Bernoulli’s method finds a primary solvent of an unilateral quadratic matrix equation \(AX^{2}+BX+C=0\) when some conditions are given on its coefficients. When a stricter condition is given, diagonal update method improves the performances of Bernoulli method, both in terms of number of iterations and in computational time. In this paper, we suggest generalized diagonal update method which improves Bernoulli’s method without the stricter condition. We also define optimal constant for the generalized diagonal update method, which guarantees monotone convergence to the primary solvent. We compare iteration numbers of the generalized diagonal update method with pure Bernoulli’s method, also compare computational time of them with the Newton’s method and the cyclic reduction. Furthermore, we show that this generalized diagonal update method is useful to solve unilateral quadratic matrix equations with the form \(AX^2+\epsilon BX+C=0\) .
