# BLUPF90

## Summary

The documentation for blupf90 is available separately as a PDF document.
See PREGSF90 with genotypes (SNP) for options.

## Options

OPTION conv_crit 1e-12

Set convergence criteria (deault 1e-12).

OPTION maxrounds 10000

Set maximum number of rounds (default 5000).

OPTION solv_method FSPAK

Selection solutions by FSPAK, SOR or PCG (default PCG).

OPTION r_factor 1.6

Set relaxation factor for SOR (default 1.4).

OPTION sol se

Store solutions and standard errors.

OPTION store_pev_pec 6

Store triangular matrices of standard errors and its covariances for correlated random effects such as direct-maternal effects and random-regression effects in “pev_pec_bf90”.

OPTION missing -999

Specify missing observations (default 0) in integer.

OPTION residual

y-hat and residual will be included in “yhat_residual”.

OPTION blksize 3

Set block size for preconditioner (default 1).

OPTION prior_solutions

Using the previous solution file to start the iteration. An additional software is required to use this option.

OPTION stdresidual

y-hat and student residuals will be included in “yhat_student_residual”.

OPTION check_levels 0

Check levels (default 1 = true).

OPTION use_yams

Run the program with YAMS (modified FSPAK).

OPTION SNP_file snp

Specify the SNP file name to use genotype data.

OPTION snp_p_value

Computes the elements of the inverse of the Mixed Model Equations that are needed for exact GWAS with p-values using postGSf90. This requires quite a lot of memory and time. For details see https://doi.org/10.1101/555243.

### 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.