This document discusses several numerical analysis methods for finding roots of equations or solving systems of equations. It describes the bisection method for finding roots of continuous functions, the method of false positions for approximating roots between two values with opposite signs of a function, Gauss elimination for transforming a system of equations into triangular form, Gauss-Jordan method which further eliminates variables in equations below, and iterative methods which find solutions through successive approximations rather than direct computation.

1. Numerical
Analysis
M.Mohana Malar M.Sc.,M.Phil,
N.Malathi M.Sc.,

2. If f(x) is a quadratic expression, then we
have a simple formula for the roots of f(x)=0.
For instance, if 𝑓 𝑥 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 then
𝑥 =
−𝑏± (𝑏2−4𝑎𝑐)
2𝑎
are the roots of the above
equation. The function f(x) may be algebraic
or transcendental

3. Bisection method
Let the function f(x) be continuous between a and b. Let f(a) be –ve and f(b) be
+ve. Then there is a root of f(x)=0, lying between a and b. Let its approximately
value be given by x0 =
𝑎+𝑏
2
. If 𝑓 𝑥 𝑜 = 0, it means that 𝑥0 is a root of 𝑓 𝑥 =
0. Otherwise the root lies between 𝑥 𝑜and b or between a and 𝑥0, according as
𝑓 𝑥 𝑜 is -ve or +ve. Then ,as before, we bisect the interval and continue the
process untill the root is known to the desired accuracy

4. Result
Let 𝑥 = 𝛼 be a root of 𝑓 𝑥 = 0 which is equivalent
to 𝑥 = 𝜑(𝑥). Let I be an Interval containing the
point 𝑥 = 𝛼. If 𝜑′
(𝑥) < 1 for all x in I, the
sequence of approximations 𝑥0, 𝑥1, 𝑥2 … … 𝑥 𝑛 will
converge to the root 𝛼, provides that the Initial
approximation 𝑥0 is chosen in I.

5. The method of iteration
will be particularly useful
for finding the real roots
of an equation given in
the form of an infinite
series.

6. Method of False Positions
Consider the equations 𝑓 𝑥 = 0 and let 𝑥1, 𝑥2 be two value
of x such that 𝑓(𝑥1) and 𝑓 𝑥2 are of opposite signs. Also
let 𝑥1 < 𝑥2. The graph of 𝑦 = 𝑓(𝑥) will meet the x-axis at
some point between 𝑥1 and 𝑥2.
𝑥 𝑛 = 𝑥 𝑛+1 −
𝑓 𝑥 𝑛+1
𝑓 𝑥 𝑛+2 − 𝑓 𝑥 𝑛+1
(𝑥 𝑛+2 − 𝑥 𝑛+1)

7. Gauss Elimination
Method
This direct method is
based on the elimination
of the unknowns, one by
one , and transforming
the given set of equations
to a triangular form

8. Gauss Jordan’s Method
After eliminating one variable by Gauss’ Method,
in the subsequent stage, the elimination is
performed not only in the equations below but
also in the equations . This is Jordan’s modification
of the gaussian elimination and called the Gauss
Jordan method.

9. Iterative Methods
All the preceeding methods of solving systems of
simultaneous linear equations are known as direct methods, which
involve a certain amount of fixed computation. We shall now describe
the iterative or indirect methods of solving these equations. Indirect
methods are those in which the solution is got by successive
approximations. Thus in an indirect or iterative method the amount of
computations depends on the degree of accuracy required.

10. Thank You