# BLUPF90

## Options

### Omit A-inverse

  OPTION omit_ainv

This option prohibits the program from creating $\mathbf{A}^{-1}$. It is especially useful for GBLUP. For example, if you would like to perform the exact GBLUP, you can put the following options to your parameter file.

  OPTION omit_ainv
OPTION TauOmega 1.0 0.0
OPTION AlphaBeta 0.95 0.05

With the above options, the program doesn't create $\mathbf{A}^{-1}$ but calculates $\tau\mathbf{G}^{-1}-\omega\mathbf{A}_{22}^{-1}$. When the omega ($\omega$) is zero, only $\mathbf{G}^{-1}$ will be included in the equations. $\mathbf{G}$ is blended with $\mathbf{A}_{22}$ as $\alpha\mathbf{G}+\beta\mathbf{A}_{22}$ before the inversion ($\alpha=0.95$ and $\beta=0.05$ in this case).

#### Details

Assuming a single-trait ssGBLUP, the mixed model equations are as follows.

$\left[ \begin{array}{ll} \mathbf{X}'\mathbf{X} & \mathbf{X}'\mathbf{Z}\\ \mathbf{Z}'\mathbf{X} & \mathbf{Z}'\mathbf{Z} + \lambda\mathbf{H}^{-1} \end{array} \right] \left[ \begin{array}{c} \mathbf{\hat{b}}\\ \mathbf{\hat{u}} \end{array} \right] = \left[ \begin{array}{c} \mathbf{X}'\mathbf{y}\\ \mathbf{Z}'\mathbf{y} \end{array} \right]$

where $\mathbf{H}$ is a matrix combining additive genetic relationship matrices and a genomic relationship matrix.

$\mathbf{H}^{-1} = \mathbf{A}^{-1} + \left[ \begin{array}{cc} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \tau\mathbf{G}^{-1}-\omega\mathbf{A}_{22}^{-1} \end{array} \right]$

If we omit $\mathbf{A}^{-1}$ and $\mathbf{A}_{22}^{-1}$, the equations are equivalent to GBLUP. GBLUP by BLUPF90 was not so easy because the program creates $\mathbf{A}^{-1}$ by default and there was no way to avoid this behavior. The new option removes $\mathbf{A}^{-1}$ from the equations so GBLUP will be easily performed.