Furthermore, some polynomials are more "difficult" than others: see for instance Wilkinson's polynomial. In extreme cases, the computed roots can be quite far from the exact roots. This is especially a problem for polynomials of high degree (how much high really depends). Numerical errors introduced in the deflation step add up and usually leads to poor accuracy of the subsequent roots.There are ways to adapt Newton's method to deal with this. If there are multiple roots, convergence there is much slower.See Newton-Kantorovitch theorem and basins of attraction of the Newton method. Not all values of X0 will guarantee convergence, as Newton-Raphson is a local method.To find complex roots, you then have to start from a complex X0, and of course use complex arithmetic. If your polynomial has real coefficients and X0 is real, you will only find a real root, if there is any.Restart from the beginning if the quotient has degree > 1.Divide P by (X-R): the division is exact (up to numerical error) since R is a root.Start from some X0 and find a root R, using Newton's algorithm.Therefore, convergence is achieved after 4 iterations which is much faster than the 9 iterations in the fixed-point iteration method.A possible algorithm to find all roots of the polynomial P consists in: The approximate relative error is given by:įor the second iteration the vector and the matrix have components:įor the third iteration the vector and the matrix have components:įinally, for the fourth iteration the vector and the matrix have components: Therefore, the new estimates for and are: The components of the vector can be computed as follows: If, then it has the following form:Īssuming an initial guess of and, then the vector and the matrix have components: In addition to requiring an initial guess, the Newton-Raphson method requires evaluating the derivatives of the functions and. Use the Newton-Raphson method with to find the solution to the following nonlinear system of equations: If is invertible, then, the above system can be solved as follows: Where is an matrix, is a vector of components and is an -dimensional vector with the components. Setting, the above equation can be written in matrix form as follows: If the components of one iteration are known as:, then, the Taylor expansion of the first equation around these components is given by:Īpplying the Taylor expansion in the same manner for, we obtained the following system of linear equations with the unknowns being the components of the vector :īy setting the left hand side to zero (which is the desired value for the functions, then, the system can be written as: Assume a nonlinear system of equations of the form: The derivation of the method for nonlinear systems is very similar to the one-dimensional version in the root finding section. Many engineering software packages (especially finite element analysis software) that solve nonlinear systems of equations use the Newton-Raphson method. The Newton-Raphson method is the method of choice for solving nonlinear systems of equations. Newton-Raphson Method Newton-Raphson Method Open Educational Resources Nonlinear Systems of Equations: Derivatives Using Interpolation Functions.High-Accuracy Numerical Differentiation Formulas.Basic Numerical Differentiation Formulas.Linearization of Nonlinear Relationships.Convergence of Jacobi and Gauss-Seidel Methods.Cholesky Factorization for Positive Definite Symmetric Matrices.
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