### Full Example: Optimization with Lazy Buffer Cache

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

This comprehensive example demonstrates using GeneralLazyBufferCache within an optimization workflow. It sets up an ODE problem, defines a negative log-likelihood function that utilizes the cached integrator, creates an OptimizationProblem, and solves it. This showcases how to integrate lazy caching into a complex scientific computing task.

```julia
using Random, DifferentialEquations, LinearAlgebra, Optimization, OptimizationNLopt,
      OptimizationOptimJL, PreallocationTools

lbc = GeneralLazyBufferCache(function (p)
    DifferentialEquations.init(ODEProblem(ode_fnc, y₀, (0.0, T), p), Tsit5(); saveat = t)
end)

Random.seed!(2992999)
λ, y₀, σ = -0.5, 15.0, 0.1
T, n = 5.0, 200
Δt = T / n
t = [j * Δt for j in 0:n]
y = y₀ * exp.(λ * t)
yᵒ = y .+ [0.0, σ * randn(n)...]
ode_fnc(u, p, t) = p * u
function loglik(θ, data, integrator)
    yᵒ, n, ε = data
    λ, σ, u0 = θ
    integrator.p = λ
    reinit!(integrator, u0)
    solve!(integrator)
    ε = yᵒ .- integrator.sol.u
    ℓ = -0.5n * log(2π * σ^2) - 0.5 / σ^2 * sum(ε .^ 2)
end
θ₀ = [-1.0, 0.5, 19.73]
negloglik = (θ, p) -> -loglik(θ, p, lbc[θ[1]])
fnc = OptimizationFunction(negloglik, Optimization.AutoForwardDiff())
ε = zeros(n)
prob = OptimizationProblem(fnc,
    θ₀,
    (yᵒ, n, ε),
    lb = [-10.0, 1e-6, 0.5],
    ub = [10.0, 10.0, 25.0])
solve(prob, LBFGS())

```

--------------------------------

### Full Example with GeneralLazyBufferCache for Optimization

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

A comprehensive example demonstrating the use of GeneralLazyBufferCache within an optimization problem. It involves defining an objective function that uses a cached ODE integrator and then solving the optimization problem.

```julia
using Random, DifferentialEquations, LinearAlgebra, Optimization, OptimizationNLopt,
      OptimizationOptimJL, PreallocationTools

lbc = GeneralLazyBufferCache(function (p)
    DifferentialEquations.init(ODEProblem(ode_fnc, y₀, (0.0, T), p), Tsit5(); saveat = t)
end)

Random.seed!(2992999)
λ, y₀, σ = -0.5, 15.0, 0.1
T, n = 5.0, 200
Δt = T / n
t = [j * Δt for j in 0:n]
y = y₀ * exp.(λ * t)
yᵒ = y .+ [0.0, σ * randn(n)...]
ode_fnc(u, p, t) = p * u
function loglik(θ, data, integrator)
    yᵒ, n, ε = data
    λ, σ, u0 = θ
    integrator.p = λ
    reinit!(integrator, u0)
    solve!(integrator)
    ε = yᵒ .- integrator.sol.u
    ℓ = -0.5n * log(2π * σ^2) - 0.5 / σ^2 * sum(ε .^ 2)
end
θ₀ = [-1.0, 0.5, 19.73]
negloglik = (θ, p) -> -loglik(θ, p, lbc[θ[1]])
fnc = OptimizationFunction(negloglik, Optimization.AutoForwardDiff())
ε = zeros(n)
prob = OptimizationProblem(fnc,
    θ₀,
    (yᵒ, n, ε),
    lb = [-10.0, 1e-6, 0.5],
    ub = [10.0, 10.0, 25.0])
solve(prob, LBFGS())
```

--------------------------------

### Example Usage of LazyBufferCache

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

Demonstrates the use of `LazyBufferCache` within an ODE problem. The cache is automatically created and managed as needed during the ODE solve.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools
function foo(du, u, (A, lbc), t)
    tmp = lbc[u]
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end
prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), LazyBufferCache()))
solve(prob, TRBDF2())

```

--------------------------------

### DiffCache Direct Usage Example

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Demonstrates direct usage of DiffCache for automatic differentiation with ForwardDiff.jl. The cache size is inferred from the input matrix.

```julia
using ForwardDiff, PreallocationTools
randmat = rand(5, 3)
sto = similar(randmat)
stod = DiffCache(sto)

function claytonsample!(sto, τ, α; randmat = randmat)
    sto = get_tmp(sto, τ)
    sto .= randmat
    τ == 0 && return sto

    n = size(sto, 1)
    for i in 1:n
        v = sto[i, 2]
        u = sto[i, 1]
        sto[i, 1] = (1 - u^(-τ) + u^(-τ) * v^(-(τ / (1 + τ))))^(-1 / τ) * α
        sto[i, 2] = (1 - u^(-τ) + u^(-τ) * v^(-(τ / (1 + τ))))^(-1 / τ)
    end
    return sto
end

ForwardDiff.derivative(τ -> claytonsample!(stod, τ, 0.0), 0.3)
ForwardDiff.jacobian(x -> claytonsample!(stod, x[1], x[2]), [0.3; 0.0])
```

--------------------------------

### FixedSizeDiffCache Constructor Example

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Shows the constructor for FixedSizeDiffCache, which preallocates dual numbers in its cache. It is suitable for non-Array types and when the chunk size is known.

```julia
FixedSizeDiffCache(u::AbstractArray, chunk_size = Val{ForwardDiff.pickchunksize(length(u))})
FixedSizeDiffCache(u::AbstractArray, chunk_size::Integer)
```

--------------------------------

### Display Package Status

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

This snippet shows how to display the current status of installed packages using Pkg.status(). It's typically used for debugging or understanding project dependencies.

```julia
using Pkg # hide
Pkg.status() # hide
```

--------------------------------

### GeneralLazyBufferCache for ODE Integrator Caching

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Example of using GeneralLazyBufferCache to cache ODE integrator objects. The cache is defined as a function of the parameter `p` to build the integrator, and subsequent calls with the same `p` reuse the cached integrator.

```julia
lbc = GeneralLazyBufferCache(function (p)
    DifferentialEquations.init(ODEProblem(ode_fnc, y₀, (0.0, T), p), Tsit5(); saveat = t)
end)
```

--------------------------------

### Get Temporary Cache from DiffCache

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Retrieves the appropriate temporary cache from a DiffCache object. Returns the dual version if `u` contains dual numbers, otherwise returns the standard cache.

```julia
get_tmp(tmp::DiffCache, u)
```

--------------------------------

### Show enlargediffcache! Documentation

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/internals.md

Displays the documentation for the internal `enlargediffcache!` function. This is an internal helper function and may change without notice.

```julia
```@docs
PreallocationTools.enlargediffcache!
```
```

--------------------------------

### Display Julia Version Information

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

This snippet displays detailed information about the Julia environment, including the version, operating system, and architecture. It's useful for reproducibility and debugging.

```julia
using InteractiveUtils # hide
versioninfo() # hide
```

--------------------------------

### Display Manifest and Project File Status

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

This snippet displays the status of all dependencies, including those in the manifest file. It provides a comprehensive view of the project's environment, useful for ensuring reproducibility.

```julia
using Pkg # hide
Pkg.status(; mode = PKGMODE_MANIFEST) # hide
```

--------------------------------

### ODE Integration with DiffCache (Initial Attempt)

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

An initial attempt at defining an ODE function that fails due to the temporary array `tmp` not being compatible with dual numbers during automatic differentiation.

```julia
using LinearAlgebra, OrdinaryDiffEq
function foo(du, u, (A, tmp), t)
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end
prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), zeros(5, 5)))
solve(prob, TRBDF2())
```

--------------------------------

### DiffCache for ODEs with Default Chunk Size

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Illustrates using DiffCache with an ODEProblem where the chunk size for automatic differentiation is determined by default settings. The DiffCache is initialized without an explicit chunk size.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools
function foo(du, u, (A, tmp), t)
    tmp = get_tmp(tmp, u)
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end
chunk_size = 5
prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), DiffCache(zeros(5, 5))))
solve(prob, TRBDF2())
```

--------------------------------

### DiffCache for Nested AD in Optimization

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Demonstrates using a multi-level DiffCache for an optimization problem involving nested automatic differentiation, such as calculating Hessians in ODE-based optimization. The cache is initialized with `levels = 3`.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools, Optimization, OptimizationOptimJL
function foo(du, u, p, t)
    tmp = p[2]
    A = reshape(p[1], size(tmp.du))
    tmp = get_tmp(tmp, u)
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end

coeffs = -collect(0.1:0.1:0.4)
cache = DiffCache(zeros(2, 2), levels = 3)
prob = ODEProblem(foo, ones(2, 2), (0.0, 1.0), (coeffs, cache))
realsol = solve(prob, TRBDF2(), saveat = 0.0:0.1:10.0, reltol = 1e-8)

function objfun(x, prob, realsol, cache)
    prob = remake(prob, u0 = eltype(x).(prob.u0), p = (x, cache))
    sol = solve(prob, TRBDF2(), saveat = 0.0:0.1:10.0, reltol = 1e-8)

    ofv = 0.0
    if any((s.retcode != :Success for s in sol)) 
        ofv = 1e12
    else
        ofv = sum((sol .- realsol) .^ 2)
    end
    return ofv
end
fn(x, p) = objfun(x, p[1], p[2], p[3])
optfun = OptimizationFunction(fn, Optimization.AutoForwardDiff())
optprob = OptimizationProblem(optfun, zeros(length(coeffs)), (prob, realsol, cache))
solve(optprob, Newton())
```

--------------------------------

### DiffCache for ODEs with Explicit Chunk Size

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Shows how to use DiffCache with an ODEProblem when automatic differentiation requires a specific chunk size. The cache is explicitly initialized with the chunk size.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools
function foo(du, u, (A, tmp), t)
    tmp = get_tmp(tmp, u)
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end
chunk_size = 5
prob = ODEProblem(foo, 
    ones(5, 5),
    (0.0, 1.0),
    (ones(5, 5), DiffCache(zeros(5, 5), chunk_size)))
solve(prob, TRBDF2(chunk_size = chunk_size))
```

--------------------------------

### GeneralLazyBufferCache Constructor

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Constructs a GeneralLazyBufferCache. The function `f` generates the cache matching the type of `u`, and subsequent indexing reuses that cache if that type of `u` has already been seen. This is the most general but slowest preallocation method.

```julia
GeneralLazyBufferCache(f = identity)
```

--------------------------------

### Using LazyBufferCache in ODEProblem

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Demonstrates using LazyBufferCache to preallocate temporary arrays within an ODE solver. The cache automatically creates and manages temporary arrays needed by the `foo` function.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools
function foo(du, u, (A, lbc), t)
    tmp = lbc[u]
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end
prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), LazyBufferCache()))
solve(prob, TRBDF2())
```

--------------------------------

### Nested AD Calls in Optimization with DiffCache

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

Solves an optimization problem for coefficients in a differential equation using nested automatic differentiation. The DiffCache is configured with `levels = 3` to handle three levels of automatic differentiation.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools, Optim, Optimization
function foo(du, u, p, t)
    tmp = p[2]
    A = reshape(p[1], size(tmp.du))
    tmp = get_tmp(tmp, u)
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end

coeffs = -collect(0.1:0.1:0.4)
cache = DiffCache(zeros(2, 2), levels = 3)
prob = ODEProblem(foo, ones(2, 2), (0.0, 1.0), (coeffs, cache))
realsol = solve(prob, TRBDF2(), saveat = 0.0:0.1:10.0, reltol = 1e-8)

function objfun(x, prob, realsol, cache)
    prob = remake(prob, u0 = eltype(x).(prob.u0), p = (x, cache))
    sol = solve(prob, TRBDF2(), saveat = 0.0:0.1:10.0, reltol = 1e-8)

    ofv = 0.0
    if any((s.retcode != :Success for s in sol))
        ofv = 1e12
    else
        ofv = sum((sol .- realsol) .^ 2)
    end
    return ofv
end
fn(x, p) = objfun(x, p[1], p[2], p[3])
optfun = OptimizationFunction(fn, Optimization.AutoForwardDiff())
optprob = OptimizationProblem(optfun, zeros(length(coeffs)), (prob, realsol, cache))
solve(optprob, Newton())

```

--------------------------------

### Initialize DiffCache

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Constructs a DiffCache object for pre-allocated vectors compatible with ForwardDiff.jl. Specify the chunk size `N` to match ForwardDiff's chunk size or provide an array for nested differentiation levels.

```julia
DiffCache(u::AbstractArray, N::Int = ForwardDiff.pickchunksize(length(u)); levels::Int = 1)
DiffCache(u::AbstractArray, N::AbstractArray{<:Int})
```

--------------------------------

### ODE Integration with DiffCache (Corrected)

Source: https://github.com/sciml/preallocationtools.jl/blob/master/docs/src/index.md

Corrected ODE function using DiffCache to ensure the temporary array `tmp` is compatible with dual numbers during automatic differentiation. This version correctly handles the dual number cache.

```julia
using LinearAlgebra, OrdinaryDiffEq, PreallocationTools
function foo(du, u, (A, tmp), t)
    tmp = get_tmp(tmp, u)
    mul!(tmp, A, u)
    @. du = u + tmp
    nothing
end
chunk_size = 5
prob = ODEProblem(foo,
    ones(5, 5),
    (0.0, 1.0),
    (ones(5, 5), DiffCache(zeros(5, 5), chunk_size)))
solve(prob, TRBDF2(chunk_size = chunk_size))
```

--------------------------------

### LazyBufferCache Constructor

Source: https://github.com/sciml/preallocationtools.jl/blob/master/README.md

Constructs a LazyBufferCache. The function `f` maps `size_of_cache = f(size(u))`, which by default creates cache arrays of the same size. An optional initializer function can be supplied.

```julia
LazyBufferCache(f::F = identity)
```

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