CS475 / CM375 Lecture 11: Oct 18, 2011 QR Factorization and Gram - - PDF document

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CS475 / CM375 Lecture 11: Oct 18, 2011 QR Factorization and Gram - - PDF document

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19/10/2011 CS475 / CM375 Lecture 11: Oct 18, 2011 QR Factorization and Gram Schmidt Orthogonalization Reading: [TB] Chapters 7, 8 CS475/CM375 (c) 2011 P. Poupart & J. Wan 1 Gram Schmidt Orthogonalization QR factorization algorithm

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CS475 / CM375 Lecture 11: Oct 18, 2011

QR Factorization and Gram‐Schmidt Orthogonalization Reading: [TB] Chapters 7, 8

CS475/CM375 (c) 2011 P. Poupart & J. Wan 1

Gram‐Schmidt Orthogonalization

  • QR factorization algorithm

– ( orthogonal and upper ∆) – Picture:

  • At the step

– is orthogonal to , … , –

  • 1

CS475/CM375 (c) 2011 P. Poupart & J. Wan 2

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Gram‐Schmidt Orthogonalization

  • Consider ∑
  • Since 0
  • ∑
  • 1, … , 1
  • ∴
  • 1

⟹ ∑

  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

3

Gram‐Schmidt Orthogonalization

  • Normalize 
  • Hence
  • ⋮

∑

  • Where

,

∑

  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

4

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Gram‐Schmidt Algorithm

For 1,2, … , for 1, … , 1

  • end
  • end

CS475/CM375 (c) 2011 P. Poupart & J. Wan 5

Modified Gram‐Schmidt

  • Change: “

”  “ ” (more stable)

  • In the ‐loop, changes for each

1:

  • 2:

⋮ 1:

∑

  • At ,
  • ∑
  • , … ,
  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

6

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Complexity of Gram‐Schmidt

  • Consider the ‐loop:
  • r
  •  mult, 1 adds

 mult, subs

∴ ∽ 4

  • Total flops ∑

∑ 4

  • ∑

1 4 ∼ 4 ∑

  • ∼ 2
  • Note: when , then 2

3

CS475/CM375 (c) 2011 P. Poupart & J. Wan 7

Example

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Householder triangularization

  • More stable than Gram‐Schmidt
  • Idea: …

∈ orthogonal matrices

  • Similar to GE, each will make the entries of col

become zero

  • Picture:

CS475/CM375 (c) 2011 P. Poupart & J. Wan 9

Householder reflectors

  • Define
  • 1

1

  • is chosen to be a Householder reflector
  • Picture

CS475/CM375 (c) 2011 P. Poupart & J. Wan 10

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Householder Reflector

  • Suppose
  • ⋮
  • then

| | ⋮

  • “reflects” across hyperplane orthogonal to
  • The orthogonal projector of onto :
  • Since is a reflector, it should go twice as far:

2

  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

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Householder Reflectors

  • Two possibilities:
  • For stability reason, the further one is chosen

– i.e.,

CS475/CM375 (c) 2011 P. Poupart & J. Wan 12

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Another Derivation

  • Let 2
  • . Find s.t. ∈ .
  • 2
  • ∈ ⟺ ∈ ,
  • Let

2

CS475/CM375 (c) 2011 P. Poupart & J. Wan 13

Derivation Continued

  • ∴
  • Hence

and ∓

CS475/CM375 (c) 2011 P. Poupart & J. Wan 14

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Example

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