CS475 / CS675 Lecture 18: June 30, 2016 QR Method with Shifts - - PowerPoint PPT Presentation

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Feb 07, 2023 266 likes •429 views

SLIDE 1

CS475 / CS675 Lecture 18: June 30, 2016

QR Method with Shifts Google Page Rank Reading: [TB] Chapter 29

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Reduction to Hessenberg Algorithm

For 1,2, … , 2 1: , /| | for , 1, … ,

1: , 1: , 2

1: ,

end for 1,2, … ,

, 1: , 1: 2 , 1:

end

CS475/CS675 (c) 2016 P. Poupart & J. Wan

SLIDE 3

Symmetric Case

, then

is also symmetric
A symmetric Hessenberg matrix

tridiagonal matrix

Two‐phase process:

SLIDE 4

Shift QR Algorithm

QR algorithm is both simultaneous iteration and

simultaneous inverse iteration

– Can apply shift technique

Algorithm (Shifted QR)

For 1,2, … Pick a shift ← (QR factorization) End

SLIDE 5

Shift QR Algorithm

Similar to regular QR, we can show that
where
Derivation:

SLIDE 6

Shift QR Algorithm

We can also show that
Derivation:

SLIDE 7

Shift QR Algorithm

Continued derivation:
If the shifts are good eigenvalue estimates, the last

column of

converges quickly to an eigenvector.

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Rayleigh quotient shift

To estimate the eigenvalue corresponding to the

eigenvector approximated by the last column of

Equivalent to applying RQI on

– i.e., QR algo has cubic convergence to that eigenvector

Note:
comes for free!

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Google PageRank

Problem: give a ranking, PageRank, to all webpages.
Idea: surfing the web is like a random walk

 a Markov chain or Markov process.

– PageRank = the limiting probability that an infinitely dedicated random surfer visits any particular page. – A page has high rank if other pages with high rank link to it.

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Google PageRank

Example:

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Google PageRank

Define connectivity matrix

by

1 if ∃ a link from page to

therwise
The column of

shows the links on the page.

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Google PageRank

Let
prob. that the random walk follows a link

and

prob. that an arbitrary page is chosen

– Typically 0.85

Define
∑
to be the prob. of jumping from page to page

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Google PageRank

Properties of A:

– Entries between 0 and 1: 0 1 – Columns sum to 1: ∑

∑
∑
1
∑ 1
1 1
By Ferron‐Frobenius theorem, a matrix

with the above properties admits a vector such that

i.e., is the eigenvector corresponding to eigenvalue 1

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Google PageRank

Normalize such that
. Then is the state

vector of the Markov chain & is Google’s PageRank!

The elements of are all positive and less than 1.
In our example,

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Google PageRank

To compute PageRank:

CS475 / CS675 Lecture 18: June 30, 2016

QR Method with Shifts Google Page Rank Reading: [TB] Chapter 29

Reduction to Hessenberg Algorithm

For 1,2, … , 2 1: , /| | for , 1, … ,

end for 1,2, … ,

end

Symmetric Case

, then

tridiagonal matrix

Shift QR Algorithm

simultaneous inverse iteration

– Can apply shift technique

For 1,2, … Pick a shift ← (QR factorization) End

Shift QR Algorithm

Shift QR Algorithm

Shift QR Algorithm

column of

Rayleigh quotient shift

eigenvector approximated by the last column of

– i.e., QR algo has cubic convergence to that eigenvector

Google PageRank

 a Markov chain or Markov process.

– PageRank = the limiting probability that an infinitely dedicated random surfer visits any particular page. – A page has high rank if other pages with high rank link to it.

Google PageRank

Google PageRank

by

1 if ∃ a link from page to

shows the links on the page.

Google PageRank

and

– Typically 0.85

Google PageRank

– Entries between 0 and 1: 0 1 – Columns sum to 1: ∑

with the above properties admits a vector such that

i.e., is the eigenvector corresponding to eigenvalue 1

Google PageRank

vector of the Markov chain & is Google’s PageRank!

Google PageRank

– Setup – Compute largest eigenvector by: