BandSPDLinLapackSolver

system BandSPD

This command is used to construct a BandSPDSOE linear system of equation object. As the name implies, this class is used for symmetric positive definite matrix systems which have a banded profile. The matrix is stored as shown below in a one dimensional array of size equal to the (bandwidth/2) times the number of unknowns. When a solution is required, the Lapack routines DPBSV and DPBTRS are used.

Theory

An n×n matrix A=(a_i,j) is a symmmetric banded matrix if all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k:

\[ a_{i,j}=0 \quad\texttt{if}\quad j<i-k \quad\texttt{ or }\quad j>i+k; \quad k \ge 0.\]

\[a_{i,j} = a_{j,i}\]

\[y^T A y \ne 0 \]

for all non-zero vectors y with real entries (\(y \in \mathbb{R}^n\)),

The bandwidth of the matrix is k + k + 1.

For example, a symmetric 6-by-6 matrix with a right bandwidth of 2:

\[\begin{bmatrix} A_{11} & A_{12} & A_{13} & 0 & \cdots & 0 \\ & A_{22} & A_{23} & A_{24} & \ddots & \vdots \\ & & A_{33} & A_{34} & A_{35} & 0 \\ & & & A_{44} & A_{45} & A_{46} \\ & sym & & & A_{55} & A_{56} \\ & & & & & A_{66} \end{bmatrix}.\]

This matrix is stored as the 6-by-3 matrix:

\[\begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{22} & A_{23} & A_{24} \\ A_{33} & A_{34} & A_{35} \\ A_{44} & A_{45} & A_{46} \\ A_{55} & A_{56} & 0 \\ A_{66} & 0 & 0 \end{bmatrix}. \]

C++ Interface

A BandSPDLinLapackSolver object can be constructed to solve a BandSPDLinSOE object. It obtains the solution by making calls on the the LAPACK library. The class is defined to be a friend of the BandSPDLinSOE class.

Public Methods

A unique class tag (defined in <classTags.h>) is passed to the BandSPDLinSolver constructor.

Does nothing.

The solver first copies the B vector into X and then solves the BandSPDLinSOE system by calling the LAPACK routines dpbsv(), if the system is marked as not having been factored, and dpbtrs() if system is marked as having been factored. If the solution is successfully obtained, i.e. the LAPACK routines return \(0\) in the INFO argument, it marks the system has having been factored and returns \(0\), otherwise it prints a warning message and returns INFO. The solve process changes \(A\) and \(X\).

Does nothing but return \(0\).

Does nothing but return \(0\).

Does nothing but return \(0\).

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Code Developed by: fmk