Homework 1
(Due: Jan 24, 2012)
1. Consider the following interpolating points for the function f (x) = sin x:
xi
f (xi )
1
0
1.3
-0.80902
1.6
-0.95106
(a) Construct a linear interpolating polynomial P1 (x) to approximate f (1.4). Compute P1 (1.4) and erro
Homework 11
(Due: May 1, 2012)
1. Use Lagrange polynomial based on tn+1 , tn , tn1 to derive the Adams-Moulton twostep method
wn+1 = wn +
h
[5f (tn+1 , wn+1 ) + 8(tn , wn ) f (tn1 , wn1 )]
12
for dierential equation y = f (t, y ).
2. Consider a two-step m
Homework 9
(Due: Apr 12, 2012)
1. Section 5.1 #4.
2. Consider the initial value problem
2
y = y + t2 et ,
t
1 t 2,
y (1) = 0,
with exact solution y (t) = t2 (et e).
(a) Use Eulers method with h = 0.5 to calculate this problem by hand. Show all
your steps.
Homework 8
(Due: Mar 27, 2012)
1. Use the Gaussian quadratures with n = 2 and n = 3 to approximate the following
integrals, and calculate the errors.
1
(a)
1
1
1
dx.
x2 + 1
x2 ex dx.
(b)
0
2. Determine the constants xi and ci in the Gaussian quadrature wi
Homework 7
(Due: Mar 20, 2012)
1. Use Simpsons rule with the error term to derive the following composite Simpsons
rule
n
n
1
2
2
b
h
h4 (4)
f ( )(b a),
f (x0 ) + 4
f (x)dx =
f (x2i1 ) + 2
f (x2i ) + f (xn )
3
180
a
i=1
i=1
where h = (b a)/n, xi = a + (i
Homework 6
(Due: Mar 13, 2012)
1. Approximate
1
x4 dx
0.5
using the trapezoidal rule, Simpsons rule, and the midpoint rule. Find the error
bounds for each approximation, and compare with the acutual errors.
2. Section 4.3 #16
3. Section 4.3 #18
4. Section
Homework 4
(Due: Feb 14, 2012)
1. (Section 3.5 #11) A natural cubic spline S on [0,2] is dened by
S (x) =
S0 (x) = 1 + 2x x3 ,
if 0 x < 1,
S1 (x) = 2 + b(x 1) + c(x 1)2 + d(x 1)3 , if 1 x 2.
Find b, c, and d.
2. (Section 3.5 #12) A clamped cubic spline s
Homework 3
(Due: Feb 7, 2012)
1. The Hermite polynomial H (x) (of degree at most 2n + 1) that agrees with f and f
at interpolating points x0 , x1 , , xn is given by
n
n
f (xj )Hj (x),
f (xj )Hj (x) +
H (x) =
i=0
j =0
where
Hj (x) = [1 2(x xj )L (xj )]L2
Homework 2
(Due: Jan 31, 2012)
1. The following table is generated when the Nevilles method is used to approximate
f (1.5).
x0 = 0
x1 = 0.5
x2 = 1
x3 = 2
P0
P1
P2
P3
=0
=1
=3
=y
P0,1
P1,2
P2,3
P0,1,2
P1,2,3
P0,1,2,3 = 4
Find y . You may use the code (Nevi