
1
h2
h4 (5)
[f (x0 + h) f (x
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
Sample Test 2
20 points
February 16, 2012
1. Taylors theorem can be used to show that centered-dierence formula to approximate
f (x0 ) can be expressed with an error formula
f ( x0 ) =
1
h2
h4 (5)
[f (x0 + h) f (
Sample Test 3
40 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
March 6, 2012
1. Consider the initial value problem
y = 2y + te3t ,
0 t 1,
y (0) = 0.
(1)
(a) Use Eulers method with h = 0.5 to approximate the solution to equation (1).
(b) The exact solt
Sample Test 3
40 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
March 6, 2012
1. Consider the initial value problem
y = 2y + te3t ,
0 t 1,
y (0) = 0.
(1)
(a) Use Eulers method with h = 0.5 to approximate the solution to equation (1).
Solution: Let f (t
Sample Test 4
25 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
April 10, 2012
1. Apply the Linear Shooting method to solve
y = 4(y x), x (0, 1),
y (0) = 0, y (1) = 2
and write down the detailed algorithm using the 4th order Runge-Kutta method.
25 poin
Add. Sample Pbs
25 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
April 24, 2012
1. What straight line best ts the data
x1234
y0112
in the least-squares sense?
25 points
2. Let g0 , g1 , . . ., gn be a set of functions such that
m
gi (xk )gj (xk ) = 0,
Sample Test 1
20 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
1. Write out the cardinal functions Li (x) appropriate to the problem of interpolating the
following table, and give the Lagrange form of the interpolating polynomial:
x
f (x)
20 points
1
Sample Test 1
20 points
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
January 31, 2012
1. Write out the cardinal functions Li (x) appropriate to the problem of interpolating the
following table, and give the Lagrange form of the interpolating polynomial:
1
2
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
Sample Test 2
20 points
February 16, 2012
1. Taylors theorem can be used to show that centered-dierence formula to approximate
f (x0 ) can be expressed with an error formula
f (x0 ) =
1
h2
h4 (5)
[f (x0 + h) f (x
NUMERICAL ANALYSIS
Math 4365 (Spring 2012)
Sample Test 2
20 points
February 16, 2012
1. Taylors theorem can be used to show that centered-dierence formula to approximate
f (x0 ) can be expressed with an error formula
f ( x0 ) =
1
h2
h4 (5)
[f (x0 + h) f (