Krylov subspace accelerated inexact Newton method for linear and nonlinear equations.

Area of Science:

• Computational Chemistry
• Numerical Analysis
• Scientific Computing

Background:

• Direct inversion in iterative subspace (DIIS) is a widely used method for accelerating convergence in iterative calculations.
• Solving linear and nonlinear equations is fundamental in many scientific and engineering disciplines.
• Efficient algorithms are crucial for handling complex computational problems, such as locating minimum energy crossing points.

Purpose of the Study:

• To introduce and describe the Krylov subspace accelerated inexact Newton (KAIN) method.
• To analyze the relationship between KAIN and the established DIIS method.
• To compare the performance of KAIN and DIIS for solving equations and locating critical points on potential energy surfaces.

Main Methods:

• Development and description of the KAIN algorithm.
• Comparative analysis of KAIN and DIIS using test equations.
• Application of both methods to find minimum energy crossing points of potential energy surfaces.

Main Results:

• KAIN is comparable in implementation complexity to DIIS.
• KAIN demonstrates superior performance, especially with poor preconditioning.
• KAIN accommodates a wider range of preconditioning strategies than DIIS.
• Under ideal preconditioning, KAIN exhibits behavior very similar to DIIS.

Conclusions:

• KAIN is a robust and flexible alternative to DIIS for solving linear and nonlinear equations.
• The enhanced performance and adaptability of KAIN make it a recommended replacement for DIIS in computational applications.
• KAIN offers significant advantages in scenarios requiring effective preconditioning.