bandcholsol#

Purpose#

Solves the system of equations \(Ax = b\) for x, given the lower triangle of the Cholesky decomposition of a positive definite banded matrix \(A\).

Format#

x = bandcholsol(b, l)#

Parameters:

b (KxM matrix) – The right-hand side(s).
l (KxN compact form matrix) – A Cholesky decomposition in compact (banded) form.

Returns:

x (KxM matrix)

Remarks#

Given a positive definite banded matrix A, there exists a matrix L, the lower triangle of the Cholesky decomposition of A, such that \(A = LL'\). l is the compact form of L; see band() for a description of the format of l.

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 matrix 'A' and right-hand side 'b'
A = { 1 2 0 0,
      2 8 1 0,
      0 1 5 2,
      0 0 2 3 };
b = { 1.3, 2.1, 0.7, 1.8 };

// Create banded matrix form of 'A'
Aband = band(A, 1);

// Cholesky factorization of the banded 'A'
Lband = bandchol(Aband);

// Solve the system of equations
x = bandcholsol(b, Lband);

After the code above is run:

        0.000  1.000       1.495      1.300        1.300
Lband = 2.000  2.000  x = -0.098  b = 2.100  A*x = 2.100
        0.500  2.179      -0.110      0.700        0.700
        0.918  1.469       0.673      1.800        1.800