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| 1 | +# This file is largely inspired from Bernard Mourrain's MultivariateSeries/diagonalization.jl |
| 2 | + |
| 3 | +export NewtonTypeDiagonalization |
| 4 | + |
| 5 | +# norm of off diagonal terms of a square matrix |
| 6 | +function norm_off(M) |
| 7 | + if size(M[1], 1) > 1 |
| 8 | + return sqrt( |
| 9 | + sum( |
| 10 | + abs2(M[i, j]) + abs2(M[j, i]) for i in 1:size(M, 1) for |
| 11 | + j in i+1:size(M, 1) |
| 12 | + ), |
| 13 | + ) |
| 14 | + else |
| 15 | + return 0.0 |
| 16 | + end |
| 17 | +end |
| 18 | + |
| 19 | +""" |
| 20 | + diagonalization_iter(D::Vector{<:AbstractMatrix{T}}) where {T} |
| 21 | +
|
| 22 | +Given the vector `D` of `[F_i * E_i, F_i * M_1 * E_i, ..., F_i * M_p * E_i]`, |
| 23 | +computes the matrix `X` (resp. `Y`) corresponding to ``(I_n + X_i)`` |
| 24 | +(resp. ``(I_n + Y_i)``) of [KMY22, Theorem 5] that solves equations |
| 25 | +[KMY22, (26)-(29)]. |
| 26 | +""" |
| 27 | +function diagonalization_iter(D::Vector{<:AbstractMatrix{T}}) where {T} |
| 28 | + n = LinearAlgebra.checksquare(D[1]) |
| 29 | + s = length(D) |
| 30 | + |
| 31 | + X = fill(zero(T), n, n) |
| 32 | + Y = fill(zero(T), n, n) |
| 33 | + |
| 34 | + A = fill(zero(T), s, 2) |
| 35 | + b = fill(zero(T), s) |
| 36 | + for i in 1:n |
| 37 | + for j in 1:n |
| 38 | + if i != j |
| 39 | + for k in 1:s |
| 40 | + A[k, 1] = D[k][i, i] |
| 41 | + A[k, 2] = D[k][j, j] |
| 42 | + b[k] = -D[k][i, j] |
| 43 | + end |
| 44 | + v = A \ b |
| 45 | + X[i, j] = v[1] |
| 46 | + Y[i, j] = v[2] |
| 47 | + end |
| 48 | + end |
| 49 | + end |
| 50 | + for i in 1:n |
| 51 | + X[i, i] = 1 |
| 52 | + Y[i, i] = 1 |
| 53 | + end |
| 54 | + return X, Y |
| 55 | +end |
| 56 | + |
| 57 | +""" |
| 58 | + struct NewtonTypeDiagonalization{T,RNGT} <: AbstractMultiplicationMatricesSolver |
| 59 | + max_iter::Int |
| 60 | + ε::T |
| 61 | + tol::T |
| 62 | + rng::RNGT |
| 63 | + end |
| 64 | +
|
| 65 | +Simultaneous diagonalization of commuting matrices using the method of [KMY22, Theorem 5]. |
| 66 | +
|
| 67 | +[KMY22] Khouja, Rima, Mourrain, Bernard, and Yakoubsohn, Jean-Claude. |
| 68 | +*Newton-type methods for simultaneous matrix diagonalization.* |
| 69 | +Calcolo 59.4 (2022): 38. |
| 70 | +""" |
| 71 | +struct NewtonTypeDiagonalization{T,RNGT} <: AbstractMultiplicationMatricesSolver |
| 72 | + max_iter::Int |
| 73 | + ε::T |
| 74 | + tol::T |
| 75 | + rng::RNGT |
| 76 | +end |
| 77 | +# These were the values in MultivariateSeries/diagonalization.jl |
| 78 | +function NewtonTypeDiagonalization(max_iter, ε, tol) |
| 79 | + return NewtonTypeDiagonalization(max_iter, ε, tol, Random.GLOBAL_RNG) |
| 80 | +end |
| 81 | +NewtonTypeDiagonalization() = NewtonTypeDiagonalization(10, 1e-3, 5e-2) |
| 82 | + |
| 83 | +function _eigvecs(M::AbstractMatrix{BigFloat}) |
| 84 | + ev = LinearAlgebra.schur(Float64.(M)).vectors |
| 85 | + # `eigvecs` is failing some tests with a non-invertible `ev` |
| 86 | + #ev = LinearAlgebra.eigvecs(Float64.(M)) |
| 87 | + return convert(Matrix{BigFloat}, ev) |
| 88 | +end |
| 89 | +_eigvecs(M::AbstractMatrix) = LinearAlgebra.eigvecs |
| 90 | + |
| 91 | +function _solve_multiplication_matrices(M, λ, solver::NewtonTypeDiagonalization) |
| 92 | + @assert length(M) == length(λ) |
| 93 | + n = length(λ) |
| 94 | + r = LinearAlgebra.checksquare(M[1]) |
| 95 | + |
| 96 | + M1 = sum(λ .* M) |
| 97 | + E = _eigvecs(M1) |
| 98 | + |
| 99 | + # With `eigvecs`, we should do `inv` but with `schur` we can just transpose |
| 100 | + #F = inv(E) |
| 101 | + F = E' |
| 102 | + |
| 103 | + D = vcat( |
| 104 | + # Add one matrix for the equation `F_i * E_i = I` |
| 105 | + # constraining `E_i` to be invertible |
| 106 | + [Matrix{eltype(M[1])}(I, r, r)], |
| 107 | + [F * M[i] * E for i in eachindex(M)], |
| 108 | + ) |
| 109 | + err = sum(norm_off.(D)) |
| 110 | + Δ = sum(norm.(D)) |
| 111 | + |
| 112 | + nit = 0 |
| 113 | + |
| 114 | + if err / Δ > solver.tol |
| 115 | + Δ = err |
| 116 | + while nit < solver.max_iter && Δ > solver.ε |
| 117 | + err0 = err |
| 118 | + X, Y = diagonalization_iter(D) |
| 119 | + # From [KMY22, Theorem 5] |
| 120 | + # Z_{i,k} + ∑_{i,k} |
| 121 | + # = F_i * M_k * E_i |
| 122 | + # = (I_n * Y_i) * (F_{i-1} * M_k * E_{i-1}) * (I_n * X_i) |
| 123 | + # = (I_n * Y_i) * D[i] * (I_n * X_i) |
| 124 | + # = Y * D[i] * X |
| 125 | + D = [Y * D[i] * X for i in eachindex(D)] |
| 126 | + # E_{i+1} = E_i * (I_n * X_i) from [KMY22, Theorem 5] |
| 127 | + E = E * X |
| 128 | + # F_{i+1} = (I_n * Y_i) * F_i from [KMY22, Theorem 5] |
| 129 | + F = Y * F |
| 130 | + nit += 1 |
| 131 | + err = sum(norm_off.(D)) |
| 132 | + Δ = err0 - err |
| 133 | + end |
| 134 | + end |
| 135 | + |
| 136 | + return [[D[j+1][i, i] / D[1][i, i] for j in 1:n] for i in 1:r] |
| 137 | +end |
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