Курсовая работа_ф_3 / FreematPrimerV4e1

Figure 143: The original graph (in blue) with pulses (in red) showing where the maxima and minima occur on the original graph.

We can add additional code to our zeroCross.m script that will go through the the vector of pulses and determine their approximate x-axis positions. We can then use that knowledge to calculate the location

of these maxima and minima more precisely.

% Script to demonstrate how to calculate where zero crossings occur.

% zeroCross.m clear all; close('all');

f=@(x) (x.*cos(x.^2)); % Equation to graph x=linspace(-2,3,1024);

y=f(x); % graph of original equation dx=x(1:end-1);

diffX=x(2)-x(1); dy=diff(y)/diffX;

ys=circshift(dy,[0,-1]); % this creates the shifted version of the

% original graph.

yz=(dy.*ys)<0; % This looks for only points that are negative (less % than zero).

yz=yz(1:end-1); % This removes the wrap-around point on the end. dx=dx(1:length(yz)); % This ensures the x-axis is the same length

% as the y-axis. plot(x,y,'linewidth',2);

% plot zero crossings in red line(dx,yz,zeros(1,length(dx)),'color','r'); legend('Original Graph','Maxima / Minima Points','location',

'southwest');

c=0; for(ii=1:length(yz)-1); if(yz(ii)>0);

c=c+1; % If the pulse goes positive, increase the counter. xz(c)=x(ii); % Add the x-axis position to the array.

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end end

xz % Print out the array of points.

We can use these points as starting points for finding more precise values of the maxima and minima. We can use the Newton-Raphson method outlined in Topic 9.1.2: The Newton-Raphson Method on page 163. However, we have to use a slightly different version. For our problem finding the maxima and minima, we have to find the roots of the first differential. This means we need to calculate the first differential, and then calculate the slope of that to home in on the more precise value of the maxima and minima using the second differential. In other words, need to calculate the slope of the slope. We can generate a new function, which we'll call sloperich2, that will allow us to calculate the second numerical differentiation. This is identical to the original sloperich.m function in that it uses the fivepoint differentiation method along with Richardson extrapolation to achieve good precision. All we're doing here is substituting the slope calculations for the calculations of the original function. In other

words, we're replacing f(x) with sloperich(f(x)).

% The purpose of this function is to calculate the slope % of an equation at a particular point. This is the

% second derivative of the equation at that point.

% This uses the five-point differentiation method along

% with Richardson's extrapolation.

% filename = sloperich2.m function returnValue=slope(f,x); if(nargin<2)

printf('Error: This function requires at least a function and point as input.\n');

return end

w=0.1; % starting value for points above and below x % to calculate the slope.

s1=(sloperich(f,x-2*w)-8*sloperich(f,x-w)+8*sloperich(f,x+w)- sloperich(f,x+2*w))/(12*w);

w=w*0.5; s05=(sloperich(f,x-2*w)-8*sloperich(f,x-w)+8*sloperich(f,x+w)- sloperich(f,x+2*w))/(12*w);

w=w*0.5; s025=(sloperich(f,x-2*w)-8*sloperich(f,x-w)+8*sloperich(f,x+w)- sloperich(f,x+2*w))/(12*w);

w=w*0.5; s0125=(sloperich(f,x-2*w)-8*sloperich(f,x-w)+8*sloperich(f,x+w)- sloperich(f,x+2*w))/(12*w);

w=w*0.5; s00625=(sloperich(f,x-2*w)-8*sloperich(f,x-w)+8*sloperich(f,x+w)- sloperich(f,x+2*w))/(12*w);

s21=(4*s05-s1)/3; s205=(4*s025-s05)/3; s2025=(4*s0125-s025)/3; s20125=(4*s00625-s0125)/3; s31=(16*s205-s21)/15; s305=(16*s2025-s05)/15; s3025=(16*s20125-s2025)/15;

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s41=(64*s305-s31)/63; s405=(64*s3025-s305)/63; s51=(256*s405-s41)/255; returnValue=s51;

We can use this in our previous script to calculate the precise point of the maxima and minima.

% Script to demonstrate how to calculate where zero crossings occur. % zeroCross.m

clear all; close('all');

f=@(x) (x.*cos(x.^2)); % Equation to graph x=linspace(-2,3,1024);

y=f(x); % graph of original equation dy=slope1(f,x);

ys=circshift(dy,[0,-1]); % this creates the shifted version of the

% original graph.

yz=(dy.*ys)<0; % This looks for only points that are negative (less % than zero).

yz=yz(1:end-1); % This removes the wrap-around point on the end. dx=x(1:length(yz)); % This ensures the x-axis is the same length

% as the y-axis.

% Calculate the approximate points where the maxima and minima occur. % This is possible by looking at which points have a positive impulse. c=0;

for(ii=1:length(x)-1); if(yz(ii)>0);

c=c+1; % If the pulse goes positive, increase the counter. xz(c)=x(ii); % Add the x-axis position to the array.

end end

printf('Approximate maxima and minima points'); xz'

% Refine the location of the maxima and minima by using the

% Newton-Raphson method. This is done by starting with the zero-

% crossing points of the first differential and calculating the % slope of them using the second differential. Then use these two

% components in the Newton-Raphson method to converge on a (fairly % accurate) solution of where the maxima and minima occur. epsilon=1e-5;;

for(ii=1:c); clear p;

k=1; % Start the counter at 1.

p(k)=xz(ii); % This is the initial guess for the NR method. fp=slope1(f,p(k));

fp2=slope2(f,p(k)); p(k+1)=p(k)-fp/fp2; while(abs(p(k+1)-p(k))>epsilon) k=k+1;

fp=slope1(f,p(k));

fp2=slope2(f,p(k)); p(k+1)=p(k)-fp/fp2; end

xz(ii)=p(end); end

printf('Precise points'); xz'

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Running this script, we get this:

--> format long --> zeroCross

Approximate maxima and minima points ans =

-1.81915933528837 -0.81231671554252

0.80547409579668

1.81231671554252

2.52101661779081 Precise points

ans = -1.81447238096425

-0.80825193293577

0.80825193293577

1.81447238101386

2.52222528574970

This gives us the relatively precise points where the maxima and minima occur.

Topic 9.3.2.4: Telling if its Maxima or Minima

The next question we can ask is this: did we find a maxima or a minima. It's going to be one or the other. Sure, we can just look at the graph and tell. But wouldn't it be nice if Freemat could figure it out for us? There's a way to do that. Look back at Figure 142 on page 210. Notice how the graph of the first differential has a particular slope depending on whether the original graph is a maxima or minima? The graph of the first differential will have a positive slope for a minima, and a negative slope for the maxima. This means that if we calculate the second differential at the maxima or minima, it will be negative for the maxima and positive for the minima. From the previous script, we already have the variable xz that has stored all of the x-axis points where the maxima / minima occur. We can create another short script that will go through each value, use the sloperich2 function we created above, and

determine if the peak at that point is a maxima or a minima.

% Determine if point is a maxima or a minima % filename=maxMinDetermine.m

% NOTE: This script requires the function

% zeroCross.m be run before this script is run. for(ii=1:length(xz)); if(sloperich2(f,xz(ii))<0);

% If the second differential is negative, its a maxima. printf('The point at %f is a maxima.\n',xz(ii));

else

% Otherwise, its a minima.

printf('The point at %f is a minima.\n',xz(ii)); end

end

Let's give it a try.

--> maxMinDetermine

The point at -1.814472 is a maxima.

The point at -0.808252 is a minima.

The point at 0.808252 is a maxima.

The point at 1.814472 is a minima.

The point at 2.522225 is a maxima.

Comparing these results with the graph shown in Figure 142, it appears that this works just fine.

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Appendix A: Special Function to Resize Plots with Titles and Labels

As of this writing, there are a couple of bugs in way that Freemat handles figures and plots. One bug is the position of the text when using the xlabel command (as shown in Topic 6.3.3.2 Setting X-Axis

(Horizontal) & Y-Axis (Vertical) Labels). Another bug is the fact with the command: get(gca,'position')

This command works fine when entered manually in the Command Window. However, if you attempt to use it in either a script or a function, it returns incorrect data. Until these are corrected, here's a function that you can save as the file labelSet.m somewhere in your path. And you can use it whenever you create a graph that has a title and/or labels (x-axis, y-axis or both). While it is far from perfect,

your graph should have a decent enough appearance for publishing.

% This function will maximize the graph size

% based on the size of any titles or labels % on the graph.

% THIS FUNCTION SHOULD BE RUN AFTER ALL TITLES % AND LABELS HAVE BEEN ADDED!

function labelSet

xaxisMarks=24; % This is the space needed for the x-axis numeric labels.

yaxisMarks=40; % This is the space needed for the y-axis numeric labels.

marginX=20; % Extra padding for the x-axis. marginY=20; % Extra padding for the y-axis. curHand=gca; % Get the current figure handle.

curSize=get(gcf,'figsize'); % Get the size of the current figure. curTitle=get(curHand,'title'); % Get the handle of the title (if any). if(isempty(curTitle)); % If there is no title, skip this section. titleFontsize=0;

else titleFontsize=1.5*get(curTitle,'fontsize'); set(curTitle,'verticalalignment','middle'); set(curTitle,'horizontalalignment','center'); if(titleFontsize<20)

titleFontsize=20; end

end curXlabel=get(curHand,'xlabel'); if(isempty(curXlabel)); xFontsize=0;

else xFontsize=2.5*get(curXlabel,'fontsize');

set(curXlabel,'horizontalalignment','center');

set(curXlabel,'verticalalignment','bottom');

if(xFontsize<20);

xFontsize=20; end

end curYlabel=get(curHand,'ylabel'); if(isempty(curYlabel)); yFontsize=0;

else

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Appendix B: The Histogram Function

As of this writing, there is a problem with the hist function. It does not appear to operate correctly. Until it is corrected, you can use this function, saving it as hist.m in one of the folders in your path. It

function [rv r]=hist(input_array,bins,normal,partial) if(nargin>4);

printf('ERROR: The hist function requires no more than 4 arguments.\n');

return end

if(nargin<4)

partial='full'; end if(nargin==1); bins=10; normal=0;

max_amp=max(input_array); min_amp=min(input_array); bin_val=linspace(min_amp,max_amp,bins+1); w=bin_val(2)-bin_val(1); r=min_amp+w/2:w:max_amp-w/2; elseif(nargin==2);

normal=0;

s=size(bins);

if(s(1)>s(2))

bins=bins'; end

if(sum(diff(diff(s)))>0)

printf('The bin array must be equally spaced.\n'); return

end if(isscalar(bins))

max_amp=max(input_array); min_amp=min(input_array); bin_val=linspace(min_amp,max_amp,bins+1); w=(max_amp-min_amp)/bins; r=(min_amp+w/2):w:(max_amp-w/2);

else r=bins;

w=bins(2)-bins(1); max_amp=max(bins)+w/2; min_amp=min(bins)-w/2;

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