[documentation] Sensitivity analysis

[amontoison]

I should add an example in the documentation to explain how we can do sensitivity analysis with Krylov.jl.

using ChainRulesCore
using Krylov
using Test
using LinearAlgebra
using SparseArrays
using ForwardDiff
import ForwardDiff: Dual, partials, value

include("test/test_utils.jl")

A, b = sparse_laplacian(FC=Float64)

function Krylov.cg(_A::SparseMatrixCSC{Dual{T, V, N}, Int64}, _b::Vector{Dual{T, V, N}}; options...) where {T, V, N}
  A = SparseMatrixCSC(_A.m, _A.n, _A.colptr, _A.rowval, value.(_A.nzval))
  dA = SparseMatrixCSC(_A.m, _A.n, _A.colptr, _A.rowval, partials.(_A.nzval, 1))
  b = value.(_b)
  db = partials.(_b,1)
  x, stats = cg(A,b)
  nb = db - dA*x
  dx, dstats = cg(dA,nb)
  return (Dual.(x, dx), stats)
end

x, stats = cg(A,b)


dA = ForwardDiff.Dual.(A, A)
db = ForwardDiff.Dual.(b, b)

It solves the normal value system Ax = b and the tangent system
A*dx + dA*x= db (or A*dx = b - dA*x).