What causes `AssertionError` when asserting a diagonalized vector in Julia? - julia

Question

I've been doing some exercises in julia, and I am currently trying to #assert a vector that has been diagonalized into a matrix against a "solution matrix" given in the exercise notebook. However, I get an AssertionError when asserting my code against the provided solution. Example of my code:
julia> using LinearAlgebra
julia> A =
[
140 97 74 168 131
97 106 89 131 36
74 89 152 144 71
168 131 144 54 142
131 36 71 142 36
]
5×5 Matrix{Int64}:
140 97 74 168 131
97 106 89 131 36
74 89 152 144 71
168 131 144 54 142
131 36 71 142 36
julia> A_eigv = eigen(A).values
5-element Vector{Float64}:
-128.49322764802145
-55.887784553057
42.752167279318854
87.16111477514494
542.4677301466137
julia> A_diag = Diagonal(A_eigv)
5×5 Diagonal{Float64, Vector{Float64}}:
-128.493 ⋅ ⋅ ⋅ ⋅
⋅ -55.8878 ⋅ ⋅ ⋅
⋅ ⋅ 42.7522 ⋅ ⋅
⋅ ⋅ ⋅ 87.1611 ⋅
⋅ ⋅ ⋅ ⋅ 542.468
julia> #assert A_diag == [-128.493 0.0 0.0 0.0 0.0;
0.0 -55.8878 0.0 0.0 0.0;
0.0 0.0 42.7522 0.0 0.0;
0.0 0.0 0.0 87.1611 0.0;
0.0 0.0 0.0 0.0 542.468]
AssertionError: A_diag == [-128.493 0.0 0.0 0.0 0.0; 0.0 -55.8878 0.0 0.0 0.0; 0.0 0.0 42.7522 0.0 0.0; 0.0 0.0 0.0 87.1611 0.0; 0.0 0.0 0.0 0.0 542.468]
Stacktrace:
[1] top-level scope
# In[90]:1
[2] eval
# ./boot.jl:360 [inlined]
[3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)
# Base ./loading.jl:1094
My inital assumption was that a difference in number of decimals was the cause of the error. I therefore replaced == with ≈ (\approx). However, as the code example below shows, the error persists:
julia> #assert A_diag ≈ #\approx
[-128.493 0.0 0.0 0.0 0.0;
0.0 -55.8878 0.0 0.0 0.0;
0.0 0.0 42.7522 0.0 0.0;
0.0 0.0 0.0 87.1611 0.0;
0.0 0.0 0.0 0.0 542.468]
AssertionError: A_diag ≈ [-128.493 0.0 0.0 0.0 0.0; 0.0 -55.8878 0.0 0.0 0.0; 0.0 0.0 42.7522 0.0 0.0; 0.0 0.0 0.0 87.1611 0.0; 0.0 0.0 0.0 0.0 542.468]
Stacktrace:
[1] top-level scope
# In[97]:1
[2] eval
# ./boot.jl:360 [inlined]
[3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)
# Base ./loading.jl:1094
I've been reading through my code multiple times now, and I am at a loss. The values in my diagonal matrix (A_diag) are seemingly identical to the solution matrix. Furthermore, setting the statement to approximately equal (\approx) renders the same error, so I assume I can count out decimal error.
My main question is: What causes the AssertionError?

Answer 1

No, the dots are treated as 0 for the purpose of testing equality.
julia> Diagonal(1:2) == [1 0; 0 2]
true
Your problem is that your arrays are actually not equal; -128.49322764802145 is not the same as -128.493. (The pretty-printed version of the array truncates floats for display, but that's not the true underlying value!).
[Edit:]
Using ≈ (\approx) will also fail in this case. The reason for this is explained in the documentation for isapprox()
The binary operator ≈ is equivalent to isapprox with the default arguments,
if an atol > 0 is not specified, rtol defaults to the square root of eps of the type of x or y, whichever is bigger (least precise).
Essentially, what this means is that ≈ will test for approximate equality with a relative tolerance of √eps() which is approximately equal to 1.5e-8, or 0.0000015%. This tolerance is waaay too low, and increasing the tolerance will resolve the issue. E.g.:
# Option 1: Absolute tolerance. Set to a reasonable max deviation:
julia> isapprox(A_diag, sol_mat, atol = 1e-3)
true
# Option 2: Relative tolerance. Setting rtol = 1e-n, n is the number of significant digits in either matrices will work in most cases.
julia> isapprox(A_diag, sol_mat, rtol = 1e-6)
true
Since the solution matrix provides the values in six significant digits, another alternative is to round the values in A_diag to this number og digits and test for equality. E.g.:
julia> round.(A_diag, RoundNearestTiesUp, sigdigits=6) ==
[-128.493 0.0 0.0 0.0 0.0;
0.0 -55.8878 0.0 0.0 0.0;
0.0 0.0 42.7522 0.0 0.0;
0.0 0.0 0.0 87.1611 0.0;
0.0 0.0 0.0 0.0 542.468]
true