Bez tytulu slajdu

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A First Example

• Using the Interactive or Calculator Mode

gtgt tlinspace(0,2,100) gtgt foft-t.exp(-4t.2)
gtgt plot(t,foft) gtgt xlabel('t') gtgt
ylabel('f(t)') gtgt title('Example Plot')
Sequential, Interpreted, Variable-Based,
Algebraic, Graphic,...
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Polynomial Manipulation

• Matlab uses a row vector of coefficients to
  represent a polynomial
• Many functions to apply...
• roots()
• poly()
• conv()
• , -
• polyval()
• lookfor polynomial

p(s)(s1)(s-1)(s-2) s3-2s2-s2
prts-1 1 2 ppoly(prts) p 1 -2
-1 2 polyval(p,prts) ans 0 0
0 slinspace(-2,3,100) pofspolyval(p,s)
plot(s,pofs)
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Curves in 3-D

• 3-D "Static" Curves3-D "Animation"

z00.15(2pi) fexp(jz) xreal(f)
yimag(f) figure(1),plot3(x,y,z)
figure(2),comet3(x,y,z)
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Surfaces in 3-D

• Simple
• Nifty
• Really Cool

X,Y,Zpeaks(25) figure(gcf),mesh(X,Y,Z)
figure(gcf),surf(X,Y,Z)
figure(gcf),surfl(X,Y,Z) shading('interp')
colormap('copper')
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A Foundation for Formulation of MATLAB 3-D Plots

• The Traditional Approach is to Create A Data Grid
  (Uniformly Sampling)

x-pi0.1pi yx X,Ymeshgrid(x,y)
Zcos(X).sin(Y) surfl(X,Y,Z) shading
interp colormap bone rotate3d on
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Visualizing Numerical Solutions of the PDEs
(ODEs, etc)

• Traditional 2-D Plots
• 3-D Possibilities

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Summary of Basic Commands
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Basic Looping

• The for loop is an old friend?
• Columnar assignment at initialization and at each
  loop.
• Large collections of these can be very, very
  slow.
• A clever concept, the empty matrix?

sum0 for n110 sumsumn end sum
sumzeros(3,1) for nrand(3,5)
sumsumn end sum
net for n110 netnet n2 end net
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Win the Prize, "Matricize"(now that I've shown
you some bad habits...)

• A large composition of for loops will run much
  slower than matrix algebraic formulations

x for a120, tmp for b110
tmptmp a end xx tmp end x
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
3 3 3 3 3 ... 9 9 9 9 9 9 9 9 9 9 ...
x120'ones(1,10)
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More Basic Looping

• The while loop is a favorite construct for
  iteration.

sum0 n1 while abs(1-sum)gteps
sumsum(1/2)n nn1 end sum, n
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Basic Conditional Tests

• The conditional if should be familiar

• Beware of the nesting construct, end terminates
  the most inner test

xrand(1) if xgt0.5 disp('Larger') else
disp('Smaller') end
xrand(1) if xgt0.5 disp('Larger') elseif
xgt0.25 Versus else if disp('Medium') else
disp('Smaller') end
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Advanced Conditional Tests

• You've never tested data like this before!

• switch/case is similar to other language
  implementations.

xrand(1,100) if any(xgt0.5)
x(xgt0.5)0.5 end plot(x)
xceil(100rand(1)) disp(sprintf('d is
',x)) switch rem(x,2) case 0,
disp('EVEN') case 1, disp('ODD')
otherwise, disp('VERY ODD') end
xrand(1,100) if any(xlt0.3)
x(xlt0.3) end plot(x)
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Summary of Basic Control Commands
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Extending the First Example

• Putting Your Work Into A MATLAB Script

MYFILE Example script tlinspace(0,2,100) foft
-t.exp(-4t.2) figure(gcf) plot(t,foft) xlabel(
't') ylabel('f(t)') title('Example Plot') End
of myfile.m
gtgt myfile
Extensible, Seamless Graphics, ...
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Review Basics of Function M-Files

• The all important interface line
• Match name of function with that of m-file (VERY
  STRONGLY SUGGESTED)
• Fully understand whether vectorization is
  desired (the dot)

function thevalfname(x) FNAME Function
m-file thevalfname(x) theval1./x
Function m-file fname.m
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Some Numeric Tasks

• Plotting of a function can be automated
• Numerical integration
• Optimization (zeros, local minima)

ezplot(1/x) fplot(fname,0.1
10) quad(fname,1,10) log(10) fmin(fname,
1,10)
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Summary of FunFun Functions
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Taking the First Example Farther

• Putting Your Work Into A MATLAB Function

function foftmyfunc(t) MYFUNC Example
function foft-t.exp(-4t.2) figure(gcf) plot(t
,foft) xlabel('t') ylabel('f(t)') title('Example
Plot') End of myfunc.m
gtgt myfunc( 00.011 )
Fully Structured Language, (selection and
repetition structures also) .
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Applying Some Numerical Tools

• Standard MATLAB Toolbox Functions

fminbnd('myfunc',0,1) ans 0.3535
fzero('myfunc',0.1) ans -2.1156e-19
quad('myfunc',0,2) ans -0.1250
Accurate, Efficient Numerical Methods, ...
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The Typical MATLAB Project

• A Mix of Script and Function m-files, Figures and
  Data

RUNFILE Script m-file to command the
overall modeling and solution process ...
function resafunca(p1,p2) FUNCA Model or
utility function ...
function resb1,resb2funcb(p1,p2,p3) FUNCB
Model or utility function ...
... Input and/or output data in text, binary,
worksheet or other forms...
Simple Text Files, No Compiler to Deal With,
Rapid Modeling, ...
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Systems of Linear Equations

• Formulate in Linear Algebraic Terms

6x14x2-3x312 5x1-10x2x38 x112x2-4x31
A6 4 -3 5 -10 1 1 12 -4 b12 8 1
xinv(A)b x 3.5714 1.5000 5.1429

• Other More Robust Methods

A6 4 -3 5 -10 1 b12 8
xpinv(A)b x 1.7182 0.0032 -0.5593
6x14x2-3x312 5x1-10x2x38
LINPACK and EISPACK are at your disposal
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ODEs (Initial Value Problems)

• Formulate as a Set of Coupled First Order ODEs

d2v/dt2(R/L)(dv/dt)(1/LC)v(1/LC)sin(wt),
v(0)1, dv(0)/dt0
x1(t) v(t) x2(t) dv(t)/dt
dx1(t) /dt x2(t) dx2(t)/dt (1/LC)
sin(wt)-(R/L)x2(t)-(1/LC)x1(t)

• Code as a Function m-file in ODE Solver Format

function xdotmyode(t,x) MYODE Model of second
order systemglobal R L C w xdotzeros(2,1) xdot(
1)x(2) xdot(2)(1/(LC))sin(wt)-(R/L)x(2)-(1/
(LC))x(1) End of function m-file myode.m
Runge-Kutta, Adams-Bashforth, Adaptive Stepsize,
Stiff Solvers...
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ODEs (Initial Value Problems)

• Write a Script to Drive the Solution and Results

MYODERUN Numerical solution of second order
systemglobal R L C w R1000 L0.001 C1e-6
w2pi315 x01 0 td0 0.01 t,xode45('my
ode',td,x0) plotyy(t,x(1,),t,x(2,)) xlabel('t,
sec'), title('Numerical System Solution') End
of script m-file myoderun.m

• Run the Script

gtgt myoderun
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Simple Polynomial Fitting

• Typical setting is the task of fitting measured
  data with an analytical function (a favorite is a
  polynomial)
• Things to Try...
• Increase the order of the specified fit (larger
  is not always better)
• Decrease the order of the specified fit ("smooth"
  the fit)
• Add more "noise" to the data which is being "fit"

PD A "test" of the curve fitting capabilities
of MATLAB xrough0110 xsmooth00.110 yp
olyval(1 -11 31 -21,x)5randn(size(xrough)) pc
oefspolyfit(xrough,y,3) plot(xrough,y,'ko',xsmoot
h,polyval(pcoefs,xsmooth),'b-') End of script
m-file pd.m
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Spline Fits (Piecewise Polynomials)

• With local attention (piecewise), much better
  behavior
• Things to Note...
• The piecewise polynomials are available
  viappspline(xrough,y)
• The re-evaluation of the fit can then be carried
  out viaysmoothppval(pp,xsmooth)

SP A "test" of the spline curve fitting
capabilities of MATLAB xrough0110
xsmooth00.110 ypolyval(1 -11 31
-21,x)5randn(size(xrough)) ysmoothspline(xrou
gh,y,xsmooth) plot(xrough,y,'ko',xsmooth,ysmooth,
'b-') End of script m-file sp.m
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Advanced Fits Via Parameter Searching

• Utilize the "local" minimum search capabilities
  of MATLAB's fminsearch() Example y(x)A
  exp(ax)sin(wxf) where A, a, w
  and f are unknown
• Algorithm
• Write an objective function which computes the
  "distance" between measured y(x) and that
  obtained by a parameter guess.
• Let fminsearch() "search" for a set of parameters
  which "locally" minimize the error.

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Advanced Fits Continued
function thediffwierdfunc(plist) WIERDFUNC
Example function for nonlinear fit
solution global xraw yraw yfitplist(1)exp(plist
(2)xraw).sin(plist(3)xrawplist(4)) thediffno
rm(yraw-yfit) End of function m-file
wierdfunc.m
WIERDRUN Driver for nonlinear fit
solution global xraw yraw xrawlinspace(0,5,100)
A2 alpha-1 omega2pi phipi/4 yrawAexp(a
lphaxraw).sin(omegaxrawphi)0.1randn(size(xra
w)) slistfminsearch('wierdfunc',3 -2 pi
0.2) yfitslist(1)exp(slist(2)xraw).sin(slist(
3)xrawslist(4)) plot(xraw,yraw,'k.',xraw,yfit,'
b-') End of script m-file wierdrun.m
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Advanced Fits Continued
gtgt wierdrun slist 2.0845 -1.0308
6.3030 0.8202
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