Solves the banded linear system Ax = b.

Syntax

x = LSolveB(M, kl, ku, b)

Arguments

Name

Description

M

Compressed diagonal storage matrix form of the banded square matrix A.  Use BandMatrix to create it and BandMatrixIndices to populate it.

kl

Number of sub-diagonals in the lower band of the original matrix.  It should include off-diagonals with zero entries that are in between diagonals with non zero entries.

ku

Number of super-diagonals in upper band of the original matrix.  It should include off-diagonals with zero entries that are in between diagonals with non zero entries.

b

The right-hand side column vector or matrix.  If it is a matrix, each column produces a separate solution to the system.  The number of rows must be equal to the row/column size of the square matrix A.

Output

Name

Description

x

The solution(s) to the system(s).  It will have the same dimensions as the right-hand side argument b.  For each column in b, there will be a solution in the corresponding column in x.

Example

Solve a banded linear system Ax=b.

Syntax

A = [1,2,0;41,15,6;0,8,9]; // banded 3x3 matrix

B = [1,2;11,22;21,42]; // the right hand side matrixcolumn

kl=1; ku=1; // Number of off diagonals

M = BandMatrix(3, kl, ku); // This will hold the compressed matrix

// For each band entry (i,j) of A do the following two operations

row, col = BandMatrixIndices(i,j,3,kl,ku);

M(row,col) = A(i,j);

// Now solve

x = LSolveB(M,B)

Result

The solution to the system in a column vector.

[Matrix] 3 x 2

-0.21659   -0.43318

0.60829    1.2166

1.7926     3.5853

Comments

For efficiency, the matrix M should be generated directly without creating the matrix A.  The number of rows of M is 2*kl+ku+1 and the number of columns is the same as in A. Hence, the narrower the band compared to the size of A, the smaller the size of M in comparison to A.

LSolveB calls dgbsv from LAPACK.

See Also:

BandMatrix

BandMatrixIndices

LSolve

LSolveSPDB

LSolveT