Apply isapprox elementwise

[thomasfaingnaert]

Applying isapprox elementwise rather than on the entire matrix catches incorrect output better, since by default, isapprox on matrices uses the Frobenius ($L_2$) norm, $$||M|| = \sqrt{\sum_{i=1}^N \sum_{j=1}^N |a_{ij} |^2}$$, which scales with the matrix size, so larger matrices tolerate larger errors with a relative tolerance:

julia> A = rand(Float32, (4, 4));

julia> B = copy(A);

julia> isapprox(A, B)
true

julia> B[1] *= 1.01;

julia> isapprox(A, B)
false

julia> A = rand(Float32, (2048, 2048));

julia> B = copy(A);

julia> isapprox(A, B)
true

julia> B[1] *= 1.01;

julia> isapprox(A, B)
true

We could use absolute tolerance, or switch to the $L_\infty$-norm instead $$||M|| = \max_{i,j} |a_{ij}|$$

by passing a custom norm function to isapprox, which does handle inaccuracies better:

julia> using LinearAlgebra

julia> A = rand(Float32, (2048, 2048));

julia> B = copy(A);

julia> isapprox(A, B)
true

julia> isapprox(A, B; norm=M->LinearAlgebra.norm(M, Inf))
true

julia> B[1] *= 1.01;

julia> isapprox(A, B)
true

julia> isapprox(A, B; norm=M->LinearAlgebra.norm(M, Inf))
false

But let's just go for the easiest approach and apply isapprox elementwise instead.

[codecov]

Codecov Report

Attention: 53 lines in your changes are missing coverage. Please review.

Comparison is base (3052b52) 34.94% compared to head (3d45196) 32.04%.
Report is 5 commits behind head on master.

Files	Patch %	Lines
src/operator.jl	6.25%	30 Missing ⚠️
src/config.jl	8.00%	23 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #185      +/-   ##
==========================================
- Coverage   34.94%   32.04%   -2.90%     
==========================================
  Files          11       11              
  Lines         933      958      +25     
==========================================
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