the resulting approximation for the first 16 faces when r 16 using plotFaces b

The resulting approximation for the first 16 faces

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the resulting approximation for the first 16 faces when r = 16 (using plotFaces ). (b) Plot each of the basis vectors for the subspace identified in the previous part when r = 16 (again using plotFaces ). (c) How large does r need to be to ensure that the relative error between the original dataset and our approximation is less than 5%? How does this relate to the singular values of the original matrix? (d) Comment on the qualitative differences between the original dataset and the approxi- mation produced by PCA and how this changes as we vary r . 3. Let A be a tri-diagonal matrix: A = d 1 c 1 0 0 0 · · · 0 f 1 d 2 c 2 0 0 · · · 0 0 f 2 d 3 c 3 0 · · · 0 0 0 f 3 d 4 c 4 · · · 0 . . . . . . . . . . . . . . . . . . 0 0 · · · f N - 2 d N - 1 c N - 1 0 0 0 · · · f N - 1 d N 1 Last updated 0:19, November 14, 2019
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(a) Argue that the LU factorization of A has the form A = * 0 0 0 · · · 0 * * 0 0 · · · 0 0 * * 0 · · · 0 0 0 * * · · · 0 . . . . . . . . . . . . 0 0 · · · * * * * 0 0 0 · · · 0 0 * * 0 0 · · · 0 0 0 * * 0 · · · 0 . . . . . . . . . 0 0 · · · * * 0 0 · · · 0 * , where * signifies a non-zero term. (b) Write down an algorithm that computes the LU factorization of A , meaning the { i } , { u i } , and { g i } below A = 1 0 0 0 · · · 0 1 1 0 0 · · · 0 0 2 1 0 · · · 0 0 0 3 1 · · · 0 . . . . . . . . . . . . 0 0 · · · N - 1 1 u 1 g 1 0 0 0 · · · 0 0 u 2 g 2 0 0 · · · 0 0 0 u 3 g 3 0 · · · 0 .
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