Math Program

[MeBeatYou]

Sorry lol this is my second topic in one day but it came to me. I need a program that can solve 4 x 4 systems of equations. Does anyone know if there is a Cramer's Rule program or anything that can help me solve these? Thanks a lot

[Weregoose]

Two excellent sources: MathWorld and Wikipedia

For solving a system of linear equations, look up the command [font="courier new"]rref()

[Jeremiah Walgren]

TI has also created applications (which are rather pointless in my opinion) that will allow you to enter systems of equations and solve them (plus a couple other things, I think). You can find this app by starting here, clicking on your calculator model in the list to the right and then finding the "Polynomial Root Finder and Simultaneous Equation Solver" app.

Personally, I prefer rref( on the calculator.

[Brazucs]

[Jeremiah Walgren]

What if each one of those topics contains a totally different subject? (I could do that if I had enough time, you know. But I don't, so I won't.)

[MeBeatYou]

well i made my own program today to solve 3 x 3 systems by
A.) Elimination
B.) Substitution
C.) Cramer's Rule
D.) Matrices

Now, i need the same thing on 4 x 4 systems only. I want to input all the vars and then just have it tell me the w, x, y, z points.

[aforsy]

someone should move this to the spiffy new math help forum.

[Fr0sty]

Isn't this just the make a matrix and [Matrix]^-1? It's been such a long time since I've solved system of equations though, but isn't that what it is?

[thornahawk]

If anyone's interested in my take on methods to solve linear equations:

Cramer's Rule: Useful for hand calculation with small systems (<4), but otherwise a profligate waste of computing time. Let me put it this way: the effort in solving an n variable equation in this manner would take ~ n! operations. Baaaad.

Compute matrix inverse before multiplying: About ~n^3 operations are needed for this. I'd use it were it not for the fact that there are (slightly) faster ways, and that this is prone to roundoff error.

rref() (a.k.a. Gaussian elimination). The standard method in all high-end linear equation solvers. Also ~n^3 operations, but this is slightly faster than multiplying the inverse, and less prone to roundoff besides.

I deliberately made my explanations sketchy, because this post would be insanely long if I put in all the details. But if you ask for 'em... I could recommend a few nice books. :)

thornahawk

P.S. Although, if you're not going beyond 10 variables, I suppose the distinctions I gave earlier wouldn't really matter much.

[Darth Android]

I use the computed inverse method and can't see how a round off error would occur unless you rounded your multiplication (TI handles 14 digits of precision)

[chipmaster]

[Darth Android]

Yup. And because the calc uses the same number format for rref(), those limitations will affect it as well.

[thornahawk]

I was expecting just that kind of question. :)

The effect of roundoff is actually more of a problem-by-problem basis. Simply put, some problems are well-conditioned: small changes in the input correspond to small changes in the output; others are ill-conditioned. You get my drift. ;)

In the theory of solving linear equations, there is such a quantity called the "condition number" that is a measure of just how much of a badass problem were you asking the algorithm to solve. Larger values mean your problem is ill conditioned. There's a facility for computing it in Matlab and Mathematica, and I have written a TI-BASIC program that will estimate it. I will post upon request. :)

ANYWAY, here's the meat: TI calculators, to my knowledge, use Gaussian elimination both for rref() and matrix inversion. Matrix inversion of, say matrix [A], would then be equivalent to rref(augment([A], identity(N))) where N is the dimension of the square matrix [A], and extracting the submatrix to the right. Now, when you're computing the inverse for solving equations, you have two possible sources of roundoff: computing the inverse and multiplying with the appropriate vector. On the other hand, with rref(), you're directly working with the vector, so there's only one way that roundoff can foul up your computation.

Now, what does this have to do with my rambling above regarding the condition number? Simply put, if your matrix is "well-conditioned", there shouldn't be much problem with both methods. But, if you're unlucky enough to be faced with stuff like Hilbert matrices, it's slightly better to use rref() than computing the inverse.

My discussion above would be too short to fully explain the theory of Gaussian elimination, but I hope the gist is there. I must mention that most books on numerical linear algebra devote at least one whole chapter to Gaussian elimination, and why it works well for some matrices and fouls up on others.

thornahawk

P.S. Besides, computing the inverse before multiplying needs more operations than doing rref() on the augmented matrix.

[MeBeatYou]

Incase anyone was in the same boat as me, I figured out how to do this. Im not going to go through my code just give a basic run down.

First, go to Matrix and edit matrix [A] (or any other you wish). If you have a 4 x 4 system you would make it a 4 x 4 box and enter the numbers as they go. If you have a 3 x 3 system, you would make [A] a 3 x 3 matrix... well you get the point.

Then you make a second matrix for everything that comes after the equals sign. In a 4 x 4 system, the second matrix would be a (in terms of [B]) 4 x 1 matrix. In this, you would then enter the numbers as they go.

Once you have finished your two matrices, you go to Matrix, Names, and select your first matrix ([A] in my tutorial). Then you hit the (x-1) button on your calculator. Next, go into the matrix again and select your second matrix ([B] in my tutorial). Press enter and you now have your values for all variables in order.

I also have a program on my calculator to solve a X by X system of equations by either matrices or cramers rule if anybody interested?

Thanks for the help

[elfprince13]

[MeBeatYou]

Wow... thanks :hmpf:

I've been trying to do that forever and we haven't learned about matrices until this year and I didn't make the connection between 2 x 2 systems and ? x ? systems heh