Question

For the cubic equation $a^{3}x+b^{2}x+cx+d=0$ , where a, b, c and d are real input coefficients. Write a MATLAB function root.m of the form:

function [largestRoot] = root(a, b, c, d)

% a: Coefficient of x^3

% b: Coefficient of x^2

% c: Coefficient of x

% d: Coefficient of 1

% largestRoot: The largest real root of the cubic

to find the largest real root of this equation accurate to within a relative error $10^{-6}$ using any methods such as Newton's, Horner's.... Any of your choice. And should not use the MATLAB functions fzero, roots or eig.

I got some root comparison code:

%%%%%
function [a, b, c, d] = rootsToCoeffs(r1, r2, r3)
    a = 1;
    b = -(r1 + r2 + r3);
    c = r1 * r2 + r2 * r3 + r3 * r1;
    d = -(r1 * r2 * r3);
end
%%%%%%%%
function maxRelErr = compareRoots(expected, actual)
    maxRelErr = -Inf;
    if expected == 0
        if actual != 0
            % If you wanted 0 but didn't have it, infinite error.
            maxRelErr = Inf;
        end
    else
        relErr = abs(expected - actual) / abs(expected);
        % Make sure error of 0.0 isn't a bad thing.
        if relErr != 0
            maxRelErr = relErr;
        end
    end
end

Please write in a basic way if possible. I'm just a beginner to MATLAB. Thank you!

Answer 1

Answer: function return f-val at x for given coefficients function kval-find?(a.biS.dix.) Afval end function to find thelargest cubic roots %bisection method is used to find the largest root function [largestRootroot (a, b, c, d) %t ! erance accur = 1e-6; &number of steps nSteps 1e-5; %starting value start 1e-3; Sending x value %loop untill we win the largest root with said tolerance value while (ed- start >= nsteps II ( abs( findF (arbr.ce start) ) accur && abs ( find a d, ed)a)) %middle of start and ed cval(start ed)/2: find fval at val and check fval is 0. % if fval is 0, we found root. %so exit from loop break %Check in what half does root falls elseif ( findE (a,bisxdstart) *findE (a val) < 0 ed-cval; else start-Cyak; end end %Get largest root if (a>b) largestRoot start; else

COPYABLE CODE:

%function return f-val at x for given coefficients

function fval=findF(a,b,c,d,x)

     %fval

     fval=a*(x.^3)+b*(x.^2)+c*x+d;

end

%function to find thelargest cubic roots

%bisection method is used to find the largest root

function [largestRoot] = root(a, b, c, d)

     %tolerance

     accur = 1e-6;

     %number of steps

     nSteps = 1e-5;

     %starting x value

     start = 1e-3;

     %ending x value

     ed = 1e3;

%loop untill we fin the largest root with said tolerance value

while (ed - start >= nSteps || ( abs( findF(a,b,c,d,start) ) >= accur && abs( findF(a,b,c,d,ed) ) >= accur ) )

          %middle of start and ed

          cval = (start + ed)/2;

          %find fVal at cval and check fval is 0.

          if ( findF(a,b,c,d,cval) == 0 )

              %If fval is 0, we found root.

              %so exit from loop

              break;

          %Check in what half does root falls

elseif ( findF(a,b,c,d,start)*findF(a,b,c,d,cval) < 0 )      

              ed = cval;

          else

              start = cval;

          end

     end

     %Get largest root

     if(a>b)

          largestRoot=start;

     else

          largestRoot=ed;

     end

%end function

end

%code to test function

%cubic function: f(x)=x^3-2

largestRoot=root(1,0,0,-2);

%display largest root

fprintf("LARGEST ROOT: %8.4f",largestRoot);