What is the efficient method to solve 10000*10000 matrix? (Non-sparse)

I want to solve Ax=b, where A is 10000*10000 matrix and b is 10000*1 coloumn vector. I am trying to solve fractional pde using rbf method. Where.I have taken 100 equidistant points on each axes. So the total number of points are 10000, hence the size of matrix is 10000*10000. I have tried A\B but because of high condition number the result is not as expected.

[John D'Errico]

Why do you think there is some magically efficient way to solve a large problem wih a high condition number? The best way it to get a really large, really fast computer. Or a magic wand, which if wielded properly can solve all problems.

In most cases, a PDE is best solved using mathods that create sparse matrices, which allow for efficient solution methods in linear algebra. If you choose to use a method that creates full matrices, then expect it to be slow. As far as the efficient solution goes, backsash IS an efficient way to solve the problem.

The condition number being high may indicate you have simply formulated the problem incorrectly. That we cannot know. Or your problem may have an inherantly high condition number. We are given no clue about the source of the high condition number.

So far, with the information provided, my best recommendation really is a magic wand. Now if only you can find someone who knows how to use one.

[ADSW121365]

You probably need to regularise your solution if the condition number is meant to be high.