LU Decomposition

Computes the partial-pivoted LU decomposition of a matrix x: $$P A = L U,$$ where \(P\) is a permutation matrix, \(L\) is unit lower triangular, and \(U\) is upper triangular.

This function returns L and U as separate matrices. Use prim_lu() to get them in packed LU form.

Usage

nv_lu(x)

Arguments

x

(arrayish)
One input, with exactly 2 axes. Can be any numeric data type: a float keeps its own, and an integer one is converted to the default float data type (see default_dtypes()). An R value materializes at its default data type and is converted in the same way.

Value

(named list of arrayish)
L and U have the input's data type – or the default float data type (see default_dtypes()) where the input was an integer one; pivots and permutation are indices at the default integer data type (see default_dtypes()).

• L – unit lower-triangular factor of shape (m, k), where (m, n) = shape(x) and k = min(m, n).

• U – upper-triangular factor of shape (k, n).

• pivots – length k, at the default integer data type (see default_dtypes()). LAPACK-style sequential row swaps as returned by getrf.

• permutation – length m, at that same data type. A permutation vector representing \(P\).

See also

prim_lu()

Examples

# `L` and `U` keep the input's data type; the pivots are the default integer
x <- nv_matrix(c(4, 3, 6, 3), nrow = 2, dtype = "f64")
nv_lu(x)
#> $L
#> AnvlArray
#>  1.0000 0.0000
#>  0.7500 1.0000
#> [ CPUf64{2,2} ] 
#> 
#> $U
#> AnvlArray
#>   4.0000  6.0000
#>   0.0000 -1.5000
#> [ CPUf64{2,2} ] 
#> 
#> $pivots
#> AnvlArray
#>  1
#>  2
#> [ CPUi32{2} ] 
#> 
#> $permutation
#> AnvlArray
#>  1
#>  2
#> [ CPUi32{2} ] 
#>