How to find the roots along a real line?

Hi guys,

I am trying to determine the first 5 eigen frequencies of a bending beam with rotational and translational spring supports. This is done by setting the determinant of the coefficient matrix equal to zero. I use vpasolve to solve the equation, as I am solving a syms variable. The first 5 values I show get are: (found using Maple)

0.00285611175

0.0279531203

0.0435503148

0.0659083580

0.0912286729

But when I try solving this in matlab I get the following output:

0.00285611175750407

0.0279531203899501

0.0435503148892885

0.091228672905353

Can someone tell me why matlab does not find the 0.0659 value?

Is there a Matlab function to find the roots along a real line? My code is not very efficient due to the fact that I must first figure out in what range the first 5 eigen values lay before I can use the function I made. My code is attached.

Any advice would be appreciated!

Answer 1

[Dana]

Numerical solvers are not guaranteed to find all roots of an expression. That's what's happening here. The algorithm vpasolve is using doesn't "notice" the root near 0.0659. If your start/end range bracketed that value, it would find it however:

>> vpasolve(freq,B, [.99,1.01]*0.0659)
ans =
0.06590835809882684726711430520395

Unless you can find a way to bracket each root like this, you'll never be 100% sure to catch all of them. One option to increase the likelihood of finding every root is to start with a small bracket, then gradually increase the upper bound until you find a root, then shift your bracket over. So for example, if you instead make your FindBeta function something like:

function Bet = FindBeta(FrequencyEquation, StartAt, EndAt, NumberOfBetas)
    
    Bet = zeros(NumberOfBetas,1);
    syms B

    incStrt = 1e-9; % increment from previous root to start search for next root
    incEnd = 1e-2; % increment to add to upper bound of search range each time
    StrtRg = StartAt; % initial lower bound of search range
    for i = 1:NumberOfBetas
        EndRg = StrtRg + incEnd; % set initial upper bound of search range
        NoRoot = true;  % = true when we still haven't found the i-th root
        while NoRoot
            % try to get a root:
            Beti = vpasolve(FrequencyEquation,B, [StrtRg; EndRg]);
            
            if isempty(Beti)    % if we didn't find a root
                if EndRg > EndAt    % if we've already exceeded EndAt
                    Beti = NaN;         % indicate no such root exists using NaN
                    NoRoot = false;     % stop searching
                end
                
                EndRg = EndRg + incEnd; % increase upper bound of search range
                                        %   and then try again
            else                % if we found a root
                if Beti > EndRg     % if the root exceeds EndAt
                    Beti = NaN;         % indicate no such root exists using NaN
                end
                NoRoot = false;     % flag that a root has been found
            end
        end
        
        if isnan(Beti)  % if the upper bound has been exceeded
            Bet(i:end) = NaN;   % set all remaining roots to NaN
            return;             % and then exit function
        else            % otherwise
            Bet(i) = Beti;              % store found root
            StrtRg = Beti + incStrt;    % increase the start range and repeat
        end
    end
end

Then:

Beta =
       0.00285611175750407
        0.0279531203899501
        0.0435503148892885
        0.0659083580988268

The parameter incEnd is a key one that must be tuned somehow. Smaller values make it more likely that you'll find all the roots, but at the cost of increasing the amount of time it takes to do so.

[KT]

Hi Dana!

Thank so much for the reply! This really help!