# 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 โ Type
PositiveNumbers <: AbstractManifold{โ}

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

Constructor

PositiveNumbers()

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

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ManifoldDiff.riemannian_Hessian โ Method
riemannian_Hessian(M::SymmetricPositiveDefinite, p, G, H, X)

The Riemannian Hessian can be computed as stated in Eq. (7.3) [Ngu23]. Let $\nabla f(p)$ denote the Euclidean gradient G, $\nabla^2 f(p)[X]$ the Euclidean Hessian H. Then the formula reads

$$$\operatorname{Hess}f(p)[X] = p\bigl(โ^2 f(p)[X]\bigr)p + X\bigl(โf(p)\bigr)p$$$
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Manifolds.PositiveArrays โ Method
PositiveArrays(nโ, nโ, ..., nแตข; parameter::Symbol=:type)

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

parameter: whether a type parameter should be used to store n. By default size is stored in a type parameter. Value can either be :field or :type.

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Manifolds.PositiveMatrices โ Method
PositiveMatrices(m::Integer, n::Integer; parameter::Symbol=:type)

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

parameter: whether a type parameter should be used to store n. By default size is stored in a type parameter. Value can either be :field or :type.

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ManifoldsBase.change_metric โ Method
change_metric(M::PositiveNumbers, E::EuclideanMetric, p, X)

Given a tangent vector $X โ T_p\mathcal M$ 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

$$$โจZ,Yโฉ = XY = \frac{c(Z)c(Y)}{p^2} = g_p(c(Y),c(Z))$$$

and hence $C(X) = pX$.

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ManifoldsBase.change_representer โ Method
change_representer(M::PositiveNumbers, E::EuclideanMetric, p, X)

Given a tangent vector $X โ T_p\mathcal M$ 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

$$$โจX,Yโฉ = XY = \frac{ZY}{p^2} = g_p(YZ)$$$

and hence $Z = p^2X$

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ManifoldsBase.check_vector โ Method
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.inner โ 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.parallel_transport_to โ Method
parallel_transport_to(M::PositiveNumbers, p, X, q)

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

$$$\mathcal P_{q\gets p}(X) = Xโ \frac{q}{p}.$$$
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