library(package="MatrixModels")
 
dat <- data.frame( y=c(1.5, 2.35, 3.4, 2.31, 1.53),
                  s1=c("AA", "Aa", "Aa", "AA", "aa"),
                  s2=c("Bb", "BB", "bb", "BB", "bb"),
                  s3=c("cc", "Cc", "Cc", "cc", "CC"),
                  s4=c("Dd", "Dd", "DD", "dd", "Dd"))
 
cols <- paste("s", 1:4, sep="")
dat[, cols] <- lapply(dat[, cols], function(z) as.integer(z) - 1)
 
lambda <- 1
 
(y <- dat$y)
(X <- model.Matrix(y ~  1,                     data=dat, sparse=TRUE))
(Z <- model.Matrix(y ~ -1 + s1 + s2 + s3 + s4, data=dat, sparse=TRUE))
 
XX <- crossprod(X)
XZ <- crossprod(X, Z)
ZZ <- crossprod(Z)
 
Xy <- crossprod(X, y)
Zy <- crossprod(Z, y)
 
(LHS <- rBind(cBind(XX,    XZ),
              cBind(t(XZ), ZZ + Diagonal(n=dim(ZZ)[1]) * lambda)))
 
(RHS <- rBind(Xy,
              Zy))
 
(sol <- solve(LHS, RHS))
 
(GEBV <- Z %*% sol[-1])

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Copyright (C) 2012 The R Foundation for Statistical Computing
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> 
> library(package="MatrixModels")
Loading required package: Matrix
Loading required package: lattice
 
Attaching package: ‘Matrix’
 
The following object(s) are masked from ‘package:base’:
 
    det
 
 
Attaching package: ‘MatrixModels’
 
The following object(s) are masked from ‘package:stats’:
 
    getCall
 
> 
> dat <- data.frame( y=c(1.5, 2.35, 3.4, 2.31, 1.53),
+                   s1=c("AA", "Aa", "Aa", "AA", "aa"),
+                   s2=c("Bb", "BB", "bb", "BB", "bb"),
+                   s3=c("cc", "Cc", "Cc", "cc", "CC"),
+                   s4=c("Dd", "Dd", "DD", "dd", "Dd"))
> 
> cols <- paste("s", 1:4, sep="")
> dat[, cols] <- lapply(dat[, cols], function(z) as.integer(z) - 1)
> 
> lambda <- 1
> 
> (y <- dat$y)
[1] 1.50 2.35 3.40 2.31 1.53
> (X <- model.Matrix(y ~  1,                     data=dat, sparse=TRUE))
"dsparseModelMatrix": 5 x 1 sparse Matrix of class "dgCMatrix"
  (Intercept)
1           1
2           1
3           1
4           1
5           1
@ assign:  0 
@ contrasts:
named list()
> (Z <- model.Matrix(y ~ -1 + s1 + s2 + s3 + s4, data=dat, sparse=TRUE))
"dsparseModelMatrix": 5 x 4 sparse Matrix of class "dgCMatrix"
  s1 s2 s3 s4
1  2  1  .  1
2  1  2  1  1
3  1  .  1  2
4  2  2  .  .
5  .  .  2  1
@ assign:  1 2 3 4 
@ contrasts:
named list()
> 
> XX <- crossprod(X)
> XZ <- crossprod(X, Z)
> ZZ <- crossprod(Z)
> 
> Xy <- crossprod(X, y)
> Zy <- crossprod(Z, y)
> 
> (LHS <- rBind(cBind(XX,    XZ),
+               cBind(t(XZ), ZZ + Diagonal(n=dim(ZZ)[1]) * lambda)))
5 x 5 sparse Matrix of class "dgCMatrix"
            (Intercept) s1 s2 s3 s4
(Intercept)           5  6  5  4  5
s1                    6 11  8  2  5
s2                    5  8 10  2  3
s3                    4  2  2  7  5
s4                    5  5  3  5  8
> 
> (RHS <- rBind(Xy,
+               Zy))
5 x 1 Matrix of class "dgeMatrix"
             [,1]
(Intercept) 11.09
s1          13.37
s2          10.82
s3           8.81
s4          12.18
> 
> (sol <- solve(LHS, RHS))
5 x 1 Matrix of class "dgeMatrix"
                   [,1]
(Intercept)  1.65468864
s1           0.05223443
s2           0.08655678
s3          -0.05223443
s4           0.45586081
> 
> (GEBV <- Z %*% sol[-1])
5 x 1 Matrix of class "dgeMatrix"
       [,1]
1 0.6468864
2 0.6289744
3 0.9117216
4 0.2775824
5 0.3513919
> 
> proc.time()
   user  system elapsed 
  2.330   0.070   2.412