cs 357: numerical methods lecture 14: orthogonal iteration ... · lecture 14: orthogonal iteration...
TRANSCRIPT
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CS 357: Numerical Methods
Lecture 14:
Orthogonal Iteration
Singular Value Decomposition
Eric Shaffer
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Finding Eigenvectors
Can we simultaneously (sort of) find all the eigenvectors of A?
What about this algorithm
X0 = arbitrary n x p matrix of rank p
for k=1,2,…
Xk=AXk-1
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Orthogonal Iteration
How about this?
X0= n x p matrix of rank p
for k=1,2,….
#compute reduced QR factorization
Qk+1Rk+1=Xk
Xk+1=AQk
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Orthogonal Iteration
X0= n x p matrix of rank pfor k=1,2,….
#compute reduced QR factorizationQk+1Rk+1=Xk
Xk+1=AQk
![Page 5: CS 357: Numerical Methods Lecture 14: Orthogonal Iteration ... · Lecture 14: Orthogonal Iteration Singular Value Decomposition ... Q k+1 R k+1 =X k X k+1 =AQ k. Orthogonal Iteration](https://reader033.vdocument.in/reader033/viewer/2022060907/60a208849911c902104df380/html5/thumbnails/5.jpg)
The Schur Form: Finding Eigenvalues
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The Schur Form: Finding Eigenvectors
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Singular Value Decomposition (SVD)
𝐴 = 𝑈Σ𝑉𝑇
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Singular Value Decomposition (SVD)
𝐴 = 𝑈Σ𝑉𝑇
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Inversion using SVD
𝐴 = 𝑈Σ𝑉𝑇
Assume A is an n x n matrix
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The Pseudo Inverse
When A is not square we can compute the pseudo-inverse
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Pseudo Inverse and Least Squares
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Notes on Computing the SVD