Okay, there is no rule. There are many possibilities as we will see. There is some calculus that is useful in determining a good one for simple systems.

But there are other questions to answer first.

For a system of equations.

A general rule here is to look for the largest coefficient and solve for that variable for each equation. Using that we get,

We will use

after 7 iterations,

the roots are x = 2, y = -1.5, z = 4 / 3.

]]>In real life the number of equations is in the hundreds or thousands with that many variables. Intelligent guessing is out unless you can plot in thousands of dimensions. We would just pick 1 or 0 for all the initial conditions.

Wow,that converges at 1.92318... And 0.54894...,why?

It does not converge it will keep being repelled by the root, going further and further away.

]]>(a)for non linear equations how do I find other answers?

(b)can I pick any value for x1?]]>

If you try that one you will get a surprise.

If we start with an initial condition of 1 we get the following diagram called a cobweb. See fig 1.

The arrow shows where the point ends up after 5 iterations. Notice it is getting further away from the root. We say it being repelled. To show that not even a closer guess will help, the second figure shows an initial condition of 1.3. After 12 iterations the arrow shows where it ended up. Further away from the root then when it started.

So although there zillions of iterative forms not all of them converge on a root. A little calculus is needed to know when convergence is assured and how fast it will be.

Try this one on your own,

]]>Welcome to the forum.

Mathematicians say that the formula used here converges because successive iterations lead to values of x that get closer and closer to a solution to the equation.

I have made a graph that shows what is going on.

You put the equation

on a graph (blue in my picture)

and also the line

shown red.

You choose an initial guess (x1) and draw a vertical line until it meets the blue curve. This is at the point (x1,y1).

Then you draw horizontally until you meet the red line. This is the point (y1,y1). But rename it (x2,x2)

Now use x2 as the new value to try for x.

Draw a vertical line until it meets the blue curve. This is the point (x2,y2)

Draw horizontally from here to meet the red line at (y2,y2). Rename this (x3,x3).

and so on.

I have shown these movements as a green path. If this path spirals inwards towards a point, then the iteration is converging.

Where the blue curve and the red line cross, is the solution.

Sometimes the iterations do not move inwards towards the solution because the iteration isn't converging. So you have to find the right iterative equation.

Surprisingly, all it depends on is the gradient of the blue curve at the point it crosses the red line.

Bob

]]>By the way the answer to 50 places of the above root is

1.3688081078213726352274143300213255395424355414875...

There are many other iterative forms for that equation. For instance,

If you try that one you will get a surprise.

]]>There are tests and things to do to determine whether convergence but for the sake of this example we will overlook everything.

]]>