Jacobi-free and complex-free method for finding simultaneously all zeros of polynomials having only real zeros

There are many methods for finding simultaneously all zeros of polynomials. However, a complex arithmetic or Jacobian matrix is needed in these methods even if the polynomial is real and only has real zeros. In this paper, a new method is proposed for finding simultaneously all zeros of real polynomials having only real zeros that does not require any complex arithmetic and Jacobian matrix within iteration. It is based on Vietaʹs theorem and Broydenʹs method. Numerical experiences show that the new method is very effective for the test polynomials.