Heath - Scientific Computing (523150), страница 2
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.6.2.3 Newton’s Method . . . . . . . . . . . . .6.2.4 Safeguarded Methods . . . . . . . . . .6.3 Multidimensional Unconstrained Optimization6.3.1 Direct Search Methods . . . . . . . . . .6.3.2 Steepest Descent Method . . . . . . . .6.3.3 Newton’s Method . . .
. . . . . . . . . .6.3.4 Quasi-Newton Methods . . . . . . . . .6.3.5 Secant Updating Methods . . . . . . . .6.3.6 Conjugate Gradient Method . . . . . . .6.3.7 Truncated Newton Methods . . . . . . .6.4 Nonlinear Least Squares . . . . . . . . . . . . .6.4.1 Gauss-Newton Method .
. . . . . . . . .6.4.2 Levenberg-Marquardt Method . . . . .6.5 Constrained Optimization . . . . . . . . . . . .6.5.1 Linear Programming . . . . . . . . . . .6.6 Software for Optimization . . . . . . . . . . . .6.7 Historical Notes and Further Reading . . . . .................................................................................................................................................................................................................................................................................................................................................................................................1831831841851861861861881891911911911911931951961971991992002012022052072087 Interpolation7.1 Interpolation . .
. . . . . . . . . . . . . . .7.1.1 Purposes for Interpolation . . . . . .7.1.2 Interpolation versus Approximation7.1.3 Choice of Interpolating Function . .................................................................2192192192202205.35.45.55.2.4 Secant Method . . . . . . . . .5.2.5 Inverse Interpolation . . .
. . .5.2.6 Linear Fractional Interpolation5.2.7 Safeguarded Methods . . . . .5.2.8 Zeros of Polynomials . . . . . .Systems of Nonlinear Equations . . . .5.3.1 Fixed-Point Iteration . . . . . .5.3.2 Newton’s Method . . . . . . . .5.3.3 Secant Updating Methods . . .5.3.4 Broyden’s Method . . . . . . .5.3.5 Robust Newton-Like Methods .Software for Nonlinear Equations . .
.Historical Notes and Further Reading............................................................CONTENTS7.27.37.47.5ix7.1.4 Basis Functions . . . . . . . . . . . . . . . . . . .Polynomial Interpolation . . . . . . . . . . . . . . . . . .7.2.1 Evaluating Polynomials . . . . .
. . . . . . . . .7.2.2 Lagrange Interpolation . . . . . . . . . . . . . . .7.2.3 Newton Interpolation . . . . . . . . . . . . . . .7.2.4 Orthogonal Polynomials . . . . . . . . . . . . . .7.2.5 Interpolating a Function . . . . . . . . . . . . . .7.2.6 High-Degree Polynomial Interpolation . . .
. . .7.2.7 Placement of Interpolation Points . . . . . . . .Piecewise Polynomial Interpolation . . . . . . . . . . . .7.3.1 Hermite Cubic Interpolation . . . . . . . . . . .7.3.2 Cubic Spline Interpolation . . . . . . . . . . . . .7.3.3 Hermite Cubic versus Cubic Spline Interpolation7.3.4 B-splines . . . . . . . . . . . . . . . .
. . . . . .Software for Interpolation . . . . . . . . . . . . . . . . .7.4.1 Software for Special Functions . . . . . . . . . .Historical Notes and Further Reading . . . . . . . . . .8 Numerical Integration and Differentiation8.1 Numerical Quadrature . . . . . . . .
. . . . . . . . . .8.1.1 Quadrature Rules . . . . . . . . . . . . . . . .8.2 Newton-Cotes Quadrature . . . . . . . . . . . . . . . .8.2.1 Newton-Cotes Quadrature Rules . . . . . . . .8.2.2 Method of Undetermined Coefficients . . . . .8.2.3 Error Estimation . . . . . . . . . .
. . . . . . .8.2.4 Polynomial Degree . . . . . . . . . . . . . . . .8.3 Gaussian Quadrature . . . . . . . . . . . . . . . . . . .8.3.1 Gaussian Quadrature Rules . . . . . . . . . . .8.3.2 Change of Interval . . . . . . . . . . . . . . . .8.3.3 Gauss-Kronrod Quadrature Rules . . . . . . .8.4 Composite and Adaptive Quadrature .
. . . . . . . . .8.4.1 Composite Quadrature Rules . . . . . . . . . .8.4.2 Automatic and Adaptive Quadrature . . . . . .8.5 Other Integration Problems . . . . . . . . . . . . . . .8.5.1 Integrating Tabular Data . . . . . . . . . . . .8.5.2 Infinite Intervals . . . . . . . . . .
. . . . . . .8.5.3 Double Integrals . . . . . . . . . . . . . . . . .8.5.4 Multiple Integrals . . . . . . . . . . . . . . . .8.6 Integral Equations . . . . . . . . . . . . . . . . . . . .8.7 Numerical Differentiation . . . . . . . . . . . . . . . .8.7.1 Finite Difference Approximations . . . . . . . .8.7.2 Automatic Differentiation . .
. . . . . . . . . .8.8 Richardson Extrapolation . . . . . . . . . . . . . . . .8.9 Software for Numerical Integration and Differentiation8.10 Historical Notes and Further Reading . . . . . . . . ..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................221222224224225229230231231232233233234236238239239..........................245245246246246247249250251251253254255255256257257257257258259261262263263266267x9 Initial Value Problems for ODEs9.1 Ordinary Differential Equations .
. . . . .9.1.1 Initial Value Problems . . . . . . .9.1.2 Higher-Order ODEs . . . . . . . .9.1.3 Stable and Unstable ODEs . . . .9.2 Numerical Solution of ODEs . . . . . . . .9.2.1 Euler’s Method . . . . . . . . . . .9.3 Accuracy and Stability . . . . . . . . . . .9.3.1 Order of Accuracy . . . . . . . .
.9.3.2 Stability of a Numerical Method .9.3.3 Stepsize Control . . . . . . . . . .9.4 Implicit Methods . . . . . . . . . . . . . .9.5 Stiff Differential Equations . . . . . . . . .9.6 Survey of Numerical Methods for ODEs .9.6.1 Taylor Series Methods . . . . . . .9.6.2 Runge-Kutta Methods . . . . . . .9.6.3 Extrapolation Methods . . . . . .9.6.4 Multistep Methods . . .
. . . . . .9.6.5 Multivalue Methods . . . . . . . .9.7 Software for ODE Initial Value Problems9.8 Historical Notes and Further Reading . .CONTENTS........................................10 Boundary Value Problems for ODEs10.1 Boundary Value Problems . . . . . . . . . . .10.2 Shooting Method . . . . . . . . . . . . .
. . .10.3 Superposition Method . . . . . . . . . . . . .10.4 Finite Difference Method . . . . . . . . . . .10.5 Finite Element Method . . . . . . . . . . . .10.6 Eigenvalue Problems . . . . . . . . . . . . . .10.7 Software for ODE Boundary Value Problems10.8 Historical Notes and Further Reading . . . .............................................................................................................................................11 Partial Differential Equations11.1 Partial Differential Equations . . .
. . . . . . . . . . .11.1.1 Classification of Partial Differential Equations .11.2 Time-Dependent Problems . . . . . . . . . . . . . . . .11.2.1 Semidiscrete Methods Using Finite Differences11.2.2 Semidiscrete Methods Using Finite Elements .11.2.3 Fully Discrete Methods . . . . . . . . . . . . .11.2.4 Implicit Finite Difference Methods .
. . . . . .11.2.5 Hyperbolic versus Parabolic Problems . . . . .11.3 Time-Independent Problems . . . . . . . . . . . . . . .11.3.1 Finite Difference Methods . . . . . . . . . . . .11.3.2 Finite Element Methods . . . . . . . . . . . . .11.4 Direct Methods for Sparse Linear Systems . . . . . . .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................275275276276277280280282282284285286288290290291293293297299300........309309310312312314318319319............325325325326327328329332333335335337337CONTENTS11.511.611.711.8xi11.4.1 Sparse Factorization Methods .
. . . . .11.4.2 Fast Direct Methods . . . . . . . . . . .Iterative Methods for Linear Systems . . . . . .11.5.1 Stationary Iterative Methods . . . . . .11.5.2 Jacobi Method . . . . . . . . . . . . . .11.5.3 Gauss-Seidel Method . . . . . . . . . . .11.5.4 Successive Over-Relaxation .
. . . . . .11.5.5 Conjugate Gradient Method . . . . . . .11.5.6 Rate of Convergence . . . . . . . . . . .11.5.7 Multigrid Methods . . . . . . . . . . . .Comparison of Methods . . . . . . . . . . . . .Software for Partial Differential Equations . . .11.7.1 Software for Initial Value Problems . . .11.7.2 Software for Boundary Value Problems .11.7.3 Software for Sparse Linear Systems . . .Historical Notes and Further Reading .
. . . .12 Fast Fourier Transform12.1 Trigonometric Interpolation . . . . . .12.1.1 Continuous Fourier Transform12.1.2 Fourier Series . . . . . . . . . .12.1.3 Discrete Fourier Transform . .12.2 FFT Algorithm . . . . . . . . . . . . .12.2.1 Limitations of the FFT . . . .12.3 Applications of DFT . . . . . . . . . .12.3.1 Fast Polynomial Multiplication12.4 Wavelets . . . . . .
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