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Numerical Analysis Lab
Roots of Equations
LAB 3
ROOTS OF EQUATIONS
1.0 OBJECTIVE
To find roots of equations using Matlab software.
2.0 INTRODUCTION
Numerical methods are used to solve problems on computers or calculators by numerical
calculation, giving a table of numbers and/or graphical representation (figures). The steps
from a given situation (in engineering, economics, etc) to the final answer are usually as
follows.
1. Modeling. We set up a mathematical model of the problem, such as an
integral, a system of equations, or a differential equation.
2. Choice of mathematical methods, perhaps together with a preliminary error
estimation, a choice of step sizes, etc.
3. Programming. From an algorithm we write a program, say, in FORTRAN, C,
or C++, and/or select suitable routines from a software system. Or we may
decide to use a computing environment, such as MAPLE, MATLAB, and
MATHCAD.
4. Doing the computation
5. Interpretation of results in physical or other terms, including decisions to
rerun if further results are needed.
In this lab we will use MATLAB software to solve the problems that involving
engineering applications.
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Numerical Analysis Lab
Roots of Equations
Figure 3.0: Symbols used in flowchart
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Numerical Analysis Lab
Roots of Equations
3.0 SUMMARY OF THEORY
Many engineering analyses require the determination of the value(s) of the
variable x that satisfies a nonlinear equation:
f ( x)  0
(3.1)
The values of x that satisfy Eq.(1) are known as the root of Equation (1); the number of
roots may be finite or infinite and may be real or complex depending on the nature of the
equation and the physical problem. The function f(x) may or may not be available in
explicit form. If it is available in explicit form, it may be in the form of a general
nonlinear equation, or a polynomial or a transcendental equation:
x 4  80 x  120  0 .(polynomial equation
(3.2)
and
tan x  tanh x  0. (transcendental equation).
(3.3)
The roots of Eq. (3.1) are also known as the zeros of the function f(x). Thus, a function is
said to have zeros, while an equation is said to have roots. A transcendental function is
one whose value cannot be determined for any specified value of the argument by a finite
number of additions, subtraction, multiplications, or divisions. Exponential, logarithmic,
trigonometric, and hyperbolic functions are examples of transcendental functions. Any
equation containing transcendental function is called a transcendental equation.
There are two major classes of methods available to find the approximate values
of the roots in nonlinear equation. They are


Bracketing method. As the name implies, these are based on two initial guesses
that ‘’bracket’’ the root-that is, are on either side of the root.
Open methods. These methods can involve one or more initial guesses, but there
is no need for them to bracket the root.
For well-posed problems, the bracketing methods always work but converge
slowly (i.e., they typically take more iterations to home in on the answer). In
contrast, the open methods do not always work (i.e., they can diverge), but when they
do they usually converge quicker. In both cases, initial guesses are required.
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Numerical Analysis Lab
Roots of Equations
Stopping Criteria
Suppose {xn} is a sequence, converging to a limit x*. The limit x* has the property
f(x*)=0. Let tol be a positive number
Absolute error (in x)
|xn – x*| < tol
Absolute error in f
|f(xn)| < tol
Relative error (in x)
|xn – x*| / |x*| < tol
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Numerical Analysis Lab
Roots of Equations
MATLAB Function
1) fzero – is designed to find the real root of a single equation. A simple
representation of its syntax is
fzero(function,x0)
where function is the name of the function being evaluated, and x0 is the initial
guess. Note that two guesses that bracket the root can be passed as a vector:
fzero(function,[x0,x1]
where x0 and x1 are guesses that bracket a sign change.
Example
To find the root of equation f ( x)  x 2  9
>> x = fzero(inline(‘x^2-9’),-4)
x = -3
>> x = fzero(inline(‘x^2-9’),[0 4])
x=3
2) roots – is designed to find the real and complex roots of a polynomial equations.
The roots function has the syntax,
x = roots (c)
where x is a column vector containing the roots and c is a row vector containing
the polynomial’s coefficients.
Example
Determine all the roots of polynomial:
f ( x)  x 5  3.5x 4  2.75x 3  2.15x 2  3.875x  1.25
>> a = [ 1 -3.5 2.75 2.125 -3.875 1.25]
>> x = roots (a)
x=
2.0000
-1.0000
1.0000 + 0.5000i
1.0000 - 0.5000i
0.5000
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Numerical Analysis Lab
Roots of Equations
3) poly – is designed to find the polynomial’s coefficients. It syntax is
c = poly (r)
where r is a column vector containing the roots and c is a row vector containing
the polynomial’s coefficients.
Example
>> a=poly(x)
a=
1.0000 -3.5000
2.7500
2.1250 -3.8750
1.2500
4) polyval – is designed to evaluate of a polynomial of n degree at x. It syntax is
y = polyval ( p, x)
where p is a column vector containing the polynomial’s coefficients, x is/are
column vector containing the element(s) of x, and y is/are a column vector
containing the answer(s) from evaluation.
Example
>> p = [ 1 -3.5 2.75 2.125 -3.875 1.25]
p=
1.0000 -3.5000
2.7500
2.1250 -3.8750
>> y=polyval(p,[3 4])
y=
42.5000 323.7500
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1.2500
Numerical Analysis Lab
Roots of Equations
1.0 Bisection Method
In order to find the roots of the equations f(x) = 0 using the bisection method, the
function f(x) is first evaluated at equally spaced intervals of x until two successive
function values are found with opposite signs.
Let a = xk and b = xk+1 be the values of x at which the function values of x at
which the function values f(a) and f(b) have apposite signs. This implies that the
function has a root between a = xk and b = xk+1.
x mid 
ab
2
(1.0)
The interval (xk, xk+1), in which the root is expected to lie, is called the interval of
uncertainty. The midpoint of the current interval of uncertainty (a, b) is computed as
and the function value f(xmid) is determined. If f(xmid) = 0, xmid will be a root of f(x)
= 0. If f(xmid) ≠ 0, then the sign of f(xmid) will coincide with that of f(a) or f(b) . If the
signs of f(xmid) and f(a) coincide, then a is replaced by xmid . Otherwise (that is, if the
signs of f(xmid) and f(b) coincide), b is replaced by xmid.
Thus the interval of uncertainty is reduced to half of its original value. Again the
midpoint of the current interval of uncertainty is computed using Eq.(3.0), and the
procedure is repeated until a specified convergence criterion is satisfied. The
reduction of the interval of uncertainty (i.e., the progress of the iterative process) is
shown in (). The following convergence criterion can be used to stop the iterative
procedure:
f ( xmid )  
or
bi  ai  
(2.1)
Here ε is a specified small number.
Assuming that the values of a and b, at which and
have opposite signs, are
known, the iterative procedure used to find the roots of = 0 can be summarized as
follows:
1. Set a(1) = a, b(1) = b, and i = 0.
2. Set iteration number i = i + 1.
a (i )  b (i )
3. Find xmid =
.
2
4. If xmid satisfies the convergence criterion
f ( xmid ) ≤ε or bi  ai  
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Numerical Analysis Lab
Roots of Equations
take the desired root as xroot = xmid and stop the procedure. Otherwise, go to
step 5.
5. If f(xmid). f(a(i)) > 0, both f(xmid) and f(a(i)) will have the same sign, hence, set
a(i+1) = xmid and b(i+1) = b(i), and go to step 2.
6. If f(xmid). f(a(i)) < 0, f(xmid) and f(a(i)) will have opposite signs, hence, set
b(i+1) = xmid and a(i+1) = a(i), and go to step 2.
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Numerical Analysis Lab
Roots of Equations
Flowchart of Bisection Method
Start
iteration number =n
tolerance = ε
Set
a(i) = a, b(i) = b
Yes
f(a(i))*f(b(i)) > 0
No
Yes
i=0
i>n
STOP
i=i+1
No
xmid =
a (i )  b (i )
, f(xmid)
2
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Numerical Analysis Lab
a(i+1) = xmid
b(i+1) = b(i)
Yes
Roots of Equations
sign f(xmid) = = sign f(a(i))
No
b(i+1) = xmid
a(i+1) = a(i)
sign f(xmid) ~ = sign f(a(i))
b(i+1), a(i+1)
Yes
STOP
abs (a(i+1)-b(i+1)) < ε
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No
Numerical Analysis Lab
Roots of Equations
Example-Bisection Method
Find the root of equation f ( x)  e  x  x using the bisection method with x1 = 0, x2 = 1,
and ε =10-3.
MATLAB Programming
% Using bisection to find the roots f(x)=exp(-x)-x.
function xm =Bisection(xleft,xright,n,esp)
%
Synopsis:
x = Bisection(xleft,xright)
%
x = Bisection(xleft,xright,n)
%
%
Input:
xleft,xright = left and right brackets of the root
%
n = (optional) number of iterations
%
esp= value of tolerance
%
%
Output:
x = estimate of the root
a = xleft;
b =xright;
fa = exp(-a)-a;
fb = exp(-b)-b;
% Initial values of f(a) and f(b)
if fa*fb > 0
disp ('There are not roots in interval')
return
end
fprintf(' k
f(xmid)\n');
a
for k=1:n
xm = (a + b)/2;
fm = exp(-xm)-xm;
fprintf('%3d
xmid
b
%computing the midpoint
% f(x) at midpoint
%12.8f
%12.8f
%12.8f
if sign(fm) == sign(fa)
a = xm;
fa = fm;
else
b = xm;
fb = fm;
end
if abs(b-a)< esp
break
end
end
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%12.3e\n',k,a,xm,b,fm);
Numerical Analysis Lab
Roots of Equations
2.0 Fixed-Point Iteration
In this method, the equation f(x) = 0 is rewritten in the form
x = g(x),
And an iterative procedure is adopted using the relation
xi+1 = g(xi); i = 1, 2, 3, ………
where a new approximation to the root, xi+1, is found using the previous
approximation, xi (x1 denotes the initial guess). The iterative process can be stopped
whenever the convergence criterion
xi 1  xi  
is satisfied, where ε is a small number on the order of 10-3 to 10-6.
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Numerical Analysis Lab
Roots of Equations
Flowchart of Fixed-Point Iteration Method
Start
iteration number =n
tolerance = ε
Set
xi(initial guess)
Yes
STOP
i=0
i>n
i=i+1
No
g(xi)
xi+1 = g(xi)
STOP
Yes
abs (xi+1-xi) < ε
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No
Numerical Analysis Lab
Roots of Equations
Example-Fixed Point Iteration Method
Find the root of equation f ( x)  e  x  x using the Fixed Point Iteration method with x1
= 1 and ε =10-3
MATLAB Programming
function xm =FixedPoint(x0,n)
% Using fixed Point Iteration to find the roots of f(x) = exp(-x)-x
% Synopsis: x =Newton(x0,left,right)
%
x =Newton(x0,left,right,n)
%
% Input:
x0 = initial guess
%
n = (optional) number of iterations
%
% Output: x = estimate of the root
x=x0;
g=exp(-x);
fprintf(' k
x
g(x)
abs(x(k+1)-x(k))\n\n');
for k=1:n
g = exp(-x);
diff = abs(g - x);
fprintf('%3d %12.6f %12.6f %18.5e\n',k-1,x,g,diff);
x
= g;
if diff < esp
break
end
end
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Numerical Analysis Lab
Roots of Equations
3.0 Newton-Raphson Method
By neglecting the higher order terms, the Taylor’s series expansion of the function
f(x) about an arbitrary point x1 is approximated as
f ( x)  f ( x1 )  ( x  x1 ) f ' ( x1 )
(3.0)
In order to find the root of f(x) = 0, we set f(x) equal to zeros in Eq. (1.0) to obtain
f ( x1 )  ( x  x1 ) f ' ( x1 )  0
(3.1)
Since the higher order derivative terms were neglected in the approximation of f(x) in
Eq. (1.0), the solution of Eq. (1.1) yields the next approximation to the root (instead
of the exact root) as
x  x2  x1 
f ( x1 )
f ' ( x1 )
(3.2)
where x2 denotes an improved approximation to the root. To further improve the root,
we use x2 in place of x1 on the right-hand side of Eq. (1.2) to obtain x3. This iterative
procedure can be generalized as
xi 1  xi 
f ( xi )
: i  1,2,....
f ' ( xi )
(3.3)
The procedure is shown graphically in Fig.1. Assuming a real root for the equation
f(x) = 0. If xi is the initial guess for the root of f (xi) = 0, the point of intersection of the
tangent to the curve at xi with the x axis gives the next approximation to the root, xi+1.
The convergence of the procedure to the exact root can also be seen in Fig. 1.
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Numerical Analysis Lab
Roots of Equations
f(x)
x f x 
f(xi)
i,
i
f(xi+1)

xi+2
xi+1
xi
X
Figure 1. Geometrical illustration of the Newton-Raphson method
The iterative process can be stopped whenever the convergence criterion
xi 1  xi   or f ( xi 1 )  
is satisfied, where ε is a small number on the order of 10-3 to 10-6.
Algorithm
The steps to apply Newton-Raphson method to find the root of an equation f(x) = 0 are
1.
Evaluate f ' ( x) symbolically
2.
Use an initial guess of the root, xi, to estimate the new value of the root xi+1 as
xi 1 = xi 3.
f(xi )
f'(x i )
Find the absolute relative approximate , a as
xi 1  xi   or f ( xi 1 )  
4.
Also check if the number of iterations has exceeded the maximum number of
iterations
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Numerical Analysis Lab
Roots of Equations
Flowchart of Newton-Raphson Method
Start
iteration number =n
tolerance = ε
Set
xi(initial guess)
Yes
STOP
i=0
i>n
i=i+1
No
f(xi), f’(xi)
xi 1  xi 
f ( xi )
f ' ( xi )
Yes
STOP
No
abs (xi+1-xi) < ε
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Numerical Analysis Lab
Roots of Equations
Example-Newton-Raphson Method
Find the root of equation f ( x)  e  x  x using the Newton-Raphson method with starting
point x1 = 0.0 and the convergence criterion ε =10-3.
MATLAB Program
function xm =Newton(x0,n,esp)
% Using Newton method to find the roots of f(x)= exp(-x)-x
%
% Synopsis: x =Newton(x0)
%
x =Newton(x0,n)
%
% Input:
x0 = initial guess
%
n = (optional) number of iterations; default: n =8
%
% Output: x = estimate of the root
xr=x0;
fprintf(' k
% Initial Guess
f(x)
dfdx
x(k+1)\n');
for k=1:n
xold = xr;
f = exp(-xr)-xr;
dfdx = -exp(-xr)-1;
xr = xr-f/dfdx;
fprintf('%3d %12.3e %12.3e %18.15f\n',k-1,f,dfdx,xr);
if abs(xr-xold)< esp
break
end
end
4.0 Secant Method
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Numerical Analysis Lab
Roots of Equations
The secant method is similar to the Newton’s method, but is different in that the
derivative f’ is approximated by using two consecutive iterative values of f. The
derivative f ' ( xi ) is approximated as
f ' ( xi ) 
f ( xi )  f ( xi 1 )
xi  xi 1
The general expression for the iterative process can then be written as
xi 1  xi 
f ( xi )
f ( xi )[ xi  xi 1 ]
 xi 
; i  2,3,4,...
'
f ( xi )  f ( xi 1 )
f ( xi )
Note that the Secant Method requires two initial guesses x1 and xi-1 to start iterative
process.
The following iterative process can be used to implement the secant method
1. Start with two initial approximations x1 and x2 for the root of f ( x)  0 and a
small number ε to test the convergence of the process. Set i= 2.
2. Find the new approximation, xi 1 , as
xi 1  xi 
f ( xi )( xi  xi 1 )
f ( xi )  f ( xi 1 )
3. Verify the convergence of the process. If
f ( xi 1 )   , or xi 1  xi  
stop the process by taking xi 1 as the root. Otherwise, update the iteration
number as i = i+1 and go to step 2.
Flowchart of Secant Method
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Numerical Analysis Lab
Roots of Equations
Start
iteration number =n
tolerance = ε
Set
xi-1 = a, xi = b
f(xi-1), f(xi)
No
Yes
i>n
STOP
i=0
i=i+1
No
xi 1  xi 
f ( xi )( xi  xi 1 )
, f(xi+1)
f ( xi )  f ( xi 1 )
xi 1  xi
xi  xi 1
Yes
STOP
abs (xi+1-xi) < ε
Example-Secant Method
20
No
Numerical Analysis Lab
Roots of Equations
Find the root of equation f ( x)  e  x  x using the bisection method with x1 = 0, x2 = 1,
and ε =10-3.
MATLAB Programming
function xm =Secant(xleft,xright,n)
% Using secant method to find the roots of f(x)=exp(-x)-x.
%
% Synopsis: x =Secant(xleft,xright)
%
x =Secant(xleft,xright,n)
%
% Input:
xleft,xright = left and right brackets of the root
%
n = (optional) number of iterations
%
esp= value of tolerance
%
% Output: x = estimate of the root
a = xleft; b =xright;
% Copy original bracket to local variables
fa = exp(-a)-a;
% Initial values of f(a) and f(b)
fb = exp(-b)-b;
fprintf(' k
a
b
xmid
f(xmid)\n');
for k=1:n
xm = b - fb*((b-a)/(fb-fa)); % Computes the new value of x
fm = exp(-xm)-xm; % f(x) at new x value
fprintf('%3d %12.8f %12.8f %12.8f %12.3e\n',k,a,b,xm,fm);
a = b; % replace the old values
fa = fb;
b = xm;
fb = fm;
if abs(b-a)< esp
break
end
end
Exercises
21
Numerical Analysis Lab
Roots of Equations
1. Determine all the roots with MATLAB of f ( x)  x 3  6 x 2  11x  6.1
2. Use bisection to determine the mass of the bungee jumper with a drag coefficient
of 0.25 kg/m to have a velocity of 36 m/s after 4 s of free fall. Note: The
acceleration of gravity is 9.81 m/s2. Start with initial guesses of xlower = 50 kg and
xupper = 200 kg and iteration until the approximate absolute error fall below 0.001.
f (m) 
gcd
gm
tanh(
t)  v
cd
m
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