Scientific Computing with MATLAB

1. What can't be ignored.- 1.1 Real numbers.- 1.1.1 How do we represent them.- 1.1.2 How do we operate with floating-point numbers.- 1.2 Complex numbers.- 1.3 Matrices.- 1.3.1 Vectors.- 1.4 Real functions.- 1.4.1 The zeros.- 1.4.2 Polynomials.- 1.4.3 Integration and differentiation.- 1.5 To err is not only human.- 1.5.1 Talking about costs.- 1.6 A few more words about MATLAB.- 1.6.1 MATLAB statements.- 1.6.2 Programming in MATLAB.- 1.7 What we haven't told you.- 1.8 Exercises.- 2. Nonlinear equations.- 2.1 The bisection method.- 2.2 The Newton method.- 2.3 Fixed point iterations.- 2.3.1 How to terminate fixed point iterations.- 2.4 What we haven't told you.- 2.5 Exercises.- 3. Approximation of functions and data.- 3.1 Interpolation.- 3.1.1 Lagrangian polynomial interpolation.- 3.1.2 Chebyshev interpolation.- 3.1.3 Trigonometric interpolation and FFT.- 3.2 Piecewise linear interpolation.- 3.3 Approximation by spline functions.- 3.4 The least squares method.- 3.5 What we haven't told you.- 3.6 Exercises.- 4. Numerical differentiation and integration.- 4.1 Approximation of function derivatives.- 4.2 Numerical integration.- 4.2.1 Midpoint formula.- 4.2.2 Trapezoidal formula.- 4.2.3 Simpson formula.- 4.3 Simpson adaptive formula.- 4.4 What we haven't told you.- 4.5 Exercises.- 5. Linear systems.- 5.1 The LU factorization method.- 5.2 The technique of pivoting.- 5.3 How accurate is the LU factorization?.- 5.4 How to solve a tridiagonal system.- 5.5 Iterative methods.- 5.5.1 How to construct an iterative method.- 5.6 When should an iterative method be stopped?.- 5.7 Richardson method.- 5.8 What we haven't told you.- 5.9 Exercises.- 6. Eigenvalues and eigenvectors.- 6.1 The power method.- 6.1.1 Convergence analysis.- 6.2 Generalization of the power method.- 6.3 How to compute the shift.- 6.4 Computation of all the eigenvalues.- 6.5 What we haven't told you.- 6.6 Exercises.- 7. Ordinary differential equations.- 7.1 The Cauchy problem.- 7.2 Euler methods.- 7.2.1 Convergence analysis.- 7.3 The Crank-Nicolson method.- 7.4 Zero-stability.- 7.5 Stability on unbounded intervals.- 7.5.1 Absolute stability controls perturbations.- 7.6 High order methods.- 7.7 The predictor-corrector methods.- 7.8 Systems of differential equations.- 7.9 What we haven't told you.- 7.10 Exercises.- 8. Numerical methods for boundary-value problems.- 8.1 Approximation of boundary-value problems.- 8.1.1 Approximation by finite differences.- 8.1.2 Approximation by finite elements.- 8.2 Finite differences in 2 dimensions.- 8.2.1 Consistency and convergence.- 8.3 What we haven't told you.- 8.4 Exercises.- 9. Solutions of the exercises.- 9.1 Chapter 1.- 9.2 Chapter 2.- 9.3 Chapter 3.- 9.4 Chapter 4.- 9.5 Chapter 5.- 9.6 Chapter 6.- 9.7 Chapter 7.- 9.8 Chapter 8.- Index of MATLAB Programs.