# FiniteDifferences.jl: Finite Difference Methods

FiniteDifferences.jl estimates derivatives with finite differences.

See also the Python package FDM.

#### FiniteDiff.jl vs FiniteDifferences.jl

FiniteDiff.jl and FiniteDifferences.jl are similar libraries: both calculate approximate derivatives numerically. You should definately use one or the other, rather than the legacy Calculus.jl finite differencing, or reimplementing it yourself. At some point in the future they might merge, or one might depend on the other. Right now here are the differences:

- FiniteDifferences.jl supports basically any type, where as FiniteDiff.jl supports only array-ish types
- FiniteDifferences.jl supports higher order approximation and step size adaptation
- FiniteDiff.jl supports caching and in-place computation
- FiniteDiff.jl supports coloring vectors for efficient calculation of sparse Jacobians

#### FDM.jl

This package was formerly called FDM.jl. We recommend users of FDM.jl update to FiniteDifferences.jl.

#### Contents

## Scalar Derivatives

Compute the first derivative of `sin`

with a 5th order central method:

```
julia> central_fdm(5, 1)(sin, 1) - cos(1)
-2.4313884239290928e-14
```

Finite difference methods are optimised to minimise allocations:

```
julia> m = central_fdm(5, 1);
julia> @benchmark $m(sin, 1)
BenchmarkTools.Trial:
memory estimate: 0 bytes
allocs estimate: 0
--------------
minimum time: 486.621 ns (0.00% GC)
median time: 507.677 ns (0.00% GC)
mean time: 539.806 ns (0.00% GC)
maximum time: 1.352 μs (0.00% GC)
--------------
samples: 10000
evals/sample: 195
```

Compute the second derivative of `sin`

with a 5th order central method:

```
julia> central_fdm(5, 2)(sin, 1) - (-sin(1))
-8.767431225464861e-11
```

To obtain better accuracy, you can increase the order of the method:

```
julia> central_fdm(12, 2)(sin, 1) - (-sin(1))
5.240252676230739e-14
```

The functions `forward_fdm`

and `backward_fdm`

can be used to construct forward differences and backward differences respectively.

### Dealing with Singularities

The function `log(x)`

is only defined for `x > 0`

. If we try to use `central_fdm`

to estimate the derivative of `log`

near `x = 0`

, then we run into `DomainError`

s, because `central_fdm`

happens to evaluate `log`

at some `x < 0`

.

```
julia> central_fdm(5, 1)(log, 1e-3)
ERROR: DomainError with -0.02069596546590111
```

To deal with this situation, you have two options.

The first option is to use `forward_fdm`

, which only evaluates `log`

at inputs greater or equal to `x`

:

```
julia> forward_fdm(5, 1)(log, 1e-3) - 1000
-3.741856744454708e-7
```

This works fine, but the downside is that you're restricted to one-sided methods (`forward_fdm`

), which tend to perform worse than central methods (`central_fdm`

).

The second option is to limit the distance that the finite difference method is allowed to evaluate `log`

away from `x`

. Since `x = 1e-3`

, a reasonable value for this limit is `9e-4`

:

```
julia> central_fdm(5, 1, max_range=9e-4)(log, 1e-3) - 1000
-4.027924660476856e-10
```

Another commonly encountered example is `logdet`

, which is only defined for positive-definite matrices. Here you can use a forward method in combination with a positive-definite deviation from `x`

:

```
julia> x = diagm([1.0, 2.0, 3.0]); v = Matrix(1.0I, 3, 3);
julia> forward_fdm(5, 1)(ε -> logdet(x .+ ε .* v), 0) - sum(1 ./ diag(x))
-4.222400207254395e-12
```

A range-limited central method is also possible:

```
julia> central_fdm(5, 1, max_range=9e-1)(ε -> logdet(x .+ ε .* v), 0) - sum(1 ./ diag(x))
-1.283417816466681e-13
```

### Dealing with Numerical Noise

It could be the case that the function `f`

you'd like compute the derivative of suffers from numerical noise. For example, `f(x)`

could be computed through some iterative procedure with some error tolerance `ε`

. In such cases, finite difference estimates can fail catastrophically. To illustrate this, consider `sin_noisy(x) = sin(x) * (1 + 1e-6 * randn())`

. Then

```
julia> central_fdm(5, 1)(sin_noisy, 1) - cos(1)
-0.027016678790599657
```

which is a terrible performance. To deal with this, you can set the keyword argument `factor`

, which specifies the level of numerical noise on the function evaluations relative to the machine epsilon. In this example, the relative error on the function evaluations is `2e-6`

(`1e-6 * randn()`

roughly produces a number in `[-2e-6, 2e-6]`

) and the machine epsilon is `eps(Float64) ≈ 2.22e-16`

, so `factor = 2e-6 / 2e-16 = 1e10`

should be appropriate:

```
julia> central_fdm(5, 1; factor=1e10)(sin_noisy, 1) - cos(1)
-1.9243663490486895e-6
```

As a rule of thumb, if you're dealing with numerical noise and `Float64`

s, `factor = 1e6`

is not a bad first attempt.

### Richardson Extrapolation

The finite difference methods defined in this package can be extrapolated using Richardson extrapolation. This can offer superior numerical accuracy: Richardson extrapolation attempts polynomial extrapolation of the finite difference estimate as a function of the step size until a convergence criterion is reached.

```
julia> extrapolate_fdm(central_fdm(2, 1), sin, 1)[1] - cos(1)
1.6653345369377348e-14
```

Similarly, you can estimate higher order derivatives:

```
julia> extrapolate_fdm(central_fdm(5, 4), sin, 1)[1] - sin(1)
-1.626274487942503e-5
```

In this case, the accuracy can be improved by making the contraction rate closer to `1`

:

```
julia> extrapolate_fdm(central_fdm(5, 4), sin, 1, contract=0.8)[1] - sin(1)
2.0725743343774639e-10
```

This performs similarly to a `10`

th order central method:

```
julia> central_fdm(10, 4)(sin, 1) - sin(1)
-1.0301381969668455e-10
```

If you really want, you can even extrapolate the `10`

th order central method, but that provides no further gains:

```
julia> extrapolate_fdm(central_fdm(10, 4), sin, 1, contract=0.8)[1] - sin(1)
5.673617131662922e-10
```

In this case, the central method can be pushed to a high order to obtain improved accuracy:

```
julia> central_fdm(20, 4)(sin, 1) - sin(1)
-5.2513549064769904e-14
```

### A Finite Difference Method on a Custom Grid

```
julia> method = FiniteDifferenceMethod([-2, 0, 5], 1)
FiniteDifferenceMethod:
order of method: 3
order of derivative: 1
grid: [-2, 0, 5]
coefficients: [-0.35714285714285715, 0.3, 0.05714285714285714]
julia> method(sin, 1) - cos(1)
-3.701483564100272e-13
```

## Multivariate Derivatives

Consider a quadratic function:

```
julia> a = randn(3, 3); a = a * a'
3×3 Matrix{Float64}:
4.31995 -2.80614 0.0829128
-2.80614 3.91982 0.764388
0.0829128 0.764388 1.18055
julia> f(x) = 0.5 * x' * a * x
```

Compute the gradient:

```
julia> x = randn(3)
3-element Vector{Float64}:
-0.18563161988700727
-0.4659836395595666
2.304584409826511
julia> grad(central_fdm(5, 1), f, x)[1] - a * x
3-element Vector{Float64}:
4.1744385725905886e-14
-6.611378111642807e-14
-8.615330671091215e-14
```

Compute the Jacobian:

```
julia> jacobian(central_fdm(5, 1), f, x)[1] - (a * x)'
1×3 Matrix{Float64}:
4.17444e-14 -6.61138e-14 -8.61533e-14
```

The Jacobian can also be computed for non-scalar functions:

```
julia> a = randn(3, 3)
3×3 Matrix{Float64}:
0.844846 1.04772 1.0173
-0.867721 0.154146 -0.938077
1.34078 -0.630105 -1.13287
julia> f(x) = a * x
julia> jacobian(central_fdm(5, 1), f, x)[1] - a
3×3 Matrix{Float64}:
2.91989e-14 1.77636e-15 4.996e-14
-5.55112e-15 -7.63278e-15 2.4758e-14
4.66294e-15 -2.05391e-14 -1.04361e-14
```

To compute Jacobian–vector products, use `jvp`

and `j′vp`

:

```
julia> v = randn(3)
3-element Array{Float64,1}:
-1.290782164377614
-0.37701592844250903
-1.4288108966380777
julia> jvp(central_fdm(5, 1), f, (x, v)) - a * v
3-element Vector{Float64}:
-7.993605777301127e-15
-8.881784197001252e-16
-3.22519788653608e-14
julia> j′vp(central_fdm(5, 1), f, x, v)[1] - a'x
3-element Vector{Float64}:
-2.1316282072803006e-14
2.4646951146678475e-14
6.661338147750939e-15
```