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 P
MATH - 4446 Homework 3
DUE: July 18, 2017 at 11:00am
Instructions:
You must show all work for full credit.
You may collaborate with others (including your teacher), but the work submitted must be yo
MATH - 4446 Homework 2
DUE: July 12, 2017 at 5:00 pm
Instructions:
You must show all work for full credit.
You may collaborate with others (including your teacher), but the work submitted must be yo
MATH - 4446 Homework 5
DUE: July 27, 2017 at 5:00pm
Instructions:
You may collaborate with others (including your teacher), but the work submitted must be your own.
Your solutions must be submitted
MATH - 4446 Homework 1
DUE: July 10, 2017 at 11:00 am
Instructions:
You must show all work for full credit.
You may collaborate with others (including your teacher), but the work submitted must be y
MATH - 4446 Homework 4
DUE: July 24, 2017 at 5:00 pm
Instructions:
You may collaborate with others (including your teacher), but the work submitted must be your own.
Your solutions must be submitted
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
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. Determin
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 ) +
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 e
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
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) +
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
Notes for Numerical Analysis
Math 5466
by
S. Adjerid
Virginia Polytechnic Institute
and State University
(A Rough Draft)
1
2
Contents
1 Polynomial Interpolation
1.1 Review . . . . . . . . . . . . . .