Description Usage Arguments Value References See Also Examples

`pdPolynomial`

generates intrinsic polynomial curves in the manifold of HPD matrices
equipped with the affine-invariant Riemannian metric (see \insertCiteB09pdSpecEst[Chapter 6]
or \insertCitePFA05pdSpecEst) according to the numerical integration procedure in \insertCiteHFJ14pdSpecEst.
Given an initial starting point `p0`

(i.e., a HPD matrix) in the Riemannian manifold and covariant
derivatives up to order *k - 1* at `p0`

, `pdPolynomial`

approximates the uniquely existing
intrinsic polynomial curve of degree *k* passing through `p0`

with the given covariant derivatives up
to order *k - 1* and vanishing higher order covariant derivatives.

1 | ```
pdPolynomial(p0, v0, delta.t = 0.01, steps = 100)
``` |

`p0` |
a |

`v0` |
a |

`delta.t` |
a numeric value determining the incrementing step size in the numerical integration procedure.
A smaller step size results in a higher resolution and therefore a more accurate approximation of the polynomial curve,
defaults to |

`steps` |
number of incrementing steps in the numerical integration procedure, defaults to |

A `(d, d, length(steps))`

-dimensional array corresponding to a generated (approximate)
intrinsic polynomial curve in the space of *(d,d)*-dimensional HPD matrices of degree *k*
passing through `p0`

with the given covariant derivatives `v0`

up to order *k - 1*
and vanishing higher order covariant derivatives.

1 2 3 4 5 6 7 8 9 | ```
## First-order polynomial
p0 <- diag(3) ## HPD starting point
v0 <- array(H.coeff(rnorm(9), inverse = TRUE), dim = c(3, 3, 1)) ## zero-th order cov. derivative
P.poly <- pdPolynomial(p0, v0)
## First-order polynomials coincide with geodesic curves
P.geo <- sapply(seq(0, 1, length = 100), function(t) Expm(p0, t * Logm(p0, P.poly[, , 100])),
simplify = "array")
all.equal(P.poly, P.geo)
``` |

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