bandsolpd#

Purpose#

Solves the system of equations \(Ax = b\) for x, where A is a positive definite banded matrix stored in compact form. Banded matrices arise in spline interpolation, finite difference methods, and time series models where each variable depends only on nearby neighbors.

Format#

x = bandsolpd(b, A)#

Parameters:

b (KxM matrix) – right-hand side vector or matrix. If b has multiple columns, the system is solved for each column independently.
A (KxN compact form matrix) – positive definite banded matrix in compact form, where N is the number of bands (including the diagonal). See band() for how to convert a full matrix to compact form.

Returns:

x (KxM matrix) – the solution vector(s). Each column of x is the solution corresponding to the matching column of b.

Remarks#

A is a positive definite banded matrix in compact form. See band() for a description of the format of A.

b can have more than one column. If so, \(Ax = b\) is solved for each column. That is,

\[A*x[.,i] = b[.,i]\]

Examples#

// Create a 4x4 tridiagonal positive definite system
// In compact banded form:
//   col 1 = sub-diagonal elements (first element is 0, no element above row 1)
//   col 2 = main diagonal elements
A_compact = { 0 4,
              1 5,
              1 6,
              1 7 };

// Right-hand side vector
b = { 8, 11, 13, 14 };

// Solve Ax = b
x = bandsolpd(b, A_compact);
print x;

The above code produces: