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A. Solution to Algebraic &Transcendental Equations. Algebraic functions. The general form of an Algebraic function:. f i = an i -th order polynomial. Example :. f 3. f 2. f 0. Polynomials are a simple class of algebraic function. a i ’s are constants. Transcendental functions.

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algebraic functions
Algebraic functions

The general form of an Algebraic function:

fi = an i-th order polynomial.

Example :

f3

f2

f0

Polynomials are a simple class of algebraic function

ai’s are constants.

transcendental functions
Transcendental functions

A transcendental function is non-algebraic.

May include trigonometric, exponential, logarithmic functions

Examples:

equation solving
Equation Solving

Given an approximate location (initial value)

find a single real root

Root

Finding

non-linear

Single variable

Open

Methods

Brackting

Methods

Iterative

Newton-

Raphson

Secant

Bisection

False-

position

problem
Problem
  • Find the root of

f(x) = e-x – x

  • There is no exact or analytic solution
  • Numerical solution:
iterative solution
Iterative Solution
  • Start with a guess say x1=1,
  • Generate
    • x2=e-x1= e-1= 0.368
    • x3=e-x2= e-0.368 = 0.692
    • x4=e-x3= e-0.692=0.500

In general:

After a few more iteration we will get

convergence examples
Convergence Examples

Convergent spiral pattern

Convergent staircase pattern

divergence example
Divergence Example

Divergent spiral pattern

Divergent staircase pattern

existence of root
Existence of Root

There exists one and only one root if

L is Lipschitz constant,

convergence
Convergence?

If x=a is a solution then,

error reduces at each step

i.e. iteration will converge

If magnitude of 1st at x=a

derivative is less than 1

k th order convergence
kth Order Convergence
  • Pervious iterative method has linear (1st order) convergence, since:
  • For kth order convergence we have:
  • Now consider a 2nd order method.

Aitken’s 2 process

aitken s process15
Aitken’s process
  • If  is a root of the equation i.e., =g() then,
  • Now if we use
algorithm
Algorithm

  guess_value;

while (!   g()) {

}

slide19
http://www.buet.ac.bd

/cse/users/faculty/reazahmed/cse317.php