Secant method

About: Secant method is a research topic. Over the lifetime, 1002 publications have been published within this topic receiving 21347 citations.

Numerical methods for unconstrained optimization and nonlinear equations

Abstract: Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.

Solving Nonlinear Equations with Newton's Method

Abstract: Preface How to Get the Software 1 Introduction 2 Finding the Newton Step with Gaussian Elimination 3 Newton-Krylov Methods 4 Broyden's Method Bibliography Index

A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on some of the Variables

Abstract: We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. This method applies to a general (indefinite) quadratic function, for which a local minimizer subject to bounds is required, and is particularly suitable for the large-scale problem. Our new method exhibits strong convergence properties, global and quadratic convergence, and appears to have significant practical potential. Strictly feasible points are generated. Experimental results on moderately large and sparse problems support the claim of practicality for large-scale problems.

Applied Numerical Methods Using MATLAB

Abstract: Preface. 1. MATLAB Usage and Computational Errors. 1.1 Basic Operations of MATLAB. 1.1.1 Input/Output of Data from MATLAB Command Window. 1.1.2 Input/Output of Data Through Files. 1.1.3 Input/Output of Data Using Keyboard. 1.1.4 2-D Graphic Input/Output. 1.1.5 3-D Graphic Output. 1.1.6 Mathematical Functions. 1.1.7 Operations on Vectors and Matrices. 1.1.8 Random Number Generators. 1.1.9 Flow Control. 1.2 Computer Errors Versus Human Mistakes. 1.2.1 IEEE 64-bit Floating-Point Number Representation. 1.2.2 Various Kinds of Computing Errors. 1.2.3 Absolute/Relative Computing Errors. 1.2.4 Error Propagation. 1.2.5 Tips for Avoiding Large Errors. 1.3 Toward Good Program. 1.3.1 Nested Computing for Computational Efficiency. 1.3.2 Vector Operation Versus Loop Iteration. 1.3.3 Iterative Routine Versus Nested Routine. 1.3.4 To Avoid Runtime Error. 1.3.5 Parameter Sharing via Global Variables. 1.3.6 Parameter Passing Through Varargin. 1.3.7 Adaptive Input Argument List. Problems. 2. System of Linear Equations. 2.1 Solution for a System of Linear Equations. 2.1.1 The Nonsingular Case (M = N). 2.1.2 The Underdetermined Case (M N): Least-Squares Error Solution. 2.1.4 RLSE (Recursive Least-Squares Estimation). 2.2 Solving a System of Linear Equations. 2.2.1 Gauss Elimination. 2.2.2 Partial Pivoting. 2.2.3 Gauss-Jordan Elimination. 2.3 Inverse Matrix. 2.4 Decomposition (Factorization). 2.4.1 LU Decomposition (Factorization): Triangularization. 2.4.2 Other Decomposition (Factorization): Cholesky, QR, and SVD. 2.5 Iterative Methods to Solve Equations. 2.5.1 Jacobi Iteration. 2.5.2 Gauss-Seidel Iteration. 2.5.3 The Convergence of Jacobi and Gauss-Seidel Iterations. Problems. 3. Interpolation and Curve Fitting. 3.1 Interpolation by Lagrange Polynomial. 3.2 Interpolation by Newton Polynomial. 3.3 Approximation by Chebyshev Polynomial. 3.4 Pade Approximation by Rational Function. 3.5 Interpolation by Cubic Spline. 3.6 Hermite Interpolating Polynomial. 3.7 Two-dimensional Interpolation. 3.8 Curve Fitting. 3.8.1 Straight Line Fit: A Polynomial Function of First Degree. 3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher Degree. 3.8.3 Exponential Curve Fit and Other Functions. 3.9 Fourier Transform. 3.9.1 FFT Versus DFT. 3.9.2 Physical Meaning of DFT. 3.9.3 Interpolation by Using DFS. Problems. 4. Nonlinear Equations. 4.1 Iterative Method Toward Fixed Point. 4.2 Bisection Method. 4.3 False Position or Regula Falsi Method. 4.4 Newton(-Raphson) Method. 4.5 Secant Method. 4.6 Newton Method for a System of Nonlinear Equations. 4.7 Symbolic Solution for Equations. 4.8 A Real-World Problem. Problems. 5. Numerical Differentiation/Integration. 5.1 Difference Approximation for First Derivative. 5.2 Approximation Error of First Derivative. 5.3 Difference Approximation for Second and Higher Derivative. 5.4 Interpolating Polynomial and Numerical Differential. 5.5 Numerical Integration and Quadrature. 5.6 Trapezoidal Method and Simpson Method. 5.7 Recursive Rule and Romberg Integration. 5.8 Adaptive Quadrature. 5.9 Gauss Quadrature. 5.9.1 Gauss-Legendre Integration. 5.9.2 Gauss-Hermite Integration. 5.9.3 Gauss-Laguerre Integration. 5.9.4 Gauss-Chebyshev Integration. 5.10 Double Integral. Problems. 6. Ordinary Differential Equations. 6.1 Euler's Method. 6.2 Heun's Method: Trapezoidal Method. 6.3 Runge-Kutta Method. 6.4 Predictor-Corrector Method. 6.4.1 Adams-Bashforth-Moulton Method. 6.4.2 Hamming Method. 6.4.3 Comparison of Methods. 6.5 Vector Differential Equations. 6.5.1 State Equation. 6.5.2 Discretization of LTI State Equation. 6.5.3 High-Order Differential Equation to State Equation. 6.5.4 Stiff Equation. 6.6 Boundary Value Problem (BVP). 6.6.1 Shooting Method. 6.6.2 Finite Difference Method. Problems. 7. Optimization. 7.1 Unconstrained Optimization [L-2, Chapter 7]. 7.1.1 Golden Search Method. 7.1.2 Quadratic Approximation Method. 7.1.3 Nelder-Mead Method [W-8]. 7.1.4 Steepest Descent Method. 7.1.5 Newton Method. 7.1.6 Conjugate Gradient Method. 7.1.7 Simulated Annealing Method [W-7]. 7.1.8 Genetic Algorithm [W-7]. 7.2 Constrained Optimization [L-2, Chapter 10]. 7.2.1 Lagrange Multiplier Method. 7.2.2 Penalty Function Method. 7.3 MATLAB Built-In Routines for Optimization. 7.3.1 Unconstrained Optimization. 7.3.2 Constrained Optimization. 7.3.3 Linear Programming (LP). Problems. 8. Matrices and Eigenvalues. 8.1 Eigenvalues and Eigenvectors. 8.2 Similarity Transformation and Diagonalization. 8.3 Power Method. 8.3.1 Scaled Power Method. 8.3.2 Inverse Power Method. 8.3.3 Shifted Inverse Power Method. 8.4 Jacobi Method. 8.5 Physical Meaning of Eigenvalues/Eigenvectors. 8.6 Eigenvalue Equations. Problems. 9. Partial Differential Equations. 9.1 Elliptic PDE. 9.2 Parabolic PDE. 9.2.1 The Explicit Forward Euler Method. 9.2.2 The Implicit Backward Euler Method. 9.2.3 The Crank-Nicholson Method. 9.2.4 Two-Dimensional Parabolic PDE. 9.3 Hyperbolic PDE. 9.3.1 The Explicit Central Difference Method. 9.3.2 Two-Dimensional Hyperbolic PDE. 9.4 Finite Element Method (FEM) for solving PDE. 9.5 GUI of MATLAB for Solving PDEs: PDETOOL. 9.5.1 Basic PDEs Solvable by PDETOOL. 9.5.2 The Usage of PDETOOL. 9.5.3 Examples of Using PDETOOL to Solve PDEs. Problems. Appendix A: Mean Value Theorem. Appendix B: Matrix Operations/Properties. Appendix C: Differentiation with Respect to a Vector. Appendix D: Laplace Transform. Appendix E: Fourier Transform. Appendix F: Useful Formulas. Appendix G: Symbolic Computation. Appendix H: Sparse Matrices. Appendix I: MATLAB. References. Subject Index. Index for MATLAB Routines. Index for Tables.