Artificial Variable
Techniques
Big Mmethod
Lecture 6
Abstract If in a starting simplex tableau,
we dont have an identity submatrix (i.e. an
obvious starting BFS), then we introduce
artificial variables to have a starting BFS.
This is known as artificial
MATRIX FORMULATION
OF THE LPps
In this lecture we shall look at the matrix
formulation of the LPPs. We see that the
Basic feasible solutions are got by solving
the matrix equation BX b
where B is a mm nonsingular submatrix
of the contraint matrix of the L
In this lecture we shall look at
some miscellaneous LPPs. Each
problem will illustrate a certain
idea which will be explained when
the problem is discussed.
Problem 6 Problem set 3.4A Page 97
Maximize
z 2 x1 4 x2 4 x3 3 x4
Subject to the constraints
x1 x2
The Simplex algorithm
Abstract: In this lecture we discuss the
computational aspects of the Simplex
algorithm. We shall see how a LPP is put
into a simplex tableau. Starting from a BFS,
we explain how to proceed step by step till
we reach the optimal solu
Algebraic Solution of LPPs  Simplex
Method
To solve an LPP algebraically, we first put it
in the standard form. This means all
decision variables are nonnegative and all
constraints (other than the nonnegativity
restrictions) are equations with nonnegati
Previous page
12.4
ORTHOGONAL COLLOCATION FOR SOLVING PDEs
In Chapters 7 and 8, we presented numerical methods for solving ODEs of
initial and boundary value type. The method of orthogonal collocation discussed
in Chapter 8 can be also used to solve PDEs.
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, Pilani
Pilani Campus
Instruction Division
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI
INSTRUCTION DIVISION
FIRST SEMESTER 20152016
Course Handout (Part II)
Date 03/08/2015
In addition to Part I (General Hand
QR Method
Dr. Anil Kumar
CC205
The QR Method:
If a matrix A is diagonal or lower or uppertriangular, its
eigenvalues are just the elements on the diagonal.
This suggests that if we can transform A to upper triangular, we
have its eigenvalues.
The Gaus
INTERPOLATION
1.
2.
3.
4.
5.
Lagrange Interpolation and error
Newton form of Interpolation
Newton's Forward and Backward form
Piecewise Polynomial Interpolation
Cubic Spline Interpolation
1
Interpolation is the process of finding the
most appropriate est
CHAPTER 4
INTERPOLATION
Lagrange Interpolation and error
2.
Newton form of Interpolation
3.
Newton's Forward and Backward form
4.
Piecewise Polynomial Interpolation
5. Cubic Spline Interpolation
1.
1
Interpolation
If we have values like (x0, y0), (x1, y1
Identifying significant digits
The rules for identifying significant digits when writing or interpreting numbers are as
follows:
All nonzero digits are considered significant. For example, 91 has two significant
digits (9 and 1), while 123.45 has five si
Eigenvalues and
Eigenvectors
Eigenvalues & Eigenvectors
Find a scalar and its corresponding nonzero vector x
for a given square matrix A such that
Ax = x
(1)
If A is n x n matrix, then the value , for which the
equation (1) has non trivial solution is cal
Eigen Values and Eigen Vectors
Instructor: Dr. Atul Gaur
April 23, 2008
0.1
Power Method
This method is used to nd an eigen value
of greatest modulus of a matrix A and its
corresponding eigen vector.
Let A be n n real matrix. Assume that
A has n linearly
Birla Institute of Technology and Science, PilaniK. K. Birla Goa Campus
Second Semester 20112012
Numerical Analysis
(AAOC C341)
Tutorial Sheet4
Answers
1. Using Newtons method, the iteration values are
p1 = 0.88034,
p2 = 0.86568.
If p0 = 0, then f (p0
Birla Institute of Technology and Science, PilaniK. K. Birla Goa Campus
Second Semester 20112012
Numerical Analysis
(AAOC C341)
Tutorial Sheet3
Answers
1. The interval is I = [2, 1] and the suitable iterative function is
g(x) = (3x 8)1/5 .
We note that
Birla Institute of Technology and Science, PilaniK. K. Birla Goa Campus
Second Semester 20112012
Numerical Analysis
(AAOC C341)
Tutorial Sheet2
Answers
1. The 3rd iteration: p3 = 0.625
2. The number of iterations are required n = 10.
3. The 8th iterati
+
G

H
Handout
Operation of the course
Common Lecture
Multiple Tutorial
Sections
Evaluation Components
Test I
: 60 min : 60 : C B
Test II
: 60 min : 60 : O B*
Online Test :
Compre
*
*
: 60 : CB
: 3 Hours : 120 : CB
Open Text Book and Class Notes only
*
Chapter 5
Time Response
Analysis
Test signals
Impulse
Step
Ramp
Test signals
Sinusoidal.
(frequency response)
Laplace Transforms
Impulse (t)
step u1 (t)
Ramp t u1 (t)
1
1/s
2
1/s
Order of
a
system is ?
Order of
a
system is order
of the differential
MATLAB
 Matrix Laboratory
Contains
base program
plus tool boxes
(e. g. Control System Tool box)
To invoke matlab
type
cd\matlab\bin
at DOS prompt
C:\
> cd\matlab\bin
C:\MATLAB\BIN
Now type
matlab
Matlab prompt
will appear as
for demo, type demo
on gettin
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
GOA CAMPUS
INSTRUCTION DIVISION
FIRST SEMESTER 20082009
Course Handout (Part II)
Date: 02/08/2008
In addition to partI (General Handout for all courses appended to the timetable) this
portion gives furth
Chapter 8
Input is sinusoidal
Steady state output
will be sinusoidal with
same frequency for
linear systems
amplitude and phase
will vary with
frequency
Example :
1
G (s) =
s+1
r (t) = sin t
R (s) = 2
s +
C(s) =
2
(s+1)(s + )
2
2
C (s) =
2
(1+ )
1
s
Chapter 9
Given open loop
frequency response ,
determine closed
loop system stability
take
(s z1) (s z2 ) .
q(s) =
(s p1) (s p2 ) .
s  plane
q(s) plane
Taking a closed
contour in s plane,
find its mapping in
q(s) plane.
s
P2
.
P3
.
P1
.
. <
P4
q (s)
Chapter 7
ROOT LOCUS
R +
E

K
G
H
C
C
KG ( s)
=
R 1 + KG ( s) H ( s)
1+ K G (s) H (s) = 0 is called
the characteristic equation
of the system. Its roots are
the closed loop poles.
Root locus is the locus
of the roots of the
characteristic
equation as K i
Chapter 4
Control
System
Components
DC Control Components
potentiometer
tachogenerator
dc servo motor
dc amplifier
amplification of
slowly varying
signals is
difficult 
hence.
ac / carrier
control
Systems
ac control system components
synchro pair
a
Chapter 2
Mathematical Modelling
Electrical Systems
R
L
C
Voltage (e), Current (i)
Mathematical Modelling
iR = e / R
iC = C de / dt
iL = (1/L) e dt
Mechanical Systems
M
K
velocity ( v)
force ( f )
B
fM = M dv/dt
fK = K v dt
fB = Bv
Force  current analogy
Chapter 6
Algebraic Criterion
for stability
Stability of a system
means
bounded input
produces bounded
output
Stability of a system
means
In the absence of input
output tends to zero
C
G
=
R 1 + GH
1+GH = 0 is called the
characteristic equation
of the s
Chapter 3
Comparing
closed loop
&
open loop
Control systems
Voltage control of a d.c.
generator
Components :
dc generator
feed back POT
reference voltage source
dc amplifier
Ref
Source
Ra
ia
Rf , Lf
if
G
eg
vt
Load
vr : reference voltage
input
vb
: feed b