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.PositiveNumbersType
PositiveNumbers <: AbstractManifold{ℝ}

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

Constructor

PositiveNumbers()

Generate the -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.expMethod
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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Manifolds.change_metricMethod
change_metric(M::PositiveNumbers, E::EuclideanMetric, p, X)

Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function with respect to the EuclideanMetric g_E, this function changes the representer into the one with respect to the positivity metric of PositiveNumbers M.

For all $Z,Y$ we are looking for the function $c$ on the tangent space at $p$ such that

\[ ⟨Z,Y⟩ = XY = \frac{c(Z)c(Y)}{p^2} = g_p(c(Y),c(Z))\]

and hence $C(X) = pX$.

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Manifolds.change_representerMethod
change_representer(M::PositiveNumbers, E::EuclideanMetric, p, X)

Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function with respect to the EuclideanMetric g_E, this function changes the representer into the one with respect to the positivity metric representation of PositiveNumbers M.

For all tangent vectors $Y$ the result $Z$ has to fulfill

\[ ⟨X,Y⟩ = XY = \frac{ZY}{p^2} = g_p(YZ)\]

and hence $Z = p^2X$

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ManifoldsBase.check_vectorMethod
check_vector(M::PositiveNumbers, p, X; 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.innerMethod
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.vector_transport_toMethod
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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