Computing log-likelihood and its derivatives for restricted maximum   likelihood methods
  Recent large scale genome wide association analysis involves large scale linear mixed models. Quantifying (co)-variance parameters in the mixed models with a restricted maximum likelihood method results in a score function which is the first derivative of a log-likelihood. To obtain a statistically efficient estimate of the variance parameters, one needs to find the root of the score function via the Newton method. Most elements of the Jacobian matrix of the score involve a trace term of four parametric matrix-matrix multiplications. It is computationally prohibitively for large scale data sets. By a serial matrix transforms and an averaged information splitting technique, an approximate Jacobian matrix can be obtained by splitting the average of the Jacobian matrix and its expected value. In the approximated Jacobian, its elements only involve Four matrix vector multiplications which can be efficiently evaluated by solving sparse linear systems with the multi-frontal factorization method. 
Shengxin Zhu
math.NA stat.CO