Setting size of DENSE set of equations.

Sparse versus Dense

ASReml partitions the terms in the linear model into two parts: a dense set and a sparse set. The partition is at the !r point unless explicitly set with the !DENSE data line qualifier or mv is included before !r, The special term mv is always included in sparse. Thus random and sparse terms are estimated using sparse matrix methods which result in faster processing. The inverse coefficient matrix is fully formed for the terms in the dense set. The inverse coefficient matrix is only partially formed for terms in the sparse set. Typically, the sparse set is large and sparse storage results in savings in memory and computing. A consequence is that the variance matrix for estimates is only available for equations in the dense portion and is printed to the .vrb if !VRB is specified. ( !CINV requests printing portion of the C inverse matrix to a .cii file.

!DENSE

ASReml normally solves the first set of fixed effect equations in the linear model in the order the terms are defined using dense equations. However, if there are more than 800 such equations, the higher model terms are moved into the sparse section. Typically these will be the higher order interactions. The sparse equations are fitted before the dense equations so putting an interaction in the sparse equations which sweep aout any terms marginal to it that appear in the dense equations.

!DENSE n sets the number of equations solved densely up to a maximum of 5000. By default, sparse matrix methods are applied to the random effects and any fixed effects listed after random factors or whose equation numbers exceed 800. Use !DENSE n to apply sparse methods to effects listed before the !r (reducing the size of the DENSE block) or if you have large fixed model terms and want them included in the ANOVA table. Individual model terms will not be split so that only part is in the dense section. n should be kept small (<100) for faster processing.

!GDENSE: Processing random terms as DENSE

Typically, all random terms and very large fixed terms are included in the SPARSE equations. However, if a random term G structure is actually dense and large as in a genomic relationship matrix, there will be a computational penalty from traeting it as SPARSE. The !GDENSE qualifier includes the first random term in the DENSE equations if the term is has a dense Ginverse matrix.
ASReml uses link list matrix methods for the sparse equations and has always included random model terms in these sparse equations. However, in the case of GRM matrices, they are typically quite large (several thousand) and dense, and is generally more efficient to process them as such. ASReml 4 has a !GDENSE qualifier (set just before the model line). If !GDENSE is set and the first random term is a GRM term, its equations will be processed as DENSE. In one example with 3226 rows in the GRM matrix, this reduced the iteration time from 196 to 175 seconds.

See Also

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