Primitive LU Decomposition

Computes the partial-pivoted LU decomposition of a matrix operand: $$P A = L U,$$ where \(P\) is a permutation matrix, \(L\) is unit lower triangular, and \(U\) is upper triangular. L (with implicit unit diagonal) and U are packed into a single LU output matching LAPACK's getrf layout. \(P\) is returned in two equivalent forms: pivots (LAPACK's sequential row-swap encoding) and permutation (an explicit permutation vector).

Usage

prim_lu(operand)

Arguments

operand

(arrayish)
Matrix of data type floating-point with exactly 2 dimensions.

Value

list of three arrayish values: LU (m, n) with the same dtype as the input; pivots (k,) of dtype i32 with k = min(m, n) (1-based row swaps such that row i was exchanged with row pivots[i] during elimination step i); and permutation (m,) of dtype i32, a 1-based permutation vector for \(P\) such that (P %*% A)[i, ] equals A[permutation[i], ].

Implemented Rules

• stablehlo

StableHLO

Lowers to a "lu" stablehlo::hlo_custom_call() (backed by LAPACK on CPU and cuSOLVER on CUDA) for LU and pivots, followed by a stablehlo::hlo_while() loop that converts pivots to permutation in-graph.

See also

nv_lu()

Examples

x <- nv_matrix(c(4, 3, 6, 3), nrow = 2, dtype = "f64")
prim_lu(x)
#> $LU
#> AnvlArray
#>   4.0000  6.0000
#>   0.7500 -1.5000
#> [ CPUf64{2,2} ] 
#> 
#> $pivots
#> AnvlArray
#>  1
#>  2
#> [ CPUi32{2} ] 
#> 
#> $permutation
#> AnvlArray
#>  1
#>  2
#> [ CPUi32{2} ] 
#>