inv_gft_broyden: Inverse GFT by Broyden's Method

In GFT: Generalized Fisher Transformation of Correlation Matrices

Inverse GFT by Broyden's Method

Description

Reconstructs the correlation matrix C with \mathrm{vecl}(\log C) = z by Broyden's method applied to the residual F(x) = \log \mathrm{diag}(e^{A[x;z]}), as in Chen, Fei and Yu (2025). Reference implementation for benchmarking against inv_gft.

Usage

inv_gft_broyden(z, x0 = NULL, tol = 1e-13, maxit = 500, warm = 1,
                globalized = FALSE)

Arguments

z	numeric vector of length n(n-1)/2.
x0	optional starting value of length n; defaults to zero.
tol	convergence tolerance on \|\mathrm{diag}(e^A) - 1\|_\infty.
maxit	maximum number of iterations.
warm	number of initial fixed-point steps before the Jacobian is formed (ignored when globalized = TRUE).
globalized	if TRUE, replace the one-step initialization with the same log-domain fixed-point phase as GFT-FP+N (fixed-point steps until \max_i \ell_i \le \log 2), and only then form the Jacobian.

Details

The exact Jacobian is computed once (an O(n^4) Hessian), then updated by rank-one Sherman-Morrison updates of its inverse, with one eigendecomposition per iteration and no line search. Without globalization the method can diverge for large \|z\|; divergence is reported gracefully via converged = FALSE.

Value

An object of class "gft_inv"; see inv_gft for the components. On divergence the result has converged = FALSE and err = Inf.

References

Chen, H., Fei, Y. and Yu, J. (2025). Multivariate stochastic volatility models based on generalized Fisher transformation. Journal of Econometrics, 251, 106041.

See Also

inv_gft.

Examples

z <- gft(0.9^abs(outer(1:5, 1:5, "-")))
r <- inv_gft_broyden(z)
r$converged

GFT documentation built on Aug. 23, 2026, 5:10 p.m.