Newton Raphson hybrid algorithm not reaching a solution

Question

Brief explanation of the problem: I use Newton Raphson algorithm for root finding in polynomials and doesn't work in some cases. why?

I took from "numerical recipes in c++" a Newton Raphson hybrid algorithm, which bisects in case New-Raph is not converging properly (with a low derivative value or if the convergence speed is not fast).

I checked the algorithm with several polynomials and it worked. Now I am testing in inside the software I have and I always got an error with an specific polynomial. My problem is that I don't know why this polynomial just doesn't get to the result, when much others do. As I want to improve the algorithm for any polynomial y need to know which one is the reason of no convergence so I can treat it properly.

Following I will post all the information I can provide about the algorithm and the polynomial in which I have the error.

The polynomial:

f(t)= t^4 + 0,557257315256597*t^3 - 3,68254086033178*t^2 +
+ 0,139389107255627*t + 1,75823776590795

It's first derivative:

 f'(t)= 4*t^3 + 1.671771945769790*t^2 - 7.365081720663563*t + 0.139389107255627

Plot:

Roots (by Matlab):

  -2.133112008595826          1.371976341295347          0.883715461977390 
  -0.679837109933505

Algorithm:

double rtsafe(double* coeffs, int degree, double x1, double x2,double xacc,double xacc2)
    {
    int j;
    double df,dx,dxold,f,fh,fl;
    double temp,xh,xl,rts;
    double* dcoeffs=dvector(0,degree);
    for(int i=0;i<=degree;i++)
        dcoeffs[i]=0.0;
    PolyDeriv(coeffs,dcoeffs,degree);
    evalPoly(x1,coeffs,degree,&fl);
    evalPoly(x2,coeffs,degree,&fh);
    evalPoly(x2,dcoeffs,degree-1,&df);
if ((fl > 0.0 && fh > 0.0) || (fl < 0.0 && fh < 0.0))
    nrerror("Root must be bracketed in rtsafe");

if (fl == 0.0) return x1;
if (fh == 0.0) return x2;

if (fl < 0.0) { // Orient the search so that f(xl) < 0.
    xl=x1;
    xh=x2;
} else {
    xh=x1;
    xl=x2;
}
rts=0.5*(x1+x2);    //Initialize the guess for root,
dxold=fabs(x2-x1);  //the "stepsize before last,"
dx=dxold;           //and the last step

evalPoly(rts,coeffs,degree,&f);
evalPoly(rts,dcoeffs,degree-1,&dx);

for (j=1;j<=MAXIT;j++) { //Loop over allowed iterations

    if ((((rts-xh)*df-f)*((rts-xl)*df-f) > 0.0) //Bisect if Newton out of range,
            || (fabs(2.0*f) > fabs(dxold*df))) { //or not decreasing fast enough.
        dxold=dx;
        dx=0.5*(xh-xl);
        rts=xl+dx;
        if (xl == rts) 
            return rts; //Change in root is negligible.
    } else {// Newton step acceptable. Take it.
        dxold=dx;
        dx=f/df;
        temp=rts;
        rts -= dx;
        if (temp == rts)
            return rts;
    }
    if (fabs(dx) < xacc) 
        return rts;// Convergence criterion
    evalPoly(rts,coeffs,degree,&f);
    evalPoly(rts,dcoeffs,degree-1,&dx);
    //The one new function evaluation per iteration.
    if (f < 0.0) //Maintain the bracket on the root.
        xl=rts;
    else
        xh=rts;

}
//As the Accuracy asked to the algorithm is really high (but usually easily reached)
//the results precission is checked again, but with a less exigent result
dx=f/df;
if(fabs(dx)<xacc2)
    return rts;
nrerror("Maximum number of iterations exceeded in rtsafe");
return 0.0;// Never get here.
}

The algorithm is called with the next variables:

x1=0.019
x2=1.05
xacc=1e-10
xacc2=0.1
degree=4
MAXIT=1000
coeffs[0]=1.75823776590795;
coeffs[1]=0.139389107255627;
coeffs[2]=-3.68254086033178;
coeffs[3]=0.557257315256597;
coeffs[4]=1.0;

the problem is that the algorithm exceeds the amximum iterations and there is an arror to the root of aproximatedly 0.15.

So my direct and abrebiated question is: Why this polynomial does not reach an accurate error when many (1000 at least) other very similar polynomials do (wuth 1e-10 of precision and few iterations!)

I know the question is difficult and it may not have a really direct answer, but I am stuck with this for some days and I don't know how to solve it. Thank you very much for taking time for reading my question.

Answer 1

I'm not sure exactly why, but the check to see if the function is decreasing fast enough doesn't appear to work in this case.

It works if I do it like this:

  double old_f = f;

  .
  .
  .

    if ((((rts-xh)*df-f)*((rts-xl)*df-f) > 0.0) //Bisect if Newton out of range,
        || (fabs(2.0*f) > old_f)) { //or not decreasing fast enough.
  .
  .
  .

    if (fabs(dx) < xacc)
      return rts;// Convergence criterion
    old_f = f;

UPDATE

It looks like there is a problem in your code:

evalPoly(rts,dcoeffs,degree-1,&dx);

should be

evalPoly(rts,dcoeffs,degree-1,&df);

Answer 2

Without having run your code, my initial guess is that you are comparing to floating point values for equality to determine if your solution has converged.

   if (xl == rts) 
        return rts; //Change in root is negligible.

Maybe you should calculate it as a ratio:

   diff = fabs(xl - rts);
   if (diff/xl <= 1.0e-8)  // pick your own accuracy value here
      return rts;