SHORT NOTES MODULE 6 : NUMERICAL METHODS
TRAPEZOIDAL RULE
to approximate definite integral that cannot be solved using direct/technique of integration
find the width of trapeziums/subintervals based on n đ?‘?−đ?‘Ž â„Ž= đ?‘›
finding b
ďƒ˛ f ( x ) dx a
with n subintervals
determine the ordinates substittute values into the formula
đ?‘Ľ0 = đ?‘Ž, đ?‘Ľ1 = đ?‘Ľ0 + â„Ž , đ?‘Ľ2 = đ?‘Ľ1 + â„Ž , ‌ , đ?‘Ľđ?‘› = đ?‘?
evaluate đ?‘“ đ?‘Ľđ?‘–
b
ďƒ˛
f ( x ) dx ď €
h ďƒŠ f ( a ) + 2f ( x1 ) + 2ďƒŤ
+ 2f ( xn−1 ) + f (b ) ďƒšďƒť
a
NEWTON-RAPHSON METHOD
to approximate root or intersection point
set equation as f(x) = 0
find f '(x)
by putting đ?‘Ľ0 as initial approximation, use formula to find đ?‘Ľ1 , đ?‘Ľ2 , ...
xn+1 = xn −
f ( xn )
f ' ( xn )
stop the process of iteration when consistent value achieved or until required accuracy.
; n = 0,1,2,