Hessian tensor of the isometric logistic map

Slice \(i\) is $$ \mathbf{H}_{\psi_i} = p_i\bigl(\boldsymbol{q}_i\boldsymbol{q}_i^{\mathsf{T}} -\mathbf{C}_V\bigr), $$ where \(\bar{\boldsymbol{v}}=\mathbf{V}^{\mathsf{T}}\boldsymbol{p}\), \(\boldsymbol{q}_i=\mathbf{V}_{i\cdot}-\bar{\boldsymbol{v}}\) and \(\mathbf{C}_V=\mathbf{V}^{\mathsf{T}}\mathbf{S}(\boldsymbol{p}) \mathbf{V}\). Array shape \((J-1)\times(J-1)\times J\).

Usage

hessian_isometric_logistic(z, V = NULL)

Arguments

z

Numeric vector \(\boldsymbol{z}\in\mathbb{R}^{J-1}\).

V

Optional ILR basis; see isometric_logistic().

Value

Numeric array used in the constrained Hessian chain rule.

Examples

hessian_isometric_logistic(c(0.2, -0.5))
#> , , 1
#> 
#>            [,1]         [,2]
#> [1,] 0.05085575  0.105113732
#> [2,] 0.10511373 -0.009675559
#> 
#> , , 2
#> 
#>            [,1]         [,2]
#> [1,]  0.0713137 -0.106489954
#> [2,] -0.1064900 -0.007291872
#> 
#> , , 3
#> 
#>              [,1]        [,2]
#> [1,] -0.122169445 0.001376222
#> [2,]  0.001376222 0.016967431
#>