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Unit 6.1.2 Overview

  • 6.1 Opening Remarks

    • 6.1.1 Whose problem is it anyway?

    • 6.1.2 Overview

    • 6.1.3 What you will learn

  • 6.2 Floating Point Arithmetic

    • 6.2.1 Storing real numbers as floating point numbers

    • 6.2.2 Error in storing a real number as a floating point number

    • 6.2.3 Models of floating point computation

    • 6.2.4 Stability of a numerical algorithm

    • 6.2.5 Conditioning versus stability

    • 6.2.6 Absolute value of vectors and matrices

  • 6.3 Error Analysis for Basic Linear Algebra Algorithms

    • 6.3.1 Initial insights

    • 6.3.2 Backward error analysis of dot product: general case

    • 6.3.3 Dot product: error results

    • 6.3.4 Matrix-vector multiplication

    • 6.3.5 Matrix-matrix multiplication

  • 6.4 Error Analysis for Solving Linear Systems

    • 6.4.1 Numerical stability of triangular solve

    • 6.4.2 Numerical stability of LU factorization

    • 6.4.3 Numerical stability of linear solve via LU factorization

    • 6.4.4 Numerical stability of linear solve via LU factorization with partial pivoting

    • 6.4.5 Is LU with Partial Pivoting Stable?

  • 6.5 Enrichments

    • 6.5.1 Systematic derivation of backward error analyses

    • 6.5.2 LU factorization with pivoting can fail in practice

  • 6.6 Wrap Up

    • 6.6.1 Additional homework

    • 6.6.2 Summary