Ordinary Differential Equations

OutlineAnnouncements:Homework II: Solutions on webHomework III: due Wed. by 5, by e-mailStructsDifferential EquationsNumerical solution of ODEsMatlabs ODE solversOptimization problemsMatlabs optimization routines

Stringscharcell={Greetings; People of Earth}disp(charcell) will output each cell on a new lineGreetingsPeople of EarthSearch lines using strmatchstrmatch('Greetings',charcell)Encapsulating dataFFTcell={a, b, dt}myfft could return a cell array in this formWould keep the related data togetherBut, user would have to know what is in each cellApplications of Cell-arrays

StructsRather than storing data in elements or cells, structs store data in fieldsBetter encapsulation than cellFFTstrct.a=a;FFTstrct.b=b;FFTstrct.dt=dt;Can have arrays of structsfor j=1:nstrctarr(j)=StrctFunc(inputs);endeach element must have same fields

Working with StructsLots of commands for working with structsfieldnames(S)--returns a cell array of strings containing names of fields in Sisfield(S,fname)--checks whether fname is a field of Srmfield(S,fname)--removes fname

Differential EquationsOrdinary differential equations (ODEs) arise in almost every fieldODEs describe a function y in terms of its derivativesThe goal is to solve for y

Example: Logistic GrowthN(t) is the function we want (number of animals)

Numerical Solution to ODEsIn general, only simple (linear) ODEs can be solved analyticallyMost interesting ODEs are nonlinear, must solve numericallyThe idea is to approximate the derivatives by subtraction

Euler Method

Euler MethodSimplest ODE scheme, but not very good1st order, explicit, multi-step solverGeneral multi-step solvers:(weighted mean of f evaluated at lots of ts)

Runge-Kutta MethodsMulti-step solvers--each N is computed from N at several timescan store previous Ns, so only one evaluation of f/iterationRunge-Kutta Methods: multiple evaluations of f/iteration:

ODE SolversLMS and RK solvers are general algorithms that can work with any ODE Need a way to pass the function f to the solverSolverN0 t0 TN1, N2,,NT

Passing FunctionsOne soln: force us to call our function fA better soln: pass the name of the function to the solverfuncname=myfsolver uses feval commanddN=feval(funcname,N,t);This is Matlab ca. 1998

Passing FunctionsNow, feval accepts function handles@myf creates a handle to myf.mSlightly faster than stringsNow, how do we solve the ODEs?

Matlabs ODE solversMatlab has several ODE solvers:ode23 and ode45 are standard RK solversode15s and ode23s are specialized for stiff problemsseveral others, check help ode23 or book

All solvers use the same syntax:[t,N]=ode23(@odefile, t, N0, {options, params })odefile is the name of a function that implements ffunction f=odefile(t, N, {params}), f is a column vectort is either [start time, end time] or a vector of times where you want NN0= initial conditionsoptions control how solver works (defaults or okay)params= parameters to be passed to odefileMatlabs ODE solvers

ParametersTo accept parameters, your ode function must be polymorphicf=odefile(t,x);f=odefile(t,x,params);

function f=odefile(t,x,params);if(nargin

Matlab functions can be polymorphic--the same function can be called with different numbers and types of argumentsExample: plot(y), plot(x,y), plot(x,y,rp);In a function, nargin and nargout are the number of inputs provided and outputs requested by the caller.Even more flexibility using varargin and varargoutnargin

Example: Lorenz equationsSimplified model of convection cells

In this case, N is a vector =[x,y,z] and f must return a vector =[x,y,z]

OptimizationFor a function f(x), we might want to knowat what value of x is f smallestlargest?equal to some value?All of these are optimization problemsMatlab has several functions to perform optimizationSame problem as with ODE: solver needs to work with YOUR functionsMust pass function handle

OptimizationSimplest functions arex=fzero(@fun, x0);%Finds f(x)==0 starting from x0x=fminbnd(@fun, [xL, xR]); %Finds minimum in interval [xL,xR]x=fminsearch(@fun,x0); %Finds minimum starting at x0All of these functions require you to write a function implementing f:function f=fun(x);%computes f(x)

Optimization ToolboxMore optimization techniques, but used in the same waySome of these functions can make use of gradient informationin higher dimensions, gradient tells you which way to go, can speed up procedurefdf/dx = 0

Optimization ToolboxTo return gradient information, your function must be polymorphic:x=fun(x) %return value[x,dx]=fun(x); %return value and gradient

function [x,dx]=fun(x);if(nargout>1)dx=endx=

Optimization OptionsNeed to tell optimization toolbox that it can use gradientUse optimset:opts=optimest(ParameterName,value);Then pass opts to optimization functionLots of options--read docs!opts=optimset(GradObj,on);%Turns gradient onx=fminunc(@fun, x0,opts);%pass opts to function

SummaryODE solvers & optimization routines in Matlab are function functionsYou must write a function returningdx/dt (ODE)f (Optimization)You pass the name of the function to the solverSolver calls it repeatedly and returns answer