Lecture 10 Householder Triangularization NLA Reading Group Spring - - PowerPoint PPT Presentation

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Lecture 10 Householder Triangularization NLA Reading Group Spring - - PowerPoint PPT Presentation

Feb 14, 2023 265 likes •491 views

Lecture 10 Householder Triangularization NLA Reading Group Spring 13 by Onur Gngr Householder and Gram-Schmidt Gram-Schmidt: triangular orthogonalization Householder: orthogonal triangularization Triangularization by Introducing Zeros

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NLA Reading Group Spring ’13

by Onur Güngör

Lecture 10 Householder Triangularization

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Householder: orthogonal triangularization

Householder and Gram-Schmidt

Gram-Schmidt: triangular orthogonalization

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Triangularization by Introducing Zeros

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Householder Reflectors

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Householder Reflectors

P is the projector onto the space H

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Householder Reflectors

Instead of We use for numerical stability.

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Householder Algorithm

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Applying Q

This will be employed while solving least squares problems using QR factorization.

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Forming Q

Q can be formed by calculating Qe1, Qe2, … and Qem.

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Operation Count

Let Each vector requires flops.

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Operation Count

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Operation Count

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NLA Reading Group Spring ’13

by Onur Güngör

Lecture 11 Least Squares Problems

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Definition

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Polynomial Interpolation

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Polynomial Least Squares Fitting

Solve by minimizing

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Orthogonal Projection

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Pseudoinverse and Normal Equations

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Least Squares via Normal Equations

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Least Squares via QR Factorization

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Least Squares via SVD