DEigen.jl

This package computes derivatives of arbitrary order for a parametric eigenvalue problem with respect to a single scalar variable.

Description

Consider a parametric square matrix $A(x)$ that depends on a single scalar variable $x$ and its associated eigenvalue problem:

$$A(x)v(x) = \lambda(x)v(x).$$

At a given point $x_0$, if all the derivatives of $A$ to the $n$-th order are known (i.e., $[A^{(k)}(x_0) \mid k=1\ldots K]$), then the derivatives of the eigenvalue $[\lambda^{(k)}(x_0) \mid k=1\ldots K]$ and the eigenvector $[v^{(k)}(x_0) \mid k=1\ldots K]$ up to $K$ can be computed.

Installation

The package can be installed using Julia's REPL

julia> import Pkg
julia> Pkg.add(url="https://github.com/ykkan/DEigen.jl.git")

or with Pkg mode (hitting ] in the command prompt)

pkg> add https://github.com/ykkan/DEigen.jl.git

Usage

Derivatives of all eigenvalues and eigenvectors

Assume that we want to evaluate the derivatives of all eigenvalues and all eigenvalues for the matrix below at $x=1$ to the 2nd order:

$$A(x) = \begin{bmatrix} x & 1 \\\ 1 & 2 \end{bmatrix}$$

using DEigen
# A0 = A(1),  A1 = A'(1), A2 = A''(1)
A0 = [1.0 1.0;
      1.0 2.0]

A1 = [1.0 0.0;
      0.0 0.0]

A2 = [0.0 0.0;
      0.0 0.0]

values_list, vectors_list = deigen(A0, A1, A2)

#=
DEigen{Float64, 2}
values_list:
2×3 Matrix{Float64}:
 0.381966  0.723607  -0.178885
 2.61803   0.276393   0.178885

vectors_list:
2×2×3 Array{Float64, 3}:
[:, :, 1] =
 -0.850651  0.525731
  0.525731  0.850651

[:, :, 2] =
 0.105146   0.17013
 0.17013   -0.105146

[:, :, 3] =
 -0.0259936   0.110111
 -0.0420585  -0.0680521
=#

The deigen function return an object of type DEigen{Float64, 2}. This obeject can be unpacked to into a tuple of two variables values_list and vectors_list. The variable values_list stores all $K$ derivatives of all $M$ eigenvalues in the way in a $M\times(K+1)$ array

$$\begin{bmatrix} \lambda^{(0)}_{1} & \lambda^{(1)}_{1} & \cdots & \lambda^{(K)}_{1}\\\ \lambda^{(0)}_{2} & \lambda^{(1)}_{2} & \cdots & \lambda^{(K)}_{2} \\\ \vdots & \vdots & \ddots & \vdots \\\ \lambda^{(0)}_{M} & \lambda^{(1)}_{M} & \cdots & \lambda^{(K)}_{M} \end{bmatrix}$$

and the variable vectors_list stores all $K$ derivatives of all $M$ eigenvectors in a $M\times M\times (K+1)$ array

$$\overset{ \mathrm{vectors\_list[:, :, 1]} }{ \begin{bmatrix} \vert & \vert & & \vert \\\ v^{(0)}_1 & v^{(0)}_2 & \cdots & v^{(0)}_M \\\ \vert & \vert & & \vert \end{bmatrix} },\, \overset{ \mathrm{vectors\_list[:, :, 2]} }{ \begin{bmatrix} \vert & \vert & & \vert \\\ v^{(1)}_1 & v^{(1)}_2 & \cdots & v^{(1)}_M \\\ \vert & \vert & & \vert \end{bmatrix} } , \ldots ,\, \overset{ \mathrm{vectors\_list[:, :, K+1]} }{ \begin{bmatrix} \vert & \vert & & \vert \\\ v^{(K)}_1 & v^{(K)}_2 & \cdots & v^{(K)}_M \\\ \vert & \vert & & \vert \end{bmatrix}. }$$

Derivatives of selected eigenvalues and eigenvectors

One can also choose to compute the derivatives of only selected eigenvalues and eigenvectors.

using DEigen
# A0 = A(1),  A1 = A'(1), A2 = A''(1)
A0 = [1.0 1.0;
      1.0 2.0]

A1 = [1.0 0.0;
      0.0 0.0]

A2 = [0.0 0.0;
      0.0 0.0]

values0, vectors0 = eigen(A0)

# the indicies for the selected eigenvalues and eigenvectors
selected_ind = 1

values_list, vectors_list = deigen(values0[selected_ind], vectors0[:,selected_ind], A0, A1, A2)

#=
DEigen{Float64, 2}
values_list:
1×3 Matrix{Float64}:
 0.381966  0.723607  -0.178885

vectors_list:
2×1×3 Array{Float64, 3}:
[:, :, 1] =
 -0.8506508083520399
  0.5257311121191335

[:, :, 2] =
 0.10514622242382668
 0.17013016167040795

[:, :, 3] =
 -0.025993575698632487
 -0.042058488969530655
=#

In this case, we select $P$ indices $[i_1,i_2,\ldots,i_P] \subseteq [1,2,\ldots,M]$, and compute the derivatives of the eigenvalues and eigenvectors corresponding to these indices. The variable values_list stores all $K$ derivatives of selected $P$ eigenvalues in the way in a $P\times(K+1)$ array

$$\begin{bmatrix} \lambda^{(0)}_{i_1} & \lambda^{(1)}_{i_1} & \cdots & \lambda^{(K)}_{i_1}\\\ \lambda^{(0)}_{i_2} & \lambda^{(1)}_{i_2} & \cdots & \lambda^{(K)}_{i_2} \\\ \vdots & \vdots & \ddots & \vdots \\\ \lambda^{(0)}_{i_P} & \lambda^{(1)}_{i_P} & \cdots & \lambda^{(K)}_{i_P} \end{bmatrix}$$

and the variable vectors_list stores all $K$ derivatives of selected $P$ eigenvectors in a $M\times P\times (K+1)$ array

$$\overset{ \mathrm{vectors\_list[:, :, 1]} }{ \begin{bmatrix} \vert & \vert & & \vert \\\ v^{(0)}_{i_1} & v^{(0)}_{i_2} & \cdots & v^{(0)}_{i_P} \\\ \vert & \vert & & \vert \end{bmatrix} },\, \overset{ \mathrm{vectors\_list[:, :, 2]} }{ \begin{bmatrix} \vert & \vert & & \vert \\\ v^{(1)}_{i_1} & v^{(1)}_{i_2} & \cdots & v^{(1)}_{i_P} \\\ \vert & \vert & & \vert \end{bmatrix} } , \ldots ,\, \overset{ \mathrm{vectors\_list[:, :, K+1]} }{ \begin{bmatrix} \vert & \vert & & \vert \\\ v^{(K)}_{i_1} & v^{(K)}_{i_2} & \cdots & v^{(K)}_{i_P} \\\ \vert & \vert & & \vert \end{bmatrix}. }$$

Reference

T. Mach and M. A. Freitag, Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion, preprint (2023) (https://arxiv.org/pdf/2302.03661)