Pseudo Eigenvalue(Under Construction!)
Pseudo Eigenvalue
http://homepages.neiu.edu/~zzeng/pseudoeig.html
Complex Eigenvalue(Under Construction!)
Complex Eigenvalue
Cauchy eigenvalue interlacing theorem Gershgorin eigenvalue circle theorem
Spectral Projection
https://www.caam.rice.edu/~caam440/chapter1.pdf
https://en.wikipedia.org/wiki/Resolvent_formalism
(A - zI) pencil (A - zI)⁻¹ resolvent / spectra det(A - zI) characteristic polynomial
(zI - A)⁻¹ = z⁻¹(I + A/z + A²/z² + ...)
特徵值shift-inverse調到無限大,環積分之後變成特徵向量外積(特徵分解外積表示法) 用C圈選隔離特徵值,得到其特徵向量。 特徵值相異,X是向量;特徵值重根,X是正規正交矩陣。 -1 ∫ --- (A - zI)⁻¹ dz = XXᵀ = P spectral projection C 2πi
A - λⱼI Pᵢ = pro ------- Lagrange interpolation j≠i λᵢ - λⱼ
專著《Eigenvalues of Matrices》
大量特徵值演算法(Sakurai-Sugiura Projection)
《A projection method for generalized eigenvalue problems using numerical integration》
大量特徵值演算法(FEAST)
《A Density Matrix-based Algorithm for Solving Eigenvalue Problems》
《PFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers》
Spectral Projection那一坨積分 用積分演算法求得(Gaussian Quadrature?) 複平面,最大最小特徵值的圓,等距取樣。 自己設定半徑,看要圈多少特徵值。 這樣就得到XXᵀ了。
P = XXᵀ不做內積分解。 而是隨便亂猜X̃,乘在後面(XXᵀ)X̃。 如果猜得準,導致相消剩下X,猜中特徵矩陣。 拿(XXᵀ)X̃來做Rayleigh-Ritz Projection,就這樣。
Quadratic Eigenvalue(Under Construction!)
Quadratic Eigenvalue
https://en.wikipedia.org/wiki/Quadratic_eigenvalue_problem
Generalized Eigenvalue(Under Construction!)
Generalized Eigenvalue
Ax = λBx. if A - zB is singular then z is an eigenvalue.
matrix pencil: A - zB. regular iff det(A - zB) ≠ 0.
B⁻¹A or AB⁻¹ has same eigenvalue.
PₗAPᵣ = λPₗBPᵣx has same eigenvalue.
Weierstrass canonical form:
[ J-zI | 0 ] Pₗ(A-zB)Pᵣ = [------+------] [ 0 | I-zJ ] below J is nilpotent. diagonal are all 0.
Generalized Schur Decomposition(QZ Decomposition)
A = QSZᵀ B = QTZᵀ
Biorthogonal Decomposition
biorthogonal decomposition / biorthogonalization
VᵀAW = B where VᵀW = I
two-side Rayleigh Quotient. generalized eigenvalue. pencil.
VᵀAW ---- VᵀW
雙正交分解演算法(Unsymmetric Lanczos Iteration)
Lanczos Iteration的時間複雜度實在太銷魂,於是有人硬是用Lanczos Iteration處理一般矩陣。
可以得到三對角線矩陣。