# bkfact

``````..  bkfact(A) -> BunchKaufman

Compute the Bunch-Kaufman [Bunch1977]_ factorization of a real symmetric or complex Hermitian matrix ``A`` and return a ``BunchKaufman`` object. The following functions are available for ``BunchKaufman`` objects: ``size``, ``\``, ``inv``, ``issym``, ``ishermitian``.

.. [Bunch1977] J R Bunch and L Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Mathematics of Computation 31:137 (1977), 163-179. `url <http://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0428694-0>`_.``````

## Examples

The `bkfact` function in Julia computes the Bunch-Kaufman factorization of a real symmetric or complex Hermitian matrix `A` and returns a `BunchKaufman` object. The `BunchKaufman` object has the following functions available: `size`, `transpose`, `inv`, `issym`, and `ishermitian`.

Here are some examples of how to use the `bkfact` function:

``````julia> A = [4.0 1.0 1.0; 1.0 5.0 2.0; 1.0 2.0 6.0];
julia> bk = bkfact(A)
BunchKaufman{Float64,Array{Float64,2}}
D factor:
3-element Array{Float64,1}:
4.0
4.0
4.0
L factor:
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.25 1.0  0.0
0.25 0.4  1.0

julia> B = [1.0+2.0im 2.0-1.0im 3.0+4.0im; 2.0+1.0im 1.0+3.0im 4.0-2.0im; 3.0-4.0im 4.0+2.0im 1.0-5.0im];
julia> bk = bkfact(B)
BunchKaufman{ComplexF64,Array{ComplexF64,2}}
D factor:
3-element Array{ComplexF64,1}:
1.0 +  2.0im
-12.0 +  0.0im
-4.0 -  8.0im
L factor:
3×3 Array{ComplexF64,2}:
1.0+0.0im  0.0+0.0im  0.0+0.0im
2.0-1.0im  1.0+0.0im  0.0+0.0im
3.0+4.0im  4.0-2.0im  1.0+0.0im``````

Note: In the examples above, the matrices `A` and `B` are used to demonstrate the usage of `bkfact` function. The resulting `BunchKaufman` object `bk` contains the D factor and L factor of the factorization.

For more information on the Bunch-Kaufman factorization, you can refer to the research paper by Bunch and Kaufman titled "Some stable methods for calculating inertia and solving symmetric linear systems" published in Mathematics of Computation in 1977.