Iterative Methods for Linear and Nonlinear Equations
Linear and nonlinear systems of equations are the basis for many, if not most, of the models of phenomena in science and engineering, and their efficient numerical solution is critical to progress in these areas. This is the first book to be published on nonlinear equations since the mid-1980s. Although it stresses recent developments in this area, such as Newton-Krylov methods, considerable material on linear equations has been incorporated. This book focuses on a small number of methods and treats them in depth. The author provides a complete analysis of the conjugate gradient and generalized minimum residual iterations as well as recent advances including Newton-Krylov methods, incorporation of inexactness and noise into the analysis, new proofs and implementations of Broyden's method, and globalization of inexact Newton methods.
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algorithm Bi-CG Broyden's method CG iteration CGNE CGNR Chapter chord method collection of MATLAB completes the proof compute condition number cost diagonalizable matrix difference approximation differential equations Dirichlet boundary conditions eigenvalues error example F(ac F(an factorization forward difference function evaluations Gauss–Seidel Givens rotation GMRES iteration Gram-Schmidt H-equation Hence implies inexact Newton initial iterate iteration converges iteration progresses iteration required iterative methods Jacobian kmaa Krylov least squares problem Lemma Let the standard line search Lipschitz continuous MATLAB codes matrix matrix-vector product million floating-point operations minimization Newton iteration Newton-GMRES Newton's method nonlinear residual nonsingular norm nsol nsola orthogonality outer iterations parameter partial differential equations Poisson solver preconditioned iteration preconditioner q-factor q-order q-superlinearly quasi-Newton method reduce relative residual reorthogonalization residual polynomial restarted result secant method sequence solution solve standard assumptions hold stationary iterative methods step storage Theorem unpreconditioned vector