An Initial Value of Newton’s Method for the Characteristic Polynomial of 4th Order PCM in Ahp

The pairwise comparison matrix plays a critical role in the analytic hierarchy process, a popular multi-criteria decision-making method based on individual subjectivity. The maximal eigenvalue of the pairwise comparison matrix can be used to measure the degree of consistency of the decision maker's judgment. Since the eigenvalues of a matrix are the roots of a characteristic equation, they can be obtained by numerical iterative methods such as Newton's method. However, the location of the initial point does not always guarantee that the maximal eigenvalue will be obtained. For the third-order pairwise comparison matrix, it has already been shown that the characteristic equations always have a single real root and that Newton's method using three as the initial point can always achieve the maximal eigenvalue. In this paper, we focus on the characteristic equation of a fourth-order pairwise comparison matrix and show that they have exactly two real roots. Furthermore, we present an initial point of Newton's method such that it always converges to the larger root, i.e., the maximal eigenvalue. Numerical experiments are also provided to compare convergence rate with other numerical iterative methods.