Chapter 7
Numerical Integration
7.1 Introduction
Numerical integration is an essential tool used by scientists and engineers to obtain
approximate values for definite integrals that cannot be solved analytically. For example, the
integral
0.5 x 2
e dx
0
h
Chapter 2
Solution of Equations in One Variable
2.1 Introduction
In Applied Mathematics, the most frequent problem is to find the values of x to satisfy the
equation
. Such values are called the roots of the equation and also known as the zeros
f ( x ) 0
Chapter 8
Solutions of Ordinary Differential Equations
8.1 Introduction
We shall consider the solution of differential equations satisfying certain conditions. Problems in
which all the initial conditions are specified at one point only are called initial
Chapter 3
Solution of System of Linear Equations
3.1 Introduction
System of linear equations occur in a variety of applications in the fields like elasticity, electrical
engineering, statistical analysis. The techniques and methods for solving system of l
Chapter 4
Interpolation & Finite Differences
4.1 Introduction
In many occasions we are given only a few discrete set of values. To study the behaviour of the
function through those points a technique known as interpolation is introduced. Polynomial is a
f
Chapter 1
Numbers, Errors and Accuracy
1.1 Introduction
Numerical Methods are widely used by scientists and engineers to solve problems. To
perform calculations in Numerical Methods, different calculating devices such as
calculators, computers are availab
Chapter 5
Curve Fitting and Spline Interpolation
5.1 Introduction
The purpose of curve fitting is to find the parameter values of the model function that closely match the data.
The fitted curves can be used to estimate the values of one variable correspo
Chapter 6
Numerical Differentiation
6.1 Introduction
Numerical differentiation is the process of finding derivatives numerically for a function
whose values are given in data form generated from an experiment. For evenly distributed
data points and if we
Fall 2015-2016
Exercise-3
1.
Solve the following system of equations by Gaussian elimination with pivoting giving
your answers to 2 decimal places
(a) 5 x 9 y 12 z 8, 20 x 4 y 7 z 10, 4 x 8 y 2 z 15
(b) 5 x 10 y 8 z 10, 15 x 6 y 7 z 20, 6 x 20 y 5 z 30
2.
Fall 2015-2016
Exercise 7
1.
Derive the trapezoidal rule on
[0, h]
of the form
which is
h
0 f ( x)dx Af (0) Bf (h)
exact for polynomial of degree
1.
Find the degree of precision of the trapezoidal rule. Estimate an error of this rule.
2.
Derive the follow
Fall 2015-2016
Exercise 5.1
1.
Find the least square line
x
y
y a bx
1
4.5
to the following data
0
6.7
2
9.2
4
11.5
5
15.6
2.
Students collected the following set of data to find the gravitational constant g. Use the
relation d = (gt2)/2, where d is the d
Fall 2015-2016
Exercise-4
Interpolation using divided differences
1.
The table below gives the values of x and f(x):
x
f(x)
-1
-12
1
-10
2
-6
3
44
5
558
(a) Construct a divided-difference table for the above data.
(b) Find the polynomial of least degree t
Fall 2015-2016
Exercise 6
1. The table below shows the values of
(a)
f (x)
at different values of x:
x
1.2
1.3
1.4
1.5
1.6
f(x)
2.624
3.237
3.947
4.763
5.697
Use two-point forward difference formula and Richardson extrapolation to estimate
a value of
.
f
Fall 2015-2016
Exercise-2
1.
Given that f ( x ) 6 sin( 2 x) 3x 1 .
a) Find the number of real roots of the equation f ( x ) 0 .
b) Apply Bisection method twice/thrice in the interval ( 1.4, 0.8) to find the new
smaller interval of this root.
c) Apply Seca
Chapter 8
1. (a) Given the initial value problem
with
.
y (2) 1.5
dy
2
2
y x 4
dx
Use the Taylor series method of order three to estimate a value of
.
y (2.1)
(b) Given the initial value problem
with
.
y (1) 1
dy
x 2 xy cos( x 1)
dx
Estimate a value of
u
AMERICAN INTERNATIONAL UNIVERSITY - BANGLADESH Dhaka, Bangladesh Faculty of Science & Information Technology
Department of Basic Science (Mathematics)
COURSE OUTLINE
Academic Term: Spring 2010-2011 ICourse Code and Title : MAT3101, Mahematical Methods of