sparse_vector.jl

using SparseArrays
using LinearSolve
using LinearAlgebra

# Constructing sparse array
function hess_sparse(x::Vector{T}) where {T}
    return [
        -sin(x[1] + x[2]) + 1,
        -sin(x[1] + x[2]),
        -sin(x[1] + x[2]),
        -sin(x[1] + x[2]) + 1.0,
        1.0,
        1.0,
        12.0 * x[5]^2 + 1.0,
        1.0
    ]
end
rowval = [1, 1, 2, 2, 3, 4, 5, 6]
colval = [1, 2, 1, 2, 3, 4, 5, 6]

# Constructing sparse vec
function grad_sparse(x::Vector{T}) where {T <: Number}
    return [cos(x[1] + x[2]), cos(x[1] + x[2]), 2 * x[3], 1 / 2, 4 * x[5]^3, 1 / 2]
end
gradinds = [1, 2, 3, 4, 5, 6]

# Forming the matrix and vector
x0 = [
    0.7853981648713337,
    0.7853981693418342,
    1.023999999999997e-7,
    -1.0,
    0.33141395338218227,
    -1.0
]
n = length(x0)
hess_mat = sparse(rowval, colval, hess_sparse(x0), n, n)
grad_vec = sparsevec(gradinds, grad_sparse(x0), n)

# Converting grad_vec to dense succeeds in solving
prob = LinearProblem(hess_mat, grad_vec)
linsolve = init(prob);
@test solve!(linsolve).u ≈ hess_mat \ Array(grad_vec)

H = hess_mat' * hess_mat
prob = LinearProblem(H, hess_mat' * grad_vec)
linsolve = init(prob, CholeskyFactorization())
VERSION >= v"1.8" && @test solve!(linsolve).u ≈ H \ Array(hess_mat' * grad_vec)