Support generic arrays

[amontoison]

close #605

using Krylov
using BlockArrays
K = rand(10,10)
B = rand(2, 10)

b = BlockVector(rand(Float64,12), [10,2])
2-blocked 12-element BlockVector{Float64}:
 0.8437110010150571  
 0.4530016441000567  
 0.686194576648592   
 0.34254162007097566 
 0.09544196921306347 
 0.5010761791056787  
 0.11907189994954914 
 0.17470076682895075 
 0.9550777571644686  
 0.4680632206046831  
 ────────────────────
 0.012863148806336877
 0.8840547265290272  

A = BlockArray(rand(Float64,12,12), [10,2], [10,2])
2×2-blocked 12×12 BlockMatrix{Float64}:
 0.116554   0.0703249  0.875842  0.506043   0.373476  0.23967    0.601902   0.860072   0.964966  0.436033    │  0.713543  0.470241
 0.799259   0.37769    0.346566  0.259132   0.635035  0.0446249  0.964017   0.92735    0.814636  0.669492    │  0.298945  0.921794
 0.816827   0.134661   0.127786  0.971469   0.840193  0.49803    0.892349   0.360627   0.160712  0.00137863  │  0.913815  0.846428
 0.20443    0.948703   0.137055  0.609844   0.164636  0.724305   0.90073    0.875758   0.862051  0.755453    │  0.145395  0.078032
 0.321294   0.687613   0.237468  0.0524443  0.833145  0.693929   0.237879   0.184222   0.625539  0.709003    │  0.515103  0.907079
 0.135291   0.879867   0.879918  0.279797   0.561364  0.963455   0.0180151  0.177779   0.994638  0.848967    │  0.21419   0.377067
 0.420909   0.864644   0.246007  0.709511   0.239352  0.279221   0.789019   0.526006   0.956405  0.581049    │  0.1384    0.642351
 0.709593   0.22967    0.274482  0.0517252  0.824723  0.934985   0.229917   0.514245   0.219793  0.583001    │  0.39413   0.196227
 0.350237   0.688092   0.994518  0.896093   0.103099  0.530616   0.420392   0.262814   0.622369  0.268439    │  0.169126  0.198431
 0.956745   0.696715   0.838499  0.576819   0.719354  0.250461   0.416511   0.963868   0.883324  0.630397    │  0.954737  0.9058  
 ────────────────────────────────────────────────────────────────────────────────────────────────────────────┼────────────────────
 0.438206   0.566904   0.495495  0.183546   0.625943  0.256114   0.116151   0.725794   0.93137   0.564458    │  0.233888  0.410463
 0.0936359  0.838417   0.210066  0.848525   0.834449  0.314861   0.849024   0.0837541  0.379385  0.96699     │  0.788724  0.133835

A[Block(1,1)] = K;
A[Block(1,2)] = B';
A[Block(2,1)] = B;

x, stats = Krylov.gmres(A, b)
([-1.6804307117446322, 1.2278956836087573, 0.7522500694970368, -1.1813882696427558, -1.7308386536458193, 1.3297218781185194, 0.7128256864623073, 0.43277609013448326, 1.819601362734177, 1.6564852399189884, -1.897804683660123, -0.6132195202643419], SimpleStats
 niter: 12
 solved: true
 inconsistent: false
 residuals: []
 Aresiduals: []
 κ₂(A): []
 status: solution good enough given atol and rtol
)

x
2-blocked 12-element BlockVector{Float64}:
 -1.6804307117446322 
  1.2278956836087573 
  0.7522500694970368 
 -1.1813882696427558 
 -1.7308386536458193 
  1.3297218781185194 
  0.7128256864623073 
  0.43277609013448326
  1.819601362734177  
  1.6564852399189884 
 ────────────────────
 -1.897804683660123  
 -0.6132195202643419 

norm(b - A * x)
2.8592284672503768e-15

[tmigot]

Because 0 is an integer, similar(S, 0) can't be used for some storage types that are not a subtype of a DenseVector. I must allocate all optional vectors required for warm-start, regularization or preconditioners in the Krylov workspaces.
Do you have a better solution?

Thanks for the details. Isn't that strange the empty vector is not defined even for sparse arrays ? In the worst-case could you do similar(S, 1) instead to avoid depending on the size?

[tmigot]

A few more comments, and then looks good to me, thanks!

[amontoison]

Because 0 is an integer, similar(S, 0) can't be used for some storage types that are not a subtype of a DenseVector. I must allocate all optional vectors required for warm-start, regularization or preconditioners in the Krylov workspaces. Do you have a better solution?

Thanks for the details. Isn't that strange the empty vector is not defined even for sparse arrays ? In the worst-case could you do similar(S, 1) instead to avoid depending on the size?

I don't understand your comment Tangi. 1 is also an integer like 0. The problem is not the value 0, I just want to create a vector that satisfy ::S and take the minimum amount of memory. Some vectors require the constructor with the axes to be initialized.

For the sparse arrays, I will add a check to insure that S is never a sparse arrays. It's for that reason that I added S <: DenseVector everywhere in the past.

[tmigot]

don't understand your comment Tangi. 1 is also an integer like 0. The problem is not the value 0, I just want to create a vector that satisfy ::S and take the minimum amount of memory. Some vectors require the constructor with the axes to be initialized.

Ok, I didn't understand the problem at first. Do you have an example that would not work?

[amontoison]

don't understand your comment Tangi. 1 is also an integer like 0. The problem is not the value 0, I just want to create a vector that satisfy ::S and take the minimum amount of memory. Some vectors require the constructor with the axes to be initialized.

Ok, I didn't understand the problem at first. Do you have an example that would not work?

Yes, I do.

using BlockArrays, Krylov

T = Float64
n = 10
x = BlockVector(rand(T,n), [n-2,2])
S = Krylov.ktypeof(x)
similar(S, 0)

[tmigot]

don't understand your comment Tangi. 1 is also an integer like 0. The problem is not the value 0, I just want to create a vector that satisfy ::S and take the minimum amount of memory. Some vectors require the constructor with the axes to be initialized.

Ok, I didn't understand the problem at first. Do you have an example that would not work?

Yes, I do.

using BlockArrays, Krylov

T = Float64
n = 10
x = BlockVector(rand(T,n), [n-2,2])
S = Krylov.ktypeof(x)
similar(S, 0)

Well... that won't really help but you could create an issue on this package and on the packages where the empty vector is not defined. To me, it looks like a very natural property for any vector type,

[gdalle]

@amontoison sorry to revive old threads, can you explain why this was closed?

[amontoison]

@gdalle I added a KrylovConstructor that can be passed to any KrylovWorkspace with a recent release.

It should handle many fancy array types.

Documentation: https://jso.dev/Krylov.jl/dev/inplace/#Krylov.KrylovConstructor

PS: Corner cases are LN / LS problems with array types that hardcode the length of the vector in the type.