I want to solve the linear equation Ax=b, with A is 40000x400000 matrix, x and b are 400000x1 array . When I use A\b in MATLAB, I have the error "out of memory". Because the size of matrix is too large, I can not use A\b in MATLAB. How can I solve my problem without using the command A\b in Matlab.
Thank you for your help
The problem is partially on Matlab and partially on your machine.
First, think very well if you actually need to solve this problem with a 40000x400000, maybe you can simplify it, maybe you can segment your problem, maybe the system is decoupled, just check very well.
For context, to store a 40000x400000 matrix of 8 byte floats, you'd need around 120gb. That's likely too much, there's a good chance you don't even have enough free disk space for it.
If this matrix has many zeros, at least much more than non-zeros, then you could exploit Matlab's sparse matrix functions. They work without storing the whole matrix, and operating only on non-zero numbers, basically.
If you are lazy but you have a very good machine (say 1TB of SSD), you might consider increasing the size of the paging file in linux (or its analogous on windows). That basically means that there is a space in the disk that you'll allow the computer to use as if it were RAM memory. While memory wouldn't crash, the operations you need to perform would take insanely long, so consider starting with a smaller matrix to gauge the execution time, which should grow with the cube of the vector length.
So case in point, try to review the problem you've got at your hands.
