This document discusses roots of equations and methods to find them. It provides a brief history of solving quadratic equations from ancient Indian, Babylonian, Chinese, Greek and Indian mathematicians. It then describes three main methods to find roots: the bisection method, Regula-Falsi method, and Newton's Rapshon method. It explains each method in 1-2 sentences and provides examples of applications for the bisection method in shot detection in video content and locating periodic orbits in molecular systems. It also provides a brief example of applying the Regula-Falsi method to predict trace quantities of air pollutants from combustion reactions.

1. Roots Of Equations
Class : TD1
Branch : Information Technology
Group Member : Jay Mehta (91600104040)
Helly Mehta (91600104030)
Pranav Patel (91600104026)
Tanmay Patel (91600104033)
Akash Ladani (91600104058)

2. History
 In the Sulba Sutras in ancient India circa 8th
century BC quadratic equations of the form ax2 =
c and ax2 + bx = c were explored using
geometric methods. Babylonian mathematicians
from circa 400 BC and Chinese mathematicians
from circa 200 BC used the method of
completing the square to solve quadratic
equations with positive roots, but did not have a
general formula.

3.  Euclid, the Greek mathematician, produced a more abstract geometrical
method around 300 BC.
 Pythagoras and Euclid used a strictly geometric approach, and found a general
procedure to solve the quadratic equation.
 In his work Arithmetica, the Greek mathematician Diophantus solved the
quadratic equation, but giving only one root, even when both roots were
positive.
 In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit
(although still not completely general) solution of the quadratic equation

4. History of Roots of Equation
 Root, in mathematics, a solution to an equation, usually expressed as
a number or an algebraic formula.
 In the 9th century, Arab writers usually called one of the equal
factors of a number jadhr (“root”), and their medieval European
translators used the Latin word radix (from which derives the
adjective radical).

5. Method To Find Roots
• In this topic we are examining equations with one independent variable.
• These equations may be linear or non-linear
• Non-linear equations may be polynomials or generally non-linear equations
• A root of the equation is simply a value of the independent variable that
satisfies the equation
• Linear: independent variable appears to the first power only, either alone or
multiplied by a constant
• Nonlinear:
– Polynomial: independent variable appears raised to powers of positive
integers only
– General non-linear: all other equations

6. Three methods to find Roots
1. Bisection Method
2. Regula-Falsi Method
3. Newton’s Rapshon method
Intermediate Value property
• If 𝑓 𝑥 is continuous in [a,b] and 𝑓 𝑎 and 𝑓 𝑏 have different signs then the
equations 𝑓 𝑎 = 0 has at least one root between x=a and x=b.

7. 1. Bisection Method
• The bisection method in mathematics is a
root finding method which repeatedly bisects
an interval and then selects a subinterval in
which a root must lie for further processing.
• It is a very simple and robust method, but it
is also relatively slow. Because of this, it is
often used to obtain a rough approximation
to a solution which is then used as a starting
point for more rapidly converging methods
.The method is also called the binary search
method or the dichotomy method
Fig 1.1 Graphical representation of
Bisection method

8. Flowchart of Bisection method to implement C Program

9. 2. Regula Falsi Method
Most numerical equation-solving methods usually converge faster than Bisection.
The price for that is that some of them (e.g. Newton’s method and Secant) can fail
to converge at all, and all of them can sometimes converge much slower than
Bisection—sometimes prohibitively slowly. None can guarantee Bisection’s
reliable and steady guaranteed convergence rate. Regula Falsi, like Bisection,
always converges, usually considerably faster than Bisection—but sometimes
much slower than Bisection.
When numerically solving an equation manually, by calculator, or when a
computer program run has to solve equations so many times that the speed of
convergence becomes important, then it could be preferable to first try a usually-
faster method, going to Bisection only if the faster method fails to converge, or
fails to converge at a useful rate.

10. It is also known as false position method.
It is same as bisection method.
In bisection method
𝑐 =
𝑎 + 𝑏
2
But in regula falsi method
𝑐 =
𝑎𝑓 𝑏 − 𝑏𝑓 𝑎
𝑓 𝑏 − 𝑓 𝑎

11. The method starts with a function f defined over the real numbers x, the
function's derivative f ′, and an initial guess x0 for a root of the function f. If the
function satisfies the assumptions made in the derivation of the formula and the
initial guess is close.
Geometrically, (x1, 0) is the intersection of the x-axis and the tangent of
the graph of f at (x0, f (x0)).
The process is repeated as
𝑥 𝑛+1 = 𝑥 𝑛 −
𝑓 𝑥 𝑛
𝑓′ 𝑥 𝑛
until a sufficiently accurate value is reached.
3. Newton’s Rapshon Method

12. Application
Bisection Method -1
Shot Detection in Video Content for Digital Video Library - The study presented the usage of
bisection method for shot detection in video content for the Digital Video Library (DVL).
DVL is a networked Internet application allowing for storage, searching, cataloguing, browsing,
retrieval, searching and unicasting video sequences.
The browsing functionality can be significantly facilitated by a fast shot detection process.
Experiments show that usage of the bisection method, allows for accelerating shot detection about
3÷150 times (related to the shot density).
At the end of the paper two possible networked applications are presented: a medical DVL
developed for eLearning purposes and a hypothetical networked news application.

13. Bisection Method -2
Locating and computing periodic orbits in molecular systems - The
Characteristic Bisection Method for finding the roots of non-linear
algebraic and/or transcendental equations is applied to LiNC/LiCN
molecular system to locate periodic orbits and to construct the
continuation/bifurcation diagram of the bend mode family. The
algorithm is based on the Characteristic Polyhedra which define a
domain in phase space where the topological degree is not zero. The
results are compared with previous calculations obtained by the Newton
Multiple Shooting algorithm. The Characteristic Bisection Method not
only reproduces the old results, but also, locates new symmetric and
asymmetric families of periodic orbits of high multiplicity.

14. REGULA FALSI METHOD
The Regula Falsi method is applied to prediction of trace quantities of
air pollutants produced by combustion reactions such as those found in
industrial point sources.
Equilibrium quantities of uncombusted fuel generally are quite small,
due to the exothermic nature of combustion reactions.
Accordingly, calculating the maximum theoretical efficiency which may
be achieved in a combustion process is difficult since the equations
describing the equilibrium state are stiff.