Compute sparse partial matrix inverse. As of 0.2.0.9010, an R
implementation of the Takahashi recursion method, unless a special build of
the fmesher package is used.
Value
A sparse symmetric matrix, with the elements of the inverse of A
for the non-zero pattern of A plus potential Cholesky in-fill locations.
Examples
A <- Matrix::Matrix(
c(2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2),
4,
4
)
# Partial inverse:
(S <- fm_qinv(A))
#> 4 x 4 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 0.8 0.6 0.4 .
#> [2,] 0.6 1.2 0.8 .
#> [3,] 0.4 0.8 1.2 0.6
#> [4,] . . 0.6 0.8
# Full inverse (not guaranteed to be symmetric):
(S2 <- solve(A))
#> 4 x 4 Matrix of class "dsyMatrix"
#> [,1] [,2] [,3] [,4]
#> [1,] 0.8 0.6 0.4 0.2
#> [2,] 0.6 1.2 0.8 0.4
#> [3,] 0.4 0.8 1.2 0.6
#> [4,] 0.2 0.4 0.6 0.8
# Matrix symmetry:
c(sum((S - Matrix::t(S))^2), sum((S2 - Matrix::t(S2))^2))
#> [1] 0 0
# Accuracy (not that S2 is non-symmetric, and S may be more accurate):
sum((S - S2)[S != 0]^2)
#> [1] 7.395571e-32