It include the basic definition of curve fitting and it's applications in mathematical and non-mathematically with the help of linear algebra and matlab.

1. PES University, Bangalore
(Established under Karnataka Act No. 16 of 2013)
MAY 2020: IN SEMESTER ASSESSMENT (ISA) B.TECH. IV SEMESTER
UE18MA251- LINEAR ALGEBRA
MINI PROJECT REPORT
ON
Linear Algebra in Curve Fitting
Submitted by
1. Name: GowthamCR SRN:PES1201801581
2. Name:NithinChandraR SRN:PES1201801340
Branch & Section : EEE 4’A
PROJECT EVALUATION
(For Official Use Only)
Sl.No. Parameter Max Marks Marks Awarded
1 Background& Framingof the problem 4
2 Approachand Solution 4
3 References 4
4 Clarityof the concepts& Creativity 4
5 Choice of examplesand understandingof
the topic
4
6 Presentationof the work 5
Total 25
Name of the Course Instructor :
Signature of the Course Instructor :

2. Curve fitting is the process of constructing a curve, or mathematical
function that has the best fit to a series of data points, possibly subject to
constraints. Curve fitting can involve either interpolation, where an exact
fit to the data is required, or smoothing, in which a "smooth" function is
constructed that approximately fits the data. A related topic is regression
analysis, which focuses more on questions of statistical inference such as
how much uncertainty is present in a curve that is fit to data observed with
random errors. Fitted curves can be used as an aid for data visualization, to
infer values of a function where no data are available, and to summarize
the relationships among two or more variables. Extrapolation refers to the
use of a fitted curve beyond the range of the observed data and is subject
to a degree of uncertainty since it may reflect the method used to construct
the curve as much as it reflects the observed data.
(Source: https://en.wikipedia.org/wiki/Curvefitting#citenote-1)
Applications:
1. Mathematical:
 To find best fitting curve for real life applications using linear
algebra.
 Any non-polynomial function can be expressed as a polynomial
function.
Application of Curve Fitting in Indian Structures:
In architecture, curves are preferred mainly based on
distinguishing element of architecture. Curves such as parabolas and
hyperbola are referred as conics. They are used in architecture to
design arches in buildings and cooling towers in power plants. Oshin
Vartanian, Psychologist of the University of Toronto compiled 200
images of interior architecture and explains about curves in
architecture. He explains that curved design in architectural
structures uses our brains to tug at our hearts. Structures flushed with

3. curved design are more beautiful as it absorbs the brain activity and
affects our feelings, which in return could drive our preference.
Curve fitting in Neogothic architectural buildings:
Analysis of the shape of the cooling towers at SIPAT thermal
power plant:
The Sipat Super Thermal Power Station is located at Sipat in
Bilaspur district in state of Chhattisgarh. The power plant is one of
the coal-based power plants of NTPC. The first unit of the plant was
commenced on August 2008. Four induced draft cooling towers are
installed at this power plant. Originally, natural draft cooling towers
were cylindrical in shape. As the design of these types of towers
evolved and the towers were made increasingly larger, the
cylindrical shape was changed to hyperbolic, since hyperbolic shape
offers superior structural strength and resistance to ambient wind
loadings.

4. (Source:https://www.krishisanskriti.org/vol_image/03Jul201502073624.pdf)
2. Non-Mathematical:
Curve fitting used in physics under the field of thermodynamics
Our approach to find the best fitting curve:

5. What is in the code?
We’ve used matrices and inverse method to find the best fitting curve –
Let’s consider the degree of polynomial m=3. And the number of co-
ordinate points, which should be greater than or equal to m+1, to be n=4.
We know that any cubic equation should be of the form
Y = Ax3+Bx2+Cx+D.
Substituting ‘x’ and ‘Y’ values of each of the co-ordinate points. We get 5
equations each with 4 common unknowns A, B, C and D. Now we use
linear algebra to find these unknowns by inserting the values of ‘Y’ in a
matrix and the co-efficient of A, B, C and D in another matrix. Let's, name
these matrixes Q and P respectively.
P*A = Q, wherein A is the matrix of Constants.
A = inverse (P)*Q || if inverse of P exists or else
A = (inverse (transpose (P)*(P)))*(transpose (P)*Q) || in which left inverse
for the matrix ‘P’ is given by (inverse (transpose (P)*(P)))*(transpose (P))
We got all the constants which are placed back in the general equation.
The required polynomial if found out and it has been plotted in the graph.
Example-
1. for the input
a. Degree of the Polynomial (m) = 3
b. Number of co-ordinate points (n) = 4
c. (0, -12), (1, -12), (-1, -6), (2, 0) are the co-ordinate points which are in
(x, y) format.
Command Window:

6. This forms the curve shown below –
2. for the input
a. Degree of the Polynomial (m) = 3

7. b. Number of co-ordinate points (n) = 10
c. (-4, 0), (-1, 0), (2, 0), (0, -2), (1, -2.5), (-2, 2), (- 3, 7), (-3, 2.5), (4,
20), (5, 40.5) are the co-ordinate points which are in (x, y) format.
Command Window:
This forms the curve shown below –
Efficiency of the method we have used:

8.  The method we have used here is much simpler when compared to
other curve fitting methods such as linear and non-linear regression.
 Output is more accurate when number of co-ordinate points is more
that is n.
 Degree of polynomial is not restricted.
Thank You