本研究主要探討二維頻率相依光子晶體的能帶結構與保結構之Arnoldi 方法。有關光子晶體之能帶結構的研究多為非頻頻率相依之材料，其所相關之頻譜問題多為線性頻譜問題，輔以離散化與的傳統特徵值演算法，如IRA, Jacobi-Davidson 等，便能求得該晶體結構與材料所對應之能帶結構的近似解。但是在頻率相依之材料中，因為其材料之介電係數與頻率為相關的，其所對應之特徵值問題多為非線性特徵值問題，故一般針對線性特徵值問題的演算法難以應用在此類問題上。C. Engstrom and M. Richter 建議了利用色散關係將原本的非線性頻譜問題轉化為線性的頻譜問題，其所對應之頻譜參數也從頻率轉變為波向量之波長。這個轉換所對應之離散型特徵值問題可以被寫為一組陀螺二次特徵值問題，其特徵值可經由保結構演算法SHIRA 來求得近似解，優於使用傳統的特徵值演算法，如IRA。但此演算法將面臨其不變子空間求解之困難。
　　本文主要在於介紹針對上述不變子空間的提取方法，與上述陀螺二次特徵值問題經由轉換後而得到之回文二次特徵值問題求解。後者將在計算上優於前者，並且無上述不變子空間求解的困難。

This study focused on the two-dimensional frequency-dependent photonic crystal band structure and structured-preserving Arnoldi method. Research related most to band structure of the frequency-independent material, it's mostly correspond to a linear spectral problem. Combined with the traditional algorithms for discrete eigenvalue problem, such as IRA, Jacobi-Davidson etal., It will be able to obtain the approximate solution of band structure corresponding to crystal structure and the material. However, on the frequency-dependent material, since the dielectric constant of the material and the frequency
is related, it corresponds to the eigenvalue problem mostly nonlinear eigenvalue problem. Therefore, the general algorithm for linear eigenvalue problem is dicult to apply in such problems. C. Engstrom and M. Richter
recommends using dispersion relation to transform the non-linear problem into a linear spectrum spectral problem. The spectral parameter also transformed from the frequency to the wavelength of the wave vector. Discrete
eigenvalue problem corresponding to the transformed spectral problem can be written as a gyroscopic quadratic eigenvalue problem. It can be solved approximately by the structured-preserving algorithm SHIRA, which will batter than the general eigensolver, such as IRA. However, this algorithm will encounter diculties of solving its invariant subspace.
This paper is focus on extracting invariant subspace described above, and the gyroscopic quadratic eigenvalue problem can be obtained to a palindromic eigenvalue problem. The latter will be better than the former on the computation, and without diculty to solve the invariant subspace.

1 Introduction 1
2 Electromagnetic Wave Model 3
2.1 Bloch solutions and spectral problems. . . . . . . . . . . . . . . . . . 3
2.2 Frequency-dependent material . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Gyroscopic Quadratic Eigenproblem 7
3.1 Nonstructured Linearization . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Structured Linearization I : SHH matrix pair . . . . . . . . . . . . . . 8
3.3 Structured Linearization II : SH matrix pair . . . . . . . . . . . . . . 9
3.4 Transform to Standard Eigenvalue Problem . . . . . . . . . . . . . . 10
4 Arnoldi Method 12
4.1 Krylov Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Arnoldi Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Arnoldi Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 The Ritz Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Implicitly Restarted Arnoldi Method 17
5.1 QR-Step with Single Shift . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 QR-Step with Double Shift . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 The Implicitly Restarted Arnoldi Method . . . . . . . . . . . . . . . . 21
6 Skew-Hamiltonian Implicitly Restarted Arnoldi Method 23
6.1 Isotropic Arnoldi Process . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.2 Skew-Hamiltonian Implicitly Restarted Arnoldi Method . . . . . . . . 26
7 Shift-Invert Technique on SHH and SH pairs 27
7.1 Matrix-vector production of R2(0;W) and R4(0;W) . . . . . . . . 28
7.2 Matrix-vector production of R(0;W) . . . . . . . . . . . . . . . . . . 30
8 An Extraction Method for Invariant Subspace 32
8.1 Extracting invariant subspace from R2(;W) . . . . . . . . . . . . . 32
8.2 Extracting invariant subspace from R(0;W) . . . . . . . . . . . . . . 35
9 Numerical Result 37
10 Discussion and Conclusions 39

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