Page 8 - Numerical methods for chemical engineering

vi Contents

Bifurcation analysis 94
MATLAB summary 98
Problems 99

3 Matrix eigenvalue analysis 104
Orthogonal matrices 104
A specific example of an orthogonal matrix 105
Eigenvalues and eigenvectors defined 106
Eigenvalues/eigenvectors of a 2 × 2 real matrix 107
Multiplicity and formulas for the trace and determinant 109
Eigenvalues and the existence/uniqueness properties of linear
systems 110
Estimating eigenvalues; Gershgorin’s theorem 111
Applying Gershgorin’s theorem to study the convergence of iterative
linear solvers 114
Eigenvector matrix decomposition and basis sets 117
Numerical calculation of eigenvalues and eigenvectors in MATLAB 123
Computing extremal eigenvalues 126
The QR method for computing all eigenvalues 129
Normal mode analysis 134
Relaxing the assumption of equal masses 136
Eigenvalue problems in quantum mechanics 137
Single value decomposition SVD 141
Computing the roots of a polynomial 148
MATLAB summary 149
Problems 149

4 Initial value problems 154
Initial value problems of ordinary differential equations
(ODE-IVPs) 155
Polynomial interpolation 156
Newton–Cotes integration 162
Gaussian quadrature 163
Multidimensional integrals 167
Linear ODE systems and dynamic stability 169
Overview of ODE-IVP solvers in MATLAB 176
Accuracy and stability of single-step methods 185
Stiff stability of BDF methods 192
Symplectic methods for classical mechanics 194
Differential-algebraic equation (DAE) systems 195
Parametric continuation 203
MATLAB summary 207
Problems 208