I am solving a system of linear equations Ax=b where A is a matrix (121*121) which contains a rational polynomial of unknown "kappa" along with floating numbers. I am using Linear algebra package with the LinearSolve command and option "solve" to find out unknown vector x in MAPLE 18. b is a vector having entries zero and 1. Maple solves for x and result is a long expression i.e. x[i] is a rational polynomial in kappa; a very long rational polynomial which I am importing as a text. The problem is that A*x-b is not close to zero for any randomly chosen value of kappa. In particular, it is seen that for kappa>=4 the error is less but it shoots up for kappa=2. Maple provides an acceptable solution for smaller size Matrix ( for example dimension of A less than 80). What is the best existing solver (Maple, Mathematica, MATLAB (symbolic)) to tackle large no of equations like this?
updates: As suggested by Matthew P., it seems that the round-off error accumulates due to floating point approximation. One way is to use symbolic calculation for as many steps as possible ( retain sqrt(2) and pi as it is, don't use 1.414213562 and 3.141592654 as a replacement ). In this way, the accuracy increases but the overall time for the computation shoots up as the number of equations reaches more than 150. [I am waiting for the results, maple is running for 40 hours]. For the reference, I have attached text files dealing with A*y=b; for 41 equations. Note kappa is replaced by x in the attachment.
Hi, I think that MAPLE gives you the general form of the solution because there is no specific information given on kappa. Using the "assume" MAPLE command you can specify for example that kappa is less or equal than 2. Solving again the linear system will perhaps lead to a more precise solution (TBC).
Dear Gaurav, in my experience in this case Matlab or Mathematica offer same advantages and troubles; do you really must have symbolic solution or numerical one can be sufficient? Gianluca
Hi Gaurav.
I think the floating point numbers in your matrix might be part of your problem. Trying to get a symbolic solution when your input involves floating point numbers can be problematic.
Ideally, if you can convert those floating point values to exact numbers (probably rational numbers) that would be best if you really want a symbolic solution. If you can't do that, then maybe try increasing Maple's precision. Set Digits:=20; at the start of your file, and re-run all the computations then see if the error is lower. If you're feeling adventurous maybe try 50 or 100 digits of precision and see if that gives a better answer.
As a follow on question, are you able to make your Maple file available to us? If I can see the file and the data, I might be able to give better advice.
Why don't you try python's sympy?
http://www.scipy-lectures.org/advanced/sympy.html
Thank you Richard . I tried the assume command but unfortunately, there was no improvement in the solution.
Thank you, Matthew. I was using 16 digits in my calculation. Considering 20 digits improves the solution. Now the solution is accurate up to 4 significant digits. Soon I will be uploading the files.
Use the command simplify to obtain the results in its simplest forms. Some linear systems are ill-conditioned. Some restrictive conditions on kappa may be given. Also you may need to increase the value of Digits and see what will happen.
Gianluca Argentini thank you for the reply. The numerical solution is easily obtained in my case but in this particular case, I am interested in a symbolic calculation.
Moawwad Elmikkawy thanks for the comment. I am already using simplify command.
- Badges
- Science topic
- Similar topics
- Vectorization