  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.
