Interface

Evaluation

Homotopy.evaluate — Function.

evaluate(H::AbstractPolynomialHomotopy, x, t)

Evaluate the homotopy H at x to time t, i.e. $H(x,t)$.

evaluate(H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Evaluate the homotopy H at x to time t using the precompuated values in cfg. Note that this is significantly faster than evaluate(H, x, t).

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Homotopy.evaluate! — Function.

evaluate!(u::Vector, H::AbstractPolynomialHomotopy, x, t)

Evaluate the homotopy H at x to time t, i.e. $H(x,t)$, and store the result in u.

evaluate!(u::AbstractVector, H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Evaluate the homotopy H at x to time t using the precompuated values in cfg and store the result in u.

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Differentiation

Homotopy.jacobian — Function.

jacobian(H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Compute the jacobian of H at x and t using the precomputed values in cfg. The jacobian is constructed w.r.t. x, i.e. it doesn't contain the partial derivatives w.r.t. t.

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Homotopy.jacobian! — Function.

jacobian!(u, H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Compute the jacobian of H at x and t using the precomputed values in cfg and store the result in u.

jacobian!(r::JacobianDiffResult, H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Compute $H(x, t)$ and the jacobian of H at x and t at once using the precomputated values in cfg and store thre result in r. This is faster than computing both values separetely.

Example

cfg = PolynomialHomotopyConfig(H)
r = JacobianDiffResult(cfg)
jacobian!(r, H, x, t, cfg)

value(r) == H(x, t)
jacobian(r) == jacobian(H, x, t, cfg)

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Homotopy.dt — Function.

dt(H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Compute the derivative of H w.r.t. $t$ at x and t using the precomputed values in cfg.

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Homotopy.dt! — Function.

dt!(u, H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Compute the derivative of H w.r.t. $t$ at x and t using the precomputed values in cfg and store the result in u.

dt!(r::DtDiffResult, H::AbstractPolynomialHomotopy, x, t, cfg::PolynomialHomotopyConfig)

Compute the derivative of H w.r.t. $t$ at x and t using the precomputed values in cfg and store the result in r. This is faster than computing both values separetely.

Example

cfg = PolynomialHomotopyConfig(H)
r = DtDiffResult(cfg)
dt!(r, H, x, t, cfg)

value(r) == H(x, t)
dt(r) == dt(H, x, t, cfg)

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Homogenization

Homotopy.homogenize — Function.

homogenize(H::AbstractPolynomialHomotopy)

Homogenize the homotopy H. This adds an additional variable. If H is already homogenized, this is the identity.

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Homotopy.dehomogenize — Function.

dehomogenize(H::AbstractPolynomialHomotopy)

Dehomogenize the homotopy H. This removes the first variable. If H is not homogenized, this is the identity.

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Homotopy.ishomogenized — Function.

ishomogenized(H::AbstractPolynomialHomotopy)

Check whether the homotopy H was homogenized.

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Homotopy.ishomogenous — Function.

ishomogenous(H::AbstractPolynomialHomotopy)

Check whether the homotopy H is homogenous. This does not imply that H was homogenized.

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Misc

Homotopy.nvariables — Function.

nvariables(H::AbstractPolynomialHomotopy)

The number of variables which H expects as input, i.e. to evaluate H(x,t) x has to be a vector of length nvariables(H).

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Homotopy.weylnorm — Function.

weylnorm(H::AbstractPolynomialHomotopy)

Creates a function with variable t that computes the Weyl norm (or Bombieri norm) of $H(x,t)$. See here for details about the Weyl norm.

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