.Modelica_LinearSystems2.Math.Matrices.choleskyUpDate

Information

Syntax

Lud = Matrices.choleskyUpDate(L, v);
Lud = Matrices.choleskyUpDate(L, v, true);

Description

This function computes the rank-1-updated Cholesky factorization Lud, with

Aud = Lud*Lud^T = A + v*v^T = L*L^T + v*v^T

from the input L, i.e. the left (lower) Cholesky factor of the original matrix A.
The approach is a transformation H*[v, L]' = [0, Lud]' with orthonormal matrix H such, that

[0, Lud] * [0, Lud]^T = [v, L]*H^T *H*[v, L]^T = [v, L]*[v, L]^T = v*v^T + A

and matrix Lud is lower (upper) triangular. The transformation is performed by n (order of A) Givens rotations. The following sequence illustrates the principle of stepwise transformation of matrix [v, L]'. Symbol "*" represents arbitrary elements. For each step the changed elements are bold.

| v' |    | * * * * |       | 0 * * * |       | 0 0 * * |       | 0 0 0 * |       | 0 0 0 0 |
|    |    | * * * * |       | * * * * |       | * * * * |       | * * * * |       | * * * * |
| L' | =  | 0 * * * |  ->   | 0 * * * |  ->   | 0 * * * |  ->   | 0 * * * |  ->   | 0 * * * |
|    |    | 0 0 * * |       | 0 0 * * |       | 0 0 * * |       | 0 0 * * |       | 0 0 * * |
|    |    | 0 0 0 * |       | 0 0 0 * |       | 0 0 0 * |       | 0 0 0 * |       | 0 0 0 * |

With the boolean input "upper" the user specifies whether the matrix L is lower or upper triangular matrix (left or right Cholesky factor). If "upper==true", the output Lud is also upper triangular. Default is "upper==false".

Interface

function choleskyUpDate
  extends Modelica.Icons.Function;
  import Modelica_LinearSystems2.Math.Matrices.LAPACK;
  input Real L[:, size(L, 1)] "Cholesky factor";
  input Real v[size(L, 1)] "Real vector A+v*v'";
  input Boolean upper = false "True if the upper triangle of A is provided and the modified upper triangle will be returned";
  output Real Lud[size(L, 1), size(L, 2)] = if upper then symmetric(L) else L "Updated Cholesky factor";
end choleskyUpDate;

Revisions