Nonlinear Solver 23 Test Problems
These benchmarks compares the runtime and error for a range of nonlinear solvers. The problems are a standard set of problems as described here. The solvers are implemented in NonlinearProblemLibrary.jl, where you can find the problem function declarations. For each problem we test the following solvers:
- NonlinearSolve.jl's Newton Raphson method (
NewtonRaphson()
). - NonlinearSolve.jl's Trust Region method (
TrustRegion()
). - NonlinearSolve.jl's Levenberg-Marquardt method (
LevenbergMarquardt()
). - MINPACK's Modified Powell method (
CMINPACK(method=:hybr)
). - MINPACK's Levenberg-Marquardt method (
CMINPACK(method=:lm)
). - NLsolveJL's Newton Raphson (
NLsolveJL(method=:newton)
). - NLsolveJL's Trust Region (
NLsolveJL()
). - NLsolveJL's Anderson acceleration (
NLsolveJL(method=:anderson)
). - Sundials's Newton-Krylov method (
KINSOL()
).
Furthermore, for NonlinearSolve.jl's Newton Raphson method we try the following Line Search options (in addition to the default):
HagerZhang
MoreThuente
BackTracking
and for NonlinearSolve.jl's Trust Region we try the following Radius Update schemes (in addition to the default):
NLsolve
NocedalWright
Hei
Yuan
Bastin
Fan
and finally for NonlinearSolve.jl's Levenberg-Marquardt method why try using both the default α_geodesic
value (0.75
) and a modified value (0.5
), and also with and without setting the CholeskyFactorization
linear solver.
For each benchmarked problem, the second, third, and fourth plots compares the performance of NonlinearSolve's Newton Raphson, Trust Region, and Levenberg-Marquardt methods, respectively. The first plot compares the best methods from each of these categories to the various methods available from other packages. At the end of the benchmarks, we print a summary table of which solvers succeeded for which problems.
Setup
Fetch required packages.
using NonlinearSolve, LinearSolve, StaticArrays, Sundials, Setfield,
BenchmarkTools, LinearAlgebra, DiffEqDevTools, NonlinearProblemLibrary, CairoMakie,
RecursiveFactorization, Enzyme
import PolyesterForwardDiff, MINPACK, NLsolve, LineSearches
const RUS = RadiusUpdateSchemes;
Declare the benchmarked solvers (and their names and plotting options).
HagerZhang() = LineSearchesJL(; method = LineSearches.HagerZhang())
MoreThuente() = LineSearchesJL(; method = LineSearches.MoreThuente())
solvers_all = [
(; pkg = :nonlinearsolve, type = :general, name = "Default PolyAlg.", solver = Dict(:alg => FastShortcutNonlinearPolyalg(; u0_len = 10, linsolve = \))),
(; pkg = :nonlinearsolve, type = :NR, name = "Newton Raphson", solver = Dict(:alg => NewtonRaphson(; linsolve = \))),
(; pkg = :nonlinearsolve, type = :NR, name = "NR (HagerZhang)", solver = Dict(:alg => NewtonRaphson(; linsolve = \, linesearch = HagerZhang()))),
(; pkg = :nonlinearsolve, type = :NR, name = "NR (MoreThuente)", solver = Dict(:alg => NewtonRaphson(; linsolve = \, linesearch = MoreThuente()))),
(; pkg = :nonlinearsolve, type = :NR, name = "NR (BackTracking)", solver = Dict(:alg => NewtonRaphson(; linsolve = \, linesearch = BackTracking()))),
(; pkg = :nonlinearsolve, type = :TR, name = "Trust Region", solver = Dict(:alg => TrustRegion(; linsolve = \))),
(; pkg = :nonlinearsolve, type = :TR, name = "TR (NLsolve Update)", solver = Dict(:alg => TrustRegion(; linsolve = \, radius_update_scheme = RUS.NLsolve))),
(; pkg = :nonlinearsolve, type = :TR, name = "TR (Nocedal Wright)", solver = Dict(:alg => TrustRegion(; linsolve = \, radius_update_scheme = RUS.NocedalWright))),
(; pkg = :nonlinearsolve, type = :TR, name = "TR (Hei)", solver = Dict(:alg => TrustRegion(; linsolve = \, radius_update_scheme = RUS.Hei))),
(; pkg = :nonlinearsolve, type = :TR, name = "TR (Yuan)", solver = Dict(:alg => TrustRegion(; linsolve = \, radius_update_scheme = RUS.Yuan))),
(; pkg = :nonlinearsolve, type = :TR, name = "TR (Bastin)", solver = Dict(:alg => TrustRegion(; linsolve = \, radius_update_scheme = RUS.Bastin))),
(; pkg = :nonlinearsolve, type = :TR, name = "TR (Fan)", solver = Dict(:alg => TrustRegion(; linsolve = \, radius_update_scheme = RUS.Fan))),
(; pkg = :nonlinearsolve, type = :LM, name = "Levenberg-Marquardt", solver = Dict(:alg => LevenbergMarquardt(; linsolve = QRFactorization()))),
(; pkg = :nonlinearsolve, type = :LM, name = "LM with Cholesky", solver = Dict(:alg => LevenbergMarquardt(; linsolve = CholeskyFactorization()))),
(; pkg = :nonlinearsolve, type = :LM, name = "LM (α_geodesic=0.5)", solver = Dict(:alg => LevenbergMarquardt(; linsolve = QRFactorization(), α_geodesic=0.5))),
(; pkg = :nonlinearsolve, type = :LM, name = "LM (α_geodesic=0.5) Chol.", solver = Dict(:alg => LevenbergMarquardt(; linsolve = CholeskyFactorization(), α_geodesic=0.5))),
(; pkg = :nonlinearsolve, type = :LM, name = "LM (no Accln.)", solver = Dict(:alg => LevenbergMarquardt(; linsolve = QRFactorization(), disable_geodesic = Val(true)))),
(; pkg = :nonlinearsolve, type = :LM, name = "LM (no Accln.) Chol.", solver = Dict(:alg => LevenbergMarquardt(; linsolve = CholeskyFactorization(), disable_geodesic = Val(true)))),
(; pkg = :nonlinearsolve, type = :general, name = "Pseudo Transient", solver = Dict(:alg => PseudoTransient(; linsolve = \, alpha_initial=10.0))),
(; pkg = :wrapper, type = :general, name = "Powell [MINPACK]", solver = Dict(:alg => CMINPACK(; method=:hybr))),
(; pkg = :wrapper, type = :LM, name = "LM [MINPACK]", solver = Dict(:alg => CMINPACK(; method=:lm))),
(; pkg = :wrapper, type = :NR, name = "NR [NLsolve.jl]", solver = Dict(:alg => NLsolveJL(; method=:newton))),
(; pkg = :wrapper, type = :TR, name = "TR [NLsolve.jl]", solver = Dict(:alg => NLsolveJL())),
(; pkg = :wrapper, type = :NR, name = "NR [Sundials]", solver = Dict(:alg => KINSOL(; linear_solver = :LapackDense, maxsetupcalls=1))),
(; pkg = :wrapper, type = :NR, name = "NR LineSearch [Sundials]", solver = Dict(:alg => KINSOL(; linear_solver = :LapackDense, globalization_strategy=:LineSearch, maxsetupcalls=1)))
];
solver_tracker = [];
wp_general_tracker = [];
Sets tolerances.
abstols = 1.0 ./ 10.0 .^ (4:12)
reltols = 1.0 ./ 10.0 .^ (4:12);
Prepares various helper functions for benchmarking a specific problem.
function set_ad_chunksize(solvers, u0)
ck = NonlinearSolve.pickchunksize(u0)
for i in eachindex(solvers)
@set! solvers[i].solver[:alg] = __set_ad_chunksize(solvers[i].solver[:alg], ck, length(u0))
end
return solvers
end
function __set_ad_chunksize(solver::GeneralizedFirstOrderAlgorithm, ck, N)
if N > ck
ad = AutoPolyesterForwardDiff(; chunksize = ck)
else
ad = AutoForwardDiff(; chunksize = ck)
end
return GeneralizedFirstOrderAlgorithm(; solver.descent, solver.linesearch,
solver.trustregion, jvp_autodiff = ad, solver.max_shrink_times, solver.vjp_autodiff,
concrete_jac = solver.concrete_jac, name = solver.name, autodiff = ad)
end
function __set_ad_chunksize(solver::NonlinearSolvePolyAlgorithm, ck, N)
algs = [__set_ad_chunksize(alg, ck, N) for alg in solver.algs]
return NonlinearSolvePolyAlgorithm(algs; solver.start_index)
end
__set_ad_chunksize(solver, ck, N) = solver
# Benchmarks a specific problem, checks which solvers can solve it and their performance
function benchmark_problem!(prob_name; solver_tracker=solver_tracker)
# Finds the problem and the true solution.
prob = nlprob_23_testcases[prob_name]
# Finds the solvers that can solve the problem
solvers_concrete = set_ad_chunksize(solvers_all, prob.prob.u0);
successful_solvers = filter(solver -> check_solver(prob, solver), solvers_concrete);
push!(solver_tracker, prob_name => successful_solvers);
# Handles the non-general cases.
solvers_NR = filter(s -> s.type==:NR, successful_solvers)
solvers_TR = filter(s -> s.type==:TR, successful_solvers)
solvers_LM = filter(s -> s.type==:LM, successful_solvers)
wp_NR = WorkPrecisionSet(prob.prob, abstols, reltols, getfield.(solvers_NR, :solver);
names=getfield.(solvers_NR, :name), numruns=100, error_estimate=:l∞,
maxiters=1000,
termination_condition = AbsNormTerminationMode(Base.Fix1(maximum, abs)))
wp_TR = WorkPrecisionSet(prob.prob, abstols, reltols, getfield.(solvers_TR, :solver);
names=getfield.(solvers_TR, :name), numruns=100, error_estimate=:l∞,
maxiters=1000,
termination_condition = AbsNormTerminationMode(Base.Fix1(maximum, abs)))
wp_LM = WorkPrecisionSet(prob.prob, abstols, reltols, getfield.(solvers_LM, :solver);
names=getfield.(solvers_LM, :name), numruns=100, error_estimate=:l∞,
maxiters=1000,
termination_condition = AbsNormTerminationMode(Base.Fix1(maximum, abs)))
# Handles the general case
solvers_general = filter(s -> s.type==:general, successful_solvers)
add_solver!(solvers_general, nothing, solvers_TR, wp_TR)
add_solver!(solvers_general, nothing, solvers_LM, wp_LM)
add_solver!(solvers_general, nothing, solvers_NR, wp_NR)
wp_general = WorkPrecisionSet(prob.prob, abstols, reltols,
getfield.(solvers_general, :solver); names=getfield.(solvers_general, :name),
numruns=100, error_estimate=:l∞, maxiters=1000,
termination_condition = AbsNormTerminationMode(Base.Fix1(maximum, abs)))
push!(wp_general_tracker, prob_name => wp_general)
fig = plot_collective_benchmark(prob_name, wp_general, wp_NR, wp_TR, wp_LM)
save(replace(lowercase(prob_name), " " => "_") * "_wpd.svg", fig)
return fig
end
# Checks if a solver can successfully solve a given problem.
function check_solver(prob, solver)
try
sol = solve(prob.prob, solver.solver[:alg]; abstol=1e-8, reltol=1e-8,
maxiters=1000000,
termination_condition=AbsNormTerminationMode(Base.Fix1(maximum, abs)))
if norm(sol.resid, Inf) < 1e-6
Base.printstyled("[Info] Solver $(solver.name) returned retcode $(sol.retcode) \
with an residual norm = $(norm(sol.resid, Inf)).\n"; color=:green)
return true
else
Base.printstyled("[Warn] Solver $(solver.name) had a very large residual \
(norm = $(norm(sol.resid, Inf))).\n"; color=:red)
return false
end
WorkPrecisionSet(prob.prob, [1e-4, 1e-12], [1e-4, 1e-12], [solver.solver];
names=[solver.name], numruns=5, error_estimate=:l∞, maxiters=1000)
catch e
Base.printstyled("[Warn] Solver $(solver.name) threw an error: $e.\n"; color=:red)
return false
end
return true
end
# Adds an additional, selected, solver to the general solver set.
function add_solver!(solvers_general, selected_solver_name, additional_solver_set, wp)
if isnothing(selected_solver_name)
isempty(wp.wps) && return
selected_idx = argmin(median.(getfield.(wp.wps, :times)))
else
selected_idx = findfirst(s -> s.name==selected_solver_name, additional_solver_set)
isnothing(selected_solver) && error("The $(selected_solver_name) was designated to \
be added to the general solver set, however, it seemed to fail on this \
problem.")
end
isnothing(selected_idx) ||
pushfirst!(solvers_general, additional_solver_set[selected_idx])
end;
Plotting related helper functions.
__log10_zero(x) = ifelse(iszero(x), -100, log10(x))
Makie.inverse_transform(::typeof(__log10_zero)) = exp10
Makie.defaultlimits(::typeof(__log10_zero)) = Makie.defaultlimits(log10)
Makie.defined_interval(::typeof(__log10_zero)) = 0.0..Inf
# Skip minor ticks for __log10_zero scale
function Makie.get_minor_tickvalues(i::IntervalsBetween, scale::typeof(__log10_zero),
tickvalues, vmin, vmax)
return []
end
tickformatter(values) = map(values) do v
e = log10(v)
if isinteger(e) && e == -100
return rich("10", superscript("-∞"))
end
sup = isinteger(e) ? Int(e) : round(e; digits=2)
return rich("10", superscript(string(sup)))
end
function __filter_nearzero((ticks, ticklabels))
if first(ticks) ≈ 1e-100
idxs = findall(x -> x ≈ 1e-100 || x ≥ 10^-40, ticks)
return ticks[idxs], ticklabels[idxs]
end
return ticks, ticklabels
end
# Plots a work-precision diagram.
function plot_collective_benchmark(prob_name, wp_general, wp_NR, wp_TR, wp_LM)
LINESTYLES = Dict(:nonlinearsolve => :solid, :simplenonlinearsolve => :dash,
:wrapper => :dot)
ASPECT_RATIO = 0.7
WIDTH = 1400
HEIGHT = round(Int, WIDTH * ASPECT_RATIO)
STROKEWIDTH = 2.5
colors = cgrad(:tableau_20, length(solvers_all); categorical = true)
cycle = Cycle([:marker], covary = true)
plot_theme = Theme(Lines = (; cycle), Scatter = (; cycle))
fig = with_theme(plot_theme) do
fig = Figure(; size = (WIDTH, HEIGHT))
axs = []
xmin, xmax, ymin, ymax = Inf, -Inf, Inf, -Inf
for i in 1:2, j in 1:2
wp = (wp_general, wp_NR, wp_TR, wp_LM)[2 * (i - 1) + j]
ax = Axis(fig[i + 1, j], ylabel = j == 1 ? L"Time $\mathbf{(s)}$" : "",
xlabelsize = 22, ylabelsize = 22,
xlabel = i == 2 ? L"Error: $\mathbf{||f(u^\ast)||_\infty}$" : "",
xscale = __log10_zero, yscale = __log10_zero,
xtickwidth = STROKEWIDTH,
ytickwidth = STROKEWIDTH, spinewidth = STROKEWIDTH,
xticklabelsize = 20, yticklabelsize = 20,
xticklabelsvisible = i == 2, yticklabelsvisible = j == 1,
xticksvisible = i == 2, yticksvisible = j == 1,)
push!(axs, ax)
ls = []
scs = []
for wpᵢ in wp.wps
idx = findfirst(s -> s.name == wpᵢ.name, solvers_all)
errs = getindex.(wpᵢ.errors, :l∞)
times = wpᵢ.times
emin, emax = extrema(errs)
tmin, tmax = extrema(times)
emin < xmin && (xmin = emin)
emax > xmax && (xmax = emax)
tmin < ymin && (ymin = tmin)
tmax > ymax && (ymax = tmax)
l = lines!(ax, errs, times; color = colors[idx], linewidth = 5,
linestyle = LINESTYLES[solvers_all[idx].pkg], alpha = 0.8,
label = wpᵢ.name)
sc = scatter!(ax, errs, times; color = colors[idx], markersize = 16,
strokewidth = 2, marker = Cycled(idx), alpha = 0.8, label = wpᵢ.name)
push!(ls, l)
push!(scs, sc)
end
legend_title = ("", "Newton Raphson", "Trust Region", "Levenberg-Marquardt")[2 * (i - 1) + j]
Legend(fig[ifelse(i == 1, 1, 4), j], [[l, sc] for (l, sc) in zip(ls, scs)],
[wpᵢ.name for wpᵢ in wp.wps], legend_title;
framevisible=true, framewidth = STROKEWIDTH,
nbanks = 3, labelsize = 16, titlesize = 16,
tellheight = true, tellwidth = false, patchsize = (40.0f0, 20.0f0))
end
linkaxes!(axs...)
xmin = max(xmin, 10^-100)
xticks = __filter_nearzero(Makie.get_ticks(LogTicks(WilkinsonTicks(10; k_min = 5)),
__log10_zero, tickformatter, xmin, xmax))
yticks = __filter_nearzero(Makie.get_ticks(LogTicks(WilkinsonTicks(10; k_min = 5)),
__log10_zero, tickformatter, ymin, ymax))
foreach(axs) do ax
ax.xticks = xticks
ax.yticks = yticks
end
fig[0, :] = Label(fig, "Work-Precision Diagram for $(prob_name)",
fontsize = 24, tellwidth = false, font = :bold)
fig
end
return fig
end
plot_collective_benchmark (generic function with 1 method)
Benchmarks
We here run benchmarks for each of the 23 models.
Problem 1 (Generalized Rosenbrock function)
benchmark_problem!("Generalized Rosenbrock function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.554312234475219e-14.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 0.0.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.0214717960366215e-10.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 9.462239880519974e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.554312234475219e-14.
[Info] Solver Trust Region returned retcode Success with an residual norm =
3.6637359812630166e-13.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 7.771561172376096e-15.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 3.6637359812630166e-13.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 2.9
976021664879227e-14.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 3.
441691376337985e-14.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
0.0.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 0.0
.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 5.613462694675775e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 5.613456810493744e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 1.3598844272877386e-10.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 1.359913293086379e-10.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 5.430651484061855e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.4307913721629575e-9.
[Warn] Solver Pseudo Transient had a very large residual (norm = 2.63808362
8039608e10).
[Warn] Solver Powell [MINPACK] had a very large residual (norm = 4.39999999
99999995).
[Warn] Solver LM [MINPACK] had a very large residual (norm = 0.000852391008
632214).
[Warn] Solver NR [NLsolve.jl] had a very large residual (norm = 1.813491901
0169324e72).
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 4.36956026916846e-10.
[Warn] Solver NR [Sundials] had a very large residual (norm = 3.83838631427
93607e6).
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 9.992007221626409e-15.
Problem 2 (Powell singular function)
benchmark_problem!("Powell singular function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 2.945100572117028e-9.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 2.945100572117028e-9.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 7.351727052878344e-9.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 2.945100572117028e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 2.945100572117028e-9.
[Info] Solver Trust Region returned retcode Success with an residual norm =
2.8429847529782356e-9.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 2.945100572117028e-9.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 2.8429847529782356e-9.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 2.7
952108150126756e-9.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 7.
530945574071419e-9.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
4.491468406067117e-9.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 2.8
551395051697775e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 7.887593900975011e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 7.887593866725585e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 9.888757034011156e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 9.888757023494699e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 5.00414635180306e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.004146355111214e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 2.9904543725837263e-9.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 2.112388311092462e-35.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
2.052303855981701e-20.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.9451005722314578e-9.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.9451005722314578e-9.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 2.9470866826953975e-9.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 2.9470866826953975e-9.
Problem 3 (Powell badly scaled function)
benchmark_problem!("Powell badly scaled function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 6.226019699795415e-12.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.5733414571172943e-11.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 8.874578849571435e-10.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 8.11542533174503e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 6.226019699795415e-12.
[Info] Solver Trust Region returned retcode Success with an residual norm =
3.1642466424841587e-12.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 1.6564527527407336e-12.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 3.1642466424841587e-12.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 1.3
68238855548043e-12.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 7.
930323064897493e-13.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
1.3252288155740644e-11.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 1.3
488543615380877e-11.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 4.4873949001100755e-10.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 4.4874393090310605e-10.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 2.687781108789977e-10.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 2.687672306933564e-10.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 3.112532453997119e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 3.112532453997119e-11.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 1.5247237916682366e-9.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 1.7124214268804394e-9.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
0.0.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.5733192526568018e-11.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 7.051692563209144e-12.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 1.5767498418028936e-11.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 6.249223361010081e-12.
Problem 4 (Wood function)
benchmark_problem!("Wood function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 2.327027459614328e-13.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 4.884981308350689e-14.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 2.1735946376111315e-11.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 8.272279083954004e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 2.327027459614328e-13.
[Info] Solver Trust Region returned retcode Success with an residual norm =
8.449333899207545e-9.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 3.6615155352137663e-13.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 3.930189507173054e-14.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 7.7
83131916738739e-10.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 3.
81392029069616e-10.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
2.0583534876550402e-13.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 1.6
49005376691548e-10.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 5.511275658065529e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 5.511484157949553e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 5.511275658065529e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 5.511484157949553e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 4.594769009713673e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 4.5968340245394756e-11.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 1.4233059175694507e-13.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 3.374256429822253e-11.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
2.375877272697835e-14.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 4.241051954068098e-14.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.1522571785604896e-10.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 1.829647544582258e-13.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 4.951594689828198e-13.
Problem 5 (Helical valley function)
benchmark_problem!("Helical valley function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 3.62004078339879e-11.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.145684605114055e-14.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.2989387343509406e-10.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 9.361802444374234e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 3.62004078339879e-11.
[Info] Solver Trust Region returned retcode Success with an residual norm =
4.7194855486336445e-11.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 2.2591098913318186e-16.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 4.7194855486336445e-11.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 2.2
20446049250313e-15.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 7.
236678234070945e-17.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
9.645878886033804e-11.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 2.9
85118509462477e-10.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.2944894081552981e-11.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.2944894080950794e-11.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 1.0641488059519515e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 1.0641488059601572e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 9.266229008643821e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 9.266229008690408e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 4.539462534003996e-14.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 2.565741632453209e-13.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
7.418578548726161e-16.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.146398379185431e-14.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.1102230246251565e-14.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 1.1459236707701688e-14.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 3.619962451002043e-11.
Problem 6 (Watson function)
benchmark_problem!("Watson function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 8.439615672983791e-11.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 4.2077452633293433e-13.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 8.543088458878856e-11.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 8.76844874486693e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 8.439615672983791e-11.
[Info] Solver Trust Region returned retcode Success with an residual norm =
1.3614664950978295e-12.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 5.617728504603292e-14.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 2.0650148258027912e-13.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 7.1
03206911551752e-13.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 4.
9960036108132044e-15.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
2.6645352591003757e-15.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 8.3
99059225894234e-12.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 5.692691873449007e-10.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 5.693838733833445e-10.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 4.2001624400711535e-11.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 4.193045910483306e-11.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 9.953426971520685e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 9.953426971520685e-11.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 5.898654675817738e-9.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 2.2196688931330755e-12.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
3.219646771412954e-15.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 4.2077452633293433e-13.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 7.24632798210223e-10.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 4.107825191113079e-15.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 4.107825191113079e-15.
Problem 7 (Chebyquad function)
benchmark_problem!("Chebyquad function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.9505604353931005e-9.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 3.685329819091976e-12.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 7.887709929654818e-10.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 7.237587085207053e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.9505604353931005e-9.
[Info] Solver Trust Region returned retcode Success with an residual norm =
4.268807529683727e-14.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 1.6653345369377348e-16.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 4.268807529683727e-14.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 8.6
8749516769185e-14.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 1.
3732182058134867e-11.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
9.381384558082573e-15.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 8.7
29128531115293e-13.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 6.712307820677665e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 6.712307765166514e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 6.500716631663295e-11.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 6.500056048963643e-11.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 2.0931367750165464e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 2.0931367750165464e-11.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 1.0070833056374795e-11.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 1.6653345369377348e-16.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
5.551115123125783e-17.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 3.685329819091976e-12.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 9.43689570931383e-16.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 3.7223002458119936e-12.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 4.052314039881821e-15.
Problem 8 (Brown almost linear function)
benchmark_problem!("Brown almost linear function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 3.1521896204367295e-11.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 9.544809387307396e-12.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 7.99860178091194e-12.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 9.866944594705274e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 3.1521896204367295e-11.
[Info] Solver Trust Region returned retcode Success with an residual norm =
7.105427357601002e-15.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 1.652681325126082e-10.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 7.105427357601002e-15.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 2.2
853652303922445e-10.
[Warn] Solver TR (Yuan) had a very large residual (norm = 5.5).
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
5.717569306895598e-9.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 1.6
140933034591853e-10.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.4030865358449773e-10.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.4029444272978253e-10.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 3.0260240979629316e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 3.026058514876695e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 7.025180437381096e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 7.025280357453312e-11.
[Warn] Solver Pseudo Transient had a very large residual (norm = 1.35899023
74321883e19).
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 6.106226635438361e-15.
[Warn] Solver LM [MINPACK] had a very large residual (norm = 1.000000000050
5507).
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 9.546030632634483e-12.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 3.3823295275681176e-9.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 3.0393909611348136e-11.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 7.4291431895190385e-9.
Problem 9 (Discrete boundary value function)
benchmark_problem!("Discrete boundary value function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 2.7755575615628914e-16.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 2.7755575615628914e-16.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.7220389003647085e-9.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 2.7755575615628914e-16.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 2.7755575615628914e-16.
[Info] Solver Trust Region returned retcode Success with an residual norm =
9.104287879146966e-10.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 2.7755575615628914e-16.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 9.104287879146966e-10.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 2.7
755575615628914e-16.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 2.
3464563625452683e-13.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
2.7755575615628914e-16.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 7.7
81601529810445e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 9.328525341079086e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 9.328519068318997e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 9.328525341079086e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 9.328519068318997e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 9.166054304454718e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 9.166054304454718e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 4.091729732813576e-11.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 1.582067810090848e-15.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
1.3877787807814457e-17.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 3.0531133177191805e-16.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 3.0531133177191805e-16.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 4.996003610813204e-16.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 4.996003610813204e-16.
Problem 10 (Discrete integral equation function)
benchmark_problem!("Discrete integral equation function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.304512053934559e-15.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.304512053934559e-15.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 5.207544603869074e-9.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 1.304512053934559e-15.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.304512053934559e-15.
[Info] Solver Trust Region returned retcode Success with an residual norm =
2.868085491325445e-10.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 1.304512053934559e-15.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 2.868085491325445e-10.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 1.3
04512053934559e-15.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 1.
304512053934559e-15.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
1.304512053934559e-15.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 4.4
40892098500626e-16.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 8.828827668949657e-10.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 8.828898723223233e-10.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 8.828827668949657e-10.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 8.828898723223233e-10.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 8.640628212930324e-10.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 8.640628212930324e-10.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 7.702256749064773e-10.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 4.1661118999059e-14.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
1.3877787807814457e-17.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.304512053934559e-15.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.304512053934559e-15.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 1.0130785099704553e-15.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 1.0130785099704553e-15.
Problem 11 (Trigonometric function)
benchmark_problem!("Trigonometric function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 7.077671781985373e-15.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 6.942141306254257e-12.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 6.83623158082014e-11.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 6.442048269827971e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 7.077671781985373e-15.
[Warn] Solver Trust Region had a very large residual (norm = 0.004295419787
4424255).
[Warn] Solver TR (NLsolve Update) had a very large residual (norm = 0.00429
5418028397016).
[Warn] Solver TR (Nocedal Wright) had a very large residual (norm = 0.00429
5415051797022).
[Warn] Solver TR (Hei) had a very large residual (norm = 0.0042954155544357
58).
[Warn] Solver TR (Yuan) had a very large residual (norm = 0.004295419303280
812).
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
8.486950447972319e-10.
[Warn] Solver TR (Fan) had a very large residual (norm = 0.0042954173747542
37).
[Warn] Solver Levenberg-Marquardt had a very large residual (norm = 0.00486
7042382490749).
[Warn] Solver LM with Cholesky had a very large residual (norm = 0.00486704
2385246448).
[Warn] Solver LM (α_geodesic=0.5) had a very large residual (norm = 0.00440
9709072016127).
[Warn] Solver LM (α_geodesic=0.5) Chol. had a very large residual (norm = 0
.004324385118965601).
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 2.772573837184211e-13.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.103298811315327e-9.
[Warn] Solver Pseudo Transient had a very large residual (norm = 31.0772041
8891476).
[Warn] Solver Powell [MINPACK] had a very large residual (norm = 0.00440555
5421797214).
[Warn] Solver LM [MINPACK] had a very large residual (norm = 0.004295514477
982321).
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 6.938394303546147e-12.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.2043231689684575e-10.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 6.954242737222671e-12.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 4.210128079495945e-11.
Problem 12 (Variably dimensioned function)
benchmark_problem!("Variably dimensioned function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.3234968676556491e-12.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.3234968676556491e-12.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 4.344763882002667e-9.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 1.3234968676556491e-12.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.3234968676556491e-12.
[Info] Solver Trust Region returned retcode Success with an residual norm =
1.3234968676556491e-12.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 1.3234968676556491e-12.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 1.3234968676556491e-12.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 1.3
234968676556491e-12.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 1.
3234968676556491e-12.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
1.3234968676556491e-12.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 1.3
234968676556491e-12.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 5.197098307263559e-11.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 5.2210791245954624e-11.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 8.205796042659586e-10.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 8.180903732224465e-10.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 5.9344907876734965e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.934479574420948e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 1.3234968676556491e-12.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 5.636824340626845e-12.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
0.0.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.868483228724017e-12.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.868483228724017e-12.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 1.3057332992616466e-12.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 1.3057332992616466e-12.
Problem 13 (Broyden tridiagonal function)
benchmark_problem!("Broyden tridiagonal function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 7.54840412398039e-10.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 7.54840412398039e-10.
[Warn] Solver NR (HagerZhang) had a very large residual (norm = 0.901578712
5870238).
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 7.54840412398039e-10.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 7.54840412398039e-10.
[Info] Solver Trust Region returned retcode Success with an residual norm =
4.9817927560980024e-12.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 7.54840412398039e-10.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 4.9817927560980024e-12.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 7.5
4840412398039e-10.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 7.
54840412398039e-10.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
7.54840412398039e-10.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 3.3
433256163561964e-12.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 2.720891512097978e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 2.720909275666372e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 2.720891512097978e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 2.720909275666372e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 3.778735546688949e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 3.778735546688949e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 2.12066397775601e-9.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 2.377050778434864e-10.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
8.881784197001252e-16.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 7.548405234203415e-10.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 7.548405234203415e-10.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 7.556142378462027e-10.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 7.556142378462027e-10.
Problem 14 (Broyden banded function)
benchmark_problem!("Broyden banded function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 9.359466091041213e-9.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 9.359466091041213e-9.
[Warn] Solver NR (HagerZhang) had a very large residual (norm = 1.199192272
3423352).
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 9.359466091041213e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 9.359466091041213e-9.
[Info] Solver Trust Region returned retcode Success with an residual norm =
9.359466091041213e-9.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 9.359466091041213e-9.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 9.359466091041213e-9.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 9.3
59466091041213e-9.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 9.
359466091041213e-9.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
9.359466091041213e-9.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 9.3
59466091041213e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.6732734642310731e-10.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.6734733043755057e-10.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 1.6732734642310731e-10.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 1.6734733043755057e-10.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 3.2234254154772657e-10.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 3.2234254154772657e-10.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 4.440892098500626e-16.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 1.406736505060735e-9.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
4.440892098500626e-16.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 9.35947180868979e-9.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 9.35947180868979e-9.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 9.36354249692073e-9.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 9.36354249692073e-9.
Problem 15 (Hammarling 2 by 2 matrix square root problem)
benchmark_problem!("Hammarling 2 by 2 matrix square root problem")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.1102230246251565e-16.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.1102230246251565e-16.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.1546319456101628e-14.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 5.146420845036914e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.1102230246251565e-16.
[Info] Solver Trust Region returned retcode Success with an residual norm =
9.998897371997112e-9.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 9.998955736891987e-9.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 9.999969447949885e-9.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 9.9
99902617651876e-9.
[Info] Solver TR (Yuan) returned retcode MaxIters with an residual norm = 9
.0466025916777e-8.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
1.5627499294623703e-12.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 9.9
99330347021732e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.6792858215097795e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.5574467271406434e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 2.119859843219274e-12.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 2.154942890797429e-12.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 2.391302489357372e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 2.1891115586925025e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 8.378686633392363e-11.
[Warn] Solver Powell [MINPACK] had a very large residual (norm = 1.67183241
95308773e-5).
[Warn] Solver LM [MINPACK] had a very large residual (norm = 1.606211426157
26e-5).
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.1102230246251565e-16.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.1102230246251565e-16.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 3.3306690738754696e-16.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 3.3306690738754696e-16.
Problem 16 (Hammarling 3 by 3 matrix square root problem)
benchmark_problem!("Hammarling 3 by 3 matrix square root problem")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.1102230246251565e-16.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.1102230246251565e-16.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.0541905126615347e-9.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 5.009677117584488e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.1102230246251565e-16.
[Info] Solver Trust Region returned retcode Success with an residual norm =
9.999020663759086e-9.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 9.999661246569013e-9.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 9.998678786868239e-9.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 9.9
94120483001654e-9.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 9.
98837892637102e-9.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
1.7390533457728452e-12.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 9.9
98200522882907e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.6792649493169165e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.0188805354971464e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 7.458966777562637e-11.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 5.413958170663591e-11.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 2.3913026003796745e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 1.8238627275124486e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 9.636735853746359e-14.
[Warn] Solver Powell [MINPACK] had a very large residual (norm = 4.24408542
72881635e-6).
[Warn] Solver LM [MINPACK] had a very large residual (norm = 1.593107358411
434e-5).
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.1102230246251565e-16.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.1102230246251565e-16.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 3.3306690738754696e-16.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 3.3306690738754696e-16.
Problem 17 (Dennis and Schnabel 2 by 2 example)
benchmark_problem!("Dennis and Schnabel 2 by 2 example")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.0977885267493548e-11.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.0977885267493548e-11.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 8.881784197001252e-16.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 1.0977885267493548e-11.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.0977885267493548e-11.
[Info] Solver Trust Region returned retcode Success with an residual norm =
8.36610780652336e-11.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 1.0977885267493548e-11.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 8.36610780652336e-11.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 0.0
.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 1.
0977885267493548e-11.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
0.0.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 2.3
772983581693552e-11.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.2820273731506404e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.2819492134497068e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 1.2147705064080583e-10.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 1.2148948513868163e-10.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 1.5793144569897777e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 1.5793588659107627e-11.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 3.97371024973836e-12.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 4.440892098500626e-16.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
2.711222357731913e-10.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.0977885267493548e-11.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.0977885267493548e-11.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 1.1375789199519204e-11.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 1.1375789199519204e-11.
Problem 18 (Sample problem 18)
benchmark_problem!("Sample problem 18")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 8.16714264178066e-9.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 8.16714264178066e-9.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.6962016015432248e-10.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 7.508960040413216e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 8.16714264178066e-9.
[Info] Solver Trust Region returned retcode Success with an residual norm =
2.882025410947776e-11.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 7.702104220860704e-9.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 2.882025410947776e-11.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 4.3
83074308503473e-9.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 8.
16714264178066e-9.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
6.113143420005196e-10.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 1.5
463449882634202e-12.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 2.197559528526869e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 2.1976003653193354e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 4.929866744169792e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 4.341279241060183e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 4.1073275074827563e-10.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 4.107327806644548e-10.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 9.503369559071248e-12.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 1.060183858689003e-19.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
2.534326130054281e-27.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 8.166545633181853e-9.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.890631918127139e-13.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 7.971648853188235e-9.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 7.971648853188235e-9.
Problem 19 (Sample problem 19)
benchmark_problem!("Sample problem 19")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 4.956692686961647e-9.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 4.956692686961647e-9.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 0.0.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 4.956692686961647e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 4.956692686961647e-9.
[Info] Solver Trust Region returned retcode Success with an residual norm =
4.956692686961647e-9.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 4.956692686961647e-9.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 4.956692686961647e-9.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 7.4
68845853882793e-9.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 3.
995240858576322e-9.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
8.338921423533208e-9.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 4.9
56692686961647e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 5.040837698179729e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 5.040837698179819e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 7.160098265449655e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 7.160098265449658e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 6.758031521862519e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 6.758031521862518e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 4.998140661041174e-9.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 9.470216925475172e-203.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
1.1957479192951021e-104.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 4.956712536099437e-9.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 4.956712536099437e-9.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 4.9568018688545355e-9.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 4.9568018688545355e-9.
Problem 20 (Scalar problem f(x) = x(x - 5)^2)
benchmark_problem!("Scalar problem f(x) = x(x - 5)^2")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 2.912771141841026e-14.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 1.939490713125329e-15.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 1.4280239299962332e-19.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 4.851357739850789e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 2.912771141841026e-14.
[Info] Solver Trust Region returned retcode Success with an residual norm =
1.1090502985101545e-11.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 0.0.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 1.1090502985101545e-11.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 0.0
.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 6.
671729349424579e-13.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
0.0.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 1.3
176112414471956e-16.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 1.8009382137169453e-9.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 1.800938213712313e-9.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 1.8009382137169453e-9.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 1.800938213712313e-9.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 1.1102230246251565e-14.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.551115123125783e-15.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 1.633059876766743e-15.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 0.0.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
0.0.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 1.9400014971535805e-15.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 0.0.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 4.015195242953235e-15.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 9.437966645577242e-17.
Problem 21 (Freudenstein-Roth function)
benchmark_problem!("Freudenstein-Roth function")
[Warn] Solver Default PolyAlg. had a very large residual (norm = 4.94895210
5524011).
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 4.964704203302972e-10.
[Warn] Solver NR (HagerZhang) had a very large residual (norm = 6.635465857
710301).
[Warn] Solver NR (MoreThuente) threw an error: LineSearches.LineSearchExcep
tion{Int64}("Invalid parameters to MoreThuente.", 0).
[Warn] Solver NR (BackTracking) had a very large residual (norm = 7.0644467
614420705).
[Warn] Solver Trust Region had a very large residual (norm = 4.948952149637
1545).
[Warn] Solver TR (NLsolve Update) had a very large residual (norm = 4.94895
209564568).
[Warn] Solver TR (Nocedal Wright) had a very large residual (norm = 4.94895
21496371545).
[Warn] Solver TR (Hei) had a very large residual (norm = 4.94895210287909).
[Warn] Solver TR (Yuan) had a very large residual (norm = 4.948952160471567
).
[Warn] Solver TR (Bastin) had a very large residual (norm = 4.9489521055240
11).
[Warn] Solver TR (Fan) had a very large residual (norm = 4.94895209647386).
[Warn] Solver Levenberg-Marquardt had a very large residual (norm = 5.22335
12793720145).
[Warn] Solver LM with Cholesky had a very large residual (norm = 5.15000802
43610284).
[Warn] Solver LM (α_geodesic=0.5) had a very large residual (norm = 5.06609
74755918815).
[Warn] Solver LM (α_geodesic=0.5) Chol. had a very large residual (norm = 5
.185988136881765).
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 5.3769895203004126e-9.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.917158318879956e-9.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 0.0.
[Warn] Solver Powell [MINPACK] had a very large residual (norm = 4.95655173
5862394).
[Warn] Solver LM [MINPACK] had a very large residual (norm = 4.948960804424
9565).
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 3.4426292927491886e-9.
[Warn] Solver TR [NLsolve.jl] had a very large residual (norm = 4.948952141
537315).
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 7.58859641791787e-12.
[Warn] Solver NR LineSearch [Sundials] had a very large residual (norm = 7.
06444669790249).
Problem 22 (Boggs function)
benchmark_problem!("Boggs function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 1.7763568394002505e-14.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 0.0.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 5.509920093903882e-10.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 9.534361966601068e-9.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 1.7763568394002505e-14.
[Info] Solver Trust Region returned retcode Success with an residual norm =
1.3361534101363759e-11.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 6.477374192570551e-12.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 1.3361534101363759e-11.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 6.4
77374192570551e-12.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 3.
3306690738754696e-16.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
6.477374192570551e-12.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 6.3
26250634458574e-11.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 2.3191071285566522e-10.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 2.319096026326406e-10.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 2.0236479159052578e-11.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 2.0236479159052578e-11.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 5.869227326371629e-10.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 5.869227326371629e-10.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 3.186340080674199e-14.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 6.772360450213455e-12.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
1.232595164407831e-32.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 6.661338147750939e-16.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.5139568204934903e-10.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 4.5099168843876214e-10.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 1.9984014443252818e-14.
Problem 23 (Chandrasekhar function)
benchmark_problem!("Chandrasekhar function")
[Info] Solver Default PolyAlg. returned retcode Success with an residual no
rm = 2.9976021664879227e-14.
[Info] Solver Newton Raphson returned retcode Success with an residual norm
= 2.9976021664879227e-14.
[Info] Solver NR (HagerZhang) returned retcode Success with an residual nor
m = 4.440892098500626e-16.
[Info] Solver NR (MoreThuente) returned retcode Success with an residual no
rm = 2.9976021664879227e-14.
[Info] Solver NR (BackTracking) returned retcode Success with an residual n
orm = 2.9976021664879227e-14.
[Info] Solver Trust Region returned retcode Success with an residual norm =
9.253242616580337e-11.
[Info] Solver TR (NLsolve Update) returned retcode Success with an residual
norm = 2.9976021664879227e-14.
[Info] Solver TR (Nocedal Wright) returned retcode Success with an residual
norm = 9.253242616580337e-11.
[Info] Solver TR (Hei) returned retcode Success with an residual norm = 2.2
20446049250313e-16.
[Info] Solver TR (Yuan) returned retcode Success with an residual norm = 1.
9977863807696394e-10.
[Info] Solver TR (Bastin) returned retcode Success with an residual norm =
2.220446049250313e-16.
[Info] Solver TR (Fan) returned retcode Success with an residual norm = 3.2
548048700675736e-9.
[Info] Solver Levenberg-Marquardt returned retcode Success with an residual
norm = 5.7264415431745874e-11.
[Info] Solver LM with Cholesky returned retcode Success with an residual no
rm = 5.72513148000553e-11.
[Info] Solver LM (α_geodesic=0.5) returned retcode Success with an residual
norm = 5.7264415431745874e-11.
[Info] Solver LM (α_geodesic=0.5) Chol. returned retcode Success with an re
sidual norm = 5.72513148000553e-11.
[Info] Solver LM (no Accln.) returned retcode Success with an residual norm
= 7.449818539839725e-11.
[Info] Solver LM (no Accln.) Chol. returned retcode Success with an residua
l norm = 7.449818539839725e-11.
[Info] Solver Pseudo Transient returned retcode Success with an residual no
rm = 1.9410728580027126e-9.
[Info] Solver Powell [MINPACK] returned retcode Success with an residual no
rm = 3.0351055002597604e-11.
[Info] Solver LM [MINPACK] returned retcode Success with an residual norm =
2.220446049250313e-16.
[Info] Solver NR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.9531932455029164e-14.
[Info] Solver TR [NLsolve.jl] returned retcode Success with an residual nor
m = 2.9531932455029164e-14.
[Info] Solver NR [Sundials] returned retcode Success with an residual norm
= 3.752553823233029e-14.
[Info] Solver NR LineSearch [Sundials] returned retcode Success with an res
idual norm = 3.752553823233029e-14.
Summary of successful solvers
Finally, we print a summary of which solvers successfully solved which problems.
solver_successes = [(solver.name in getfield.(prob[2], :name)) ? "O" : "X" for prob in solver_tracker, solver in solvers_all]
total_successes = [sum(solver_successes[:,i] .== "O") for i in 1:length(solvers_all)]
solver_outcomes = vcat(total_successes', solver_successes);
using PrettyTables
io = IOBuffer()
println(io, "```@raw html")
pretty_table(io, solver_outcomes; backend = Val(:html), header = getfield.(solvers_all, :name), alignment=:c)
println(io, "```")
Base.Text(String(take!(io)))
Default PolyAlg. | Newton Raphson | NR (HagerZhang) | NR (MoreThuente) | NR (BackTracking) | Trust Region | TR (NLsolve Update) | TR (Nocedal Wright) | TR (Hei) | TR (Yuan) | TR (Bastin) | TR (Fan) | Levenberg-Marquardt | LM with Cholesky | LM (α_geodesic=0.5) | LM (α_geodesic=0.5) Chol. | LM (no Accln.) | LM (no Accln.) Chol. | Pseudo Transient | Powell [MINPACK] | LM [MINPACK] | NR [NLsolve.jl] | TR [NLsolve.jl] | NR [Sundials] | NR LineSearch [Sundials] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22 | 23 | 20 | 22 | 22 | 21 | 21 | 21 | 21 | 20 | 22 | 21 | 21 | 21 | 21 | 21 | 23 | 23 | 20 | 18 | 17 | 22 | 22 | 22 | 22 |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | X | X | X | X | O | X | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | X | O | O | O | O | O | O | O | O | X | O | X | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | X | X | X | X | X | O | X | X | X | X | X | O | O | X | X | X | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | X | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | X | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | X | X | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | X | X | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
X | O | X | X | X | X | X | X | X | X | X | X | X | X | X | X | O | O | O | X | X | O | X | O | X |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |
Summary of General Solver Performance on All Problems
fig = begin
LINESTYLES = Dict(:nonlinearsolve => :solid, :simplenonlinearsolve => :dash,
:wrapper => :dot)
ASPECT_RATIO = 1
WIDTH = 1800
HEIGHT = round(Int, WIDTH * ASPECT_RATIO)
STROKEWIDTH = 2.5
colors = cgrad(:tableau_20, length(solvers_all); categorical = true)
cycle = Cycle([:marker], covary = true)
plot_theme = Theme(Lines = (; cycle), Scatter = (; cycle))
with_theme(plot_theme) do
fig = Figure(; size = (WIDTH, HEIGHT))
axs = Matrix{Any}(undef, 5, 5)
ls = []
scs = []
labels = []
solver_times = []
for i in 1:5, j in 1:5
idx = 5 * (i - 1) + j
idx > length(wp_general_tracker) && break
prob_name, wp = wp_general_tracker[idx]
ax = Axis(fig[i, j],
xscale = __log10_zero, yscale = __log10_zero,
xtickwidth = STROKEWIDTH,
ytickwidth = STROKEWIDTH, spinewidth = STROKEWIDTH,
title = prob_name, titlegap = 10,
xticklabelsize = 16, yticklabelsize = 16)
xmin, xmax, ymin, ymax = Inf, -Inf, Inf, -Inf
for wpᵢ in wp.wps
idx = findfirst(s -> s.name == wpᵢ.name, solvers_all)
errs = getindex.(wpᵢ.errors, :l∞)
times = wpᵢ.times
emin, emax = extrema(errs)
tmin, tmax = extrema(times)
emin < xmin && (xmin = emin)
emax > xmax && (xmax = emax)
tmin < ymin && (ymin = tmin)
tmax > ymax && (ymax = tmax)
l = lines!(ax, errs, times; color = colors[idx], linewidth = 5,
linestyle = LINESTYLES[solvers_all[idx].pkg], alpha = 0.8,
label = wpᵢ.name)
sc = scatter!(ax, errs, times; color = colors[idx], markersize = 16,
strokewidth = 2, marker = Cycled(idx), alpha = 0.8, label = wpᵢ.name)
if wpᵢ.name ∉ labels
push!(ls, l)
push!(scs, sc)
push!(labels, wpᵢ.name)
end
if wpᵢ.name ∈ first.(solver_times)
idxi = findfirst(x -> first(x) == wpᵢ.name, solver_times)
push!(solver_times[idxi][2], median(times) / length(wp.prob.u0))
else
push!(solver_times, wpᵢ.name => [median(times) / length(wp.prob.u0)])
end
end
xmin = max(xmin, 10^-100)
xticks = __filter_nearzero(Makie.get_ticks(LogTicks(WilkinsonTicks(5; k_min = 3)),
__log10_zero, tickformatter, xmin, xmax))
yticks = __filter_nearzero(Makie.get_ticks(LogTicks(WilkinsonTicks(5; k_min = 3)),
__log10_zero, tickformatter, ymin, ymax))
ax.xticks = xticks
ax.yticks = yticks
end
ordering = sortperm(median.(last.(solver_times)))
fig[0, :] = Label(fig, "Work-Precision Diagram for 23 Test Problems",
fontsize = 24, tellwidth = false, font = :bold)
fig[:, 0] = Label(fig, "Time (s)", fontsize = 20, tellheight = false, font = :bold,
rotation = π / 2)
fig[end + 1, :] = Label(fig,
L"Error: $\mathbf{||f(u^\ast)||_\infty}$",
fontsize = 20, tellwidth = false, font = :bold)
Legend(fig[5, 4:5], [[l, sc] for (l, sc) in zip(ls[ordering], scs[ordering])],
labels[ordering], "Solvers";
framevisible=true, framewidth = STROKEWIDTH, orientation = :horizontal,
titlesize = 20, nbanks = 9, labelsize = 20, halign = :center,
tellheight = false, tellwidth = false, patchsize = (40.0f0, 20.0f0))
return fig
end
end
save("summary_wp_23test_problems.svg", fig)
CairoMakie.Screen{SVG}
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/NonlinearProblem","nonlinear_solver_23_tests.jmd")
Computer Information:
Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonlinearProblem/Project.toml`
[2169fc97] AlgebraicMultigrid v1.0.0
[6e4b80f9] BenchmarkTools v1.6.0
[13f3f980] CairoMakie v0.13.4
[2b5f629d] DiffEqBase v6.167.2
[f3b72e0c] DiffEqDevTools v2.48.0
[a0c0ee7d] DifferentiationInterface v0.6.52
⌃ [7da242da] Enzyme v0.13.37
[40713840] IncompleteLU v0.2.1
[b964fa9f] LaTeXStrings v1.4.0
[d3d80556] LineSearches v7.3.0
[7ed4a6bd] LinearSolve v3.8.0
[4854310b] MINPACK v1.3.0
[2774e3e8] NLsolve v4.5.1
[b7050fa9] NonlinearProblemLibrary v0.1.3
[8913a72c] NonlinearSolve v4.6.0
[ace2c81b] PETSc v0.3.1
[98d1487c] PolyesterForwardDiff v0.1.3
[08abe8d2] PrettyTables v2.4.0
[f2c3362d] RecursiveFactorization v0.2.23
[31c91b34] SciMLBenchmarks v0.1.3
[efcf1570] Setfield v1.1.2
[727e6d20] SimpleNonlinearSolve v2.2.2
[9f842d2f] SparseConnectivityTracer v0.6.17
[0a514795] SparseMatrixColorings v0.4.18
[f1835b91] SpeedMapping v0.3.0
[860ef19b] StableRNGs v1.0.2
[90137ffa] StaticArrays v1.9.13
[c3572dad] Sundials v4.28.0
[0c5d862f] Symbolics v6.38.0
Info Packages marked with ⌃ have new versions available and may be upgradable.
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.
And the full manifest:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonlinearProblem/Manifest.toml`
[47edcb42] ADTypes v1.14.0
[a4c015fc] ANSIColoredPrinters v0.0.1
[621f4979] AbstractFFTs v1.5.0
[1520ce14] AbstractTrees v0.4.5
[7d9f7c33] Accessors v0.1.42
[22286c92] AccurateArithmetic v0.3.8
[79e6a3ab] Adapt v4.3.0
[35492f91] AdaptivePredicates v1.2.0
[2169fc97] AlgebraicMultigrid v1.0.0
[66dad0bd] AliasTables v1.1.3
[27a7e980] Animations v0.4.2
[4fba245c] ArrayInterface v7.18.0
[4c555306] ArrayLayouts v1.11.1
[67c07d97] Automa v1.1.0
[13072b0f] AxisAlgorithms v1.1.0
[39de3d68] AxisArrays v0.4.7
[6e4b80f9] BenchmarkTools v1.6.0
[e2ed5e7c] Bijections v0.1.9
[d1d4a3ce] BitFlags v0.1.9
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
[70df07ce] BracketingNonlinearSolve v1.1.3
[fa961155] CEnum v0.5.0
[2a0fbf3d] CPUSummary v0.2.6
[159f3aea] Cairo v1.1.1
[13f3f980] CairoMakie v0.13.4
[d360d2e6] ChainRulesCore v1.25.1
[fb6a15b2] CloseOpenIntervals v0.1.13
[944b1d66] CodecZlib v0.7.8
[a2cac450] ColorBrewer v0.4.1
[35d6a980] ColorSchemes v3.29.0
[3da002f7] ColorTypes v0.12.1
[c3611d14] ColorVectorSpace v0.11.0
[5ae59095] Colors v0.13.0
[861a8166] Combinatorics v1.0.2
[38540f10] CommonSolve v0.2.4
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.0.0
[34da2185] Compat v4.16.0
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.3
[f0e56b4a] ConcurrentUtilities v2.5.0
[8f4d0f93] Conda v1.10.2
[187b0558] ConstructionBase v1.5.8
[d38c429a] Contour v0.6.3
[a2441757] Coverage v1.6.1
[c36e975a] CoverageTools v1.3.2
[adafc99b] CpuId v0.3.1
[a8cc5b0e] Crayons v4.1.1
[9a962f9c] DataAPI v1.16.0
[864edb3b] DataStructures v0.18.22
[e2d170a0] DataValueInterfaces v1.0.0
[927a84f5] DelaunayTriangulation v1.6.4
[2b5f629d] DiffEqBase v6.167.2
[f3b72e0c] DiffEqDevTools v2.48.0
[77a26b50] DiffEqNoiseProcess v5.24.1
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.15.1
[a0c0ee7d] DifferentiationInterface v0.6.52
[b4f34e82] Distances v0.10.12
[31c24e10] Distributions v0.25.119
[ffbed154] DocStringExtensions v0.9.4
[e30172f5] Documenter v1.10.1
[35a29f4d] DocumenterTools v0.1.20
[5b8099bc] DomainSets v0.7.15
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⌃ [0bca4576] SciMLBase v2.84.0
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[9fa8497b] Future
[b77e0a4c] InteractiveUtils
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[37e2e46d] LinearAlgebra
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.