Page 482 - Numerical Methods for Chemical Engineering

Index 471

rank 44, 142 Example. 1-D laminar flow of shear-thinning fluid
real, symmetric matrix 10, 119 85–88
Schur decomposition 119 Example. multiple steady states in a nonisothermal
similar matrices 118 CSTR 204–206
Singular Value Decomposition (SVD) (see also Example. steady-state CSTR for polycondensation
Singular Value Decomposition) 141–148 89–94
sparse (see also Sparse matrix) 50, 51, 52, 53 Example. steady-state CSTR with two reactions
spectral radius 113 71–72, 85, 88–89
square matrix 8 homotopy 88, 203
submatrix 44 Jacobian matrix 73
symmetric 10, 119 solving a single equation
trace 110 bracketing and bisection 70
transpose 9 MATLAB fzero 70, 99
tridiagonal 50 solving by Newton’s method (see also Newton’s
unitary matrix 119 method)63
Matrix inverse solving by secant method 69
calculation by Cramer’s rule 36 solving multiple equations
definition 36 MATLAB fsolve 83, 98
pseudo (generalized) inverse 145 reduced-step line search 79, 80
numerical calculation 37 solving by Newton’s method (see also Newton’s
MCMC (Markov Chain Monte Carlo) simulation (see method)72
Monte Carlo) trust-region Newton method 81
Mean Norm
MATLAB mean 364 MATLAB norm, normest 113
of a random variable (see also Expectation) 322 matrix 44, 113
Metric vector 6
for vector space 6 2-norm (length) 6
Metropolis Monte Carlo method 353–357 infinity norm 7
Michaelis-Menten kinetics 58 p-norm 6
Monte Carlo Normal distribution (see also Gaussian distribution)
Bayesian Markov Chain Monte Carlo (MCMC) 331–332
simulation 403–411, 419–421 Normal matrix 119
Example. Monte Carlo simulation of 2-D Ising eigenvalue properties 121–123
lattice 356–357 Normal mode analysis 134
integration method 168, 360–361 Null
kinetic Monte Carlo 369 vector 5
Markov chain 354 space (kernel) 29, 144
Markov process 353
Metropolis algorithm 353–357 Optimal control 245–251
Bellman function 248
Newton’s method closed loop problem 250
for interpolation 157 cost functional 246
for optimization (see also Optimization) 223–227 dynamic programming 248–251
Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hamilton-Jacobi-Bellman (HJB) equation 249
formula 224 numerical solution by finite differences 275
Newton line search method 223–225 horizon time 246
Newton trust-region method 225–227 open loop method 247–248
for solving nonlinear algebraic systems Optimization 212–218
Broyden’s method 77 applied to parameter estimation 388–389,
demonstrated performance 74–76 417–419
finding “false” solutions 80 augmented Lagrangian method 231–240
quadratic convergence 69 complementary condition 238
quasi-Newton method 77 conjugate gradient (CG) method 218–223
reduced-step line search 79, 80 MATLAB pcg 223, 285
single equation systems 63; demonstrated performance for quadratic cost functions
performance 64–67; Jacobian matrix 73; 220–223
MATLAB fzero 70, 99; MATLAB fsolve constrained problems 231–245
83, 98; multiple equation systems 71, 72 cost function 212
trust-region Newton method 81 deterministic local methods 212–251
Nonlinear algebraic systems 61–99 discrete parameter optimization 361–364
arc length continuation 203 dogleg method 225–227
bifurcation point 94 equality constraints 232–235
complex solutions 70 Example. finding closest points on two ellipses 235