Band matrices

Band matrices are special matrices that have many zero elements away from the diagonal. These zero elements are not stored in memory, which can save a lot of space, and they are not used in calculations, which can save a lot of compute time. On the other hand, the algorithms for band matrices are more complex and slower than for dense matrices, so it is important to know that the matrix is actually banded to use this type.

It is implemented in the class band_matrix in the matpack namespace. The implementation is just a wrap of Matrix with special item access. The only operation it supports is solving \(\mathrm{A}\vec{x} = \vec{b}\) for \(\vec{x}\), where \(\mathrm{A}\) is a band matrix and \(\vec{b}\) is a vector. It implements this solution by wrapping LAPACK’s dgbsv. Please see its documentation for more details on the layout of the band matrix.

The band matrix is not available via the python interface but used solely for internal solvers.

Like Sparse matrices, this is a very simple class that has very specialized use cases. It is not recommended using this class for general purpose linear algebra, but it can be very useful for specific problems.

Relevant files

The relevant files for the band matrix are:

• matpack/matpack_mdspan_helpers_band_matrix.h - the header file for the band matrix.

• matpack/matpack_mdspan_helpers_band_matrix.cpp - the implementation file for the band matrix.