solve-banded 
Solve a system of banded linear equations
Introduction
This module accepts javascript Arrays or typed arrays representing the bands of a banded matrix and computes the solution using the Thomas Algorithm. A system with one subdiagonal and two superdiagonal bands, for example, looks like:
![\left[
\begin{matrix}
{b_0} & {c_0} & {d_0} & { } & { } & { } & { 0 } \\
{a_1} & {b_1} & {c_1} & {d_1} & { } & { } & { } \\
{ } & {a_2} & {b_2} & \cdot & \cdot & { } & { } \\
{ } & { } & \cdot & \cdot & \cdot & {d_{n-4}} & { } \\
{ } & { } & { } & \cdot & {b_{n-3}} & {c_{n-3}} & {d_{n-3}} \\
{ } & { } & { } & { } & {a_{n-2}} & {b_{n-2}} & {c_{n-2}}\\
{ 0 } & { } & { } & { } & { } & {a_{n-1}} & {b_{n-1}}\\
\end{matrix}
\right]
\left[
\begin{matrix}
{x_0 } \\
{x_1 } \\
\cdot \\
\cdot \\
\cdot \\
{x_{n-2} } \\
{x_{n-1} } \\
\end{matrix}
\right]
=
\left[
\begin{matrix}
{e_0 } \\
{e_1 } \\
\cdot \\
\cdot \\
\cdot \\
{e_{n-2} } \\
{e_{n-1} } \\
\end{matrix}
\right].](https://raw.githubusercontent.com/scijs/solve-banded/HEAD/images/left-beginmatrix-b_0-c_0-d_0-0-a_1-b_1-c_1-d_-28211f3a7e.png)
The solver will fail if the matrix is singular and may not succeed if the matrix is not diagonally dominant. If the solver fails, it will log a console message and return false. Note that the solver accepts a stride and offset for the input/output vector (x, below), so that a system can be solved in-place in an ndarray. The coefficient matrix does not currently accept strides or offsets and must be passed as individual vectors.
Example
Consider the solution of the tridiagonal system
![\left[
\begin{matrix}
2 & 1 & 0 \\
-1 & 7 & 4 \\
0 & 2 & -3 \\
\end{matrix}
\right]
\left[
\begin{matrix}
{x_0 } \\
{x_1 } \\
{x_2 } \\
\end{matrix}
\right]
=
\left[
\begin{matrix}
{4} \\
{25} \\
{-5} \\
\end{matrix}
\right].](https://raw.githubusercontent.com/scijs/solve-banded/HEAD/images/left-beginmatrix-2-1-0-1-7-4-0-2-3-endmatrix--f1451b965d.png)
var solveBanded = var e = 4 25 -5 console// => e = [ 1, 2, 3 ]Installation
$ npm install solve-bandedAPI
require('solve-banded')(diagonals, nsub, nsup, x, nx [, ox = 0 [, sx = 1]])
Arguments:
diagonals: an array of diagonal bands, starting with with the subdiagonal-most band (a in the example matrix above) and proceeding to the superdiagonal-most band (d in the example matrix above). Each vector must be a javascriptArrayor typed array of lengthnx.nsub: an integer representing the number of subdiagonal bands, excluding the diagonal.nsup: an integer representing the number of superdiagonal bands, excluding the diagonal.x: a javascriptArrayor typed array of lengthnxrepresenting the known vector (e in the example matrix above). On successful completion, this vector will contain the solution.nx: an integer representing the number of equations, i.e. the length ofx.ox(optional): an integer representing the offset of data inx. If not provided, assumed equal to zero.sx(optional): an integer representing the stride of data inx. If not provided, assumed equal to one.
Returns: True on successful completion, false otherwise.
License
© 2016 Ricky Reusser. MIT License.