I have a linear system of the form $\mathbf{A}\mathbf{x} = \mathbf{f}$. If the length of the vector $\mathbf{x}$ is $N$, meaning that there are $N$ unknowns, then the matrix $\mathbf{A}$ has seven bands and the matrix is sparse. The matrix is not symmetric. It looks like this:$\pmatrix{a_1&a_2&a_3&a_4&&&&&&\\b_1&b_2&b_3&b_4&b_5&&&&&\\c_1&c_2&c_3&c_4&c_5&c_6&&&&\\d_1&d_2&d_3&d_4&d_5&d_6&d_7&&&\\&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&&\\&&&&&&\\&&&k_1&k_2&k_3&k_4&k_5&k_6&k_7&&&\\&&&&l_1&l_2&l_3&l_4&l_5&l_6&l_7&&&\\&&&&&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&&\\&&&&&&\\&&&&&&&w_1&w_2&w_3&w_4&w_5&w_6&w_7\\&&&&&&&&x_1&x_2&x_3&x_4&x_5&x_6\\&&&&&&&&&y_1&y_2&y_3&y_4&y_5\\&&&&&&&&&&z_1&z_2&z_3&z_4\\}$ and the vector $\mathbf{f}$ looks like this:$\pmatrix{f_1\\f_2\\f_3\\f_4\\\vdots\\\\f_i\\f_{i+1}\\\vdots\\\\f_{N-3}\\f_{N-2}\\f_{N-1}\\f_{N}}$I know that the general method is to use a LU decomposition and as the matrix is constant for me, I can pre-compute its decomposition and store it and use it to solve for the unknowns. But I would like to know if there is any method to solve such systems faster than the LU decomposition. Methods may not be restricted to direct methods only.
Fast solution of a heptadiagonal linear system
matrices;linear solver;banded matrix