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Sep 29, 2025
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Initial guess #52

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initial guess
ocots Sep 29, 2025
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ocots Sep 29, 2025
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2 changes: 1 addition & 1 deletion docs/make.jl
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Expand Up @@ -67,4 +67,4 @@ makedocs(;
plugins=[links],
)

deploydocs(; repo=repo_url * ".git", devbranch="main")
deploydocs(; repo=repo_url * ".git", devbranch="main", push_preview=true)
16 changes: 8 additions & 8 deletions docs/src/tutorial-goddard.md
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Expand Up @@ -253,7 +253,7 @@ println("\nNorm of the shooting function: ‖s‖ = ", norm(s), "\n")
We aggregate the data to define the initial guess vector.

```@example main-goddard
ξ = [p0..., t1, t2, t3, tf] # initial guess
ξ_guess = [p0..., t1, t2, t3, tf] # initial guess
```

### MINPACK.jl
Expand Down Expand Up @@ -291,18 +291,18 @@ Let us define the problem to solve.

```@example main-goddard
# auxiliary function with aggregated inputs
nle!(s, ξ) = shoot!(s, ξ[1:3], ξ[4], ξ[5], ξ[6], ξ[7])
shoot!(s, ξ) = shoot!(s, ξ[1:3], ξ[4], ξ[5], ξ[6], ξ[7])

# Jacobian of the (auxiliary) shooting function
jnle!(js, ξ) = jacobian!(nle!, similar(ξ), js, backend, ξ)
jshoot!(js, ξ) = jacobian!(shoot!, similar(ξ), js, backend, ξ)
nothing # hide
```

We are now in position to solve the problem with the `hybrj` solver from MINPACK.jl through the `fsolve` function, providing the Jacobian. Let us solve the problem and retrieve the initial costate solution.

```@example main-goddard
# resolution of S(ξ) = 0
indirect_sol = fsolve(nle!, jnle!, ξ, show_trace=true)
indirect_sol = fsolve(shoot!, jshoot!, ξ_guess, show_trace=true)

# we retrieve the costate solution together with the times
ξ = indirect_sol.x
Expand All @@ -326,8 +326,8 @@ println("\nNorm of the shooting function: ‖s‖ = ", norm(s), "\n")
Alternatively, we can use the [NonlinearSolve.jl](https://docs.sciml.ai/NonlinearSolve) package to solve the shooting equation. The code is similar, but we use the `solve` function instead of `fsolve`. Let us define the problem.

```@example main-goddard
nle!(s, ξ, λ) = shoot!(s, ξ[1:3], ξ[4], ξ[5], ξ[6], ξ[7])
prob = NonlinearProblem(nle!, ξ)
shoot!(s, ξ, λ) = shoot!(s, ξ[1:3], ξ[4], ξ[5], ξ[6], ξ[7])
prob = NonlinearProblem(shoot!, ξ_guess)
nothing # hide
```

Expand Down Expand Up @@ -359,7 +359,7 @@ println("\nNorm of the shooting function: ‖s‖ = ", norm(s), "\n")
The results found for by the two solvers are extremely close, so now, lets benchmark these two resolutions to compare their performances.

```@example main-goddard
@benchmark fsolve(nle!, jnle!, ξ; tol=1e-8, show_trace=false) #MINPACK
@benchmark fsolve(shoot!, jshoot!, ξ_guess; tol=1e-8, show_trace=false) #MINPACK
```

```@example main-goddard
Expand All @@ -382,7 +382,7 @@ We plot the solution of the indirect solution (in red) over the solution of the
f = f1 * (t1, fs) * (t2, fb) * (t3, f0) # concatenation of the flows
flow_sol = f((t0, tf), x0, p0) # compute the solution: state, costate, control...

plot!(plt, flow_sol, label="indirect")
plot!(plt, flow_sol; label="indirect", color=2)
```

## References
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