Eigen efficient inverse of symmetric positive definite matrix

Question

In Eigen, if we have symmetric positive definite matrix A then we can calculate the inverse of A by

A.inverse();

or

A.llt().solve(I);

where I is an identity matrix of the same size as A. But is there a more efficient way to calculate the inverse of symmetric positive definite matrix?

For example if we write the Cholesky decomposition of A as A = LL^{T}, then L^{-T} L^{-1} is an inverse of A since A L^{-T} L^{-1} = LL^{T} L^{-T} L^{-1} = I (and where L^{-T} denotes the inverse of the transpose of L).

So we could obtain the Cholesky decomposition of A, calculate its inverse, and then obtain the cross-product of that inverse to find the inverse of A. But my instinct is that calculating these explicit steps will be slower than using A.llt().solve(I) as above.

And before anybody asks, I do indeed need an explicit inverse - it is a calculation for part of a Gibbs sampler.

Answer 1

With A.llt().solve(I), you assumes A to be a SPD matrix and apply Cholesky decomposition to solve the equation Ax=I. The mathematical procedure of solving the equation is exactly same as your explicit way. So the performance should be same if you do every step correctly.

On the other hand, with A.inverse(), you are doing general matrix inversion, which uses LU decomposition for large matrix. Thus the performance should be lower than A.llt().solve(I);.