Time complexity of Cholesky Decomposition for the LDL form

Question

There are two different forms for Cholesky Decomposition:

A = M * ctranspose (M)

and the LDL form

A = L * D * ctranspose (L)

where ctranspose is the complex transpose.

I want to know the number of floating point operations for each form. Wikipedia references a paper Matrix Inversion Using Cholesky Decomposition which says

When efficiently implemented, the complexity of the LDL decomposition is same (sic) as Cholesky decomposition.

The paper says Cholesky decomposition requires n^3/6 + O(n^2) operations. However, Wikipedia says the number of floating point operations is n^3/3 and my own calculation gets that as well for the first form. It basically comes down to the sum of triangle numbers which is:

n*(n+1)*(n+2)/6.  

That's where I think the paper gets n^3/6. But for the first form the term in the inner most triple sum is a[i][k]*a[j][k] which is basically a dot product. That's 2*n floating point operations in the sum. So the floating pointer operations should be n^3/3 + O(n^2). And if you look at the LDL form the term in the innermost sum is a[i][k]*a[j][k]*d[k]. That's 3*n floating pointer operations (2 multiplications and 1 addition). So the floating point operations should ben^3/2 + O(n^2).

In other words the LDL form needs 50% more floating point operations. Am I correct? I think the paper is wrong (though they don't define what they mean by an operation). This is important because I'm implementing a modified form of Choleksy Decomposition based on the LDL form and I want to estimate the efficiency of my algorithm.

Maybe this question is more suitable for https://math.stackexchange.com/

Answer 1

That statement considers the overall complexity of Cholesky decomposition including (an implementation of) inverse square root, and is what is left of a section that detailed the whole algorithm on a DSP.

I do see now that out of context the statement is misleading. So, to compute the term within the brackets (in the paper), LDL requires more operations than Cholesky decomposition (an operation is a complex MAC).