Apply Newton Raphson method to find root for logx-cosx=0 near x=1.5
Answers
Engineering Mathematics - III 2
1.1 Solution of Algebraic and Transcendental Equations
1.1.1 Introduction
A polynomial equation of the form
f (x) = pn (x) = a0 x
n–1 + a1 x
n–1 + a2 x
n–2 + … + an–1 x + an = 0 …..(1)
is called an Algebraic equation. For example,
x
4
– 4x
2
+ 5 = 0, 4x
2
– 5x + 7 = 0; 2x
3
– 5x
2
+ 7x + 5 = 0 are algebraic equations.
An equation which contains polynomials, trigonometric functions, logarithmic functions,
exponential functions etc., is called a Transcendental equation. For example,
tan x – e
x
= 0; sin x – xe
2x
= 0; x e
x
= cos x
are transcendental equations.
Finding the roots or zeros of an equation of the form f(x) = 0 is an important problem in
science and engineering. We assume that f (x) is continuous in the required interval. A root of
an equation f (x) = 0 is the value of x, say x = for which f () = 0. Geometrically, a root of
an equation f (x) = 0 is the value of x at which the graph of the equation y = f (x) intersects the
x – axis (see Fig. 1)
Fig. 1 Geometrical Interpretation of a root of f (x) = 0
A number is a simple root of f (x) = 0; if f () = 0 and f ( α ) 0 ' . Then, we can write
f (x) as,
f (x) = (x – ) g(x), g() 0 …..(2)
A number is a multiple root of multiplicity m of f (x) = 0, if f () = f 1
() = .... = f (m–1) () = 0
and f m () = 0.
Then, f (x) can be writhen as,
f (x) = (x – )
m g (x), g () 0 …..(3)