A comparison of point and ball iterations in the contractive mapping case

In applications, one of the basic problems is to solve the fixed point equation x=Tx with T a contractive mapping. Two theorems which can be implemented computationally to verify the existence of a solution x^* to the equation and to obtain a convergent approximate solution sequence {x_n} are the classical Banach contraction mapping theorem and the newly established global convergence theorem of the ball algorithms in You, Xu and Liu [16]. These two theorems are compared on the basis of sensitivity, precision, computational complexity and efficiency. The comparison shows that except for computational complexity, the latter theorem is of far greater sensivity, precision and computational efficiency. This conclusion is supported by a number of numerical examples.