FiniteDifferenceMethod{G<:AbstractVector, C<:AbstractVector, E<:Function}

A finite difference method.

Fields

• grid::G: Multiples of the step size that the function will be evaluated at.
• q::Int: Order of derivative to estimate.
• coefs::C: Coefficients corresponding to the grid functions that the function evaluations will be weighted by.
• bound_estimator::Function: A function that takes in the function and the evaluation point and returns a bound on the magnitude of the length(grid)th derivative.

FiniteDifferenceMethod(
    grid::AbstractVector,
    q::Int;
    condition::Real=DEFAULT_CONDITION
)

Construct a finite difference method.

Arguments

• grid::Abstract: The grid. See FiniteDifferenceMethod.
• q::Int: Order of the derivative to estimate.
• condition::Real: Condition number. See DEFAULT_CONDITION.

Returns

• FiniteDifferenceMethod: Specified finite difference method.

function estimate_step(
    m::FiniteDifferenceMethod,
    f::Function,
    x::T;
    factor::Real=1,
    max_step::Real=0.1 * max(abs(x), one(x))
) where T<:AbstractFloat

Estimate the step size for a finite difference method m. Also estimates the error of the estimate of the derivative.

Arguments

• m::FiniteDifferenceMethod: Finite difference method to estimate the step size for.
• f::Function: Function to evaluate the derivative of.
• x::T: Point to estimate the derivative at.

Keywords

• factor::Real=1. Factor to amplify the estimated round-off error by. This can be used to force a more conservative step size.
• max_step::Real=0.1 * max(abs(x), one(x)): Maximum step size.

Returns

• Tuple{T, <:AbstractFloat}: Estimated step size and an estimate of the error of the finite difference estimate.

forward_fdm(
    p::Int,
    q::Int;
    adapt::Int=1,
    condition::Real=DEFAULT_CONDITION,
    geom::Bool=false
)

Contruct a finite difference method at a forward grid of p linearly spaced points.

Arguments

• p::Int: Number of grid points.
• q::Int: Order of the derivative to estimate.

Keywords

• adapt::Int=1: Use another finite difference method to estimate the magnitude of the pth order derivative, which is important for the step size computation. Recurse this procedure adapt times.
• condition::Real: Condition number. See DEFAULT_CONDITION.
• geom::Bool: Use geometrically spaced points instead of linearly spaced points.

Returns

• FiniteDifferenceMethod: The specified finite difference method.

central_fdm(
    p::Int,
    q::Int;
    adapt::Int=1,
    condition::Real=DEFAULT_CONDITION,
    geom::Bool=false
)

Contruct a finite difference method at a central grid of p linearly spaced points.

Arguments

• p::Int: Number of grid points.
• q::Int: Order of the derivative to estimate.

Keywords

• adapt::Int=1: Use another finite difference method to estimate the magnitude of the pth order derivative, which is important for the step size computation. Recurse this procedure adapt times.
• condition::Real: Condition number. See DEFAULT_CONDITION.
• geom::Bool: Use geometrically spaced points instead of linearly spaced points.

Returns

• FiniteDifferenceMethod: The specified finite difference method.

backward_fdm(
    p::Int,
    q::Int;
    adapt::Int=1,
    condition::Real=DEFAULT_CONDITION,
    geom::Bool=false
)

Contruct a finite difference method at a backward grid of p linearly spaced points.

Arguments

• p::Int: Number of grid points.
• q::Int: Order of the derivative to estimate.

Keywords

• adapt::Int=1: Use another finite difference method to estimate the magnitude of the pth order derivative, which is important for the step size computation. Recurse this procedure adapt times.
• condition::Real: Condition number. See DEFAULT_CONDITION.
• geom::Bool: Use geometrically spaced points instead of linearly spaced points.

Returns

• FiniteDifferenceMethod: The specified finite difference method.

extrapolate_fdm(
    m::FiniteDifferenceMethod,
    f::Function,
    x::T,
    h::Real=0.1 * max(abs(x), one(x));
    power=nothing,
    breaktol=Inf,
    kw_args...
) where T<:AbstractFloat

Use Richardson extrapolation to refine a finite difference method.

Takes further in keyword arguments for Richardson.extrapolate. This method automatically sets power = 2 if m is symmetric and power = 1. Moreover, it defaults breaktol = Inf.

Arguments

• m::FiniteDifferenceMethod: Finite difference method to estimate the step size for.
• f::Function: Function to evaluate the derivative of.
• x::T: Point to estimate the derivative at.
• h::Real=0.1 * max(abs(x), one(x)): Initial step size.

Returns

• Tuple{<:AbstractFloat, <:AbstractFloat}: Estimate of the derivative and error.

assert_approx_equal(x, y, ε_abs, ε_rel[, desc])

Assert that x is approximately equal to y.

Let eps_z = eps_abs / eps_rel. Call x and y small if abs(x) < eps_z and abs(y) < eps_z, and call x and y large otherwise. If this function returns True, then it is guaranteed that abs(x - y) < 2 eps_rel max(abs(x), abs(y)) if x and y are large, and abs(x - y) < 2 eps_abs if x and y are small.

Arguments

• x: First object to compare.
• y: Second object to compare.
• ε_abs: Absolute tolerance.
• ε_rel: Relative tolerance.
• desc: Description of the comparison. Omit or set to false to have no description.

FiniteDifferences.DEFAULT_CONDITION

The default condition number used when computing bounds. It provides amplification of the ∞-norm when passed to the function's derivatives.

grad(fdm, f, xs...)

Compute the gradient of f for any xs for which to_vec is defined.

jacobian(fdm, f, x...)

Approximate the Jacobian of f at x using fdm. Results will be returned as a Matrix{<:Real} of size(length(y_vec), length(x_vec)) where x_vec is the flattened version of x, and y_vec the flattened version of f(x...). Flattening performed by to_vec.

jvp(fdm, f, xẋs::Tuple{Any, Any}...)

Compute a Jacobian-vector product with any types of arguments for which to_vec is defined. Each 2-Tuple in xẋs contains the value x and its tangent ẋ.

j′vp(fdm, f, ȳ, x...)

Compute an adjoint with any types of arguments x for which to_vec is defined.

to_vec(x)

Transform x into a Vector, and return the vector, and a closure which inverts the transformation.