Optimizing (Reducing run time) of a sparse matrix division

Hi, so I have a sparse matrix as shown below:

[ap ae 0 an 0 0 0 0 0 ...

[aw ap ae 0 an 0 0 0 0 ...

[0 aw ap 0 0 an 0 0 0 ...

[as 0 0 ap ae 0 an 0 0 0 ....

[0 as 0 aw ap ae 0 an 0 0 0 0 ...

and this matrix is multiplied by

[T1,1]

[T1,2]

[T1,3]

[T2,1]

T[2,2]

.

.

.

This is used to solve Ax=T

To find x, I use A\T,

but when the A matrix is really large (1000 x 1000 or even more), the matrix division starts taking up a lot of time, but most of the entries are 0.

I was wondering if there is any way to optimize this to reduce run time, or since it doesnt follow a particular pattern it can't be done?

[Torsten]

And you already work with A, T in sparse format ?

[Steven Lord]

What is "a lot of time" in this context?

Are aw, ap, ae, and/or T variables containing numbers or are they symbolic variables? If the latter, try substituting values for those variables before solving the system.

[Walter Roberson]

If MATLAB is not able to detect one of the special sparse forms such as a bandwidth-limited matrix, then it uses solvers that typically end up needing dense matrices.

[Bjorn Gustavsson]

If the standard methods doesn't "perform", then you can try the different iterative solvers available. You can check the documentation for Systems of Linear Equations for a list of those, or the help and documentation for pcg, bicg, bicgstab, bicgstabl, cgs, gmres, lsqr, minres, qmr, symmlq, or tfqmr.

[Santosh Fatale]

What is the sparsity of the first matrix (Matrix A) in your case?

What are the dimesion for matrix A and matrix T?

As pointed out Steven, are the variables in matrix A are of type syms?

If possible could you provide the data for matrix A and matrix T?

[Walter Roberson]

sparse() matrices cannot hold symbolic variables.