Topic : Algebraic and Transcendental equations
– General Iterative Method
Introduction
• A problem of great importance in engineering is that of
determining the roots of an equation of the form 𝑓 𝑥 = 0.
• An equation of the form 𝑓 𝑥 = 0 is called an algebraic
equation if 𝑓(𝑥) is a polynomial. Example: 𝑥 3 − 5𝑥 + 1 = 0
• An equation of the form 𝑓 𝑥 = 0 is called a transcendental
equation if it is not an algebraic equation.
Examples : cosx − xex = 0 , 𝑙𝑛𝑥 − 𝑥 + 3 = 0
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Definition1 (Root) :
A number 𝜉 is called a root (or zero or solution) of 𝑓 𝑥 = 0
if 𝑓(𝜉) ≡ 0.
Definition2 (Simple and Multiple roots):
A number 𝜉 is called a multiple root of 𝑓 𝑥 = 0 with
multiplicity 𝑚, if the function 𝑓 𝑥 can be written in the form
𝑥 − 𝜉 𝑚 𝑔 𝑥 , where 𝑔(𝑥) is bounded and 𝑔 𝜉 ≠ 0.
If 𝑚 = 1, then the root 𝜉 is called a simple root.
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Note : If the function 𝑓 𝑥 in the
equation 𝑓 𝑥 = 0 has a multiple root 𝜉
with multiplicity 𝑚, then the function
𝐹 𝑥 = 𝑓 𝑥
1
𝑚
in the equation
𝐹 𝑥 = 0 has a simple root at 𝜉.
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Alternative Definition (for Simple and Multiple roots):
A root 𝜉 of an equation 𝑓 𝑥 = 0 is said to be a multiple root
with multiplicity 𝑚, if 𝑓 𝑥 and its first 𝑚 − 1 derivatives
are zero at 𝜉, and 𝑓
𝑚
(𝜉) ≠ 0.
Note :
• If 𝑚 = 1, then 𝜉 is called a simple root.
• If 𝑚 = 2, then 𝜉 is called a double root.
• If 𝑚 =3, then 𝜉 is called a triple root, and so on.
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Direct methods
Methods to find the roots of 𝑓 𝑥 = 0
Iterative methods
(or Indirect methods)
1) Direct methods provide the exact value of the roots of 𝑓 𝑥 = 0
in a finite number of steps. These methods give all the roots simultaneously.
Example : The roots of a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are
−𝑏 ± 𝑏2 − 4𝑎𝑐
𝑥=
2𝑎
2) Iteration methods are based on the idea of successive approximations
and give only one real root of the equation 𝑓 𝑥 = 0 at a time.
.
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Definition3 : A sequence of iterates 𝑥0 , 𝑥1 , 𝑥2 , … is said to converge
to the root 𝜉, if
lim 𝑥𝑘 − 𝜉 = 0 or lim 𝑥𝑘 = 𝜉.
𝑘→∞
𝑘→∞
For each 𝑘, the error, denoted ∈𝑘 , is defined by ∈𝑘 = 𝑥𝑘 − 𝜉.
-----------------------------------------------------------------------------------We stop the iteration for some 𝑘 if either 𝑓 𝑥𝑘 < 𝜖 or
𝑥𝑘 − 𝑥𝑘−1 < 𝜖, for a pre-defined desired accuracy or error tolerance 𝜖.
In this case, the approximation 𝑥𝑘 is the root correct to the desired decimal
places determined by the accuracy 𝜖.
Examples :
• For two decimal accuracy, 𝜖 = 0.005.
• For three decimal accuracy, 𝜖 = 0.0005, and so on.
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Initial Approximation for an iteration method :
• For a polynomial equation 𝑓 𝑥 = 0, the number of positive roots cannot
exceed the number of changes of signs in the coefficients of 𝑓(𝑥).
• For a polynomial equation 𝑓 𝑥 = 0, the number of negative roots cannot
exceed the number of changes of signs in the coefficients of 𝑓(−𝑥).
The following theorem of calculus is used to determine an initial approximation.
• Intermediate value theorem : If 𝑓(𝑥) is continuous on some
interval [𝑎, 𝑏] and 𝑓 𝑎 𝑓 𝑏 < 0, then the equation 𝑓 𝑥 = 0 has at
least one real root or an odd number of real roots in (𝑎, 𝑏).
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Definition4 (Fixed point) :
A point 𝛼 is called a fixed point of a function 𝑓 𝑥 if 𝛼 = 𝑓(𝛼)
Definition5 (Simple iteration method) :
If 𝑥0 is an initial approximation to the root, then a simple iteration method
is defined as 𝑥𝑘+1 = ∅ 𝑥𝑘 , where ∅ is the iteration function. The root
of a simple iteration method is a fixed point of the iteration function ∅.
Example : Fixed point method, Newton-Raphson method
Definition6 (Multipoint iteration method) :
If 𝑥𝑘 , 𝑥𝑘−1 , … , 𝑥𝑘−𝑚+1 are ′𝑚′ initial approximations to the root, then a
multipoint iteration method is defined as
𝑥𝑘+1 = ∅ 𝑥𝑘 , 𝑥𝑘−1 , … , 𝑥𝑘−𝑚+1 , where ∅ is the iteration function.
Example : Secant method, Regula-falsi method
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CONVERGENCE OF A SIMPLE ITERATION METHOD
Theorem : Let ∅(𝑥) be a continuous function in some interval
[𝑎, 𝑏] that contains the root with ∅′(𝑥) ≤ 𝑐 < 1. Then, for
any choice of 𝑥0 ∈ (𝑎, 𝑏), the sequence of iterates 𝑥𝑘
obtained using the iteration scheme
𝑥𝑘+1 = ∅(𝑥𝑘 ) , 𝑘 = 0,1,2,3, …
converges to the root of the equation 𝑥 = ∅(𝑥).
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Iterative Methods for simple roots
• Fixed point iteration method
(or method of successive approximations)
• Newton-Raphson method or Newton’s method
• Secant method
Iterative Methods for multiple roots
• Modified Newton-Raphson method
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Fixed point Iteration method :
Algorithm: The equation 𝑓 𝑥 = 0 is converted into the
form 𝑥 = ∅(𝑥). Using the simple iteration scheme
𝑥𝑘+1 = ∅ 𝑥𝑘 , 𝑘 = 0,1,2, …
with some initial approximation 𝑥0 , we obtain a sequence
of approximations or iterates 𝑥𝑘 which converges to the
root 𝜉 of 𝑥 = ∅(𝑥) correct to the desired accuracy. This
root 𝜉 being a fixed point of ∅ is the root of 𝑓 𝑥 = 0.
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GRAPHICAL REPRESENTATION OF SIMPLE ITERATION METHOD OR FIXED POINT METHOD
𝜉
𝑓 𝑥 = 0 ⇒ 𝑥 = ∅(𝑥) ⇒ 𝑥𝑘+1 = ∅(𝑥𝑘 )
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Example : Consider a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
expressed in the form 𝑥 = ∅(𝑥) for which a simple iteration
scheme is given by 𝑥𝑘 = ∅ 𝑥𝑘 , 𝑘 = 1,2, ….
𝑥=
−𝑐−𝑎𝑥 2
𝑏
𝑥=
−𝑐−𝑏𝑥
𝑎𝑥
𝑥=
−𝑐
𝑎𝑥+𝑏
⇒ 𝑥𝑘+1 =
⇒ 𝑥𝑘+1 =
⇒ 𝑥𝑘+1 =
−𝑐−𝑎 𝑥𝑘2
𝑏
−𝑐−𝑏 𝑥𝑘
𝑎 𝑥𝑘
−𝑐
𝑎 𝑥𝑘 +𝑏
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Worked Problem1
Find a smallest positive real root of the equation
𝑥 3 + 𝑥 2 − 1 = 0 , correct to 4 decimal places using
simple (or fixed point) iteration method.
Solution:
Let 𝑓(𝑥) = 𝑥 3 + 𝑥 2 − 1. 𝑓 0 = −1 and 𝑓(1) = 1
Hence a smallest positive real root lies in the interval (0,1).
1
1
𝑥 +𝑥 −1 =0⇒𝑥 𝑥+1 =1⇒𝑥 =
⇒𝑥=
𝑥+1
𝑥+1
3
2
2
The iteration function is ∅ 𝑥 =
2
1
.
𝑥+1
Now, ∅′ 𝑥 =
Clearly ∅′ 𝑥 is continuous in (0,1) and ∅′ 𝑥
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−1
2
1
𝑥+1
< 1 in (0,1).
3
2
.
Let the initial approximation be 𝑥0 = 0.50000.
The sequences of iterates are obtained from the iteration scheme
𝑥𝑘+1 = ∅ 𝑥𝑘 =
1
,
𝑥𝑘 +1
𝑘 = 0,1,2, ….
Now,
𝑥1 = 𝜑 𝑥0 =
𝑥2 = 𝜑 𝑥1 =
𝑥3 = 𝜑 𝑥2 =
1
1 + 0.5
= 0.81650
1
1 + 0.81650
1
1 + 0.74196
= 0.74196
= 0.75767
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𝑥4 = 𝜑 𝑥3 =
𝑥5 = 𝜑 𝑥4 =
𝑥6 = 𝜑 𝑥5 =
𝑥7 = 𝜑 𝑥6 =
1
1 + 0.75767
1
1 + 0.75428
1
1 + 0.75501
1
1 + 0.75485
= 0.75428
= 0.75501
= 0.75485 ≈ 0.7549
= 0.75488 ≈ 0.7549
After the 7th iteration, 𝑥7 − 𝑥6 = 0.00003 < 0.00005 for a
four decimal accuracy. Hence the iteration is stopped and the
root to the desired accuracy is 0.7549.
Prepared by Dr. V. Prabhakar, Professor, VIT Chennai
Prepared by Dr. V. Prabhakar, Professor, VIT Chennai