Continuation Method for Multiparameter Eigenvalue Problem

Abstract

Differential equations often arise in the mathematical formulation of problems in science and engineering. When solving certain boundary value problems in partial differential equations using the separation of variables method, one often encounters multiparameter eigenvalue problems. There has been limited numerical research on multiparameter eigenvalue problems, making further study an urgent necessity. This paper focuses on obtaining numerical solutions to multiparameter eigenvalue problems through the continuation method. Finally, numerical results are presented to illustrate the effectiveness of the proposed method.

Introduction

The multiparameter eigenvalue problem (MEP) is a generalization of the classical one-parameter eigenvalue problem and arises naturally in many areas of mathematical physics. Applications include the analysis of homogeneous beams under load, vibrating membranes, and electrically conducting fluid flows between non-parallel planes. Such problems commonly appear when the method of separation of variables is applied to solve boundary value problems derived from partial differential equations. In matrix form, an MEP consists of a system of coupled eigenvalue equations involving multiple eigenvalues and corresponding eigenvectors. The study focuses on symmetric real matrices and particularly considers right-definite and diagonal right-definite multiparameter eigenvalue problems, where specific determinant conditions ensure the existence and uniqueness of solutions. These problems can also be transformed into an equivalent system of one-parameter eigenvalue problems using operator determinants and tensor products.

The primary objective of the work is to develop a numerical solution method for right-definite multiparameter eigenvalue problems using the Homotopy Continuation Method. Homotopy continuation is a powerful numerical technique widely used for solving nonlinear systems of equations. Instead of solving a difficult problem directly, the method constructs a continuous family of problems that gradually transforms a simple problem with a known solution into the original complex problem. A homotopy function H(x,t)H(x,t)H(x,t), where the parameter ttt varies from 0 to 1, is defined such that H(x,0)H(x,0)H(x,0) represents an easily solvable problem and H(x,1)H(x,1)H(x,1) corresponds to the original multiparameter eigenvalue problem. The solution is obtained by incrementally increasing the parameter ttt and using the solution from the previous step as the initial approximation for the next step.

The study extends homotopy continuation methods previously developed for standard, generalized, and two-parameter eigenvalue problems to right-definite multiparameter eigenvalue problems. A homotopy formulation is constructed by introducing auxiliary symmetric matrices whose associated eigenvalue problems possess distinct eigenvalues. When t=0t=0t=0, the homotopy reduces to these simpler auxiliary eigenvalue problems, while at t=1t=1t=1 it becomes the original multiparameter eigenvalue problem. Throughout the continuation process, the right-definite property is preserved, ensuring numerical stability and convergence.

The proposed algorithm discretizes the continuation parameter into equally spaced intervals between 0 and 1. First, the auxiliary eigenvalue problem at t=0t=0t=0 is solved to obtain initial eigenvalues and eigenvectors. These solutions serve as starting approximations for solving the homotopy equations at the next value of ttt. At each continuation step, Newton's method is applied to refine the solution. This predictor-corrector process is repeated until t=1t=1t=1, at which point the obtained solution corresponds to the original multiparameter eigenvalue problem. The gradual continuation from a simple system to the target system improves convergence and avoids many of the numerical difficulties associated with solving highly nonlinear eigenvalue problems directly.

To demonstrate the effectiveness of the proposed approach, the paper presents a three-parameter eigenvalue problem involving three coupled matrix equations. Appropriate auxiliary matrices are selected to construct the homotopy, and the continuation parameter is divided into six equally spaced values (t=0,0.2,0.4,0.6,0.8,t = 0, 0.2, 0.4, 0.6, 0.8,t=0,0.2,0.4,0.6,0.8, and 111). The auxiliary problems at t=0t=0t=0 are solved first to determine the initial eigenvalues and eigenvectors. These initial solutions are then successively updated using Newton's method as the continuation parameter increases until the final eigenvalues of the original problem are obtained at t=1t=1t=1. The numerical example illustrates the practicality of the homotopy continuation approach for solving multiparameter eigenvalue problems efficiently while maintaining solution accuracy and numerical stability.

Conclusion

Since the successive differences between the computed eigenvalues tend to zero, the method converges rapidly to the exact eigenvalue. Thus, it has been demonstrated that the method is convergent, and it can therefore be extended to solve three-parameter or higher-order eigenvalue problems. Another advantage of this approach is that it does not require an initial guess, as the eigenvalues corresponding to t=0 serve as the initial approximations. Furthermore, the method yields both eigenvalues and eigenvectors simultaneously. However, if the solution path passes through a singular point, switching between homotopy curves may occur. To prevent such switching in the case of two-parameter right-definite eigenvalue problems, Plestenjak [14] derived a bound ?that is independent of t. The present work does not discuss that procedure in detail. Instead, by appropriately selecting W_1, W_2 and W_3 at t=0, the singularity at t=0 is avoided.

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Copyright © 2026 Dr. Songita Boruah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.