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How to find constants when 2 equations are equal?
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Hi there! I have a function of (x) (and it's polynomial,degree=5) and I want to make it equal to "c1*exp(c2*x)". How can I find these two constants???? I would appreciate your helps...
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Star Strider
2016-5-31
编辑:Star Strider
2016-5-31
You can fit your polynomial with the core MATLAB funciton fminsearch and an extra line of code:
pf = @(x) x.^4 - 2*x.^3 + 3*x.^2 - 5*x + 4; % Polynomial Function
ef = @(c,x) c(1).*exp(c(2)*x); % Exponential Function
x = linspace(0, 5, 10); % Vector Of ‘x’ Values (Arbitrary)
SSECF = @(c) sum((pf(x) - ef(c,x)).^2); % Sum Squared Error Cost Function
c0 = [1; 1]; % Initial Parameter Estimates
[c, SSE] = fminsearch(SSECF, c0) % Estimate ‘c’, Return Sum-Squared-Error At Convergence
figure(1)
plot(x, pf(x), 'bp')
hold on
plot(x, ef(c,x), '-r')
hold off
grid
Substitute your own polynomial function. The plot is optional.
EDIT — You need to include a y-offset term. With that, a single exponential fits very well:
d = load('Amin Moosavi a3.mat');
x = d.locs;
y = d.pks;
ef = @(c,x) c(1).*exp(c(2)*x) + c(3); % Exponential Function
SSECF = @(c) sum((y - ef(c,x)).^2); % Sum Squared Error Cost Function
c0 = [100; -1; 400]; % Initial Parameter Estimates
[c, SSE] = fminsearch(SSECF, c0) % Estimate ‘c’, Return Sum-Squared-Error At Convergence
figure(1)
plot(x, y, 'bp')
hold on
plot(x, ef(c,x), '-r', 'LineWidth',1)
hold off
grid
text(150, 470, sprintf('f(x) = %.1f\\cdote^{%.4f\\cdotx} + %.1f', c))
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11 个评论
Star Strider
2016-6-1
As always, my pleasure!
I wish the same for you.
I doubt I’m going to do any traveling anytime soon, but thank you for the invitation.
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Walter Roberson
2016-5-31
You cannot get out any meaningful results if you are working with a single x. In such a case the exact solution is
c2 = ln(a5*x^5+a4*x^4+a3*x^3+a2*x^2+a1*x+a0-c1)/x
so as you change c1, c2 changes as well, and for any given x you can force c2 to become imaginary (by using a large enough c1 that the polynomial minus c1 becomes negative, leading to ln() of a negative number.
For any two given x = x1, x2, you would have
c1 = a5*x2^5+a4*x2^4+a3*x2^3+a2*x2^2+a1*x2-exp(RootOf(exp(Z)-a5*x2^5-a4*x2^4-a3*x2^3-a2*x2^2-a1*x2+a5*x1^5+a4*x1^4+a3*x1^3+a2*x1^2+a1*x1-exp(Z*x1/x2)),Z)+a0
c2 = RootOf(exp(Z)-a5*x2^5-a4*x2^4-a3*x2^3-a2*x2^2-a1*x2+a5*x1^5+a4*x1^4+a3*x1^3+a2*x1^2+a1*x1-exp(Z*x1/x2),Z)/x2
here, RootOf(f(Z),Z) means the set of Z such that f(Z) becomes 0 -- the roots of the expression. You can see that the values you calculate would be very much tied up to the exact values of x1 and x2.
If you were hoping to take a series of values for the polynomial and fit the best c1, c2 for the entire series, then the result is going to depend a lot on which values you fit against.
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