cholrank1 update with LDL decomposition

Question

I have cholrank1 update procedure (wikipedia) for the symmetric positive definite (SPD) matrix.

function [L] = cholupdate(L,x)
    p = length(x);
    for k=1:p
        r = sqrt(L(k,k)^2 + x(k)^2);
        c = r / L(k, k);
        s = x(k) / L(k, k);
        L(k, k) = r;
        L(k+1:p,k) = (L(k+1:p,k) + s*x(k+1:p)) / c;
        x(k+1:p) = c*x(k+1:p) - s*L(k+1:p,k);
    end
end

It works with LL decomposition. I try to fix procedure to work with LDL decomposition (ie without calling sqrt) like this:

function [L] = cholupdate_ldl(L,x)
    p = length(x);
    for k=1:p
        r = L(k,k) + x(k)^2;
        c = r / L(k, k);
        s = x(k) / L(k, k);
        L(k, k) = r;
        L(k+1:p,k) = (L(k+1:p,k) + s*x(k+1:p)) / c;
        x(k+1:p) = sqrt(c)*(x(k+1:p) - x(k)*L(k+1:p,k));
    end
end

It works fine but I was forced to use sqrt. How can I update LDL decomposition without using sqrt at all?

Answer 1

There are a number of ways. See Gill, Golub, Murray and Saunders (1974): Methods for Modifying Matrix Factorizations in Mathematics of Computation. To summarize your question formally, I quote from the paper:

Finally we get to the algorithm:

And here is my implementation in MATLAB:

function [L1,D1] = ldlt_update(L0,D0,z)

n = size(L0,1) ;
D1 = zeros(n,n) ;
L1 = zeros(n,n) ;

a = 1 ;
w = z ;
for jj = 1:n
    p = w(jj) ;
    D1(jj,jj) = D0(jj,jj) + a*p^2 ;
    b = p*a/D1(jj,jj) ;
    a = D0(jj,jj)*a/D1(jj,jj) ;
    for r = jj+1:n
        w(r) = w(r) - p*L0(r,jj) ;
        L1(r,jj) = L0(r,jj) + b*w(r) ;
    end
end
end

There is an alternative algorithm in the paper cited above and in Gill, Murray, and Wright (1982): Practical Optimization. Brian Borchers has a complete set of MATLAB code for working with real symmetric positive definite LDLT factorizations as defined in Golub and Van Loan (2013): Matrix Computations and on his web site.