Solve cannot find solutions where plotting shows they exist

I am trying to find points in a function, after fitting to some data, where the slope is zero. The obvious (to me) way to do this is to take the derivative of the equation and find the zero-crossings. The equation I am fitting is a difference of Gaussians (with shared mean but different standard deviations) on a linear slope. I am using a mixture of standard and symbolic functions because I am not sure how to do everything using one or the other. Basically, I

• Define my function using a function handle and using symbolic math
• Pull some reasonable initial parameters from the data
• Optimize the fit of the function
• Compute the derivative of the optimized function
• Solve the derivative for y = 0.

The following code looks, from the graph produced, like it should have four solutions for this particular data-set, but I get none.

    load yData.mat

    % MATLAB Function
    DoGoLS = @(Params, xData) ...
        Params(1) + ...
        Params(2) * exp(-.5 * ((xData - Params(3))/Params(4)).^2) - ...
        Params(5) * exp(-.5 * ((xData - Params(3))/Params(6)).^2) + ...
        Params(7) * xData

    % Symbolic Function
    syms y0 a1 x0 b1 a2 b2 c x
    SymDoGoLS(y0, a1, x0, b1, a2, b2, c, x) = ...
        y0 + ...
        a1 * exp(-.5*((x - x0)/b1)^2) - ...
        a2 * exp(-.5*((x - x0)/b2)^2) + ...
        c * x

    % Initial parameters
    DoGoLSInit = ...
        [median(yData), ...                                %y0
        max(yData) - min(yData), ...                       %a1
        size(yData, 1) / 2, ...                            %x0
        35, ...                                            %b1
        max(yData) - min(yData), ...                       %a2
        350, ...                                           %b2
        0];                                                %c

    % Optimal parameters
    DoGoLSFit = lsqcurvefit(...
        DoGoLS, ...                                        % handle
        DoGoLSInit, ...                                    % initial params
        (1:size(yData, 1))', ...                           % xData
        yData);                                            % yData

    % Compute and plot derivative of solved function
    SymDoGoLSdYdX(x) = diff(SymDoGoLS(...
        DoGoLSFit(1), ...
        DoGoLSFit(2), ...
        DoGoLSFit(3), ...
        DoGoLSFit(4), ...
        DoGoLSFit(5), ...
        DoGoLSFit(6), ...
        DoGoLSFit(7), x), x);
    subplot(2, 1, 1)
    plot((1:size(yData, 1)), yData, 'b', ...
        (-750:1750), DoGoLS(DoGoLSFit, (-750:1750)), '--r')
    axis auto; title('Data and Fit')
    subplot(2, 1, 2)
    ezplot('0', [-750, 1750]);
    hold on
    ezplot(matlabFunction(SymDoGoLSdYdX), [-750, 1750]);
    axis auto; title('Fit dY/dX')

    % Find zero-crossings
    ZCs = solve(SymDoGoLSdYdX(x) == 0, x)

Can someone explain what I am doing wrong? Or is there some better way to obtain the solution? I've attached the data

Answer 1

[John D'Errico]

No. Solve could not find an ANALYTICAL solution. The expression you have has no such solution. That a numerical solver could not find a solution is not true.

ZCs = solve(SymDoGoLSdYdX(x) == 0, x)
Warning: Cannot find explicit solution. 
> In solve (line 316) 
ZCs =
Empty sym: 0-by-1

ZCs = vpasolve(SymDoGoLSdYdX(x) == 0, x,500)
ZCs =
522.06599522322937335889187954597

[James Akula]

> The expression you have has no such solution.

Yes, I can solve it numerically using fsolve or vpasolve or other approaches, but those each only identify a single solution, which you show. If I know other starting points, I can find alternative solutions, but I don't have any a priori way of knowing where to start guessing, and with different input data, the locations will be different.

In this particular case, the correct answer has four solutions, and they are certainly "determinate" from the formula, somehow. That is, I expect the expression does have an ANALYTICAL solution. But perhaps there is something happening here I don't understand.

Why wouldn't it?