ADVANCED MATHEMATICS FORM 6 – NUMERICAL METHOD

SECANT METHOD
The secant method requires two initials values X₀ and X₁

Line AB is a secant line on the curve f(x)



We find the roots of this, the value of x such as that y=0

 

 


 



In general secant formula is given


Comparison with Newton’s method
-Newton’s converges faster (order 2 against   ≈1.6)
– Newton’s requires the evaluation of f and f₁ at every step
-Secant method only requires the evaluation of f

Example
Calculate in 3 iteration the root of the function f(x)= x²-4x+2 which his between
X₀=0 and X ₁=1
Solution


 



 


 



 





NUMERICAL INTEGRATION

Definite integral    is used to determine the area between y= f(x), the x -axis and the ordinates x = a and x = b

An approximate value for the integral can be found by estimating this area by another two methods

A.   Trapezium rule

 

 







Example
Estimate to 4 decimal places       
Using five ordinates by the trapezium rule

Solution

Taking five ordinates from X = 0 to X = 1
5 – 1 = 4 number of strips


 





 
 

Simpson’s rule

Simpson’s rule is another method which can be used to find the area under the curve y= f(x) between x = a and x = b

A quadratic equation is fitted (parabola) passing through the three points i.e through A,B,C

Then

Example