7.1 Naïve Gaussian Elimination 8.1 The LU Factorization
Transcript of 7.1 Naïve Gaussian Elimination 8.1 The LU Factorization
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7.1 Naïve Gaussian Elimination8.1 The LU Factorization
• Motivating Ax=b: Newton's method for systems of nonlinear equations (pp. 96-99) • C&K 7.1: Naive Gaussian Elimination
– See answered problems (1, 2, 3ab, 4, 5, 6, 7abc)
• C&K 8.1: Matrix Factorizations (see pp. 293-302 on LU factorization)– See answered problems (1-8)
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A x = b
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Motivation: Newton’s Method (for systems of nonlinear equations, pp.96-99)
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Motivation: Newton’s Method (for systems of nonlinear equations, pp.96-99)
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Motivation: Newton’s Method (for systems of nonlinear equations, pp.96-99)
Example 2 (p. 99)
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Cha
pter
7.1
(p.
245
)
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A x = b
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A x = b
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A x = b
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Naïve Gaussian Elimination
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Naïve Gaussian Elimination
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Naïve Gaussian Elimination
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Naïve Gaussian Elimination
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Back Substitution
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Naïve Gaussian Elimination
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Naïve Gaussian Elimination
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Naïve Gaussian Elimination
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Continuing to second equation…
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kth step of forward elimination
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Naïve GE: Pseudocode
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Storing Multipliers (instead of zeros)
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Back Substitution
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Back Substitution
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Back Substitution
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LU Factorization Example(supplements chalkboard)
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Matrix Interpretation
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Column 1
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Column 2
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Column 3
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Recognizing “L”
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Verifying LU=A