Continuous newton method and its modification

Abstract

The Continuous Newton method CNM and its modification MCNM are studied. MCNM permits one to replace the inversion of the derivative operator at every step of iterations by its inversion only at the initial approximation point. Then the extended system of the differential equations in Hilbert space, introduced in this paper, permits the realization of the iterative process with the simultaneous calculation of the inverse derivative operator. For CNM and MCNM the theorems establishing convergence with exponential rate are proved. The convergence theorem for MCNM is proved under almost the same conditions as for CNM.

Keywords:

• Continuous Newton method
• Frechet derivative
• exponential convergence
• integral inequality

AMS:

• 65J15
• 58C15
• 47H17