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
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.
pdPolynomial(p0, v0, delta.t = 0.01, steps = 100)
a (d, d)-dimensional HPD matrix specifying the starting point of the polynomial curve.
a (d, d, k)-dimensional array corresponding to a sequence of (d,d)-dimensional Hermitian matrix-valued
covariant derivatives from order zero up to order k - 1 at the starting point
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,
number of incrementing steps in the numerical integration procedure, defaults to
(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
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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