A Method to Accelerate the Convergence of the Secant Algorithm

Algorithm 2

define x(p,  i)

 if p  <  −1 or i  <  0 or i  >  imax(p)

    return error

 else if p  =  −1

    return x₋₁

 else if p  =  0

    return x₀

 else

    if i  =  0              Secant step

      x1 = x(p  −  1, imax(p  −  1))

      x2 = x(p  −  2, imax(p  −  2))

      return x1  −  f(x1)  *  (x1  −  x2)/(f(x1)  −  f(x2))

    else              ith order approximant

      x1 = x(p  −  1, i  −  1)

      x2 = x(p  −  1, imax(p  −  1))

      x3 = x(p, i  −  1)

      x4 = x(p  −  i  −  2, imax(p  −  i  −  2))

      return x3  +  ((x2  −  x3)(x1  −  x3))/(x1  +  x2  −  x3  −  x4)