Question

use Fixed Point iteration, to solve the nonlinear equations.
u(x,y)=x^2+xy-10=0
v(x,y)=3xy^2+y-57=0
use initial guess
x=1, y=2

Answer 1

we wont use jacobian matrix here ha? that was for newton..

Answer 2

i have no idea how to solve this .
i can solve in single variable using Fixed point iteration.

Answer 3

\(x^2+xy-10=0\) isolate x --> \(x=\sqrt{10-xy}\)
\(3xy^2+y-57=0\) isolate y --> \(y=\sqrt{\frac{57-y}{3x}}\)

so forming fixed iteration equations like this
\[x_{n+1}=\sqrt{10-x_ny_n}\]\[y_{n+1}=\sqrt{\frac{57-y_n}{3x_n}}\]solve and tell me if it converges or not?

Answer 4

\(x_0=1\) , \(y_0=2\)

Answer 5

oops its not working... @amistre64 will help us

Answer 6

are those 2 seperate equations to work out? or is this a system of equations?

Answer 7

if i recall correctly, we determine a sequence generated by a function such that:\[x_{n}=f(x_{n-1})\]
if the sequence converges, we assume that to be our answer

Answer 8

its system.

Answer 9

yes .

Answer 10

mukushla seems to have worked up some functions to play with, my question is, do we have to form these into sqrts ?

y = (10-x^2)/x
x = (57-y)/(2y^2)

Answer 11

not necessarily sqrts...that one will work i think.

Answer 12

for Fixed point iteration
xn=f(xn-1)
it should be in such form that if derivative is taken then it comes out Less than 1for all values.

Answer 13

@mukushla i think you got it right i am getting convergence after 7th iteration x is approaching 2 and y is approaching 3.0459

Answer 14

but it turns to negatives... under radical !!

Answer 15

my book agrees with the answers i got above !.
thanks for your help :)

Answer 16

my idea doesnt seem to converge :)

Answer 17

lol...im still wondering how she got it..
http://www.wolframalpha.com/input/?i=x%5E2%2Bxy-10%3D0+and+3xy%5E2%2By-57%3D0+

Answer 18

X0=1
Y0=2
X1=2.828
Y1=2.546
X2=1.673
Y2=3.293
X3=2.119
Y3=2.9066
X4=1.9598
Y4=3.033

Answer 19

\[x_1=\sqrt{10-x_0y_0}=\sqrt{10-2}=\sqrt{8}\]

Answer 20

if you want i can do more iteration.?

Answer 21

my y1 is 4.28

Answer 22

if i code in mukus equations i get

Answer 23

oh oh ... i think i got it ... u put x1 for calculating y1 instead of x0