Chapter 2 Solution of Equations in One Variable 2.1 Introduction In Applied Mathematics, the most frequent problem is to find the values ofx to satisfy the equation 0 ) ( = x f. Such values are called the roots of the equation and also known as the zeros of ) ( x f. Some equations are easy to solve. The linear equation such as 0 2 3 = − x can be solved easily. Quadratic equations can be solved by factorization or by the standard formula. It is possible to solve polynomial equations of higher degree, if they are factorisable, otherwise it is difficult to solve. Many equations involve sines, cosines, exponential and other transcendental functions and it i s difficult to solve them precisely. In fact, majority of equations cannot be solved in any precise manner and so we have to solve them by using iterative procedures. An iterative procedure is a repeative process that produces a sequence of approximations to the equation. In this chapter, we shall consider some of the important approximate methods in finding the roots of the equations in one variable. 2.2 Roots and its Location A polynomial inx of degree n is of the form n n x a x a x a a x p + ⋅ ⋅ ⋅ + + + = 2 2 1 0 ) ( where a’s are constants and n is a positive integer. A polynomial equation 0 ) ( = x p of degree n has exactly n roots. Some of them are real and others are complex. For a non polynomial equation 0 ) ( = x f, there is no such rule offinding the number of roots. Geometrically, if the graph of) ( x fy = crosses the x-axis at a x = , then a x = is a real root of 0 ) ( = x f. Now we shall consider graphically to find the the number of real roots and its location. 2.2.1 Number of Real Roots by Graphical Method Rewrite the equation 0 ) ( = x fas ) ( ) ( 2 1 x fx f= . At the point of intersection 1 x x = (say) of the graphs of) ( 1 x fy = and ) ( 2 x fy = we have ) ( ) ( 1 2 1 1 x fx f= Thus the number of intersections of the two graphs will be the number of real roots of0 ) ( = x f.
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In Applied Mathematics, the most frequent problem is to find the values of x to satisfy the
equation 0)( = x f . Such values are called the roots of the equation and also known as the
zeros of )( x f . Some equations are easy to solve. The linear equation such as 023 =− x
can be solved easily. Quadratic equations can be solved by factorization or by the standard
formula. It is possible to solve polynomial equations of higher degree, if they arefactorisable, otherwise it is difficult to solve. Many equations involve sines, cosines,
exponential and other transcendental functions and it is difficult to solve them precisely. In
fact, majority of equations cannot be solved in any precise manner and so we have to solve
them by using iterative procedures. An iterative procedure is a repeative process that
produces a sequence of approximations to the equation. In this chapter, we shall consider
some of the important approximate methods in finding the roots of the equations in one
variable.
2.2 Roots and its Location
A polynomial in x of degree n is of the formn
n xa xa xaa x p +⋅⋅⋅+++= 2210)(
where a’s are constants and n is a positive integer.
A polynomial equation 0)( = x p of degree n has exactly n roots. Some of them are real
and others are complex. For a non polynomial equation 0)( = x f , there is no such rule of
finding the number of roots. Geometrically, if the graph of )( x f y = crosses the x-axis at
a x = , then a x = is a real root of 0)( = x f . Now we shall consider graphically to find the
the number of real roots and its location.
2.2.1 Number of Real Roots by Graphical Method
Rewrite the equation 0)( = x f as )()( 21 x f x f = . At the point of intersection 1 x x = (say)of the graphs of
)(1 x f y = and )(2 x f y =
we have
)()( 1211 x f x f =
Thus the number of intersections of the two graphs will be the number of real roots of
CHAPTER 2: Solutions of Equations in One Variable 15
y
x α
))(,( a f a
O
))(,( b f b
x0a
b
Fig. False Position Method
],[ 00 ba is the starting interval and2
000
ba x
+= is the midpoint.
],[ 11 ba is the second interval which brackets the root and 1 x is the midpoint.
At each step relabel the interval which brackets the root and find the midpoint by
using
2
nn
n
ba
x
+
= .
2.4 False Position Method
This method is also known as the Regula Falsi or Linear Approximation. The method is
very similar to Bisection method. In bisection method midpoint of the interval ],[ ba is
used for the next iterate. In this method we replace the midpoint formula by the value of
the x-intercept of the line joining the points ))(,( a f a and ))(,( b f b .
The straight line through ))(,( a f a and ))(,( b f b is
)()()(
)( a xab
a f b f a f y −−
−
=−
On the x-axis 0= y and let 0 x x = , then
)()()(
)( 0 a xab
a f b f a f −
−
−=−
Solving for 0 x ,
)()()(
)(0 a f
a f b f
aba x
−
−−= .
Also note that )()()(
)(0 b f
a f b f
abb x
−
−−= .
If 0)( 0 = x f , then 0 x is the root. Otherwise the root lies either between 0 x and b or
between a and 0 x depending on whether )()( 0 x f a f is positive or negative. By designatingthe new interval of root as ],[ 11 ba , we can then calculate the next iterate 1 x by the formula
similar to above. Repeat the process until ε ≤−+ nn x x 1 , where ε is the specified
accuracy.
2.5 Order of Convergence
Let nε be the error in the nth iteration for a root α of 0)( = x f , then
α ε −= nn x
If
=+
∞→ Rn
n
n ε
ε 1lim const
then the order of convergence of the sequence }{ n x is R.
CHAPTER 2: Solutions of Equations in One Variable 29
6. The equation 0324423704923 =+−+ x x x has a double root near x = 1. Use
modified Newton-Raphson method to find the root correct to 3 decimal places.
7. The equation 043 46 =+− x x has two double roots. One near 5.1−= x and another
near 5.1= x . Find the roots correct to 5 decimal places using the Newton-Raphson
method.
8. The equation 0625100045027 24 =−+− x x x has a multiple root of multiplicity three
near x = 2. Use Newton-Raphson method to find the root correct to 5 decimal
places.
9. Consider the solution of 09523 =−+ x x in [1, 2] based on the following iterative
formulae
(a) 59
21 −=+
n
n x
x (b)
2 / 13
15
9
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −=+
nn
x x (c)
2 / 1
15
9⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+=+
n
n x
x
Perform few iterations and comment on the different iterative formulas.
10. The equation 2 cos x – x = 0 has a root near x = 1.1. Among others, the following
iteration formulae are suggested to estimate the root.
(i) nn x x cos21 =+ (ii) nnn x x x2
1cos1 +=+ (iii) )(cos
3
21 nnn x x x +=+
Apply an appropriate test to determine, for each rearrangement, whether or not the
corresponding iteration converges to the root.
Using whichever of these iteration formulae you consider most appropriate, find the
root correct to 4 significant figures.
11. The equation 0345 23 =−+− x x x has a root near 4= x , which is to be computed
by the iteration
k
x x xk x nnn
n
32
15)4(3 −+−+
=+ and 40 = x
(i) Determine which value of k will give the fastest convergence.
(ii) Using this value of k, iterate three times and estimate the error in 3 x .
12. Given that 2)( 2 −+= x xe x f x
.
(a) Show that the equation 0)( = x f has only two real roots.
(b) Use the false position method twice in [0, 1] to find the root of 0)( = x f to 2
decimal places.
(c) Use Newton-Raphson method thrice to find the root in ]1,2[ −− of the equation0)( = x f giving your answer to 3 decimal places. Asses the accuracy of the root
without further calculation.
(d) An iterative formula )2(2
1 −++=+ n x
nnn xe xk x x n , k ≠ 0, can be used to
estimate the roots of f ( x) = 0. Find the range of values of k for which the
iterative formula will converge to the root near 0.7.
Use this iterative formula with a suitable value of k to estimate the root correct to