# Solving system of equations by using simple iteration method

I have a problem:

$$\begin{cases} \sin(x) + 2y = 2 \\ \cos(y - 1) + x = 0.7 \end{cases}$$

with margin of error 0.00001

And I need to solve this by using Fixed-point iteration method.

Can someone help me with that? Algorithm for full solution might be the best thing ever, since I gave to solve this in Maple later..

P.S. one more question. What is q in this method and how to find it?

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newton-raphson method en.wikipedia.org/wiki/Newton%27s_method – Ömer Oct 7 '13 at 17:37
@user1075413: Not sure what you mean by that, but you could solve the first equation for $y$, substitute into the second equation and then use Newton's Method (iteration) to find $x$. Lastly, substitute into equation one to find $y$. I get $((x, y) = (-0.289809, 1.14288)$. – Amzoti Oct 7 '13 at 17:42
A simpler method than Newton to start with could be gradient descent – Keeran Brabazon Oct 7 '13 at 17:49
This question can't really be answered without knowing which method you mean. In my opinion, "Simple Iteration Method" sounds like fixed-point iteration. Is any of the suggestions correct? – JiK Oct 7 '13 at 18:06
@JiK it seems that it actually is "fixed-point iteration" method. Can you help with that? – domakas Oct 7 '13 at 18:10

Try this iteration:

$\qquad x_{n+1}=0.7-\cos(y_n-1)$

$\qquad y_{n+1}=(2-\sin(x_n))/2$

Taking $x_0=y_0=0$ works fine for me. I get that

$\qquad x^*=-0.28980932884903, \quad y^*= 1.1428847552227$

is a fixed point of $F(x,y)=(0.7-\cos(y-1),2-\sin(x))/2)$ after 27 steps, but it only takes 11 steps to get an error below $10^{-6}$. The error is measured by $\| F(x,y)-(x,y) \|_1$, where $\|(x,y)\|_1 = \max(|x|,|y|)$.

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What is the condition while checking for error? I use abs(x[n+1]-x[n]) < (1-q)*e/q, however I don't know how to calculate q and I only check for x (do I need to check y?) – domakas Oct 7 '13 at 18:31
Can you explain what ∥F(x,y)−(x,y)∥ is? And few stupid questions: what ∥ ∥ means? And what F(x,y) is called in English? – domakas Oct 7 '13 at 18:46
Still don't understand what condition should be checked while checking for precision. ∥F(x,y)−(x,y)∥ should be smaller that e or what? – domakas Oct 7 '13 at 19:48
@user1075413, yes, compare that with say $10^{-6}$. – lhf Oct 7 '13 at 19:51
Can't get it right.. after first iteration ∥F(x,y)−(x,y)∥ = 1.447760103... and after 10th iteration 1.432694311... it decreases very very slowly, so it's never lower than 10^-6 – domakas Oct 7 '13 at 19:57