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| Preface | |
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| Linear algebra | |
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| Linear systems of algebraic equations | |
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| Review of scalar, vector, and matrix operations | |
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| Elimination methods for solving linear systems | |
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| Existence and uniqueness of solutions | |
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| The determinant | |
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| Matrix inversion | |
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| Matrix factorization | |
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| Matrix norm and rank | |
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| Submatrices and matrix partitions | |
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| Example. Modeling a separation system | |
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| Sparse and banded matrices | |
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| MATLAB summary | |
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| Problems | |
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| Nonlinear algebraic systems | |
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| Existence and uniqueness of solutions to a nonlinear algebraic equation | |
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| Iterative methods and the use of Taylor series | |
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| Newton's method for a single equation | |
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| The secant method | |
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| Bracketing and bisection methods | |
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| Finding complex solutions | |
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| Systems of multiple nonlinear algebraic equations | |
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| Newton's method for multiple nonlinear equations | |
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| Estimating the Jacobian and quasi-Newton methods | |
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| Robust reduced-step Newton method | |
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| The trust-region Newton method | |
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| Solving nonlinear algebraic systems in MATLAB | |
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| Example. 1-D laminar flow of a shear-thinning polymer melt | |
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| Homotopy | |
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| Example. Steady-state modeling of a condensation polymerization reactor | |
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| Bifurcation analysis | |
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| MATLAB summary | |
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| Problems | |
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| Matrix eigenvalue analysis | |
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| Orthogonal matrices | |
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| A specific example of an orthogonal matrix | |
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| Eigenvalues and eigenvectors defined | |
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| Eigenvalues/eigenvectors of a 2 x 2 real matrix | |
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| Multiplicity and formulas for the trace and determinant | |
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| Eigenvalues and the existence/uniqueness properties of linear systems | |
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| Estimating eigenvalues; Gershgorin's theorem | |
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| Applying Gershgorin's theorem to study the convergence of iterative linear solvers | |
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| Eigenvector matrix decomposition and basis sets | |
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| Numerical calculation of eigenvalues and eigenvectors in MATLAB | |
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| Computing extremal eigenvalues | |
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| The QR method for computing all eigenvalues | |
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| Normal mode analysis | |
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| Relaxing the assumption of equal masses | |
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| Eigenvalue problems in quantum mechanics | |
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| Single value decomposition SVD | |
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| Computing the roots of a polynomial | |
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| MATLAB summary | |
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| Problems | |
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| Initial value problems | |
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| Initial value problems of ordinary differential equations (ODE-IVPs) | |
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| Polynomial interpolation | |
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| Newton-Cotes integration | |
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| Gaussian quadrature | |
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| Multidimensional integrals | |
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| Linear ODE systems and dynamic stability | |
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| Overview of ODE-IVP solvers in MATLAB | |
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| Accuracy and stability of single-step methods | |
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| Stiff stability of BDF methods | |
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| Symplectic methods for classical mechanics | |
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| Differential-algebraic equation (DAE) systems | |
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| Parametric continuation | |
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| MATLAB summary | |
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| Problems | |
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| Numerical optimization | |
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| Local methods for unconstrained optimization problems | |
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| The simplex method | |
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| Gradient methods | |
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| Newton line search methods | |
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| Trust-region Newton method | |
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| Newton methods for large problems | |
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| Unconstrained minimizer fminunc in MATLAB | |
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| Example. Fitting a kinetic rate law to time-dependent data | |
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| Lagrangian methods for constrained optimization | |
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| Constrained minimizer fmincon in MATLAB | |
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| Optimal control | |
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| MATLAB summary | |
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| Problems | |
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| Boundary value problems | |
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| BVPs from conservation principles | |
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| Real-space vs. function-space BVP methods | |
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| The finite difference method applied to a 2-D BVP | |
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| Extending the finite difference method | |
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| Chemical reaction and diffusion in a spherical catalyst pellet | |
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| Finite differences for a convection/diffusion equation | |
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| Modeling a tubular chemical reactor with dispersion; treating multiple fields | |
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| Numerical issues for discretized PDEs with more than two spatial dimensions | |
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| The MATLAB 1-D parabolic and elliptic solver pdepe | |
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| Finite differences in complex geometries | |
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| The finite volume method | |
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| The finite element method (FEM) | |
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| FEM in MATLAB | |
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| Further study in the numerical solution of BVPs | |
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| MATLAB summary | |
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| Problems | |
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| Probability theory and stochastic simulation | |
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| The theory of probability | |
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| Important probability distributions | |
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| Random vectors and multivariate distributions | |
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| Brownian dynamics and stochastic differential equations (SDEs) | |
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| Markov chains and processes; Monte Carlo methods | |
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| Genetic programming | |
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| MATLAB summary | |
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| Problems | |
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| Bayesian statistics and parameter estimation | |
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| General problem formulation | |
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| Example. Fitting kinetic parameters of a chemical reaction | |
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| Single-response linear regression | |
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| Linear least-squares regression | |
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| The Bayesian view of statistical inference | |
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| The least-squares method reconsidered | |
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| Selecting a prior for single-response data | |
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| Confidence intervals from the approximate posterior density | |
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| MCMC techniques in Bayesian analysis | |
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| MCMC computation of posterior predictions | |
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| Applying eigenvalue analysis to experimental design | |
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| Bayesian multi response regression | |
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| Analysis of composite data sets | |
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| Bayesian testing and model criticism | |
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| Further reading | |
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| MATLAB summary | |
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| Problems | |
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| Fourier analysis | |
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| Fourier series and transforms in one dimension | |
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| 1-D Fourier transforms in MATLAB | |
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| Convolution and correlation | |
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| Fourier transforms in multiple dimensions | |
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| Scattering theory | |
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| MATLAB summary | |
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| Problems | |
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| References | |
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| Index | |