CHAPTER 11
Numerical Differentiation
and Integration
Differentiation and integration are basic mathematical operations with a wide
range of applications in many areas of science. It is therefore important to have
good methods to compute and manipulate der

1. The computation of the expression
G
1 8 1
2
2
involves the difference of small numbers when < 1. Obtain the value of G for = 0.001
and estimate the corresponding relative error, performing operations by rounding all
mantissas to six decimals (use doub

1. Find a root of the following equation using Muellers method to an approximate error of
:
a 0.1%
x 4 sin x e x 0
Take three starting values as 1, 2 and 3.
2. Find the root of the polynomial using Bairstows method using = 0.01%.
x3 - 21x2 + 129x - 234 =

1. For the circuit shown in the figure, find the currents through the elements using (a) Gauss
elimination (b) Gauss Jordan and (c) Gauss Seidel methods. (Use R 1=10 Ohm; R2=20 Ohm;
R3=40 Ohm; VA-VC = 200 Volt; VB-VC = 100 Volt)
R1
A
R2
i1
R3
i3
B
i2
C
2.

1. Consider the following Matrix:
7 2 1
2 10 2
1 2 7
a) Obtain the Equation of the Characteristic polynomial using Fadeev-Leverrier Method.
b) Perform one complete iteration of the Bairstows method with the starting values as r =
18 and s = 10. Compu

1. The mass of a radioactive substance is measured at 2-day intervals till 8 days.
Unfortunately, the reading could not be taken at 6 days due to equipment malfunction. The
following table shows the other readings:
Time (d)
0
2
4
8
Mass (g)
1.000
0.7937
0

1. Derive a finite difference approximation for
f j
in terms of
fj
,
f j 1
and
f j 2
. What is
the order of accuracy of this approximation?
2. The location of an object at various times was measured as follows:
Time (min)
Distance (cm)
0
0
1
3
2
14
3
39
4

1.
The cost of fuel consumed by a truck was assumed to be linearly related to the travel
distance and the load carried. Over a certain period, the following data was recorded by the
driver. Obtain the underlying relationship (add the constraint that there

1. The amount of lowering of water level, s, in a well at a time t, due to pumping from
groundwater is governed by an equation of the form s=A W(u), where A is a constant
(proportional to the discharge), W is called the Well Function, and u is inversely p

27. Solve the differential equation dy/dt = 100 y + 99 e t with the initial condition y(0)=2
using the Eulers method to obtain the value of y at t=0.1. Use time steps of (a) 0.01 (b) 0.02
and (c) 0.025. Find the analytical solution and compare the errors

PLASTIC DEFORMATION
Dislocations and their role in
plastic deformation
What are dislocations?
Dislocations are line defects that exist in
metals
There are two types of dislocations:
edge and screw
The symbol for a dislocation is
The dislocation density in

ESO 208a: Computational Methods in Engineering
Programming Assignment I
Date of Circulation: 22.08.2014
Evaluation Start on: 25.08.2014
Write a program (using C or MATLAB) for the solution of a system of linear
equations using Gauss Elimination. Incorpora

ESO 208a: Computational Methods in Engineering
Programming Assignment II
Date of Circulation: 21.09.2014
Evaluation Start on: 26.09.2014
Write a program (using C or MATLAB) for a given data set to obtain the cubic spline.
The user should have the followin

ESO 218: Computational Methods in Engineering
Programming Assignment III
Date of Circulation: 26.10.2014
Evaluation Start on: 30.10.2014
Write two programs (using C or MATLAB) for the following problems:
(a) Solve the following IVP using 4th order Runge K

ESO 208a: Computational Methods in Engineering
2014-15, 1st Semester, Second Quiz
Name:
Roll No.:
Section no.:
Time 30 minutes
Full Marks 10
1. Using Gauss Seidel method, find the solution of the following equations with a relative
error less than 2%. Use

i
i
i
rjlfdm
2007/6/1
page 149
i
Copyright 2007 by the Society for Industrial and Applied Mathematics
This electronic version is for personal use and may not be duplicated or distributed.
Chapter 7
Absolute Stability for
Ordinary Differential
Equations
7.

Lecture 5
Successive Overrelaxation Method (SOR)
Jinn-Liang Liu
2011/4/10
The SOR is devised by applying extrapolation to GS. This extrapolation
takes the form of a weighted average between the previous iterate and the
current GS iterate successively for

Applied Mathematics, 2012, 3, 1583-1592
http:/dx.doi.org/10.4236/am.2012.311218 Published Online November 2012 (http:/www.SciRP.org/journal/am)
An Algorithm to Optimize the Calculation of the Fourth
Order Runge-Kutta Method Applied to the Numerical
Integr

Numerical Integration
Romberg Extrapolation
Acceleration
The term acceleration is a term sometimes used in numerical analysis that refers
to how you can improve the results of an iterative algorithm by applying another
algorithm to it.
In the example we u

Chebyshev Polynomials
Reading
Problems
Differential Equation and Its Solution
The Chebyshev differential equation is written as
(1 x2 )
d2 y
dx2
x
dy
dx
+ n2 y = 0
n = 0, 1, 2, 3, . . .
If we let x = cos t we obtain
d2 y
dt2
+ n2 y = 0
whose general solut

2.3 ERROR ANALYSIS AND NORMS
37
like this at each step we can eliminate all variables both below and above the diagonal.
The end result is an equivalent diagonal system, thus eliminating the need to backsolve
to obtain the x;'s. This is called the Gauss-J

1
PSO201A: Quantum Physics
Semester II, 2013-14; IIT Kanpur
Homework # 5
(Due in class on Monday, March 31st, 2014)
Problem 5.1: Essential Concepts (5+5+5+5+5+5+5+10=45 marks)
(a) Do the following three vectors form a basis:
1
|1i = 1 ;
0
1
|2i = 0 ;
1
3

Online travel portals MakeMyTrip and ibibo announce merger
Makemytrip and Ibibo have agreed to merge in a stock transaction, representing the coming together of Indias
largest travel booking portals and presaging what many believe is an impending consolid

Birla Institute of Technology & Science, Pilani
Work Integrated Learning Programmes Division
M.S. Software Engineering at Wipro Technologies (WASE)
First Semester 2015 2016
Mid Semester Examination (Regular)
Course Number
Course Title
Type of Exam
Weighta

F12
Birla Institute of Technology & Science, Pilani
Work Integrated Learning Programmes Division
M. S (Software Engineering) at Wipro Technologies (WASE)
I Semester 2015 - 2016
MID Semester Examination (Regular)
Course Number:
Course Title:
Type of Exam:

IS616FinalPaper
Thefinalpaperistoactasasummationofthetermsstudy.Wehavecoveredalargenumberof
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