Having issues with precision output of symbolic expression for a Legendre polynomial.

I am trying to determine the roots of a 3rd degree Legendre Polynomial so I can create a Gaussian Quadrature function. I am having issues with the output precision on 2021b and 2022a. I have tried both versions (updating to 2022a did not resolve my issue). The symbolic or double output only goes to 4 decimals, regardless of the formatting, or method of solving. I have spent a few hours reading through vpa, solve, symbolic, double, and format documentation and have not been able to resolve my issue. MATLAB online does not have issues with the output, but I do not understand why the desktop version (either) does not work properly. If anyone could help explain why the desktop version might be doing this, and provide a solution, that would be great!

I have tried every combination of double() or vpa() that I can think of to try and make this work. I have posted what I am getting stuck on below, with the expected results vs what I am getting. Changing the degree of the polynomial does not resolve my precision issue either. I have also browsed through my variable settings and tried changing default array format to long and long g.

%   Method 1 - numerically solve
syms x
vpasolve(legendreP(3, x) == 0)

%   Method 2 - symbolically solve and convert to double
syms x
l = legendreP(3, x);
vpa(solve(l, x))
    %   double(solve(l, x)) does not work either
    
%   Method 3 - format output
format long
format long g
syms x
vpasolve(legendreP(3, x) == 0)

%   Method 4 - anonymous function
l = @(x) legendreP(3, x);
solve(l(x), x)
vpasolve(l(x), x)   %   this does not work either

% Expected results:

-0.77459666924148337703585307995648

0

0.77459666924148337703585307995648

% Actual results:

-0.7746

0

0.7746

Answer 1

[Walter Roberson]

sympref('FloatingPointOutput', 0)

[Nathan Biggers]

Thanks so much! It works now - if you are able to for my own understanding, could you explain what this sympref does?

[Walter Roberson]

When the opposite is in effect, symbolic output is passed through vpa(Expression, 4)before being output.