New quasi-Newton method for solving systems of nonlinear equations

We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of t...

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Bibliographic Details
Published inApplications of mathematics (Prague) Vol. 62; no. 2; pp. 121 - 134
Main Authors Lukšan, Ladislav, Vlček, Jan
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2017
Springer Nature B.V
Subjects
Analysis
Applications of Mathematics
Classical and Continuum Physics
Iterative methods
Jacobi matrix method
Jacobian matrix
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinear equations
Nonlinear programming
Nonlinear systems
Optimization
Quasi Newton methods
Theoretical
trust-region method
numerical algorithm
quasi-Newton method
65K10
adjoint Broyden method
nonlinear equation
system of equations
numerical experiment
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Summary:We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires O ( n 2 ) arithmetic operations per iteration in contrast with the Newton method, which requires O ( n 3 ) operations per iteration. Computational experiments confirm the high efficiency of the new method.
ISSN:0862-7940
1572-9109
DOI:10.21136/AM.2017.0253-16