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, NUMERICAL METHODS I – COS 233-8
Interval bisection method
Strategy Find two values of x, that is, a and b, which bracket the root by checking whether f ( a ) × f (b) < 0. Then
successively divide the interval in half and replace one endpoint with the midpoint so that the root is
bracketed again.
Requirements The function f(x) must be continuous in the interval.
There should not be multiple roots in the interval.
Advantages The number of iterations to achieve a specified accuracy is known in advance.
It is the method that is recommended for finding a first approximation to the root.
Convergence Slow because the estimate of the root may be better at an earlier iteration than at later ones.
Order of convergence 1
(b − a )
Error formula e=
2n
Note This does not mean that each error is smaller than the previous one.
Secant method
Strategy Choose two values of x, that is, x 0 and x1 , which are close to the root. Draw a straight line through the
points (these points can either be on the same side or on opposites sides of the root). The intersection of
the line with the x-axis should be close to the root. Repeat the process by always using the last two
computed values.
Requirements The function f(x) must be continuous.
The function f(x) must not be far from linear in the vicinity of the root.
Convergence Intermediate because the error is proportional to the product of the previous two errors. It is therefor
faster than a linear method but slower than a quadratic method.
Order of convergence 1.62
f ( xn )
Iteration formula x n +1 = x n − ( x n − x n −1 )
f ( x n ) − f ( x n −1 )
g ′′( ξ1 , ξ2 )
Error formula en +1 = (e n )(en −1 )
2
1+ 5
Note The order of convergence is = 1.62 .
2
EXAM
PACK 2023
QUESTIONS WITH ANSWERS
EMAIL:
,COS2633
EXAM
PACK 2023
LATEST
QUESTIONS
AND ANSWERS
For queries or any assignment help
Email:
,COS2633
EXAM PACK
Revision PACK
Questions. Answers
, NUMERICAL METHODS I – COS 233-8
Interval bisection method
Strategy Find two values of x, that is, a and b, which bracket the root by checking whether f ( a ) × f (b) < 0. Then
successively divide the interval in half and replace one endpoint with the midpoint so that the root is
bracketed again.
Requirements The function f(x) must be continuous in the interval.
There should not be multiple roots in the interval.
Advantages The number of iterations to achieve a specified accuracy is known in advance.
It is the method that is recommended for finding a first approximation to the root.
Convergence Slow because the estimate of the root may be better at an earlier iteration than at later ones.
Order of convergence 1
(b − a )
Error formula e=
2n
Note This does not mean that each error is smaller than the previous one.
Secant method
Strategy Choose two values of x, that is, x 0 and x1 , which are close to the root. Draw a straight line through the
points (these points can either be on the same side or on opposites sides of the root). The intersection of
the line with the x-axis should be close to the root. Repeat the process by always using the last two
computed values.
Requirements The function f(x) must be continuous.
The function f(x) must not be far from linear in the vicinity of the root.
Convergence Intermediate because the error is proportional to the product of the previous two errors. It is therefor
faster than a linear method but slower than a quadratic method.
Order of convergence 1.62
f ( xn )
Iteration formula x n +1 = x n − ( x n − x n −1 )
f ( x n ) − f ( x n −1 )
g ′′( ξ1 , ξ2 )
Error formula en +1 = (e n )(en −1 )
2
1+ 5
Note The order of convergence is = 1.62 .
2
