Question

Help : : : Question from numerical analysis.
Give an example with proper justification to show that the condition of convergence of fixed point iteration for numerical solution of equations is sufficient and by no means necessary.

Answer 1

@amistre64 @ganeshie8 @eliassaab

Answer 2

i must say that this is not going to be workable to me. i simple do not have the proper thrms and or justifications (vocabulary) to assess it.

Answer 3

You just give me the example @amistre64 and a brief outline, i can manage the rest.:)

Answer 4

@gorv

Answer 5

it will be too long

Answer 6

bt i can give u idea

Answer 7

like u have to iterate ant function

Answer 8

Yeah thats what i need.:)

Answer 9

when u itrate let the ans is 0.2345098655

Answer 10

then keep iterating

Answer 11

at one point when the ans will come like ....four point after decimal are similar thn that will be your ans

Answer 12

@Princer_Jones u got me ??

Answer 13

@gorv

Answer 14

ohh u have any function??

Answer 15

we first find \[\phi(x)\]

Answer 16

no thats what we need to construct , by the question. If i already had the function, then i would have completed solving it.:)

Answer 17

such that it derivative is less than one

Answer 18

Derivative will be grater than 1... if it is less than 1, then it will always converge, since the condition is sufficient

Answer 19

let x^4-x-10=0

Answer 20

ok should we proceed??

Answer 21

yes

Answer 22

in fixed point iteration , the equation becomes x=x^4-10 Hence here fi(x)=x^4-10 whose derivative is 4x^3

Answer 23

whic is greater than 1

Answer 24

x^4-x=10

Answer 25

x(x^3-1)=10

Answer 26

x=10/(x^3-1)

Answer 27

now??

Answer 28

oh you took different expression for fi(x)?.

Answer 29

Here fi(x)=10/(x^3-1)?

Answer 30

yeah..and its derivative will be <1

Answer 31

that how we select the function ..ok gotcha??

Answer 32

hmm but we need the derivative to be greater than 1 and show that it still converges...

Answer 33

u can go with any condition

Answer 34

this one will be smallest

Answer 35

u can go with
x=x^4-10

Answer 36

but it will take longer

Answer 37

The condition of convergence is that the derivative is less than 1. But we have to construct a function such that the derivative is greater than 1 , but it still converges, and then only it will prove that going against the condition of convergence too we have a solution, that means the condition is sufficient and no longer necessary

Answer 38

well thn u can go with your function

Answer 39

x^4-10?

Answer 40

@ganeshie8 help please//