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Chapter 1: Introduction

• Floating point in Julia

• Absolute and relative accuracy

• Surprising arithmetic

• Horner’s rule

• Instability of the quadratic formula

• Stable alternative to the quadratic formula

• Polynomial roots

Chapter 2: Square linear systems

• Interpolating the population of China

• Vectors and matrices in Julia

• Backslash for solving linear systems

• Solving triangular systems

• Gaussian elimination

• Row operations

• Using LU factorization

• Timing and flop counts

• Timing matrix-vector multiplication

• Timing of LU factorization

• Failure of plain LU factorization

• Row pivoting

• Vector norms

• Matrix norms

• Condition number bounds on error

• Banded matrices

• Timing for banded matrices

• Symmetric LU

• Cholesky factorization

Chapter 3: Overdetermined linear systems

• Interpolating temperature data

• Fitting temperature data

• Convergence to \pi

• Instability in the normal equations

• QR factorizations in Julia

• Householder QR

Chapter 4: Roots of nonlinear equations

• Roots of Bessel functions

• Conditioning of roots

• Fixed point iteration

• Convergence of fixed point iteration

• The idea of Newton’s method

• Convergence of Newton’s method

• Usage of newton

• Using the secant line

• Convergence of the secant method

• Inverse quadratic interpolation

• The Newton idea for systems

• Usage of newtonsys

• Using levenberg

• Michaelis–Menten fitting

Chapter 5: Piecewise interpolation

• Poor conditioning in polynomial interpolation

• Piecewise polynomial interpolation

• Polynomial cardinal functions

• Hat functions

• Piecewise linear interpolation

• Convergence of PL interpolation

• Cubic splines

• FD at arbitrary nodes

• Convergence of finite differences

• Comparison of 1st and 2nd order

• Roundoff in finite differences

• Numerical integration

• Convergence of trapezoidal integration

• Extrapolation for numerical integration

• An oscillatory integrand

• Adaptive integration

Chapter 6: Initial-value problems for ODEs

• Numerical IVP solution

• Using an IVP solution

• IVP with singularity

• Conditioning of IVPs

• Convergence of Euler’s method

• Matrix exponential

• Predator-prey system

• Convergence of RK methods

• Adaptive RK

• Adaptivity at a singularity

• Convergence of AB4

• Accuracy isn’t everything

• Instability in a multistep method

Review: Linear algebra Chapter 1 (Introduction)