Berzi, P. Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates. AppliedMath2024, 4, 143-181.
Berzi, P. Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates. AppliedMath 2024, 4, 143-181.
Berzi, P. Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates. AppliedMath2024, 4, 143-181.
Berzi, P. Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates. AppliedMath 2024, 4, 143-181.
Abstract
A system of simultaneous multi-variable nonlinear equations can be solved by the Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (secant equation) doesn’t completely specify the Jacobian approximate in multi-dimensional case, so its full-rank update is not possible with classic variants of methods. The suggested new iteration strategy (“T-Secant”) allows full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates, then the Jacobian approximate can fully be updated. It is shown, that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the super-linear convergence of the secant method (φS = 1.618...) to super-quadratic (φT = φS + 1 = 2.618...) and the quadratic convergence of the Newton-method (φN = 2) to cubic (φT = φN + 1 = 3) in one dimensional case. The Broyden-type efficiency (mean convergence rate) of the suggested method in multi-dimensional case is an order higher than the efficiency of other classic low-rank update quasi-Newton methods as shown by numerical examples on a Rosenbrock-type test-function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single variable case helps explaining the basic operations and a vector-space description is also given in multi-variable case.
Computer Science and Mathematics, Applied Mathematics
Copyright:
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