Accuracy of results in symbolic variable and double

Hello there, I have wriiten a code to genrate some generalized matrix and now I am performing some matrix operations like inversion, addition over that matrix. So i observed two things:

1. If I use symbolic variable to create that particular matrix, say A, then the programm is taking too much time (as the order of matrix around 1000) to do the computaion and
2. If I don't use symbolic variables to create that matrix then I get all the results in double (which is obvious and the time taken by the code is also very less) but the results are different than the one I got from symbolic method. For example the inverse is not coming out as exact.

I understand the problem of double values but is there any way that I can avoid this problem. Is there any solution that without using the symbolic I can get the exact result?

Any help is appreciated. Thanks in the advance.

[Dana]

If I understand correctly, with option #1 you're setting up a symbolic matrix, inverting it (again, symbolically), then substituting in particular numeric values to get an inverse. With option #2, you're just setting up a numeric matrix directly, and then inverting that.

If that's correct, then let me assure you that you will not generally get a noticeably more accurate answer with option #1 than option #2, and indeed sometimes you'll find that option #2 is actually more accurate (modern computers are equipped with very efficient and stable algorithms for numerically computing matrix inverses). Given the massively longer amount of time it takes to do it via option #1, I have a hard time believing there are situations where it's worthwhile to go that route (especially for the size of the matrices you're talking about).

Put another way, if the difference in accuracy you'll get between these two methods is important for your application (and again, depending on the matrix, option #2 may actually be more precise in some cases), then you should probably consider some alternative numerical approach altogether. For example, you may want to use a data type with more precision than the double type (see, e.g., the user-written HPF Toolbox on the MATLAB exchange).

[raina RAJ]

HPF toolbox doesn't perform the matrix inversion. Si I am still at square one. I am not able to move forward. The accuracy is important for me because in the calculation I often find the error saying "badly constructed matrix" which is not exactly true. Is there anything else I can use instead of double? Please help me out. It's really important for me.

[Dana]

"Badly constructed matrix" isn't an actual error in MATLAB. Is it a "Matrix is close to singular or badly scaled" error? If so, using option #1 isn't going to get rid of that problem either: Fundamentally, you're going to end up dividing somewhere by a number x=y-z, where y and z are very close to each other. As a result, x will be very close to zero, and more importantly, highly sensitive to tiny numerical discrepancies in y and z (which will themselves be the result of some arithmetic operations that are subject to numerical degradations). So in the end, your numerical matrix inverse won't be accurate no matter which way you go.

When you get a "Matrix is close to singular or badly scaled" error, unless you have some good reason to believe otherwise, it probably means that the theoretical matrix (i.e., the matrix you're somewhat inaccurately representing numerically on your computer) is actually singular. When that's the case, you need to think carefully about why it would be singular, and then fix that underlying problem. Without knowing more about your application, though, I can't offer any guesses as to why your matrix would be singular.

[Walter Roberson]

I have encountered matrices that were non-singular in theory, but singular in practice. In some of the cases, adding more digits by using software floating point (vpa() and symbolic toolbox) was enough. A couple of times any finite precision would not have been enough (though it can then be difficult to prove that theoretically infinite precision would be enough in those cases.)

If you do use the symbolic toolbox, watch out for the fact that by default the symbolic toolbox approximates floating point numbers. For example

>> sym(pi-313*eps)
ans =
pi
>> sym(pi-314*eps)
ans =
7074237752028283/2251799813685248

It is quite common to find that the approximation done is what you would like to be done, that it ends up restoring the theoretical values that got approximated numerically to form the number in your array... but because it is an approximation, it can get it wrong. So sometimes you should sym(pi-313*eps,'f') or sym(pi-313*eps, 'd')... but if you are going to do the later, you should probably be using something like sym('3.14159265358972383808122685877606') because sym() does not get handed the number until it has already been approximated as floating point.

[John D'Errico]

No. I never wrote a matrix inverse for HPF. It would not be difficult.

Answer 1

[Walter Roberson]

No, there is no way around that problem.

The numeric approximation of your coefficients into floating-point can destroy your theoretical accuracy. https://blogs.mathworks.com/cleve/2013/02/02/hilbert-matrices/

You can reduce the numeric problems by never using inv(). Use the / or \ operators instead.