This study sets up two new merit functions, which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems. For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less, where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector. 1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues. Simultaneously, the real and complex eigenvectors can be computed very accurately. A simpler approach to the nonlinear eigenvalue problems is proposed, which implements a normalization condition for the uniqueness of the eigenvector into the eigen-equation directly. The real eigenvalues can be computed by the fictitious time integration method (FTIM), which saves computational costs compared to the one-dimensional golden section search algorithm (1D GSSA). The simpler method is also combined with the Newton iteration method, which is convergent very fast. All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

We consider a general nonlinear eigenvalue problem [

which is an eigen-equation used to determine the eigen-pair

where

The nonlinear eigenvalue problem consists of finding vector

In the gyroscopic system

In the free vibration of a

where

In the engineering application, the mass matrix

In terms of the vibration mode

which leads to a nonlinear eigen-equation for

by inserting

A lot of applications and solutions to quadratic eigenvalue problems have been proposed, e.g., the homotopy perturbation technique [

This paper develops two simple approaches to solving nonlinear eigenvalue problems. The innovation points of this paper are as follows:

1. When solving the nonlinear eigenvalue problems, they can be transformed into minimization problems regardless of real and complex eigenvalues.

2. For solving the linear equations system on the right-hand side with a zero vector, this paper presents the variable transformation to create a new nonhomogeneous linear system and merit functions.

3. When solving real or complex nonlinear eigenvalue problems, the vector variable of merit functions can search linearly in a desired range of the curve or surface by using 1D and 2D golden section search algorithms.

4. A simpler method is combined with the Newton iteration method, which is convergent very fast to solve nonlinear eigenvalue problems.

The rest of the paper’s contents are organized as follows:

We call the set of all eigenvalues λ of

It is known that if

As noticed by Liu et al. [

In order to definitely obtain a nonzero vector _{0}-th component. We can normalize it to be

where _{0}-th column of the matrix

which is obtained by moving the _{0}-th column of the eigen-equation to the right-hand side as shown by the componential form:

In the _{ij} from _{ij}.

If λ is an eigenvalue and _{0}, then we can determine the correct eigenvalue of

where _{1}. Otherwise,

The motivation by transforming

The first method (FM) for solving the nonlinear eigenvalue problems is given as follows. (i) Select _{0}. (ii) Solve

To demonstrate the new idea in

We take a large interval with _{0} = 1, and plot

Applying the 1D GSSA to solve this problem, we record the number of iterations (NI) and the number of the computations of

With

We note that the Newton type methods are not suitable for solving the minimization problem in

For a received nonlinear eigenvalue problem, the initial interval

The complex eigenvalue is assumed to be

Correspondingly, we take

Inserting

Upon letting

To determine the complex eigenvalue, instead of

The first method (FM) seeks the complex eigenvalue by (i) selecting _{0}; (ii) for each

Both the convergence criteria of the 1D and 2D GSSA are fixed to be

When

whose complex eigenvalues are

We take _{2} over the plane

If

We can prove the following result.

Combining

On both sides multiplied by

we have

Hence, we can derive

which leads to

where

On both sides multiplied by

we have

Hence, we can derive

If the coefficient matrix

In general, the GSSA requires many iterations and the evaluations of merit function to solve the minimization problem. The fictitious time integration method (FTIM) was first coined by Liu et al. [

Let us define

which is an implicit function of

For

where

Starting from an initial guess

which makes a monotonically decreasing sequence of

The second method (SM) for determining the real eigenvalue of

We revisit the generalized eigenvalue problem given by

With

NI | ||||
---|---|---|---|---|

−0.187352893196976 | 14 | 0.001 | 290 | |

1.313278952662423 | 15 | 0.001 | 220 | |

5.537956370847892 | 15 | 0.001 | 190 | |

12.0896928530668 | 17 | 0.001 | 163 | |

21.24642471661986 | 10 | 0.001 | 120 |

where

Equating the real and imaginary parts of

which can be recast to that in

When

We revisit the standard eigenvalue problem given by

To compare with the Newton method, we supplement

which is a normalized condition. Letting

At the

Then the Newton iteration method is given by

The iteration is terminated if it converges with a given criterion

In

NI | |||
---|---|---|---|

−0.1873528931969764 | 0 | 11 | |

1.313278952662422 | 1.5 | 10 | |

5.537956370847892 | 5 | 8 | |

12.0896928530668 | 12 | 8 | |

21.24642471661986 | 2 | >1000 |

We can observe that

To improve the performance, we can combine the Newton method to the SM, and solve

The iteration is terminated if the given convergence criterion

In

NI | |||
---|---|---|---|

−0.1873528931969766 | −0.2 | 7 | |

1.313278952662422 | 1.5 | 6 | |

5.537956370847891 | 5 | 7 | |

12.0896928530668 | 12 | 5 | |

21.24642471661986 | 22 | 7 |

Upon comparing

where

We apply the 1D GSSA to solve this problem. With

There are totally 24 eigenvalues as shown in

In

With

NI | ||||
---|---|---|---|---|

0.04080314176866105 | 1.5 × 10^{−16} |
12 | 0.001 | 95 |

0.7425972620277168 | 3.26 × 10^{−15} |
45 | 0.001 | 88 |

where

It describes a time-delay system.

We take _{1} = 1.5, _{2} = 1, _{3} = 0.5 and _{1} = 0.3, _{2} = 0.2, _{3} = 0.1, and there exist four complex eigenvalues. Through some manipulations, we find that

When we apply the 2D GSSA to solve this problem, with _{0} = 2, we obtain _{0} = 2, we obtain _{1} =

By taking the parameters of _{i}, _{i}, _{0} = 6, we obtain _{0} = 6; we can obtain

where

We can derive

By applying the 2D GSSA to solve this problem with _{1} =

where

We apply the SM to solve this problem with

NI | ||||
---|---|---|---|---|

−0.2328574586400297 | 30 | 0.001 | 29 | |

2.355885632295363 | 30 | 0.001 | 18.8 |

where

By taking _{11} = 1, _{12} = 1.3, _{21} = 0.1, _{22} = 1.1, _{31} = 1 and _{32} = 1.2.

We apply the SM to solve this problem with

NI | ||||
---|---|---|---|---|

0.6726432606416961 | 18 | 0.01 | 400 | |

0.9866442954682634 | 24 | 0.01 | 650 | |

1.068910201481607 | 15 | 0.0001 | 1500 |

In

NI | |||
---|---|---|---|

0.6726432606416951 | 0.6 | 8 | |

0.9866442954682592 | 0.1 | 10 | |

1.068910201481607 | 1.06 | 11 |

We can observe that

In

NI | |||
---|---|---|---|

0.6726432606416943 | 0.6 | 6 | |

0.9866442954682596 | 0.1 | 6 | |

1.068910201481607 | 1.06 | 8 |

Upon comparing

We take _{0} = 1 with the first mode being shown in

Starting from

Starting from

Starting from

Starting from _{0} = 1, and the fifth mode is shown in

This study proposes two simple approaches to solve the nonlinear eigenvalue problems, which directly implement a normalization condition for the uniqueness of the eigenvector into the eigen-equation. When the eigen-parameter runs in a desired range, the curve or surface for real and complex eigenvalues reveals local minimums of the constructed merit functions. In the merit function, the vector variable is solved from the nonhomogeneous linear system, which is available by reducing the eigen-equation by one dimension less and moving the normalized component to the right side. It is possible to quickly obtain the real and complex eigenvalues using 1D and 2D golden section search algorithms to solve the resultant minimization problems. The second method is simpler than the first by inserting the normalization condition into the eigen-equation. From the resulting nonhomogeneous linear system, the fictitious time integration method (FTIM) computes the real eigenvalues faster, which saves computation costs compared to the GSSA. The combination of the simpler method with the Newton iteration outperforms the original Newton method. Combining the simpler method to the Newton iteration without using the extra parameters is also better than the FTIM. It can obtain highly precise eigenvalues and eigenvectors very fast.

The authors would like to thank the National Science and Technology Council, Taiwan, for their financial support (Grant Number NSTC 111-2221-E-019-048).

The corresponding authors would like to thank the National Science and Technology Council, Taiwan for their financial support (Grant Number NSTC 111-2221-E-019-048).

The authors confirm contribution to the paper as follows: study conception and design: Chein-Shan Liu; data collection: Jian-Hung Shen; analysis and interpretation of results: Yung-Wei Chen; draft manuscript preparation: Chung-Lun Kuo and all authors reviewed the results and all approved the final version of the manuscript.

Not applicable.

The authors declare that they have no conflicts of interest to report regarding the present study.

In this appendix, we demonstrate the computer code to obtain the coefficient matrix [_{ij}] from [_{ij}] by

If