`gradient`, `jacobian`, `hessian`

Extracts specific partial derivatives from a TPS

Syntax

grad = GTPSA.gradient(f [, include_params=bool])
GTPSA.gradient!(grad, f [, include_params=bool])

J = GTPSA.jacobian(F [, include_params=bool])
GTPSA.jacobian!(J, F [, include_params=bool])

Jt = GTPSA.jacobiant(F [, include_params=bool])
jacobiant!(Jt, F [, include_params=bool])

H = GTPSA.hessian(f [, include_params=bool])
GTPSA.hessian!(H, f [, include_params=bool])

Description

grad = GTPSA.gradient(f) extracts the gradient from the TPS f, defined as $\nabla f$

GTPSA.gradient!(grad, f) fills grad vector in-place with the gradient extracted from the TPS f

J = GTPSA.jacobian(F) extracts the Jacobian matrix from the array of TPSs F, defined as $J_{ij} = \frac{\partial F_i}{\partial x_j}$

GTPSA.jacobian!(J, F) fills the J matrix in-place with the Jacobian extracted from the array of TPSs F

Jt = GTPSA.jacobiant(F) extracts the transpose of the Jacobian matrix from the vector of TPSs F, with the Jacobian defined as $J_{ij} = \frac{\partial F_i}{\partial x_j}$

GTPSA.jacobiant!(Jt, F) fills the Jt matrix in-place with the transpose of the Jacobian extracted from the vector of TPSs F

H = GTPSA.hessian(f) extracts the Hessian matrix from the TPS f, defined as $H_{ij} = \frac{\partial^2 f}{\partial x_i\partial x_j}$

GTPSA.hessian!(H, f) fills the H matrix in-place with the Hessian extracted from the TPS f

Optional Argument

include_params can be set to true (default is false) to include the partial derivatives with respect to the parameters in any of the extraction functions above.

Examples

julia> d = Descriptor(2,10);
julia> Δx = @vars(d);
julia> f = Δx[1] + 2*Δx[2] + 3*Δx[1]^2 + 4*Δx[1]*Δx[2] + 5*Δx[2]^2;
julia> g = 5*Δx[1] + 4*Δx[2] + 3*Δx[1]^2 + 2*Δx[1]*Δx[2] + Δx[2]^2;
julia> grad = GTPSA.gradient(f)2-element Vector{Float64}:
 1.0
 2.0
julia> J = GTPSA.jacobian([f, g])2×2 Matrix{Float64}:
 1.0  2.0
 5.0  4.0
julia> H = GTPSA.hessian(f)2×2 Matrix{Float64}:
 6.0   4.0
 4.0  10.0

Documentation

GTPSA.gradient — Function

GTPSA.gradient(t::TPS; include_params=false)

Extracts the first-order partial derivatives (evaluated at 0) from the TPS. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the first-order monomial coefficients already in the TPS.

Input

t – TPS/ComplexTPS64 to extract the gradient from
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false

Output

grad – Gradient of the TPS

source

GTPSA.gradient! — Function

GTPSA.gradient!(result, t::TPS; include_params=false, unsafe_inbounds=false)

Extracts the first-order partial derivatives (evaluated at 0) from the TPS and fills the result vector in-place. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the first-order monomial coefficients already in the TPS.

Input

t – TPS/ComplexTPS64 to extract the gradient from
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false
unsafe_inbounds – (Optional) Flag which, if false, ignores checking the size of result. The size of result is used to index the TPS. Default is false

Output

result – Vector to fill with the gradient of the TPS, must be 1-based indexing

source

GTPSA.jacobian — Function

GTPSA.jacobian(m::AbstractArray{<:TPS}; include_params=false)

Extracts the first-order partial derivatives (evaluated at 0) from the array of TPSs. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the first-order monomial coefficients already in the TPSs.

Input

m – Array of TPSs to extract the Jacobian from, must be 1-based indexing
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false

Output

J – Jacobian of m

source

GTPSA.jacobian! — Function

GTPSA.jacobian!(result, m::AbstractArray{<:TPS}; include_params=false, unsafe_inbounds=false)

Extracts the first-order partial derivatives (evaluated at 0) from the array of TPSs. and fills the result matrix in-place. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the first-order monomial coefficients already in the TPSs.

Input

m – Array of TPSs to extract the Jacobian from, must be 1-based indexing
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false
unsafe_inbounds – (Optional) Flag which, if false, ignores checking the size of result. The size of result is used to index the TPS. Default is false

Output

result – Matrix to fill with the Jacobian of m, must be 1-based indexing

source

GTPSA.jacobiant — Function

GTPSA.jacobiant(m::AbstractArray{<:TPS}; include_params=false) where {N,P,I}

Extracts the first-order partial derivatives (evaluated at 0) from the array of TPSs, as the transpose of the Jacobian. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the first-order monomial coefficients already in the TPSs.

Input

m – Array of TPSs to extract the Jacobian from, must be 1-based indexing
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false

Output

Jt – Transpose of the Jacobian of m

source

GTPSA.jacobiant! — Function

GTPSA.jacobiant!(result, m::AbstractArray{<:TPS}; include_params=false, unsafe_inbounds=false)

Input

m – Vector of TPSs to extract the Jacobian from, must be 1-based indexing
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false
unsafe_inbounds – (Optional) Flag which, if false, ignores checking the size of result. The size of result is used to index the TPS. Default is false

Output

result – Matrix to fill with the transpose of the Jacobian of m, must be 1-based indexing

source

GTPSA.hessian — Function

GTPSA.hessian(t::TPS; include_params=false)

Extracts the second-order partial derivatives (evaluated at 0) from the TPS. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the second-order monomial coefficients already in the TPS.

Input

t – TPS/ComplexTPS64 to extract the Hessian from
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false

Output

H – Hessian of the TPS

source

GTPSA.hessian! — Function

GTPSA.hessian!(result, t::TPS; include_params=false, tmp_mono::Union{Nothing,Vector{UInt8}}=nothing, unsafe_fast=false, unsafe_inbounds=false)

Extracts the second-order partial derivatives (evaluated at 0) from the TPS and fills the result matrix in-place. The partial derivatives wrt the parameters will also be extracted when the include_params flag is set to true. Note that this function is not calculating anything - just extracting the second-order monomial coefficients already in the TPS.

Input

t – TPS/ComplexTPS64 to extract the Hessian from
include_params – (Optional) Extract partial derivatives wrt parameters. Default is false
tmp_mono – (Optional) Vector{UInt8} to store the monomial, when different orders of truncation are used
unsafe_fast – (Optional) Flag to specify that "fast" indexing should be used without checking. This will give incorrect results if any variable has a TO < 2. Default is false.
unsafe_inbounds – (Optional) Flag which, if false, ignores checking the size of result. The size of result is used to index the TPS. Default is false

Output

result – Matrix to fill with the Hessian of the TPS, must be 1-based indexing

source

gradient, jacobian, hessian

Syntax

Description

Optional Argument

Examples

Documentation

`gradient`, `jacobian`, `hessian`