MATLAB Mathematics - SERC - Index of

More documents

Recommendations

Info

5 Differential Equations Problem Size, Memory Use, and Computation Speed Question How large a problem can I solve with the ODE suite? Answer The primary constraints are memory and time. At each time step, the solvers for nonstiff problems allocate vectors of length n, where n is the number of equations in the system. The solvers for stiff problems but also allocate an n-by-n Jacobian matrix. For these solvers it may be advantageous to use the sparse option. If the problem is nonstiff, or if you are using the sparse option, it may be possible to solve a problem with thousands of unknowns. In this case, however, storage of the result can be problematic. Try asking the solver to evaluate the solution at specific points only, or call the solver with no output arguments and use an output function to monitor the solution. I'm solving a very large system, but only care about a couple of the components of y. Is there any way to avoid storing all of the elements? What is the startup cost of the integration and how can I reduce it? Yes. The user-installable output function capability is designed specifically for this purpose. When you call the solver with no output arguments, the solver does not allocate storage to hold the entire solution history. Instead, the solver calls OutputFcn(t,y,flag) at each time step. To keep the history of specific elements, write an output function that stores or plots only the elements you care about. The biggest startup cost occurs as the solver attempts to find a step size appropriate to the scale of the problem. If you happen to know an appropriate step size, use the InitialStep property. For example, if you repeatedly call the integrator in an event location loop, the last step that was taken before the event is probably on scale for the next integration. See ballode for an example. 5-44
Initial Value Problems for ODEs and DAEs Time Steps for Integration Question The first step size that the integrator takes is too large, and it misses important behavior. Can I integrate with fixed step sizes? Answer You can specify the first step size with the InitialStep property. The integrator tries this value, then reduces it if necessary. No. Error Tolerance and Other Options Question How do I choose RelTol and AbsTol? Answer RelTol, the relative accuracy tolerance, controls the number of correct digits in the answer. AbsTol, the absolute error tolerance, controls the difference between the answer and the solution. At each step, the error e in component i of the solution satisfies |e(i)|
Page 1 and 2:
MATLAB® The Language of Technical
Page 3:
Revision History June 2004 First pr
Page 6 and 7:
2 Polynomials and Interpolation Pol
Page 8 and 9:
Introduction to Initial Value ODE P
Page 10 and 11:
vi Contents
Page 12 and 13:
1 Matrices and Linear Algebra Funct
Page 14 and 15:
1 Matrices and Linear Algebra Matri
Page 16 and 17:
1 Matrices and Linear Algebra Addin
Page 18 and 19:
1 Matrices and Linear Algebra If x
Page 20 and 21:
1 Matrices and Linear Algebra y = v
Page 22 and 23:
1 Matrices and Linear Algebra Vecto
Page 24 and 25:
1 Matrices and Linear Algebra backs
Page 26 and 27:
1 Matrices and Linear Algebra If A
Page 28 and 29:
1 Matrices and Linear Algebra You c
Page 30 and 31:
1 Matrices and Linear Algebra 0.9 0
Page 32 and 33:
1 Matrices and Linear Algebra Inver
Page 34 and 35:
1 Matrices and Linear Algebra X = p
Page 36 and 37:
1 Matrices and Linear Algebra There
Page 38 and 39:
1 Matrices and Linear Algebra The e
Page 40 and 41:
1 Matrices and Linear Algebra The L
Page 42 and 43:
1 Matrices and Linear Algebra In co
Page 44 and 45:
1 Matrices and Linear Algebra Matri
Page 46 and 47:
1 Matrices and Linear Algebra compu
Page 48 and 49:
1 Matrices and Linear Algebra Eigen
Page 50 and 51:
1 Matrices and Linear Algebra produ
Page 52 and 53:
1 Matrices and Linear Algebra Singu
Page 54 and 55:
1 Matrices and Linear Algebra 1-44
Page 56 and 57:
2 Polynomials and Interpolation Pol
Page 58 and 59:
2 Polynomials and Interpolation pol
Page 60 and 61:
2 Polynomials and Interpolation Cal
Page 62 and 63:
2 Polynomials and Interpolation whe
Page 64 and 65:
2 Polynomials and Interpolation Int
Page 66 and 67:
2 Polynomials and Interpolation int
Page 68 and 69:
2 Polynomials and Interpolation 2 G
Page 70 and 71:
Page 72 and 73:
2 Polynomials and Interpolation for
Page 74 and 75:
2 Polynomials and Interpolation Con
Page 76 and 77:
2 Polynomials and Interpolation Lat
Page 78 and 79:
2 Polynomials and Interpolation −
Page 80 and 81:
2 Polynomials and Interpolation −
Page 82 and 83:
2 Polynomials and Interpolation 7 3
Page 84 and 85:
2 Polynomials and Interpolation T =
Page 86 and 87:
2 Polynomials and Interpolation Bec
Page 88 and 89:
Page 90 and 91:
2 Polynomials and Interpolation 2-3
Page 92 and 93:
2 Polynomials and Interpolation 2-3
Page 94 and 95:
3 Fast Fourier Transform (FFT) Intr
Page 96 and 97:
3 Fast Fourier Transform (FFT) 200
Page 98 and 99:
3 Fast Fourier Transform (FFT) 2 x
Page 100 and 101:
3 Fast Fourier Transform (FFT) 500
Page 102 and 103:
3 Fast Fourier Transform (FFT) Func
Page 104 and 105:
4 Function Functions Function Summa
Page 106 and 107:
4 Function Functions You can also c
Page 108 and 109:
4 Function Functions 25 20 15 10 5
Page 110 and 111:
4 Function Functions Minimizing Fun
Page 112 and 113:
4 Function Functions To try fminsea
Page 114 and 115:
4 Function Functions Plotting the R
Page 116 and 117:
4 Function Functions The number of
Page 118 and 119:
4 Function Functions and displays t
Page 120 and 121:
4 Function Functions end function s
Page 122 and 123:
4 Function Functions States of the
Page 124 and 125:
4 Function Functions 100 80 60 40 2
Page 126 and 127:
4 Function Functions You can verify
Page 128 and 129: 4 Function Functions Troubleshootin
Page 130 and 131: 4 Function Functions A three-dimens
Page 132 and 133: 4 Function Functions Parameterizing
Page 134 and 135: 4 Function Functions “Anonymous F
Page 136 and 137: 5 Differential Equations Initial Va
Page 138 and 139: 5 Differential Equations odephas3 o
Page 140 and 141: 5 Differential Equations This secti
Page 142 and 143: 5 Differential Equations The basic
Page 144 and 145: 5 Differential Equations 2 Code the
Page 146 and 147: 5 Differential Equations function.
Page 148 and 149: 5 Differential Equations One way to
Page 150 and 151: 5 Differential Equations y0, yp0 Ve
Page 152 and 153: 5 Differential Equations For exampl
Page 154 and 155: 5 Differential Equations 1 0.8 0.6
Page 156 and 157: 5 Differential Equations 2.5 Soluti
Page 158 and 159: 5 Differential Equations To run thi
Page 160 and 161: 5 Differential Equations 2 u′ i =
Page 162 and 163: 5 Differential Equations dydt(i+1,:
Page 164 and 165: 5 Differential Equations refine = 4
Page 166 and 167: 5 Differential Equations Example: A
Page 168 and 169: 5 Differential Equations function [
Page 170 and 171: 5 Differential Equations Note The R
Page 172 and 173: 5 Differential Equations 1 Robertso
Page 174 and 175: 5 Differential Equations The MATLAB
Page 176 and 177: 5 Differential Equations Summary of
Page 180 and 181: 5 Differential Equations Error Tole
Page 182 and 183: 5 Differential Equations Troublesho
Page 184 and 185: 5 Differential Equations Function d
Page 186 and 187: 5 Differential Equations dde23 prod
Page 188 and 189: 5 Differential Equations The exampl
Page 190 and 191: 5 Differential Equations solution y
Page 192 and 193: 5 Differential Equations Example: C
Page 194 and 195: 5 Differential Equations Changing D
Page 196 and 197: 5 Differential Equations BVP Functi
Page 198 and 199: 5 Differential Equations two-point
Page 200 and 201: 5 Differential Equations The input
Page 202 and 203: 5 Differential Equations 2 Pass the
Page 204 and 205: 5 Differential Equations 3 Code the
Page 206 and 207: 5 Differential Equations Finding Un
Page 208 and 209: 5 Differential Equations vectorized
Page 210 and 211: 5 Differential Equations There is a
Page 212 and 213: 5 Differential Equations 3 Solve on
Page 214 and 215: 5 Differential Equations hold off T
Page 216 and 217: 5 Differential Equations Note The d
Page 218 and 219: 5 Differential Equations legend('An
Page 220 and 221: 5 Differential Equations Here, v(1-
Page 222 and 223: 5 Differential Equations solution v
Page 224 and 225: 5 Differential Equations Note The D
Page 226 and 227: 5 Differential Equations After disc
Page 228 and 229:
5 Differential Equations The output
Page 230 and 231:
5 Differential Equations 2 Code the
Page 232 and 233:
5 Differential Equations Note See t
Page 234 and 235:
5 Differential Equations The exampl
Page 236 and 237:
5 Differential Equations and the ri
Page 238 and 239:
5 Differential Equations u1(x,t) 1
Page 240 and 241:
5 Differential Equations Selected B
Page 242 and 243:
6 Sparse Matrices Function Summary
Page 244 and 245:
6 Sparse Matrices Function Summary
Page 246 and 247:
6 Sparse Matrices This matrix requi
Page 248 and 249:
6 Sparse Matrices S = (3,1) 1 (2,2)
Page 250 and 251:
6 Sparse Matrices Now F = full(S) d
Page 252 and 253:
6 Sparse Matrices Importing Sparse
Page 254 and 255:
6 Sparse Matrices west0479 west0479
Page 256 and 257:
6 Sparse Matrices The find Function
Page 258 and 259:
6 Sparse Matrices of the rows and c
Page 260 and 261:
6 Sparse Matrices The vertices of o
Page 262 and 263:
6 Sparse Matrices 0 10 20 30 40 50
Page 264 and 265:
6 Sparse Matrices 0 500 1000 1500 2
Page 266 and 267:
6 Sparse Matrices simply sparse(m,n
Page 268 and 269:
6 Sparse Matrices Similarly, S(:,p)
Page 270 and 271:
6 Sparse Matrices The following MAT
Page 272 and 273:
6 Sparse Matrices 0 Original 0 Reve
Page 274 and 275:
6 Sparse Matrices QR Factorization
Page 276 and 277:
6 Sparse Matrices shows that A has
Page 278 and 279:
6 Sparse Matrices Functions for Ite
Page 280 and 281:
6 Sparse Matrices set up the five-p
Page 282 and 283:
6 Sparse Matrices Manipulating Spar
Page 284 and 285:
6 Sparse Matrices Selected Bibliogr
Page 286 and 287:
Index comparing sparse and full mat
Page 288 and 289:
Index H hb1dae demo 5-35 hb1ode dem
Page 290 and 291:
Index nonstiff ODE examples rigid b
Page 292 and 293:
Index LU factorization 6-30 minimum
Page 294:
Index twobvp demo 5-63 two-dimensio
show all

MATLAB Mathematics - SERC - Index of

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?