# Non-linear Equations

17 replies to this topic

### #1 Behnoud

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Posted 22 May 2006 - 05:29 PM

How can I solve a non-linear Equation system?
I khow it is possible for a TI to solve it!
I entered equations as below:
solve({x+x*y=5,x^2+y=2},{x,y})

### #2 Daniel (CZE)

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Posted 22 May 2006 - 08:42 PM

As far as I know ClassPad with OS 2.2 can't solve non-linear equation system with solve() function. You must split calculation to several steps to get result.

### #3 Behnoud

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Posted 23 May 2006 - 05:27 PM

How?
It's not clear for me!
Do u mean it needs programming?

### #4 R00KIE

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Posted 23 May 2006 - 06:00 PM

My HP49 couldn't solve it and mathematica gave a really long solution,are you sure the TIs can solve this particular one?

### #5 unique33

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Posted 23 May 2006 - 06:18 PM

there is no solution in the main application , but you can solve it numerically using LuaNumAn Project written by PAP.

(The Broyden Method , I posted a related problem and the solution on the Engineering Applications Topic)

### #6 Daniel (CZE)

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Posted 23 May 2006 - 06:57 PM

How?
It's not clear for me!
Do u mean it needs programming?

No.

You must extract first variable from first equation using solve() then use it in second equation (with | command) and find all results manualy (as you write on paper).

I tried to solve this system but in one part CP didn't return exact expression but approximation only. So I don't if this is usable for you.

```solve(x^2+y=2,y)
{y=-x^2+2}

solve(x+x*y=5,x)|ans[1]
{x=-5/(x^2-3)}

solve(x=-5/(x^2-3),x)
// CP can't solve this. We must edit the expression.

solve(x(x^2-3)=-5,x)
{x=-2.279018786}```

And sorry for my english. I'm not from English-speaking country

### #7 TacoFred

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Posted 23 May 2006 - 10:28 PM

If you really have to solve nonlinear systems of equations, and you have time, you can just do it by hand...simplest way unless you can make the CP do it (which it doesn't without modifications, but by that time, you can just do it by hand).

### #8 PAP

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Posted 25 May 2006 - 08:05 AM

How can I solve a non-linear Equation system?
I khow it is possible for a TI to solve it!
I entered equations as below:
solve({x+x*y=5,x^2+y=2},{x,y})

A quick solution:
```require("table","LuaNumAn/Broyden")
local function F(x)
return {x[1]+x[1]*x[2]-5,x[1]^2+x[2]-2}
end
local function J(x)
return{{1+x[2],x[1]},{2*x[1],1}}
end
root=Broyden({-1,0},F,J,5E-16)
table.foreach(root,print)```
You get {-2.27901878616659,-3.19392662770025}, accurate to at least 15 decimal digits. See LuaNumAn's documentation for details .

### #9 DrCoyote

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Posted 25 May 2006 - 01:03 PM

My HP49 couldn't solve it and mathematica gave a really long solution,are you sure the TIs can solve this particular one?

My TI-92 Plus gives x = -2.27901878617, y = -3.1939266277. So yes, the TI calculators can solve it.

### #10 Orwell

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Posted 25 May 2006 - 03:33 PM

A quick solution:

```require("table","LuaNumAn/Broyden")
local function F(x)
return {x[1]+x[1]*x[2]-5,x[1]^2+x[2]-2}
end
local function J(x)
return{{1+x[2],x[1]},{2*x[1],1}}
end
root=Broyden({-1,0},F,J,5E-16)
table.foreach(root,print)```
You get {-2.27901878616659,-3.19392662770025}, accurate to at least 15 decimal digits. See LuaNumAn's documentation for details .

I don't remember having said that require() now accepts several arguments. But since you use it, that means you know it. Glad to see it's useful.
Btw, I noticed that you often use the "table" package only for the foreach() function. You could have the same result with
`for key,value in pairs(root) do print(key,value) end`
The pairs() function does not require any extra package

And again, this is another proof of the utility of CPLua and LuaNumAn

### #11 R00KIE

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Posted 25 May 2006 - 05:35 PM

i was looking for an EXACT solution not an aproximation, the HP can't solve it as a system directly because it can't factor the expression, but if i use the same method used to solve it on the classpad i get not one but three solutions
{x=1.14+.946i , x=1.14-.946i , x=-2.279} and {y=1.597-2.157i , y=1.597+2.157i , y=-3.194}, these are the rounded results (obviously).
we must use the calc together with our heads not appart. anyway, this system gives a third order system, that can still be solved exactly but calcs don't have it implemented.

### #12 Orwell

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Posted 25 May 2006 - 07:26 PM

There is no such thing as an exact solution here, you may only find some good approximations for it

### #13 Guest_Anonimo_*

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Posted 25 May 2006 - 10:12 PM

My HP49 couldn't solve it and mathematica gave a really long solution,are you sure the TIs can solve this particular one?

MMmmmm....the program name is Solvesys. For the hp 49g. Find hpcalc.org

### #14 Behnoud

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Posted 26 May 2006 - 08:25 AM

I found an eActivity for systems on classpad.org.
It solved the equations system which I post.

### #15 PAP

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Posted 26 May 2006 - 11:41 PM

I don't remember having said that require() now accepts several arguments. But since you use it, that means you know it. Glad to see it's useful.

Oh, yes, you did .

Btw, I noticed that you often use the "table" package only for the foreach() function. You could have the same result with

`for key,value in pairs(root) do print(key,value) end`
The pairs() function does not require any extra package

I know this solution, but it doesn't make the program shorter, and I usually want to print a table more than once in a program. But if the "table" package uses a lot of memory, your solution is definitely better.

### #16 PAP

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Posted 26 May 2006 - 11:59 PM

i was looking for an EXACT solution not an aproximation, the HP can't solve it as a system directly because it can't factor the expression, but if i use the same method used to solve it on the classpad i get not one but three solutions
{x=1.14+.946i , x=1.14-.946i , x=-2.279} and {y=1.597-2.157i , y=1.597+2.157i , y=-3.194}, these are the rounded results (obviously).

First, I want to ask something: What do you mean by "i use the same method used to solve it on the classpad"? As far as I know the only way to solve nonlinear systems in ClassPad is to use the "Broyden" function, included in my LuaNumAn package. Don't tell me that you have translated it for the HP ? You probably mean something else...
About the solutions you have find: Indeed, there are also complex conjugate solutions, but LuaNumAn can't handle complex numbers... yet. The main problem with complex arithmetics is that Lua doesn't support complex numbers directly. I could probably use the "cas" package to overpass this problem, but I'm afraid that it will be slow, since I will need to use plenty of CAS commands within a Lua program (even a simple addition will need the CAS, or a Lua script for adding complex numbers ). Implementing a complicated numerical method this way will be a real nightmare. For the moment, LuaNumAn contains numerical methods for real -not complex- numbers.

anyway, this system gives a third order system, that can still be solved exactly but calcs don't have it implemented.

Incorrect. In most cases, nonlinear systems cannot be solved "exactly" by any CAS system, simply because there is no exact solution, or because there is an exact solution, but it is very complicated and useless. This is when Numerical Analysis is involved. Nonlinear systems can be solved "exactly" only in very special (simple) cases. Their numerical solution is not an easy task as well: numerical methods for nonlinear systems are not easy.
In general, if you seek for "exact" solutions in real problems, you will dissapointed. But I understand, most programmers, novice to Numerical Analysis, always seek for "exact" solutions, even if there is no such a thing.

Btw, the TI probably solves this system numerically, using a numerical method which is, quite probably, a version of Broyden's method, similar to the corresponding implementation I have included in LuaNumAn.

### #17 R00KIE

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Posted 29 May 2006 - 05:11 PM

When i said "i use the same method used to solve it on the classpad" i meant i used the same method "Daniel (CZE)" used.
"Incorrect. In most cases, nonlinear systems cannot be solved "exactly" by any CAS system", i guess you should take a look at the cardano method to solve third order systems at http://mathworld.wol...bicFormula.html
also i wasn't trying in any way to say anything bad about LuaNumAn.
you are right to say that in the real world exact solutions are worthless, but it's still a good exercise to know if it is possible or not to find them and why.
my point is, you should have a math background even when solving linear or nonlinear problems with a calculator or pc, so you can at least see if the result is coherent.

### #18 PAP

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Posted 30 May 2006 - 01:13 PM

When i said "i use the same method used to solve it on the classpad" i meant i used the same method "Daniel (CZE)" used.

Ok. I guessed that you used some method, but you didn't specified which one.

i guess you should take a look at the cardano method to solve third order systems at http://mathworld.wol...bicFormula.html

No offence, but this is not correct. The web page you mentioned does not describe a way to solve third order nonlinear systems. It describes a way to solve general cubic equations, which is a completely different thing. Cardano's method involves some nonlinear systems, but they are only special cases, not an arbitrarily complex nonlinear system. I insist: there is no general method to solve nonlinear systems analytically, including third order systems; only special nonlinear systems can be solved exactly, and this is the exception, not the rule.

also i wasn't trying in any way to say anything bad about LuaNumAn.

I know that, it was obvious that you didn't wanted to be offensive.

my point is, you should have a math background even when solving linear or nonlinear problems with a calculator or pc, so you can at least see if the result is coherent.

I absolutely agree. We use calculators to perform addition, but we should know how to add numbers anyway. We use numerical methods to solve a complicated problem, but we should at least know what exactly we solve, and how. I never said that one must use numerical methods blindly.

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