Hermitian banded eigendecompositions where T <: BlasFloat

[MikaelSlevinsky]

This draft PR adds Hermitian banded eigendecompositions based on tridiagonal reduction. It made me realize that LinearAlgebra is missing a Hermitian tridiagonal type and parametric Givens rotations with different types for the sines and cosines: in the BLAS, a complex Givens rotation uses a real cosine. Having a finer grained parametrization of a Givens rotation in LinearAlgebra would probably speed up their application. This proposal is a breaking change to LinearAlgebra, so I think that means if implemented it would only be available from Julia v2.

RFC: it would be easy to destroy the HermTridiagonal type and preserve the outcomes of this PR. Probably HermTridiagonal belongs in LinearAlgebra.

Other improvements:

• the count of the Givens rotations in band reduction is hoisted out of sbtrd! so that they can be pre-allocated rather than pushed.
• the Julia sbtrd! function doesn't default to WANTQ = true, unifying calls for eigvals as well.
• only in-place eigen! and eigvals! are defined and keyword arguments may be passed so that one may ask for, e.g. eigvals(A, -2, 2), all eigenvalues of Symmetric/Hermitian banded A between -2 and 2. Also works with a unitrange.

Todo:

• Hermitian-definite banded GEP.
• Documentation.

[KlausC]

LinearAlgebra is missing a Hermitian tridiagonal type

I dont't see the need for a HermTridiagonal{<:Complex}. All matrices of this format can be easily transformed into SymTridiagonal{<:Real} by a diagonal unitary matrix (all diagonal elements are (complex) numbers with norm 1). All the transformations of Hermitian matrices to 3-Diagonals can be changed to symmetric real matrices by just replacing the subdiagonals by their absolute values; the diagonals are real by definition.

... missing ... parametric Givens rotations with different types for the sines and cosines

I agree with that statement. If you multiply givens rotations with complex arrays you can safe 1/4 of real multiplications, if one of the coefficients in Givens is real (and the other complex). That will improve efficiency of eigenvector calculation.

How is the PR coordinated with #183?

[MikaelSlevinsky]

Rest assured it will be coordinated with #183. Probably we'll merge yours before this one. Please note this is a draft, a work in progress.

[codecov]

Codecov Report

Merging #185 into master will decrease coverage by 0.62%.
The diff coverage is 51.65%.

@@            Coverage Diff             @@
##           master     #185      +/-   ##
==========================================
- Coverage   71.83%   71.21%   -0.63%     
==========================================
  Files          21       22       +1     
  Lines        2546     2710     +164     
==========================================
+ Hits         1829     1930     +101     
- Misses        717      780      +63     

Impacted Files	Coverage Δ
src/BandedMatrices.jl	50.00% <ø> (-50.00%)	⬇️
src/symbanded/symbanded.jl	86.95% <ø> (-3.05%)	⬇️
src/hermtridiag.jl	19.00% <19.00%> (ø)
src/symbanded/bandedeigen.jl	65.79% <95.18%> (+11.12%)	⬆️
src/lapack.jl	56.96% <100.00%> (-27.04%)	⬇️

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