Unit v

CONTENTS
• Matrices and Linear systems of equations
• Eigen values and eigen vectors
• Real and complex matrices and Quadratic forms
• Algebraic equations transcendental equations and
Interpolation
• Curve Fitting, numerical differentiation & integration
• Numerical differentiation of O.D.E
• Fourier series and Fourier transforms
• Partial differential equation and Z-transforms

TEXT BOOKS
• 1.Mathematical Methods, T.K.V.Iyengar, B.Krishna
Gandhi and others, S.Chand and company
• Mathematical Methods, C.Sankaraiah, V.G.S.Book
links.
• A text book of Mathametical Methods,
V.Ravindranath, A.Vijayalakshmi, Himalaya
Publishers.
• A text book of Mathametical Methods, Shahnaz
Bathul, Right Publishers.

REFERENCES
• 1. A text book of Engineering Mathematics,
B.V.Ramana, Tata Mc Graw Hill.
• 2.Advanced Engineering Mathematics, Irvin Kreyszig
Wiley India Pvt Ltd.
• 3. Numerical Methods for scientific and Engineering
computation, M.K.Jain, S.R.K.Iyengar and R.K.Jain,
New Age International Publishers
• Elementary Numerical Analysis, Aitkison and Han,
Wiley India, 3rd Edition, 2006.

UNIT HEADER
Name of the Course:B.Tech
Code No:07A1BS02
Year/Branch:I Year CSE,IT,ECE,EEE,ME,
Unit No: V
No.of slides:29

UNIT INDEX
UNIT-V
S.No. Module Lecture PPT Slide
No. No.
1 Curve Fitting L1-7 8-18

2 Numerical L8-12 19-25
Differentiation
3 Numerical Integration L13-16 26-29

UNIT – V
PART – A
CURVE FITTING

LECTURE-1
INTRODUCTION : In interpolation, We have
seen that when exact values of the function
Y=f(x) is given we fit the function using
various interpolation formulae. But sometimes
the values of the function may not be given.
In such cases, the values of the required
function may be taken experimentally.
Generally these expt. Values contain some
errors. Hence by using these experimental
values . We can fit a curve just approximately
which is known as approximating curve.

Now our aim is to find this approximating curve
as much best as through minimizing errors of
experimental values this is called best fit
otherwise it is a bad fit.
In brief by using experimental values the
process of establishing a mathematical
relationship between two variables is called
CURVE FITTING.

LECTURE-2
METHOD OF LEAST SQUARES
Let y1, y2, y3 ….yn be the experimental values of
f(x1),f(x2),…..f(x n) be the exact values of the
function y=f(x). Corresponding to the values
of x=xo,x1,x2….x n .Now error=experimental
values –exact value. If we denote the
corresponding errors of y1, y2,….yn as e1,e2,e3,
…..en , then e1=y1-f(x1), e2=y2-f(x2 )

• e3=y3-f(x3)…….en=yn-f(xn).These errors
e1,e2,e3,……en,may be either positive or
negative.For our convenient to make all
errors into +ve to the square of errors i.e
e12,e22,……en2.In order to obtain the best fit
of curve we have to make the sum of the
squares of the errors as much minimum
i.e e12+e22+……+en2 is minimum.

METHOD OF LEAST SQUARES

• Let S=e12+e22+……+en2, S is minimum.When S
becomes as much as minimum.Then we obtain
a best fitting of a curve to the given data, now
to make S minimum we have to determine the
coefficients involving in the curve, so that S
minimum.It will be possible when
differentiating S with respect to the
coefficients involving in the curve and
equating to zero.

LECTURE-3
FITTING OF STRAIGHT LINE
Let y = a + bx be a straight line
By using the principle of least squares for solving the
straight line equations.
The normal equations are
∑ y = na + b ∑ x
∑ xy = a ∑ x + b ∑ x2
solving these two normal equations we get the values
of a & b ,substituting these values in the given
straight line equation which gives the best fit.

LECTURE-4
FITTING OF PARABOLA
Let y = a + bx + cx2 be the parabola or second degree
polynomial.
By using the principle of least squares for solving the
parabola
The normal equations are
∑ y = na + b ∑ x + c ∑ x2
∑ xy = a ∑ x + b∑ x2 + c ∑ x3
∑ x2y = a ∑ x2 + b ∑ x3 + c ∑ x4
solving these normal equations we get the values of a,b
& c, substituting these values in the given parabola
which gives the best fit.

LECTURE-5
FITTING OF AN
EXPONENTIAL CURVE
The exponential curve of the form
y = a ebx
taking log on both sides we get
logey = logea + logee bx
logey = logea + bx logee
logey = logea + bx

• Y = A+ bx
• Where Y = logey, A= logea
• This is in the form of straight line equation
and this can be solved by using the straight
line normal equations we get the values of A &
b, for a =eA, substituting the values of a & b in
the given curve which gives the best fit.

EXPONENTIAL CURVE
The equation of the exponential form is of the form y =
abx
log ey = log ea +log ebx
Y = A + Bx
where Y= logey, A= logea , B= logeb
this is in the form of the straight line equation which
can be solved by using the normal equations we get
the values of A & B for a = eA
b = eB substituting these values in the equation which
gives the best fit.

LECTURE-7
FITTING OF POWER CURVE
Let the equation of the power curve be
y = a xb
log ey = logea + log exb
Y = A + Bx
this is in the form of the straight line equation which
can be solved by using the normal equations we get
the values of A & B, for a = eA b= eB, substituting these
values in the given equation which gives the best fit.

LECTURE-8
NUMERICAL DIFFERENTIATION
The process by which we can find the derivative
of a function i.e dy/dx for some particular
value of independent variable x is called
Numerical Differentiation.
The problem of numerical differentiation are to
be solved by approximating the function using
interpolation formula and then differentiating
this formula as many times as desired.

LECTURE-9
NEWTON’S FORMULA
• If the values of the argument are equally
spaced and if the derivative is required for
some values of given x lying in the begin of
the table, we can represent the function by
Newton- Gregory forward interpolation
formula.
• If the value of dy/dx is required at a point near
the end of the table, we have to use Newtons
backward formula.

LECTURE-10
CENTRAL DIFFERENCE

– If the derivative dy/dx is to be found at some point
lying near the middle of the tabulated value we
have to use central difference interpolation formula

• While applying these formulae, it must be observed
that the table of values defines the function at these
points only and does not completely define the
function and hence the function may not be
differentiable at all.

•
• Derivatives by using Forward Difference
Formula :

• The first order derivative of the function is given
by
• (dy/dx)x=xo = 1/h (∆yo – 1/2∆2yo + 1/3 ∆3yo-
¼ ∆4yo + ………)

• (d2x/dy2)x=xo= 1/h2 (∆2yo – ∆3yo + 11/12 ∆4yo- 5/6
∆5yo + ………)

LECTURE-11
Derivatives by using Backward Difference Formula :
The first order derivative of the function is given by
(dy/dx)x=xn = 1/h( yn+1/2 2yn+1/3 3yn+1/4 4yn+
∇

……………..) ∇ ∇ ∇
∇
(d2y/dx2)x=xn = 1/h( 2yn+ 3yn+11/12 4yn+5/6 yn+
5

……………..) ∇ ∇ ∇
∇

LECTURE-12
Striling’s formula is given by using central
difference is
(dy/dx)x=xo = 1/h ( ∆y0 +∆y-1/2 – 1/6 ∆3y-
1+∆3y-2 /2 + 1/30 ∆5y-2+ ∆5y-3/2 +……..)

LECTURE-13
NUMERICAL INTEGRATION
• Numerical Integration is a process of
finding the value of a definite
integral.When a function y = f(x) is not
known explicity. But we give only a set of
values of the function y = f(x)
corresponding to the same values of x.
This process when applied to a function of
a single variable is known as a
quadrature.

• For evaluating the Numerical Integration
we have three important rules i.e
• Trapezoidal Rule
• Simpsons 13 Rule
• Simpsons 38 th Rule

LECTURE-14
• Trapezoidal Rule : The Trapezoidal Rule of
the function y = f(x) is given by
• f (x ) dx= h2 ( y0 + yn ) + 2 ( y1 + y2 + y3+….+
yn-1 )
• f (x ) dx = h2 (sum of the first and last terms )
+ ( sum of the remaining terms )

LECTURE-15
• Simpson’s 13 rd Rule : The Simpson’s 13
rd Rule of the function f ( x ) is given by
• f ( x ) dx = h/3 ( y0 + yn ) + 4 ( y1 + y3 + y5 +
…..yn-1 ) + 2 ( y2 + y4 + y6 + …..)

LECTURE-16
• Simpson’s 3/8 th Rule : The Simpson’s 3/8
th rule for the function f ( x ) isgiven by
• f ( x ) dx = 3h/8 ( y0 + yn ) + 2 ( y3 + y6 + y9 +
……) + 3 ( y1 + y2 +y4 +……)
• f (x ) dx = 3h/8 ( sum of the first and the
last term) + 2 ( multiples of three ) + 3
( sum of the remaining terms )

Unit v

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