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the equation which is often non liner, their solution is not easy as like linear solution.various methods are described with numerical in this PPT.
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Numerical Solutions of Non- Linear equations
Raju Sharma Assistant Professor Department of Civil Engineering Chandigarh University
Introduction of polynomial
If f(x) is a quadratic or cubic equation then
we can find the root of the equation f(x)
=0
(1)2x2-8x+6=0
(2)2x3+x2-13x+5=0
f(x) =a0xn + a1xn-1 + …. + an-1x +an
• When f(x) is a polynomial of higher degree or an
expression involving transcendental functions
(equation contains trigonometric, logarithmic,
exponential) e.g. 1+cos-5x, xtanx-cosx, e-x – sin x
etc, algebraic methods are not available.
• To find the roots of such types of equations,
numerical methods are used.
Following are the Numerical Methods
1. Bisection Method2. Regula - falsi Method or method of false
position3. Secant Method4. Newton-Raphson Method5. Iteration Method
Bisection Method
• This method is based on the
theorem:
• If a function f(x) is continuous b/w a
& b, and f(a) , f(b) are of opposite
signs then there exist at least one
root b/w a & b.
• Suppose that a root of f(x)=0 lies in
the interval (a, b) i.e. f(a) f(b)<0. we
bisect this interval and obtain
• x1 = (a+b)/2
• If f(x1)=0, then x1 is a root of f(x)=0
• Otherwise root lies b/w a and x1 or
x1 & b
Y =
f (x
)
Y
a x2
b x1x3 x0
Then, we bisect the interval as
before and continue the
process until the root is found
to desired accuracy.
• Then the second approximation to the root is x2 =
(a+x1)/2
• We have f(x2)<0
• and f(x2) f(x1) <0
• Root lies in (x2, x1)
• If f(x2) is –ve, The root lies between x1 & x2 . Then
the third approximation to the root is
x3 = (x1+ x2 )/2
and so on…..
Q-1 Find a root of the equation x3-4x-9=0,using the bisection method correct the three decimal places• f(x) =x3-4x-9 ……(1)
• f(2)=23-4 x 2 -9=-9
• f(3)= 33-4 x 3 -9=6
• Hence root lies b/w 2 and 3
• First approximation to the
root is x1=(2+3)/2 = 2.5
• Then put x1 in eq. 1
• f(2.5)= 2.53-4 x 2.5-9=-
3.375< 0
• f(x1) f(3) < 0
• root lies b/w x1 and 3
• x2 = (2.5+3)/2 =2.75
• Then put x2 in eq. (1)
• f(2.75)= 2.753-4 x 2.75-
9=0.796 > 0
• f(x1) f(x2) < 0
• Root lies between x1 & x2
• Thus, the third approximation to the root is
• x 3 = (x1 + x2 )/2 = 2.625
• f(2.625)= 2.6253-4 x 2.625-9=-1.4121 <0
• And f(x2) f(x3) < 0
• Root lies b/w x2 & x3
Thus, the fourth approximation to the root is
x4 = (x2 + x3 )/2 = 2.6875
• By repeating the process, the successive approximation are
• x5 = 2.71875
• x6 = 2.70313
• x7 = 2.71094
• x8 = 2.70703
• x9 = 2.70508
• x10 = 2.70605
• x11 = 2.70654
• x12 = 2.70642
• Hence the required root is 2.706
Regula - falsi Method or method of false position
• Choose two points x0 & x1,
such that f(x0) & f(x1) are of
opposite signs i.e., f(x0) f(x1)
< 0
• The graph of y= f(x) crosses
the x-axis b/w these points.
• This method consist in
replacing the curve AB by the
chord AB and taking the point
of intersection of the chord
with the x-axis as an
approximation to the root
Y
XO
A[x0, f(x0)]
P(x)
x1x2x3
x0
B[x1, f(x1)]
• Equation of the chord joining the
points A[x0, f(x0)] and B[x1, f(x1)] is
• y-f(xo) =((f(x1)- f(xo)/(x1-xo))/ (x-xo)
• The abscissa of the point where the
chord cuts the x-axis (y=0) is given by
• x =x2=(xo – (x1-xo)/(f(x1)- f(xo)) f(xo)
---- (1)
• Which is the approximation to the root.
• Or x2 = (xo f(x1)-x1 f(xo))/( f(x1)- f(xo))
• If now f(xo) and f(x2) are of opposite
signs, then the root lies b/w xo and x2
• So replacing x1 by x2 and find the new
root x3
•Y
•X•O
•A[x0, f(x0)]
•P(x)
•x1
•x2
•x3
•x0
•B[x1, f(x1)]
• The procedure is repeated till the root is found to
desired accuracy. The iteration process based on
equation 1 is known as Regula Falsi Method.
• Note :- if the root lies initially in (xo, x1), then one
of the end points is fixed for all iterations.
• For example, if xo is fixed, then the method is
of the form
• xk+1= xo f(xk)- xk f(xo)/ f(xk) - f(xo)
• K=1,2,3,………
• Find a real root of the equation x3-2x-5=0, by regula –falsi method correct to three decimal places •Solution:-
•A real root of the equation f(x) = x3-5x+1=0 lies in the interval (0, 1). Perform four iterations of regular-falsi method to obtain this root
Solution:-
Secant method • This method is an improvement over the method
of false position (or Regula-falsi method ) as it
does not require the condition
• f(xo) x f(x1) < 0
• In this method two most recent approximations
to the root are used to find the next
approximation
• Let xk-1, xk be two
approximations to the root
of f(x)=0. Then p (xk-1 , f(xk-
1)) Q (xk, f (xk)) are points
on the curve y=f(x)
• the equation of straight
line PQ is given by
• y- f(xk) = (f(xk-1)- f(xk)/(xk-1-xk)) (x-
xk)
Substitute y=o, and solving
for x, we get
p((xk-1, f(xk-
1))
a(xk , f(xk))
xk+1 x
yo
•x=xk-((xk-xk-1)/f(xk)-f(xk-1)) x f(xk)
•The next approximation, xk+1 , to the root
is written as
•xk+1=(xk-1 f(xk) – xk f(xk-1))/ f(xk) - f(xk-1)
•k = 1,2,3……
Find a root of the equation x3-2x-5=0 using secant method correct to three decimal places •Solution:-
Drawback of secant method
• If at any intersection f(xn)= f(xn-1), this
method fails and shows that it does not
converge necessarily. This is a drawback
of secant method over the method of false
position which always converge. But if the
secant method once converges, its rate of
convergence is 1.6 which is faster that
that of the method of false position.
Newton-Raphson Method
•Let x0 be an approximate root of the
equation f(x)=0. if x1 = x0+h be the exact
root then f(x1)=0
•Expanding f(x0+h) by Talor’s series
• f(x0) +h f ´(x0) +(h2 /2!) f ´´ (x0) +_ _ _ _=0
•Since h is very small, neglecting h2 & higher
power of h, we get
• f(x0) +h f ´(x0) =0 h = -(f(x0)/ f ´(x0))
•We have x1 = x0+h
• x1 = x0 -(f(x0)/ f ´(x0))
•Successive approximation are given by x2, x3, _ _ _ _, xn+1
•Where •xn+1 = x0 -(f(xn)/ f ´(xn))
•Which is known as Newton-Raphson formula
Find the positive root of x4-x=10 correct to three decimal places, using newton - raphson method •Solution:-
Some deduction from Newton-Rapson formula
• Iterative formula to find 1/N is xn+1 = xn (2-Nxn)
• Iterative formula to find √N is xn+1 = ½ (xn
+N/xn)
• Iterative formula to find 1/ √N is xn+1 = ½ (xn
+1/Nxn )
• Iterative formula to find 1/ √N is xn+1 =1/ k [(k-
1) xn+ N/xnk-1)
k
• Let x = 1/N or (1/x)-N =0• Take f(x) = (1/x)-N where f(x) = -(1/x2)• Newton’s formula gives
• xn+1 = xn – f(xn)/f’ (xn)
• = xn – ((1/ xn) –N)/ -(1/xn2 )
• = xn + ((1/ xn) –N) xn2
•
• = xn + xn –N xn2
• = 2xn –N xn2
•
• xn(2-N xn)
• Hence Proved
Evaluate 1/31 correct to four decimal places by newton/s iteration method
•Solution:-
Iteration Method
• To find the root of the equation f(x) =0 …..(i)
• By successive approximation, we rewrite (i) in
the form x=Ф (x).
x
y
Y =x
Y =Ф (x)
x0 x2 x3 x1o
• Let x = xo be an initial approximation of the
desired root α
• then the first approximation x1 is given by
• x1=Ф (x0)
• Now treating x1 as the initial value, the second
approximation is x2=Ф (x1)
• Processing in this way, the nth approximation is
xn=Ф (xn-1)
Sufficient condition for the convergence of iterations
• If
I. α is a root of f(x)=0 which is equivalent to x=Ф (x)
II. I, be any interval containing the point x= α
III. | Ф´ (x)|< 1 for all x lies b/w I
Then the sequence of approximations x0, x1, x2,_
_ _ _,xn all converge to the root α provided the
initial approximation x0 in chosen I
Find the real root of the equation Cosx=3x-1.correct to three decimal places using iteration method.•Solution
Rate of convergence • Let x0,x1,x2, _ _ _ _ be the value of root (α) of an
equation at the o , 1st ,2nd ……… iterations while
its actual value is 3.5567
• The value of this root calculated by three
different methods are as given below:-Root 1st Method 2nd Method 3rd method
x0 5 5 5
x1 5.6 3.8527 3.8327
x2 6.4 3.5693 3.56834
x3 8.3 3.55798 3.55743
x4 9.7 3.55687 3.55672
x5 10.6 3.55676
x6 11.9 3.55671
Rate of convergence
Bisection Method 1/2
Regula-falsi Method 1.62
Secent method 1.6
Newton Rapson 2
The fastness of convergence in any method is represented by its rate of convergence
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