| Preface | p. ix |
| Linear algebra | p. 1 |
| Linear systems of algebraic equations | p. 1 |
| Review of scalar, vector, and matrix operations | p. 3 |
| Elimination methods for solving linear systems | p. 10 |
| Existence and uniqueness of solutions | p. 23 |
| The determinant | p. 32 |
| Matrix inversion | p. 36 |
| Matrix factorization | p. 38 |
| Matrix norm and rank | p. 44 |
| Submatrices and matrix partitions | p. 44 |
| Example. Modeling a separation system | p. 45 |
| Sparse and banded matrices | p. 46 |
| MATLAB summary | p. 56 |
| Problems | p. 57 |
| Nonlinear algebraic systems | p. 61 |
| Existence and uniqueness of solutions to a nonlinear algebraic equation | p. 61 |
| Iterative methods and the use of Taylor series | p. 62 |
| Newton's method for a single equation | p. 63 |
| The secant method | p. 69 |
| Bracketing and bisection methods | p. 70 |
| Finding complex solutions | p. 70 |
| Systems of multiple nonlinear algebraic equations | p. 71 |
| Newton's method for multiple nonlinear equations | p. 72 |
| Estimating the Jacobian and quasi-Newton methods | p. 77 |
| Robust reduced-step Newton method | p. 79 |
| The trust-region Newton method | p. 81 |
| Solving nonlinear algebraic systems in MATLAB | p. 83 |
| Example. 1-D laminar flow of a shear-thinning polymer melt | p. 85 |
| Homotopy | p. 88 |
| Example. Steady-state modeling of a condensation polymerization reactor | p. 89 |
| Bifurcation analysis | p. 94 |
| MATLAB summary | p. 98 |
| Problems | p. 99 |
| Matrix eigenvalue analysis | p. 104 |
| Orthogonal matrices | p. 104 |
| A specific example of an orthogonal matrix | p. 105 |
| Eigenvalues and eigenvectors defined | p. 106 |
| Eigenvalues/eigenvectors of a 2 x 2 real matrix | p. 107 |
| Multiplicity and formulas for the trace and determinant | p. 109 |
| Eigenvalues and the existence/uniqueness properties of linear systems | p. 110 |
| Estimating eigenvalues; Gershgorin's theorem | p. 111 |
| Applying Gershgorin's theorem to study the convergence of iterative linear solvers | p. 114 |
| Eigenvector matrix decomposition and basis sets | p. 117 |
| Numerical calculation of eigenvalues and eigenvectors in MATLAB | p. 123 |
| Computing extremal eigenvalues | p. 126 |
| The QR method for computing all eigenvalues | p. 129 |
| Normal mode analysis | p. 134 |
| Relaxing the assumption of equal masses | p. 136 |
| Eigenvalue problems in quantum mechanics | p. 137 |
| Single value decomposition SVD | p. 141 |
| Computing the roots of a polynomial | p. 148 |
| MATLAB summary | p. 149 |
| Problems | p. 149 |
| Initial value problems | p. 154 |
| Initial value problems of ordinary differential equations (ODE-IVPs) | p. 155 |
| Polynomial interpolation | p. 156 |
| Newton-Cotes integration | p. 162 |
| Gaussian quadrature | p. 163 |
| Multidimensional integrals | p. 167 |
| Linear ODE systems and dynamic stability | p. 169 |
| Overview of ODE-IVP solvers in MATLAB | p. 176 |
| Accuracy and stability of single-step methods | p. 185 |
| Stiff stability of BDF methods | p. 192 |
| Symplectic methods for classical mechanics | p. 194 |
| Differential-algebraic equation (DAE) systems | p. 195 |
| Parametric continuation | p. 203 |
| MATLAB summary | p. 207 |
| Problems | p. 208 |
| Numerical optimization | p. 212 |
| Local methods for unconstrained optimization problems | p. 212 |
| The simplex method | p. 213 |
| Gradient methods | p. 213 |
| Newton line search methods | p. 223 |
| Trust-region Newton method | p. 225 |
| Newton methods for large problems | p. 227 |
| Unconstrained minimizer fminunc in MATLAB | p. 228 |
| Example. Fitting a kinetic rate law to time-dependent data | p. 230 |
| Lagrangian methods for constrained optimization | p. 231 |
| Constrained minimizer fmincon in MATLAB | p. 242 |
| Optimal control | p. 246 |
| MATLAB summary | p. 252 |
| Problems | p. 252 |
| Boundary value problems | p. 258 |
| BVPs from conservation principles | p. 258 |
| Real-space vs. function-space BVP methods | p. 260 |
| The finite difference method applied to a 2-D BVP | p. 260 |
| Extending the finite difference method | p. 264 |
| Chemical reaction and diffusion in a spherical catalyst pellet | p. 265 |
| Finite differences for a convection/diffusion equation | p. 270 |
| Modeling a tubular chemical reactor with dispersion; treating multiple fields | p. 279 |
| Numerical issues for discretized PDEs with more than two spatial dimensions | p. 282 |
| The MATLAB 1-D parabolic and elliptic solver pdepe | p. 294 |
| Finite differences in complex geometries | p. 294 |
| The finite volume method | p. 297 |
| The finite element method (FEM) | p. 299 |
| FEM in MATLAB | p. 309 |
| Further study in the numerical solution of BVPs | p. 311 |
| MATLAB summary | p. 311 |
| Problems | p. 312 |
| Probability theory and stochastic simulation | p. 317 |
| The theory of probability | p. 317 |
| Important probability distributions | p. 325 |
| Random vectors and multivariate distributions | p. 336 |
| Brownian dynamics and stochastic differential equations (SDEs) | p. 338 |
| Markov chains and processes; Monte Carlo methods | p. 353 |
| Genetic programming | p. 362 |
| MATLAB summary | p. 364 |
| Problems | p. 365 |
| Bayesian statistics and parameter estimation | p. 372 |
| General problem formulation | p. 372 |
| Example. Fitting kinetic parameters of a chemical reaction | p. 373 |
| Single-response linear regression | p. 377 |
| Linear least-squares regression | p. 378 |
| The Bayesian view of statistical inference | p. 381 |
| The least-squares method reconsidered | p. 388 |
| Selecting a prior for single-response data | p. 389 |
| Confidence intervals from the approximate posterior density | p. 395 |
| MCMC techniques in Bayesian analysis | p. 403 |
| MCMC computation of posterior predictions | p. 404 |
| Applying eigenvalue analysis to experimental design | p. 412 |
| Bayesian multi response regression | p. 414 |
| Analysis of composite data sets | p. 421 |
| Bayesian testing and model criticism | p. 426 |
| Further reading | p. 431 |
| MATLAB summary | p. 431 |
| Problems | p. 432 |
| Fourier analysis | p. 436 |
| Fourier series and transforms in one dimension | p. 436 |
| 1-D Fourier transforms in MATLAB | p. 445 |
| Convolution and correlation | p. 447 |
| Fourier transforms in multiple dimensions | p. 450 |
| Scattering theory | p. 452 |
| MATLAB summary | p. 459 |
| Problems | p. 459 |
| References | p. 461 |
| Index | p. 464 |
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