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Unit 5.1.2 Overview

  • 5.1 Opening Remarks
    • 5.1.1 Of Gaussian elimination and LU factorization

    • 5.1.2 Overview

    • 5.1.3 What you will learn

  • 5.2 From Gaussian elimination to LU factorization
    • 5.2.1 Gaussian elimination

    • 5.2.2 LU factorization: The right-looking algorithm

    • 5.2.3 Existence of the LU factorization

    • 5.2.4 Gaussian elimination via Gauss transforms

  • 5.3 LU factorization with (row) pivoting
    • 5.3.1 Gaussian elimination with row exchanges

    • 5.3.2 Permutation matrices

    • 5.3.3 LU factorization with partial pivoting

    • 5.3.4 Solving A x = y via LU factorization with pivoting

    • 5.3.5 Solving with a triangular matrix

    • 5.3.6 LU factorization with complete pivoting

    • 5.3.7 Improving accuracy via iterative refinement

  • 5.4 Cholesky factorization
    • 5.4.1 Hermitian Positive Definite matrices

    • 5.4.2 The Cholesky Factorization Theorem

    • 5.4.3 Cholesky factorization algorithm (right-looking variant)

    • 5.4.4 Proof of the Cholesky Factorizaton Theorem

    • 5.4.5 Cholesky factorization and solving LLS

    • 5.4.6 Implementation with the classical BLAS

  • 5.5 Enrichments
    • 5.5.1 Other LU factorization algorithms

  • 5.6 Wrap Up
    • 5.6.1 Additional homework

    • 5.6.2 Summary