The given BBD sparse linear equations are as the following eq.(1), where coefficient matrix is a 2x2 block sparse matrix, x and b are both vectors.

The principle of solving the eq.(1) is as follow(eq.(1)(2)(3)(4) are shown in the picture):

- Factor sparse coefficient matrix of eq.(1) into a lower and an upper triangular matrix, as coefficient matrix of eq.(2).
- Forward substitute variable y of eq.(3) to get solution y.
- Backward substitute variable x of eq.(4) to get solution x.

Could you help recommend available libraries and APIs to solving eq.(1) according to the principle mentioned above?