Preserve the element type properly in tdvp, dmrg_x, etc.

[mtfishman]

Also change some internal uses of tdvp to alternating_update, sweep_update, sub_sweep_update, and region_update, following the names planned in ITensorNetworks.jl.

Additionally, run examples in the tests and update style of import and using for overloading functions and bringing functions and types into namespaces.

[codecov-commenter]

Codecov Report

Attention: 10 lines in your changes are missing coverage. Please review.

Comparison is base (449e3a4) 84.09% compared to head (82cfcc5) 85.87%.

Files	Patch %	Lines
src/tdvp.jl	50.00%	4 Missing ⚠️
src/alternating_update.jl	86.66%	2 Missing ⚠️
src/projmps2.jl	60.00%	2 Missing ⚠️
src/sweep_update.jl	91.66%	1 Missing ⚠️
src/tdvporder.jl	0.00%	1 Missing ⚠️

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Additional details and impacted files

@@            Coverage Diff             @@
##             main      #63      +/-   ##
==========================================
+ Coverage   84.09%   85.87%   +1.77%     
==========================================
  Files          17       17              
  Lines         522      524       +2     
==========================================
+ Hits          439      450      +11     
+ Misses         83       74       -9     

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[mtfishman]

@emstoudenmire this is mostly superficial code cleanup, hopefully we can simplify the situation with reduced/projected operators in this package with the work you are doing rewriting that code in ITensors.jl. Ideally we can get rid of ProjMPOApply in favor of an outer product operator type that will also be used by excited state DMRG, and get rid of ProjMPS2 and ProjMPO_MPS2 (which is a weird hack to the linsolve working) in favor of using existing projected operator types. That might require introducing a LinearProblem type that wraps project operators in order to define which parts are A and b of the linear problem Ax = b (similar to the design of https://github.com/SciML/LinearSolve.jl).