Table 1 Principle Matlab operators and functions Name Description Page reference

Transcription

1 Matlab Index Table 1 summarises the Matlab supplied operators and functions to which we have referred. In most cases only a few of the options available to the individual functions have been fully utilised. Full details may be found by using the on-line help system. It should be emphasised that this table represents a mere fraction of the facilities available to Matlab users. Table 1 Principle Matlab operators and functions Name Description Page reference Arithmetic operators + / Add, subtract, multiply and divide 2. Element by element matrix multiplication 12./. Raising to a power element by element 116 \ Left-division, if A X=B,X=A 1 B 4 ' Matrix transpose 3 Numerical functions : Generate numbers over a range 6 sqrt, exp, sin, cos etc. Standard functions 2 bintprog Binary integer programming 164 eig Eigenvalues and eigenvectors 229 floor Nearest integer below a given number 115 function Define a new function 10 fzero Root of a function of a single variable 66 linprog Linear programming 161 lu LU decomposition 42 max Largest element 11 fminsearch Unconstrained local minimum 191 norm Norm 4 (continued on the next page) C. Woodford, C. Phillips, Numerical Methods with Worked Examples: Matlab Edition, DOI / , Springer Science+Business Media B.V

2 250 Matlab Index Table 1 (Continued) Name Description Page reference ode45 Solutions to non-stiff differential equations 213 optimset Sets options for optimisation routines 68 pi π 3 polyfit Fit polynomial to data 92 polyval Polynomial evaluation 93 qr QR factorisation 229 quad Numerical integration 114 roots Roots of a polynomial 3 size Number of rows and columns 8 fsolve Solve systems of nonlinear equations 68 strcmp Compare two strings 10 strncmp Compare first n characters of two strings 10 trapz Numerical integration using the trapezium rule 115 Program name > Passing function names as parameters 66 && >< Relationships: and, or less than, greater than, 7 <= >= == less than or equal, greater than or equal, not, equal 7 break Terminate a loop prematurely 68 if Conditional command execution 7 clear Delete all or specified program variables 16 for loop Program section to be repeated n times 6 load Retrieve program variables from backing store 15 save Save program variables to backing store 15 while loop Program section to be repeated conditionally 8 Miscellaneous % Program comment, not regarded by Matlab as a command 6 also used in specifying formats Continuation mark 13 Ctrl C Terminate execution 8 more Page by page or continuous screen output 7 Data types cell Extension of matrix concept to elements of different type 14 complex Specify a complex number 132 global Declare a variable to be global 132 i Complex number i 132 real Extract the real part of a complex number 133 single Use 32-bit rather than 64-bit word 42 syms Symbolic variables 133 (continued on the next page)

3 Matlab Index 251 Table 1 (Continued) Name Description Page reference disp Display text string or program variable 9 ezplot Function plotter 133 fopen Access a file 14 format Define a format 12 fprintf format a string for output to a file 14 frewind File rewind 15 input Request keyboard input 9 sprintf Formatted screen output 12 textscan formatted file input 14 type File listing 14 Plotting axis Set axis limits 160 colorbar Colour chart 12 figure Start a new (plotting) figure 161 gca Get access to the current plot to change settings 245 gtext Interactive graph annotating facility 161 linspace Specify points over an interval 12 meshgrid Specify points over a grid 12 plot 2D-graph plotting 11 surf 3D-surface plotting 12 Special matrices eye Generate an identity matrix 165 diag Diagonal matrix with or without sub-diagonals 214 hilb Hilbert matrix 43 ones Generate a matrix with all elements = 1 43 zeros Generate a matrix with all elements = Statistical functions cov Covariance 245 mean Mean 11 std Standard deviation 11 bar Bar chart 11 pie Pie chart 11 hist Histogram 245 polyfit Linear regression 92 rand Random number generator 246 randint Random integer generator 246

4 Index B Backward substitution, 21 Boundary-value problem difference equations, 209 worked, 210 shooting method, 207 worked, 208 C Cauchy s theorem, 128 Chebyshev s theorem, 235 Cotes, 110 Curve fitting, 71 D Dependent variable, 18 Differences, 75 Differentiation accuracy of methods, 126 Cauchy s theorem, 128 worked, 128, 129 comparison of methods, 123, 130 five-point formula, 122 worked, 124 higher order derivative, 125 worked, 127 second-order derivative worked, 125 three-point formula, 122 worked, 123 two-point formula, 120 worked, 120, 123 E Eigenvalue, 216 characteristic polynomial, 217 worked, 217 largest, 218 worked, 218 power method, 218 simple QR method, 223 worked, 223 smallest, 221 worked, 221 upper triangular matrix worked, 223 Eigenvector, 216 normalised, 217 worked, 216 Extrapolation, 73 F Francis, 227 Frequency, 232 Frequency distribution, 232 G Gauss Seidel iteration, 37 worked, 38 Gaussian elimination, 21 multiplier, 23 partial pivoting, 26 worked, 26, 28 pivot, 26 pivotal equation, 26 row interchange, 24 worked, 21, 22 Global minimum, 170 H Hilbert, 43 Histogram, 232 C. Woodford, C. Phillips, Numerical Methods with Worked Examples: Matlab Edition, DOI / , Springer Science+Business Media B.V

5 254 Index I IEEE standards, 26 Independent variable, 18 Initial-value problem Euler s method, 200 worked, 200 Runge Kutta, 202, 204, 206 worked, 202, 205, 206 Integer programming, 136, 149 branch and bound, 150 worked, 151 continuous problem, 150 decision problem, 153 decision variable, 154 worked, 154 the machine scheduling problem, 156 worked, 156 the travelling salesman problem, 155 worked, 155 Integrand, 97 Integration adaptive quadrature, 112 worked, 112 comparison of methods, 109, 114 Gaussian quadrature, 110 worked, 111 higher order rules, 109 Monte Carlo, 244 worked, 244 Newton Cotes, 109 three-eighths rule, 109 of analytic function, 104 worked, 108 of irregularly tabulated function, 102 worked, 102 of tabulated function, 98 worked, 98 Simpson s rule, 101, 106 worked, 101 trapezium rule, 99, 101, 104 worked, 99 Integration by parts, 98 Interpolating polynomial comparison of methods, 81 Neville, 80 worked, 80 Newton, 77 worked, 78, 79 spline, 82 worked, 84 Interpolation inverse, 73 Lagrange, 108 linear, 72 piecewise, 84 worked, 72 piecewise linear, 74 polynomial, 77 piecewise (spline), 84 Iterative refinement, 35 residual, 36 worked, 35 J Josephson junction, 198 L Lagrangian, 187 Least squares approximation, 85 confidence interval, 240 linear, 86 worked, 86 normal equations, 88 polynomial, 88, 91 worked, 89 residual, 86 Legendre polynomial, 110 Linear approximation, 72 Linear dependence, 33 Linear equation comparison of methods, 38 Linear programming, 135 basic variables, 144 canonical form, 140, 141 worked, 141 constraints, 135 costs, 140 dual problem, 146 feasible region, 138 feasible vertex, 142 graphical method, 138 worked, 138 objective, 135 optimal solution, 136 problem variables, 140 simplex method, see separate entry, 142 slack variables, 141 standard form, 141 worked, 141 trivial constraints, 138 vertex, 140 Linear relation, 18 Hooke s law, 18 Kirchoff s law, 18 mercury thermometer, 18 Linear system, 18 as a model of physical processes, 20 banded, 211

6 Index 255 Linear system (cont.) portal frame, 20 ill-conditioned, 37 in matrix vector form, 19 worked, 19 infinity of solutions, 32 worked, 32 multiple right hand sides, 30 worked, 30 no solution, 32 worked, 32 no unique solution, 32 worked, 32 overdetermined, 86 singular, 33 sparse, 211 symmetric positive definite, 33 worked, 33 upper triangular system, 20 worked, 20 Local minimum, 170 comparison of methods, 174, 181 constrained variables, 184 golden section search, 172 worked, 173 grid search, 171 worked, 171 method of steepest descent, 176 worked, 176 multiplier penalty function method, 186 worked, 188 penalty function method, 185 worked, 185 rank-one method, 178 worked, 179 rank-two method, 184 single variable, 171 LU decomposition, 24 M Matrix coefficient matrix, 19 determinant, 217 diagonally dominant, 35 Jacobian, 63 permutation matrix, 28 singular, 33 sparse, 38 symmetric positive definite, 33 unitary, 229 upper triangular, 21 Mean value theorem, 60 Mixed integer programming, 136 N Natural vibration, 216 Nonlinear equation bisection method, 48 worked, 49 comparison of methods, 58 Butler Volmer, 48 Newton Raphson method problems, 58 worked, 56 rule of false position, 51 worked, 51 secant method, 52 breakdown, 54 worked, 52 Nonlinear system, 58 metal plate, 59 turbine rotor, 63 Newton s method, 58 worked, 59, 63 Normal distribution, 233 O Operations research, 159 Order of convergence, 58 Ordinary differential equation, 198 boundary-value problem, see separate entry, 199 comparison of methods, 203 first-order, 198 higher order equations, 207 initial-value problem, see separate entry, 199 local truncation error, 205 nth-order, 199 second-order, 199 P Pseudo-random number, 242 generator, 242 multiplicative congruential, 242 R Random number, see also pseudo-random number, 241 Random variable, 232 average, 233 correlation, 236 worked, 237 covariance, 236 dispersion, 234 expected value, 233

7 256 Index Random variable (cont.) mean, 233 mean and standard deviation worked, 235 standard deviation, 234 variance, 234 Raphson, 47 Regression, 239 standard error estimate, 239 worked, 239 Root, 47 Rounding error, 26 S Similarity transformation, 223 Simplex method all slack solution, 144 artificial variable, 148 basic feasible solution, 144 non-basic variables, 143 pseudo objective, 147 revised simplex method, 146 worked, 142, 144, 147, 148 Stochastic variable, 232 T Taylor series, 120, 179, 204 Z Zero (of a function), 47