Editing Cholesky decomposition (section)

==== Rank-one update ====
The specific case, where the updated matrix <math display=inline> \tilde{\mathbf{A}} </math> is related to the matrix <math display=inline>\mathbf{A}</math> by <math display=inline> \tilde{\mathbf{A}} = \mathbf{A} + \mathbf{x} \mathbf{x}^* </math>, is known as a ''rank-one update''.

Here is a function<ref>Based on: {{cite book|last=Stewart|first=G. W.|title=Basic decompositions|year=1998|publisher=Soc. for Industrial and Applied Mathematics|location=Philadelphia|isbn=0-89871-414-1}}</ref> written in [[Matlab]] syntax that realizes a rank-one update:
<syntaxhighlight lang="matlab">
function [L] = cholupdate(L, x)
    n = length(x);
    for k = 1:n
        r = sqrt(L(k, k)^2 + x(k)^2);
        c = r / L(k, k);
        s = x(k) / L(k, k);
        L(k, k) = r;
        if k < n
            L((k+1):n, k) = (L((k+1):n, k) + s * x((k+1):n)) / c;
            x((k+1):n) = c * x((k+1):n) - s * L((k+1):n, k);
        end
    end
end
</syntaxhighlight>
A ''rank-n update'' is one where for a matrix <math display=inline>\mathbf{M}</math> one updates the decomposition such that <math display=inline> \tilde{\mathbf{A}} = \mathbf{A} + \mathbf{M} \mathbf{M}^* </math>.  This can be achieved by successively performing rank-one updates for each of the columns of <math display=inline>\mathbf{M}</math>.