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 <: Manifold{ℝ}

The hyperbolic manifold of positive numbers H1H^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.

exppX=pexp(X/p).\exp_p X = p\operatorname{exp}(X/p).
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Base.logMethod
log(M::PositiveNumbers, p, q)

Compute the logarithmic map on the PositiveNumbers M.

logpq=plogqp.\log_p q = p\log\frac{q}{p}.
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ManifoldsBase.check_tangent_vectorMethod
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.distanceMethod
distance(M::PositiveNumbers, p, q)

Compute the distance on the PositiveNumbers M, which is

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

gp(X,Y)=XYp2.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.

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