This document discusses the iteration method for solving algebraic and transcendental equations. It begins by defining algebraic and transcendental equations. The iteration method works by rewriting the equation in the form x = φ(x) and finding successive approximations that converge to the root. It provides the sufficient condition for convergence being |φ'(x)| < 1. Applications of the method include finding roots of equations like cos(x) = 3x - 1 and x3 + x2 - 1 = 0. The conclusion states that iteration methods are widely used for solving differential equations.

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NATIONAL INSTITUTE OF TECHNOLOGY RAIPUR
DEPARTMENT OF MINING ENGINEERING
A TERM PROJECT
“08-SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL
EQUATION- ITERATION METHOD & ITS
APPLICATIONS”
IN
NUMERICAL ANALYSIS AND COMPUTER PROGRAMMING
PREPARED BY:
AMIT KUMAR
ROLL NO. 18121008
B.TECH. MINING ENGINEERING
UNDER GUIDANCE OF:
DR. S.N. RAW
ASSISTANT PROFESSOR
DEPARTMENT OF APPLIED MATHEMATICS

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ACKNOWLEDGEMENT
I would like to express my special thank of gratitude to respected Dr. S.N. RAW
sir for his guidance, help and useful suggestion and give us opportunity to do
this project of solution of algebraic and transcendental equation-iteration
method (NACP).

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INDEX
CONTENTS
1. ABSTRACT /SUMMERY...................................................................................4
1.1 KEYWORDS................................................................................................4
2.INTRODUCTION...............................................................................................4
3. DEVELOPMENT OF FORMULA /METHOD .......................................................5
4.ANALYSIS OF ITERATION METHOD..................................................................6
5.APPLICATIONS OF ITERATION METHOD..........................................................7
6.CONCLUSION ................................................................................................10
7.REFERENCES..................................................................................................11
8. BIODATA ABOUT WRITER.............................................................................11

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1. ABSTRACT /SUMMERY: In this project, we will discuss about Iteration
method and its applications in detail. With Iteration method, we will solve
algebraic equations and transcendental equations. This method is used finding
the root of only those equations f(x) = 0, which are expressible as x = φ(x). The
root of the equation is the same as the point of intersection of the straight-line
y = x and y = φ(x).
1.1 KEYWORDS: Iteration method, algebraic equation, Transcendental
equation, applications, root
2.INTRODUCTION: An expression of the form
f(x)= 𝑎𝑎0 𝑥𝑥 𝑛𝑛
+ 𝑎𝑎1 𝑥𝑥 𝑛𝑛−1
+ ⋯ 𝑎𝑎𝑛𝑛−1 𝑥𝑥 + 𝑎𝑎𝑛𝑛
where a’ s are constants (𝑎𝑎0 ≠ 0) and n is a positive integer, is called a
polynomial in x of degree n. The polynomial f(x) =0 is called an algebraic equation
of degree n. If f(x) contains some other functions such as trigonometric,
logarithmic, exponential etc., then f(x) = 0 is called a transcendental equation.
Def. The value of x which satisfies f(x) = 0 (1)
is called a root of f(x) = 0. Geometrically, a root of (1) is that value of x where
the graph of y = f(x) crosses the x-axis. The process of finding the roots of an
equation is known as the solution of that equation. This is a problem of basic
importance in applied mathematics.

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If f(x) is a quadratic, cubic, or a biquadratic expression,
algebraic solutions of equations are available. But the need often arises to solve
higher degree or transcendental equations for which no direct methods exist.
Such equations can best be solved by approximate methods. In this project, we
shall discuss numerical iteration method for the solution of algebraic and
transcendental equations.
3. DEVELOPMENT OF FORMULA /METHOD:
To find the roots of the equation f(x) = 0 (i)
by successive approximations, we rewrite (i) in the form x =∅(𝑥𝑥) (ii)
The roots of (i) are the same as the points of intersection of the straight line y =x
and the curve representing 𝑦𝑦 = ∅(𝑥𝑥). Figure 2.7 illustrates the working of the
iteration method which provides a spiral solution.
Let x =𝑥𝑥0 be an initial approximation of the desired root 𝛼𝛼. Then the first
approximation x1 is given by x1 =∅(x0)
Now treating x1 as the initial value, the second approximation is x2 =∅(x1)
Proceeding in this way, the nth approximation is given by 𝑥𝑥𝑛𝑛=∅𝑓𝑓(𝑥𝑥𝑛𝑛−1)

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4.ANALYSIS OF ITERATION METHOD
Sufficient condition for convergence of iterations: It is not certain whether
the sequence of approximations 𝑥𝑥1, 𝑥𝑥2..., 𝑥𝑥𝑛𝑛 always converges to the same
number which is a root of (1) or not. As such, we have to choose the initial
approximation 𝑥𝑥0 suitably so that the successive approximations 𝑥𝑥1, 𝑥𝑥2..., 𝑥𝑥𝑛𝑛
converge to the root 𝛼𝛼. The following theorem helps in making the right choice
of 𝑥𝑥0:
Theorem:
If (i) 𝛼𝛼 be a root of f (x) = 0 which is equivalent to 𝑥𝑥 = ∅(𝑥𝑥),
(ii) I, be any interval containing the point x=𝛼𝛼,
(iii) |∅′(x) | < 1 for all x in I,
then the sequence of approximations 𝑥𝑥0, 𝑥𝑥1, 𝑥𝑥2, . . . 𝑥𝑥𝑛𝑛 will converge to the root
𝛼𝛼 provided the initial approximation 𝑥𝑥0 is chosen in I.
Proof. Since 𝛼𝛼 is a root of 𝑥𝑥 = ∅(𝑥𝑥), we have 𝛼𝛼 = ∅(𝛼𝛼)
If 𝑥𝑥𝑛𝑛−1 and 𝑥𝑥𝑛𝑛 be 2 successive approximations to 𝛼𝛼, we have 𝑥𝑥𝑛𝑛 = ∅(𝑥𝑥𝑛𝑛−1)
∴ 𝑥𝑥𝑛𝑛 − 𝛼𝛼 = ∅(𝑥𝑥𝑛𝑛−1) − ∅(𝛼𝛼)
By mean value theorem,
∅(𝑥𝑥𝑛𝑛−1)−∅(𝛼𝛼)
𝛼𝛼𝑛𝑛−1−𝛼𝛼
= ∅′(𝜀𝜀) 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑥𝑥𝑛𝑛−1 < 𝜀𝜀 < 𝛼𝛼
Hence (1) becomes 𝑥𝑥𝑛𝑛 − 𝛼𝛼 = (𝑥𝑥𝑛𝑛−1 − 𝛼𝛼)∅′(𝛼𝛼)
If |∅′(𝑥𝑥𝑖𝑖) ≤ 𝑘𝑘 < 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖, then
𝑥𝑥𝑛𝑛 − 𝛼𝛼 ≤ 𝑘𝑘|𝑥𝑥𝑛𝑛−1 − 𝛼𝛼|
Similarly, 𝑥𝑥𝑛𝑛−1 − 𝛼𝛼 ≤ 𝑘𝑘|𝑥𝑥𝑛𝑛−2 − 𝛼𝛼|
i.e., 𝑥𝑥𝑛𝑛 − 𝛼𝛼 ≤ 𝑘𝑘2
|𝑥𝑥𝑛𝑛−2 − 𝛼𝛼|
Proceeding in this way, 𝑥𝑥𝑛𝑛 − 𝛼𝛼 ≤ 𝑘𝑘 𝑛𝑛
|𝑥𝑥0 − 𝛼𝛼|
As n → ∞, the R.H.S. tends to zero, therefore, the sequence of approximations
converges to the root 𝛼𝛼.
Note:
1. The smaller the value of ∅′(𝑥𝑥), the more rapid will be the convergence.
2. This method of iteration is particularly useful for finding the real roots of an
equation given in the form of an infinite series.

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5.APPLICATIONS OF ITERATION METHOD
(1) Find a real root of the equation cos x = 3x − 1 correct to three decimal
places using
(i) Iteration method
(ii) Aitken’s ∆2
method.

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(2)Using iteration method, find a root of the equation 𝑥𝑥3
+ 𝑥𝑥2
− 1 = 0 correct
to four decimal places.

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(3) Apply iteration method to find the negative root of the equation
𝑥𝑥3
− 2𝑥𝑥 + 5 = 0 correct to four decimal places.

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(4)
6.CONCLUSION
Iteration method based on the strong initial assumption. The initial
approximation is usually given in question or we can find. In actual practice there
are variety in assumptions which the iteration function ∅(𝑥𝑥) must satisfy to
ensure that the iteration approach to the root. but, to use those assumption you
would require a lot of practice.

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Iteration methods are also applied to solve simultaneous equation and
computation of approximate solutions of stationary and evolutionary problems
associated with differential equations. It is used in different cycle of software
model.
7.REFERENCES
 Numerical method in engineering and science C, C++ and MATLAB BY B.S.
Grewal
 sciencedirect.com/topic/mathematics/iteration method
8. BIODATA ABOUT WRITER
NAME – AMIT KUMAR
ROLL NUMBER – 18121008
BRANCH – MINING ENGINEERING
SEMESTER – 5TH
TOPIC – 08-SOLUTION OF ALGRBRAIC AND TRANSCENDENTAL EQUATIONS
ITERATION METHOD AND ITS APPLICATIONS
THANK YOU