Iterative Methods for Linear and Nonlinear EquationsLinear 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. |
Contents
FR16_ch1 | 3 |
FR16_ch2 | 11 |
FR16_ch3 | 33 |
FR16_ch4 | 65 |
FR16_ch5 | 71 |
FR16_ch6 | 95 |
FR16_ch7 | 113 |
FR16_ch8 | 135 |
FR16_backmatter | 153 |
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Common terms and phrases
algorithm analysis apply Bi-CG Broyden-Armijo Broyden's method CGNE CGNR Chapter chord method collection of MATLAB completes the proof compute condition number cost diagonalizable matrix difference approximation differential equations eigenvalues error example F(xc F(xn factorization forward difference function evaluations Givens rotation GMRES iteration Gram-Schmidt H-equation Hence implementation implies inexact Newton initial iterate inner iteration iteration converges iteration progresses iteration required iterative methods Jacobian kmax least squares problem Lemma Let the standard line search Lipschitz continuous MATLAB codes matrix matrix norm matrix-vector product million floating-point operations minimization Newton step Newton-GMRES Newton's method nonsingular norm nsol nsola orthogonality outer iterations parameter Pk+1 Poisson solver preconditioned iteration preconditioner q-factor q-order q-superlinearly quasi-Newton method reduce Relative Nonlinear Residual relative residual reorthogonalization residual polynomial restarted result secant method sequence solution solve standard assumptions hold storage termination TFQMR Theorem Uk+1 unpreconditioned vector Xn+1