https://github.com/JuliaLang/julia
### Introduction This PR adds a generic implementation of the Bunch-Kaufman factorization in native Julia code, and a generic implementation of a inertia calculation function. Right now Julia only support the Bunch-Kaufman factorization for `Float32`, `Float64` and their complex variants. This is because the factorization is handled by LAPACK, which only supports these types. To extend support to generic number types, I translated the LAPACK implementation to native Julia code, and the code performs the factorization in-place. I also included the function `inertia` that computes the number of positive, negative, and zero eigenvalues of an $n \times n$ Bunch-Kaufman factorized matrix in $\mathcal{O}(n)$ time. ### Changes - `bunchkaufman` and `bunchkaufman!` now work for any `AbstractFloat`, `Rational` and their complex variants. Behavior for previously supported types is not changed (LAPACK is used when possible). `bunchakaufman!` does not support `Integer` types, as in general the factorization lies in the arithmetic closure of the number type (the rationals for the integers). On the other hand, `bunchakaufman` supports `Integer` types, by making an internal conversion to `Rational{BigInt}`. - `ldiv!` for a `BunchKaufman` factorization has extended support for generic number types with type stability. - Previously, Julia extracted the diagonal factor of an $n \times n$ `BunchKaufman` object by making a copy of the matrix and then calling a LAPACK function (`dsyconvf`, `csyconvf`, etc., depending on the number type). This function also computes the triangular factor, so it runs in $\mathcal{O}(n^2)$ time. Now Julia uses a native implementation of the LAPACK function with small modifications, so it computes the diagonal factor in $\mathcal{O}(n)$ time, without making a new copy of the matrix. - Added the function `inertia` that computes the number of positive, negative and zero eigenvalues of the diagonal factor of an $n \times n$ `BunchKaufman` object, in case that the matrix is real symmetric or Hermitian. For complex symmetric matrices, `inertia` only computes the number of zero eigenvalues of the diagonal factor. `inertia` runs in $\mathcal{O}(n)$ time and only uses arithmetic and real absolute value operations. Therefore, `inertia` can be used for matrices of any generic number type, including `Rational`. In particular, for rational matrices the output of `inertia` is exact (unless a positive tolerance is specified). - Unit tests of the `BunchKaufman` library has been adapted to handle low precision number types (approximate comparisons with tolerance `sqrt(eps(Float64))` do not make sense when the floating point type is `Float16`, for example). The test-set now also runs on the following types: `Float16, Complex{Float16}, BigFloat, Complex{BigFloat}, Complex{Int}, BigInt, Complex{BigInt}, Rational{BigInt}, Complex{Rational{BigInt}}`. Unit tests for the `inertia` function have been added too.
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CITATION.bib | -rw-r--r-- | 513 bytes |
CITATION.cff | -rw-r--r-- | 940 bytes |
CONTRIBUTING.md | -rw-r--r-- | 23.4 KB |
HISTORY.md | -rw-r--r-- | 372.8 KB |
LICENSE.md | -rw-r--r-- | 1.3 KB |
Make.inc | -rw-r--r-- | 55.9 KB |
Makefile | -rw-r--r-- | 30.3 KB |
NEWS.md | -rw-r--r-- | 8.8 KB |
README.md | -rw-r--r-- | 7.4 KB |
THIRDPARTY.md | -rw-r--r-- | 3.8 KB |
VERSION | -rw-r--r-- | 11 bytes |
julia.spdx.json | -rw-r--r-- | 35.8 KB |
pkgimage.mk | -rw-r--r-- | 5.8 KB |
sysimage.mk | -rw-r--r-- | 4.2 KB |
typos.toml | -rw-r--r-- | 78 bytes |
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