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Lecture 11

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University of California - Davis

ENG 06 – Lecture 11; 2/12/2013 Numerical Calculus and Solution of Nonlinear Equations  Numerical Calculus o Matlab has good built in methods to give numerical solutions o Have to make sure answers make sense  Review: Anonymous Functions o One line function that you can use quickly  Name = @(input)statement  Quick numerical derivative estimations o Derivative is change in y over change in x o Function diff  accepts a single vector as argument  Calculates difference between each element  Diff(y)./diff(x) estimates derivative  Caution: dimensions of slope vector will not be the same as x and y vectors, there will be one less  x=0:.001:10;
  y = x.^2;
  slope = diff(y)./diff(x); o Slope is now vector of slopes at all points corresponding to x and y vectors  Find index of array closest to 3.4 and read out value of slope  [i,k]=min(abs(x‐3.4)) i=  0 k=  3401 >> slope(k)  ans = 6.8010  Forward Difference Approx. of First Derivative o To approximate derivative at x(i), find point to the right, x(i+1), and find slope between the two points  [email protected](t,delt) (func(t+delt)‐func (t))./delt;  Backward Difference Approx of First Derivative o Look at point behind x(i) this time  f’(x(i)) = (f(x(i))-f(x(i-1)))/(x(i)-(x(i-1))  [email protected](t,delt) (func(t)‐func (t‐delt))./delt;  Central Difference Approx. of First Derivative o Take points from behind and in front of x(i) o [email protected](t,delt) (func(t+delt)‐func (t‐delt))./(2.*delt);  Error Estimates o Use taylor expansion to find difference  Example o Velocity of rocket: find acceleration at 16 seconds o Use exact expression, find forward, backward, and central difference approx.. o Numerical Estimate  >> [email protected](t) 2000*log((14*10^4)./(14*10^4‐2100.*t))‐9.8.*t;  >> [email protected](t,delt) (velocity(t+delt)‐velocity(t))./delt;
  >> [email protected](t,delt) (velocity(t)‐velocity(t‐delt))./delt;
  >> [email protected](t,delt) (velocity(t+delt)‐velocity(t‐delt))./(2.*delt);  Use time increment of 2 seconds (delt = 2)  acell_forw(16,2)  ans =30.4739
  >> acell_back(16,2)  ans =28.9145
  >> acell_cent(16,2)  ans =29.6942  The exact value is 29.674 m/s2. (center approx. is closest)  You can also get a vector of acceleration by inputting delt as a vector of increments  Integration o Measures area under a function plotted on a graph o Limits of integration: x(i-1) is lower, x(i+1) is upper  Numerical Integration  In estimating integral, two options: o If you are given discrete points  Use trapz  x = 0:.001:10;  y = x.^2;
 Z = trapz(x,y);  disp(Z)  333.3333 o If you are given a function (or if you are given discrete points and use interpolation/regression to create closed form expression)  Use quad or quadl  Give a function as a function handle from anonymous or user-defined function as well as upper and lower limits  Quad and quadl will usually give the same answer  Ex.  >> y = @(x) x.^2;  Z = quad(y,0,10)
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