Newton's method and secant methods: a longstanding relationship from vectors to matrices

DOI10.4171/PM/1901zbMath1248.65051OpenAlexW1981217254MaRDI QIDQ658578

Marcos Raydan, Marlliny Monsalve

Publication date: 13 January 2012

Published in: Portugaliae Mathematica. Nova Série (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.4171/pm/1901

zbMATH Keywords

Newton's methodRiccati equationsecant methodquadratic equationmatrix exponentialmatrix inversereview papermatrix logarithmmatrix \(p\)-th rootmatrix rational equationnonlinear matrix problemssign of a matrix

Mathematics Subject Classification ID

Newton-type methods (49M15) Numerical computation of solutions to systems of equations (65H10) Methods of quasi-Newton type (90C53) Nonlinear differential equations in abstract spaces (34G20) Matrix equations and identities (15A24) Matrix exponential and similar functions of matrices (15A16)

Related Items (1)

Pseudoinverse preconditioners and iterative methods for large dense linear least-squares problems

Uses Software

• mctoolbox
• MATLAB expm
• Algorithm 432

Cites Work

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• A Schur method for the square root of a matrix
• Solving stable generalized Lyapunov equations with the matrix sign function
• Generalized Lyapunov equation and factorization of matrix polynomials
• Globally convergent inexact quasi-Newton methods for solving nonlinear systems
• A classification of quasi-Newton methods
• Algorithms for the matrix \(p\)th root
• Newton's method and boundary value problems
• A Globally Convergent Newton-GMRES Subspace Method for Systems of Nonlinear Equations
• Rayleigh Quotient Iteration for Nonsymmetric Matrices
• Convergence and Applications of Newton-type Iterations
• Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems
• A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation
• On the Local Convergence of Quasi-Newton Methods for Constrained Optimization
• An exact line search method for solving generalized continuous-time algebraic Riccati equations
• Cubically Convergent Iterations for Invariant Subspace Computation
• A Practical Algorithm for General Large Scale Nonlinear Optimization Problems
• Algorithm 432 [C2: Solution of the matrix equation AX + XB = C [F4]]
• Numerical Methods for Structured Markov Chains
• Approximation of Large-Scale Dynamical Systems

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