Positive Numbers

The manifold PositiveNumbers represents positive numbers with hyperbolic geometry. Additionally, there are also short forms for its corresponding PowerManifolds, i.e. PositiveVectors, PositiveMatrices, and PositiveArrays.

Manifolds.PositiveNumbers — Type

PositiveNumbers <: Manifold{ℝ}

The hyperbolic manifold of positive numbers $H^1$ is a the hyperbolic manifold represented by just positive numbers.

Constructor

PositiveNumbers()

Generate the ℝ-valued hyperbolic model represented by positive positive numbers. To use this with arrays (1-element arrays), please use SymmetricPositiveDefinite(1).

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Base.exp — Method

exp(M::PositiveNumbers, p, X)

Compute the exponential map on the PositiveNumbers M.

\exp_p X = p\operatorname{exp}(X/p).

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Base.log — Method

log(M::PositiveNumbers, p, q)

Compute the logarithmic map on the PositiveNumbers M.

\log_p q = p\log\frac{q}{p}.

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Manifolds.PositiveArrays — Method

PositiveArrays(n₁,n₂,...,nᵢ)

Generate the manifold of i-dimensional arrays with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.

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Manifolds.PositiveMatrices — Method

PositiveMatrices(m,n)

Generate the manifold of matrices with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.

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Manifolds.PositiveVectors — Method

PositiveVectors(n)

Generate the manifold of vectors with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.

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ManifoldsBase.check_manifold_point — Method

check_manifold_point(M::PositiveNumbers, p)

Check whether p is a point on the PositiveNumbers M, i.e. $p>0$ .

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ManifoldsBase.check_tangent_vector — Method

check_tangent_vector(M::PositiveNumbers, p, X; check_base_point, kwargs...)

Check whether X is a tangent vector in the tangent space of p on the PositiveNumbers M. For the real-valued case represented by positive numbers, all X are valid, since the tangent space is the whole real line. For the complex-valued case X [...]

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ManifoldsBase.distance — Method

distance(M::PositiveNumbers, p, q)

Compute the distance on the PositiveNumbers M, which is

d(p,q) = \Bigl\lvert \log \frac{p}{q} \Bigr\rvert = \lvert \log p - \log q\rvert.

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ManifoldsBase.injectivity_radius — Method

injectivity_radius(M::PositiveNumbers[, p])

Return the injectivity radius on the PositiveNumbers M, i.e. $\infty$ .

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ManifoldsBase.inner — Method

inner(M::PositiveNumbers, p, X, Y)

Compute the inner product of the two tangent vectors X,Y from the tangent plane at p on the PositiveNumbers M, i.e.

g_p(X,Y) = \frac{XY}{p^2}.

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ManifoldsBase.manifold_dimension — Method

manifold_dimension(M::PositiveNumbers)

Return the dimension of the PositiveNumbers M, i.e. of the 1-dimensional hyperbolic space,

\dim(H^1) = 1

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ManifoldsBase.project — Method

project(M::PositiveNumbers, p, X)

Project a value X onto the tangent space of the point p on the PositiveNumbers M, which is just the identity.

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ManifoldsBase.vector_transport_to — Method

vector_transport_to(M::PositiveNumbers, p, X, q, ::ParallelTransport)

Compute the parallel transport of X from the tangent space at p to the tangent space at q on the PositiveNumbers M.

\mathcal P_{q\gets p}(X) = X\cdot\frac{q}{p}.

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