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ptc-48545491-Visitor
1-Visitor
November 27, 2012
Question

Need assistance with symbolic matrix multiplication and plotting with 2 variables

  • November 27, 2012
  • 2 replies
  • 7398 views

I have several 8x8 matrices that need to be multiplied together like so:

[U] = [A]*[B]*[C]...etc

with [U] being the overall matrix for the system

Within the [A],[B],[C],etc., matrices are various numerical values as well as two unkown variables. The ultimate goal here is to have the final [U] matrix display these 2 unknown variables symbolically. After this, I can find the determinant of the matrix and create a large polynomial that solves for the relationship for the 2 variables so that the polynomial = 0.

I am trying to multiply these matrices together symbolically using the ---> (symbolic command) but the problem is that the matrix is becoming too large and not displaying the result. I have attempted to do each section individually and then simplify the results (for example, [1] = [A]*[B], simplify, then [2] = [1]*[C], simplify, etc.) but even that is becoming too large.

Is there any way around this so that I can see the results?

Also, once I have the results, how do I go about plotting the relationship between the 2 variables? Say I have a polynomial to the 24th power - that would mean mean I have 12 roots so 12 different curves that display the relationship so that the 2 variables drive the polynomial equation to zero. How can I not only find these values over a range, but also plot them?

Thanks

2 replies

A
AlanStevens19-Tanzanite
19-Tanzanite
November 28, 2012

Upload a worksheet with the matrices and what you've tried so far.

Alan

P
ptc-48545491-VisitorAuthor
1-Visitor
November 28, 2012

Here is what I have so far. My apologies for the size of it.

Basically I have identified [F4], [F3], [F2], and [F1] matrices, all of which have known values.

I then identified [P4], [P3], [P2], and [P1] matrices which have unkown variables "v" and "w" present.

I then try to symbolically multiply [U] = [P4][F4][P3][F3][P2][F2][P1][F1] but it becomes too large to display, so I have to do iterations of this and simplify it. For example, [U22] is the matrix of [P4][F4][P3] after it has been simplified. I conitinue this process and luckily I get a result, [U6] at the bottom which is the overall system. However, it is red and quite honestly it is difficult to work with.

Is there any easier way to do this?

I have a couple more problems with this but I'll focus on this part first.

R
RichardJ19-Tanzanite
19-Tanzanite
November 28, 2012

Your final result displays because you added a numeric evaluation to the expression, not because you did the calculation in small pieces. That is an undocumented "feature" (i.e. a bug that works to our advantage ). Actually, I didn't realize this "feature" existed until I saw your worksheet. The problem with that approach is that the result can't be used in a subsequent symbolic expression. An alternative is to look at the matrix a column at a time. Creating a polynomial function in w and n is easy, but I don't know what you are going to do with it. I don't know any way to find all the roots of a polynomial that goes to the 24th power in w and the 8th power in n.

T
tdevaraj1-Visitor
1-Visitor
November 28, 2012

Not sure I have got exactly what you are attempting. And I'm not sure which version are you using?

In Mathcad14/15 you get the warning.

But you should still be able to calculate the determinant of U, regardless of this warning as the result does get saved to U.

mc15_matrix.png

I tried this is Prime 2.0 and you can see the matrix U

prime2_matrix+solve.png

I've attached the prime 2 file and the PDF of the calculation.

I used a surface plot to show the function of x and y.

P
ptc-48545491-VisitorAuthor
1-Visitor
November 28, 2012

I am using Mathcad 15.

Also, my 8x8 matrix results are MUCH larger than the example you showed. Are you sure Prime 2.0 could handle an exponentially larger matrix?

Also, I don't need to calculate the entire determintant of the 8x8 matrix [U]. I have some boundary conditions set that will drive many of the elements to zero, and my result is a 4x4 matrix of which I need to find the determinant of.

I guess this is just a minor complaint, because with some fiddling I am getting the matrix I need. However, you did sort of go into my next question. After I have my polynomial (in my case in terms of variables w and n), how do I go about creating a plot that shows all values (roots) of w over a range of n so that it drives the polynomial to 0?

T
tdevaraj1-Visitor
1-Visitor
November 28, 2012

I removed all the imtermediate steps and converted this to Prime 2.0 and calculated the Matrix product in a single step.projectcalc_prime2.png

I've only copied part of the solution here...but it extends way off into right hand side!

Mathcad 15 & Prime use the same symbolic engine...

I have used Prime 2.0 M010 64 bit

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