Rank-one Modification of Symmetric Eigenproblem Zack 11/8/2013

Eigenproblem • 𝐵 = 𝑅𝐸𝑅 𝑈 • Q is an orthonormal matrix, 𝑅𝑅 𝑈 = 𝑅 𝑈 𝑅 = 𝐽 • D is diagonal matrix, 𝑒 1 ≤ 𝑒 2 ≤ ⋯ ≤ 𝑒 𝑜

Initial modification of eigenproblem 𝐵 = 𝐵 + 𝜏𝑣𝑣 𝑈 • • 𝐵 = 𝑅𝐸𝑅 𝑈 is known. • Compute eigenvalue decomposition of 𝐵 : 𝐵 = 𝑅 𝐸 𝑅

𝐵 = 𝐵 + 𝜏𝑣𝑣 𝑈 = 𝑅 𝐸 + 𝜏𝑨𝑨 𝑈 𝑅 𝑈 • • Define C = 𝐸 + 𝜏𝑨𝑨 𝑈 , problem comes to eigenvalue decomposition of matrix C

• The eigenvalues of the matrix C satisfy the equation: det 𝐸 + 𝜏𝑨𝑨 𝑈 − λ𝐽 = 0 • det 𝐸 + 𝜏𝑨𝑨 𝑈 − λ𝐽 = det 𝐸 − λ𝐽 det 𝐽 + 𝜏 𝐸 − λ𝐽 −1 𝑨𝑨 𝑈 𝑜 𝑜 2 𝜂 𝑗 = ( 𝑒 𝑗 − 𝜇 )(1 + σ 𝑒 𝑗 − 𝜇 ) 𝑗=1 𝑗=1

𝑜 2 𝜂 𝑗 1 + σ 𝑒 𝑗 − 𝜇 = 0 𝑗=1

𝑜 2 𝜂 𝑗 1 + σ 𝑒 𝑗 − 𝜇 = 0 𝑗=1 • Define 𝜇 = 𝑒 𝑗 + 𝜏𝑢 , 𝜀 𝑘 = (𝑒 𝑘 − 𝑒 𝑗 )/𝜏 𝑜 2 𝜂 𝑘 𝑥 𝑗 𝑢 = 1 + 𝜀 𝑘 − 𝑢 = 0 𝑢 ∈ (0, 𝜀 𝑗+1 ) 𝑘=1

Newton’s method would be an obvious choice for finding the solution. However, Newton’s method is based on a local linear approximation to the function. Since our functions are rational functions, it seems more natural to develop a method based on a local approximation via simple rational function f(t).

Step1: choose an approximate function , for example: 𝑢→0 𝑔 𝑢 = −∞ lim 𝑐 𝑑  𝑔 𝑢 = 𝑏 + 𝑢 + lim 𝑢→𝜀 𝑔 𝑢 = +∞ 𝜀−𝑢 Step2: iteration 1. Choose 𝑢 0 ∈ (0, δ) 2. solve 𝑏 0 ,𝑐 0 , 𝑑 0 , with 𝑔 𝑢 0 = 𝑥 𝑗 𝑢 0 , 𝑔′ 𝑢 0 = 𝑥 𝑗 ′ 𝑢 0 , 𝑔′′ 𝑢 0 = 𝑥 𝑗 ′′ 𝑢 0 … 3. solve 𝑔 𝑢 1 = 0

𝑐 𝑑 • 𝑔 𝑢 = 𝑏 + 𝜀−𝑢  Gragg’s method 𝑢 + Converges from any point in 0, 𝜀 with a cubic order of convergence. 2 𝜂 𝑗 𝑐 • 𝑔 𝑢 = 𝑏 − 𝜀−𝑢  fixed weight 1 method (FW1) 𝑢 + 2 𝑐 𝜂 𝑗+1 𝜀−𝑢  fixed weight 2 method (FW2) 𝑔 𝑢 = 𝑏 + 𝑢 + Converges from any point in 0, 𝜀 with quadratic order of convergence

• Divide 𝑥 𝑗 𝑢 into 2 part: 𝑗 𝑜 2 2 𝜂 𝑘 𝜂 𝑘 𝑥 𝑗 𝑢 = 1 + 𝜀 𝑘 − 𝑢 + 𝜀 𝑘 − 𝑢 𝑘=1 𝑘=𝑗+1 = 1 + 𝜔 𝑢 + 𝜚(𝑢)

𝑗 2 𝜂 𝑘 𝜔 𝑢 = 𝜀 𝑘 − 𝑢 𝑘=1 𝑜 2 𝜂 𝑘 𝜚 𝑢 = 𝜀 𝑘 − 𝑢 𝑘=𝑗+1 𝜀 1 < ⋯ < 𝜀 𝑗 = 0 < 𝜀 𝑗+1 < ⋯ < 𝜀 𝑜

• choose an approximate function for 𝜔 𝑢 ,𝜚 𝑢 𝜔 𝑢  𝑏 + 𝑐𝑢 −1 𝜚 𝑢  𝑑 + 𝑒(𝜀 − 𝑢) −1 This method is named “the middle way”. For this method, convergence cannot be guaranteed unless the starting point lies close enough to the root. In case of convergence, the order is quadratic.

“approaching from the left” (BNS1) 𝜔 𝑢  𝑏(𝑐 − 𝑢) −1 𝜚 𝑢  𝑑 + 𝑒(𝜀 − 𝑢) −1 Converge from any point in (0, 𝑢 ∗ ] and the order of convergence is quadratic. “approaching from the right” (BNS2) 𝜔 𝑢  𝑏 + 𝑐𝑢 −1 𝜚 𝑢  𝑑(𝑒 − 𝑢) −1 Converge from any point in [𝑢 ∗ ,𝜀) and the order of convergence is quadratic.

Starting points 𝑜 2 𝜂 𝑘 1 + 𝑘 − 𝑢 = 0 𝜀 𝑘=1 𝑜 2 2 2 𝜂 𝑘 1 − 𝜂 𝑗 𝜂 𝑗+1 𝑢 + 𝜀 𝑗+1 − 𝑢 + 𝑘 − 𝑢 = 0 𝜀 𝑘=1 𝑘≠𝑗,𝑗+1 𝑜 2 2 2 𝜂 𝑘 1 − 𝜂 𝑗 𝜂 𝑗+1 + + ≈ 0 𝑢 0 𝜀 𝑗+1 − 𝑢 0 𝜀 𝑘 − 𝜀 𝑗+1 𝑘=1 𝑘≠𝑗,𝑗+1

Calculating the eigenvector • Assume 𝑟 𝑗 is 𝑗 th eigenvector of matrix A 𝑟 𝑗 − 𝐵 𝑒 𝑗 𝑟 𝑗 = 0 • if 𝑨 = 𝑅 𝑈 𝑐 and 𝑦 𝑗 = 𝑅 𝑈 𝑟 𝑗 𝑒 𝑗 𝐽 + 𝜏𝑨𝑨 𝑈 𝑦 𝑗 = 0 𝐸 −

Calculating the eigenvector • Theorem. If A is invertable and 𝜍 ≠ 0 , the following statements are equivalent: 𝐵 + 𝜍𝑣𝑤 𝑈 𝑦 = 𝑐 And 1 𝑦 = 𝐵 −1 𝑐 − 𝜄𝐵 −1 𝑣 , 𝜈𝜄 = 𝑤 𝑈 𝐵 −1 𝑐 , where 𝜈 = ( 𝜍 + 𝑤 𝑈 𝐵 −1 𝑣) 𝑒 𝑗 𝐽) −1 𝑨 • Therefore, 𝑦 𝑗 = −𝜄(𝐸 −

Rank-one Modification of Symmetric Eigenproblem Zack 11/8/2013 - PowerPoint PPT Presentation

Rank-one Modification of Symmetric Eigenproblem Zack 11/8/2013 Eigenproblem = Q is an orthonormal matrix, = = D is diagonal matrix, 1 2

A divide-and-conquer algorithm for a symmetric eigenproblem Binh T. Nguyen Anh-Duc Luong-Thanh

Symmetric rank distance codes Kai-Uwe Schmidt Otto-von-Guericke University Magdeburg, Germany 1

On A Quadratic Eigenproblem Arising In The Analysis of Delay Equations Heike Fabender AG

2 3 4 5 8 9 MINNEAPOLIS MILWAUKEE MSA RANK #16 MSA RANK #39 CHICAGO MSA RANK #3

Three notions of tropical rank for symmetric matrices Dustin Cartwright and Melody Chan UC

Symmetric Indefinite Triangular Factorization Revealing the Rank Profile Matrix Jean-Guillaume

The Bruhat rank of binary symmetric staircase pattern Carlos M. da Fonseca with Zhibin Du

Parallel Solution of Symmetric Eigenvalue Problems Zack 2/21/2014 Typically, the eigenvalue

An Efficient Gauss-Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations Xin Liu

Multiple-Rank Updates to Matrix Factorizations Zack 8/30/2013 Outline u Introduction u

1 SVD applications: rank, column, row, and null spaces Rank : the rank of a matrix is equal to:

Machine learning on the symmetric group Jean-Philippe Vert ML ML ML ML What if inputs are

Game Plan modification? Lifestyle and behavioral modification involves altering long-

/k Content 2/15 1. Introduction 2. Hamming weight 3. Rank weight 4. Extended rank weight

Symmetric Key Cryptography Introduction to Symmetric Key Cryptography What can we do?

10. Learning to Rank Outline 10.1. Why Learning to Rank (LeToR)? 10.2. Pointwise, Pairwise,

Fields modification Fields modification in high Al or In content III- -nitrides nitrides in

0 4 1 3 2 No deterministic symmetric dining solution [RL81] Probabilistic symmetric

A new family of maximum rank distance codes or: Maximum rank distance codes and finite semifields

Introduction to rank-based cryptography Philippe Gaborit University of Limoges, France ASCRYPTO

On the minimum rank of a graph Jisu Jeong June 21, 2013 Jisu Jeong On the minimum rank of a

Three Air Three Airpor ports Rank at ts Rank at the Bottom the Bottom At 20 At 20 minutes FAA

Protection of metals from corrosion. 1.Modification of environment. 2.Modification of

6. The Symmetric Eigenvalue Problem A must for engineers . . . 6. The Symmetric Eigenvalue