BandGeneral SOE

This command is used to construct a BandGeneralSOE linear system of equation object. As the name implies, this class is used for matrix systems which have a banded profile. The matrix is stored as shown below in a 1dimensional array of size equal to the bandwidth times the number of unknowns. When a solution is required, the Lapack routines DGBSV and SGBTRS are used. The following command is used to construct such a system:

system BandGeneral < -det >

Theory

An \(n\times n\) matrix \(A=(a_{ij})\) is a band matrix if all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k₁ and k₂:

: \[a_{i,j}=0 \quad\texttt{if}\quad j<i-k_1 \quad\texttt{ or }\quad j>i+k_2; \quad k_1, k_2 \ge 0.\]

The quantities k₁ and k₂ are the left and right half-bandwidth, respectively. The bandwidth of the matrix is k₁ + k₂ + 1 (in other words, the smallest number of adjacent diagonals to which the non-zero elements are confined).

and matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.

For example, 6-by-6 a matrix with bandwidth 3:

\[ \begin{bmatrix} B_{11} & B_{12} & 0 & \cdots & \cdots & 0 \\ B_{21} & B_{22} & B_{23} & \ddots & \ddots & \vdots \\ 0 & B_{32} & B_{33} & B_{34} & \ddots & \vdots \\ \vdots & \ddots & B_{43} & B_{44} & B_{45} & 0 \\ \vdots & \ddots & \ddots & B_{54} & B_{55} & B_{56} \\ 0 & \cdots & \cdots & 0 & B_{65} & B_{66} \end{bmatrix} \]

is stored as the 6-by-3 matrix

\[ \begin{bmatrix} 0 & B_{11} & B_{12}\\ B_{21} & B_{22} & B_{23} \\ B_{32} & B_{33} & B_{34} \\ B_{43} & B_{44} & B_{45} \\ B_{54} & B_{55} & B_{56} \\ B_{65} & B_{66} & 0 \end{bmatrix}. \]

BandGenLinLapackSolver

This command is used to construct a BandGeneralSOE linear system of equation object. As the name implies, this class is used for 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 times the number of unknowns. When a solution is required, the Lapack routines DGBSV and SGBTRS are used.

Theory

An \(n\times n\) matrix A=(a_i,j) is a band matrix if all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k₁ and k₂:

\[a_{i,j}=0 \quad\texttt{if}\quad j<i-k_1 \quad\texttt{ or }\quad j>i+k_2; \quad k_1, k_2 \ge 0.\]

The quantities k₁ and k₂ are the left and right half-bandwidth, respectively. The bandwidth of the matrix is k₁ + k₂ + 1 (in other words, the smallest number of adjacent diagonals to which the non-zero elements are confined).

and matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.

For example, 6-by-6 a matrix with bandwidth 3:

\[ \begin{bmatrix} B_{11} & B_{12} & 0 & \cdots & \cdots & 0 \\ B_{21} & B_{22} & B_{23} & \ddots & \ddots & \vdots \\ 0 & B_{32} & B_{33} & B_{34} & \ddots & \vdots \\ \vdots & \ddots & B_{43} & B_{44} & B_{45} & 0 \\ \vdots & \ddots & \ddots & B_{54} & B_{55} & B_{56} \\ 0 & \cdots & \cdots & 0 & B_{65} & B_{66} \end{bmatrix}\] is stored as the 6-by-3 matrix

\[\begin{bmatrix} 0 & B_{11} & B_{12}\\ B_{21} & B_{22} & B_{23} \\ B_{32} & B_{33} & B_{34} \\ B_{43} & B_{44} & B_{45} \\ B_{54} & B_{55} & B_{56} \\ B_{65} & B_{66} & 0 \end{bmatrix}.\]

C++ Interface

#include <system_of_eqn/linearSOE/bandGEN/BandGenLinLapackSolver.h>

class BandGenLinLapackSolver: public BandGenLinSolver

   MovableObject
   Solver
   LinearSOESolver

A BandGenLinLapackSolver object can be constructed to solve a BandGenLinSOE object. It obtains the solution by making calls on the the LAPACK library. The class is defined to be a friend of the BandGenLinSOE class (see <BandGenLinSOE.h>).

Public Methods

A unique class tag (defined in <classTags.h>) is passed to the BandGenLinSolver constructor. Sets the size of iPiv to \(0\), iPiv being an integer array needed by the LAPACK routines.

Invokes delete on iPiv to free the memory allocated to store the array.

The solver first copies the B vector into X and then solves the BandGenLinSOE system by calling the LAPACK routines dgbsv(), if the system is marked as not having been factored, and dgbtrs() 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 \(X\).

Is used to construct a 1d integer array, iPiv that is needed by the LAPACK solvers. It checks to see if current size of iPiv is large enough, if not it deletes the cold and creates a larger array. Returns \(0\) if successful, prints a warning message and returns a \(-1\) if not enough memory is available for this new array.

Code developed by: fmk