I need a method for solving a linear system $A x = B$, where $A$ is a tridiagonal matrix of type
\begin{equation}\begin{bmatrix} x & x & \cdots & \cdots\\ x & x & x & \cdots \\ \cdots & x & x & x \\ \cdots & \cdots & x & x \end{bmatrix}\end{equation}
with (generic) elements
\begin{equation}\begin{bmatrix} x_1*(k_{11}+k_{11a}/(x_1+x_2+ \cdots x_n)) & x_2*k_{12} & \cdots & \cdots \\ x_1*k_{21} & x_2*(k_{22}+k_{22a}/(x_1+x_2+ \cdots x_n)) & x_3*k_{23} & \cdots \\ \cdots & x_2*k_{32} & x_3*(k_{33}+k_{33a}/(x_1+x_2+ \cdots x_n)) x_4*k_{34} \end{bmatrix}\end{equation}
and so on, size about $400 \times 400$ elements (sparse). I can solve by preprocessing and converting to fractional values however the total number of operations required is exceedingly high compared to those required by methods for tridiagonal systems such as Thomas algorithm and I have to solve the system many, many times.
Of course the problem is in diagonal elements
$(1,1 \ldots n,n)$
which include the term
$Knn/(x_1+x_2+ \cdots +x_n)$,
or the sum of all variables $(x_1 \ldots x_n)$ in system,
perhaps there is a efficient solution for solving similar problems and I hope an expert can point me toward such solution,
any contribute will be appreciated, thanks for help.