Fix solve_exp_ode not updating position (#748)

* Fix  solve_exp_ode doc string

Fixes #740

* Implement failing test for #744

* Fix`solve_exp_ode` does not ever update position #744

* Add changelog entry

NEWS.md

@@ -5,6 +5,11 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [0.10.3] - unreleased
+
+* Fixed `solve_exp_ode` only returning the starting position ([#744](https://github.com/JuliaManifolds/Manifolds.jl/issues/744))
+* Fixed documentation of `solve_exp_ode` function signature ([#740](https://github.com/JuliaManifolds/Manifolds.jl/issues/740))  
+
 ## [0.10.2] - unreleased
 
 ### Changed

ext/ManifoldsOrdinaryDiffEqExt.jl

@@ -61,7 +61,7 @@ function solve_exp_ode(
     params = (M,)
     prob = ODEProblem(exp_problem, u0, (0.0, t), params)
     sol = solve(prob, solver; kwargs...)
-    q = sol.u[1][(n + 1):(2 * n)]
+    q = sol.u[end][(n + 1):(2 * n)]
     return q
 end
 # also define exp! for metric manifold anew in this case

src/manifolds/ConnectionManifold.jl

@@ -375,8 +375,8 @@ function solve_exp_ode end
         M::ConnectionManifold,
         p,
         X,
-        t::Number,
-        B::AbstractBasis;
+        t::Number;
+        B::AbstractBasis = DefaultOrthonormalBasis(),
         backend::AbstractDiffBackend = default_differential_backend(),
         solver = AutoVern9(Rodas5()),
         kwargs...,

test/metric.jl

@@ -296,6 +296,23 @@ Manifolds.inner(::MetricManifold{ℝ,<:AbstractManifold{ℝ},Issue539Metric}, p,
         N2 = ConnectionManifold(E, TestConnection())
         @test exp(N2, p, X) == X
     end
+
+    # see also Issue #744 (https://github.com/JuliaManifolds/Manifolds.jl/issues/744) 
+    @testset "solve_exp_ode values" begin
+        E = TestEuclidean{3}()
+        g = TestEuclideanMetric()
+        g_scaled = TestScaledEuclideanMetric()
+        M = MetricManifold(E, g)
+        default_retraction_method(::TestEuclidean) = TestRetraction()
+        p = [1.0, 2.0, 3.0]
+        X = [2.0, 3.0, 4.0]
+        t = 2.5
+
+        # we're testing on a flat euclidean space 
+        @test exp(M, p, X) ≈ p + X
+        @test exp(M, p, X, t) ≈ p + t * X
+    end
+
     @testset "Local Metric Error message" begin
         M = MetricManifold(BaseManifold{2}(), NotImplementedMetric())
         A = Manifolds.get_default_atlas(M)