A First Course on Numerical Methods
A First Course on Numerical Methods is designed for students and researchers who seek practical knowledge of modern techniques in scientific computing. Avoiding encyclopedic and heavily theoretical exposition, the book provides an in-depth treatment of fundamental issues and methods, the reasons behind the success and failure of numerical software, and fresh and easy-to-follow approaches and techniques. The authors focus on current methods, issues and software while providing a comprehensive theoretical foundation, enabling those who need to apply the techniques to successfully design solutions to nonstandard problems. The book also illustrates algorithms using the programming environment of MATLABŪ, with the expectation that the reader will gradually become proficient in it while learning the material covered in the book. The book takes an algorithmic approach, focusing on techniques that have a high level of applicability to engineering, computer science and industrial mathematics.
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abscissae accuracy algebraic algorithm applied approximation basic basis functions calculate chapter Chebyshev coefficients column computing condition number consider convergence corresponding cubic data points defined derivative diagonal differential equations discretization eigenvalues eigenvectors exact solution Figure fixed point iteration floating point system formula Fourier function f(x Gaussian elimination given grid initial guess instance integration interval a,b Krylov subspace l2-norm Lagrange polynomials least squares problem linear system LU decomposition MATLAB matrix mesh minimizing Newton’s method nonlinear nonsingular nonzero norm normal equations Note numerical differentiation obtain ODE system optimization orthogonal piecewise plot polynomial interpolation preconditioner quadratic quadrature relative error residual resulting RK methods root rounding unit roundoff error satisfies scalar secant method Section solving Specific exercises spline step subintervals symmetric positive definite Theorem transform trigonometric polynomial truncation error vector xk+1 yields zero