How to force slope to be zero at a particular point using function PolyFit?

Q1 = 1xn;

T = 1xn;

Finding a curve fitting (T,Q1) such that slope is zero at Q1(i), where i = 1,2,...,n

Answer 1

[Teja Muppirala]

If you have LSQLIN in the Optimization Toolbox, this can be done with a little bit of effort, as described here http://www.mathworks.com/support/solutions/en/data/1-12BBUC/

% Making some random data
Q1 = 0:0.01:1;
T = cos(2.1*pi*Q1)+0.2*randn(size(Q1));

plot(Q1,T,'k.');

% Polyfit without constraints
order = 4;
C1 = polyfit(Q1,T,order);
hold on;
plot(Q1,polyval(C1,Q1))

V = bsxfun(@power,Q1(:),order:-1:0); % Make Vandermonde Matrix

% Make Constraints on the derivatives
Aleft = [(order:-1:1).*Q1(1).^(order-1:-1:0) 0]; 
Aright = [(order:-1:1).*Q1(end).^(order-1:-1:0) 0];
Aeq = [Aleft; Aright];
beq = [0;0]; %Value of the derivative is set to zero

% Call LSQLIN with options to prevent warnings
opts = optimset('lsqlin');
opts.LargeScale = 'off';
C2 = lsqlin(V,T,[],[],Aeq,beq,[],[],[],opts);

plot(Q1,polyval(C2,Q1),'r')
hold off;

legend({'Data' 'Polyfit' 'Constrained Polyfit'},'location','best');

[Harshit Jain]

thanks for the answer. I have done it through spline. It works fine and is very easy.

Answer 2

[Matt J]

If the slope is zero at all i=1...n, it means you are fitting Q1 with a constant.

p=mean(Q1);

[Harshit Jain]

Slope is not zero at all the points.

Eg - only at i = 1 and n.

Answer 3

[Image Analyst]

Your assignment says nothing about requiring the polyfit() function. There's no way to have the slope be 0 at "1" and "end" in general, unless you use Matt's solution. For specific functions, e.g. 4th order, you may luck out if your data happens to go in between the right elements, e.g. the two humps of a 4th order, but in general that won't happen. So, I'd probably use a spline. You might have to replicate the first and last value of the array to make sure that the slope is zero there.

[Harshit Jain]

Thanks. Got it

[Matt J]

You might have to replicate the first and last value of the array to make sure that the slope is zero there.

MATLAB's SPLINE command let's you specify the endslopes directly. For more general constrained splines, there is this FEX tool

http://www.mathworks.com/matlabcentral/fileexchange/24443-slm-shape-language-modeling

[Image Analyst]

Thanks for pointing that out. I didn't know that and it's nice to know.

Answer 4

[Jan]

You find in the FileExchange polynomial fits with constraints:

• http://www.mathworks.com/matlabcentral/fileexchange/38926-fit-polynomial-to-data-with-constraints
• http://www.mathworks.com/matlabcentral/fileexchange/220-polyfit2/content/polyfit2.m

[Harshit Jain]

thanks for the files.