MATLAB: Does roots have an issue with large coefficients

roots vpasolve

I have noticed that the values I obtain with roots are different than those that I find with vpasolve (closer to expected).

Best Answer

Why would you be even remotely surprised? Roots works in DOUBLE precision arithmetic. Of course big coefficients can be a problem, because floating point arithmetic will be involved.
For example, see how easy it is to trip roots up:
syms x
P = expand((x-1)*(x - 1e10 + i)*(x - 1e10 - i))
P =
x^3 - 20000000001*x^2 + 100000000020000000001*x - 100000000000000000001
vpasolve(P==0)
ans =
                  1.0
 10000000000.0 - 1.0i
 10000000000.0 + 1.0i
roots([1 -20000000001 100000000020000000001 -100000000000000000001])
ans =
        1e+10 +     147.33i
        1e+10 -     147.33i
            1 +          0i
That is not always the case of course. So if I just pick some set of large random coefficients, roots will probably do ok.
format long g
C = [1,rand(1,4)*1e15]
C =
                         1           184342604119778           162956226687924            75790019563777           436978740562955
roots(C)
ans =
           -184342604119778 +                     0i
          -1.57673059792158 +                     0i
          0.346372446939334 +      1.17619511597991i
          0.346372446939334 -      1.17619511597991i
vpasolve(x^4 + C(2)*x^3  + C(3)*x^2  + C(4)*x  + C(5) == 0)
ans =
                                              -184342604119777.5535142959570983444967693
                                              -1.576730598062861967656730050376918994998
 0.3463724470099801560767496588525252066424 - 1.176195116160874500159456451136807293786i
 0.3463724470099801560767496588525252066424 + 1.176195116160874500159456451136807293786i
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