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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.
I think he wants to know how to know which x to solve for to actually get the solutions that do not diverge...
X to a side? I do not understand what you are asking.
Thank you,but is there a rule for taking x to a side(as the second one you gave didn't solve) and you didnt write about system of linear equation.(again just curious)
Usually it is trial and error. To find other answers you try different initial conditions. For that cubic you would now try x1 = 1 + i maybe, since you know the other two answers are in the complex plane.
Yes,and about other answers,is there a iterative method for that?
Iteration is more robust than the classical methods taught. It is a general method to solve any type of equation. You can use any initial condition but not all of them will converge. You would of course try to get a good initial guess for a simple system like this.
It does not converge it will keep being repelled by the root, going further and further away.
Wow,that converges at 1.92318... And 0.54894...,why?and I have 2 other questions
Hi Bob and Still Learning;
If we start with an initial condition of 1 we get the following diagram called a cobweb. See fig 1.
hi Still Learning,
on a graph (blue in my picture)
and also the line
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.
If you try that one you will get a surprise.
Oh yes,I have tried it,it converges at 1.368808...
A system of equations is a little different. First you should see how the one we are doing is solved.
Thank you it was very helpful,how does one solve system of equations using it?(just curious)
No, when you get convergence you stop. But it may take many iterations for that to occur.