Scientific Computing and Differential Equations: An Introduction to Numerical Methods
Scientific Computing and Differential Equations: An Introduction to Numerical Methods, is an excellent complement to Introduction to Numerical Methods by Ortega and Poole. The book emphasizes the importance of solving differential equations on a computer, which comprises a large part of what has come to be called scientific computing. It reviews modern scientific computing, outlines its applications, and places the subject in a larger context.
This book is appropriate for upper undergraduate courses in mathematics, electrical engineering, and computer science; it is also well-suited to serve as a textbook for numerical differential equations courses at the graduate level.
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Chapter 9 The Curse of Dimensionality
Analysis and Differential Equations
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algorithm applied approximate solution assume banded matrix basis functions boundary conditions boundary-value problem Chapter coefficient matrix column consider corresponding cubic spline defined derivatives diagonally dominant difference equations discretization error Discussion and References eigenvalues eigenvectors Euler's method evaluated exact solution example Exercise Figure finite difference method formula Gauss–Seidel Gaussian elimination given grid points Hence Hessenberg Hessenberg matrix ill-conditioned initial conditions initial-value problem interchange iterative methods linear equations linear system LU decomposition main diagonal multiplications n x n Newton iterates Newton's method non-zero nonlinear nonsingular norm obtain operation count ordinary differential equations orthogonal orthogonal matrix partial differential equations permutation matrix piecewise polynomial of degree positive definite QR method quadratic rate of convergence requires roots rounding error Runge-Kutta method satisfies scientific computing second-order Show stability step Supplementary Discussion symmetric system of equations Theorem triangular tridiagonal matrix value problem variables vector verify yk+1