Therefore, convergence is achieved after 4 iterations which is much faster than the 9 iterations in the fixed-point iteration method.
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: The idea of Newton's method is that we linearize the system around some guess point and solve the resulting linear system. My adaptation is not the one you found through your research - it's simpler. Where is an matrix, is a vector of components and is an -dimensional vector with the components. I'll answer the question of how one can solve a system of n-1 equations with n unknowns in Matlab by adapting Newton's method. 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.
#Code matlab of newton raphson method software
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:
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