Contents 4.4 Composite Numerical Integration 203 4.5 Romberg Integration 213 4.6 Adaptive Quadrature Methods 220 4.7 Gaussian Quadrature 228 4.8 Multiple Integrals 235 4.9 Improper Integrals 250 4.10 Survey of Methods and Software 256 5 Initial-Value Problems for Ordinary Differentia Equations 259 5.1 The Elementary Theory of Initial-Value Problems 260 5.2 Euler's Method 266 5.3 Higher-Order Taylor Methods 276 5.4 Runge-Kutta Methods 282 5.5 Error Control and the Runge-Kutta-Fehlberg Method 293 5.6 Multistep Methods 302 5.7 Variable Step-Size Multistep Methods 315 5.9 Higher-Order Equations and Systems of Differential Equations 328 5.10 Stability 339 5.11 Stiff Differential Equations 348 5. 12 Survey of Methods and Software 355 6 Direct Methods for Solving Linear Systems 357 6.1 Linear Systems of Equations 358 6.2 Pivoting Strategies 372 6.3 Linear Algebra and matrix Inversion 381 6. 4 The Determinant of a matrix 396 6.5 Matrix Factorization 400 6.6 Special Types of Matrices 411 6.7 Survey of Methods and Software 428 Iterative Techniques in Matrix Algebra 431 7.1 Norms of vectors and matrices 432 7.2 Eigenvalues and Eigenvectors 443 7.3 The Jacobi and Gauss-Siedel Iterative Techniques 450 7.4 Relaxation Techniques for Solving Linear Systems 462 7.5 Error Bounds and Iterative Refinement 469 7.6 The Conjugate Gradient Method 479 7.7 Survey of Methods and Software 495 Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. May no be copied, scanned, or duplicated, in whole or in part Due tovi Contents 4.4 Composite Numerical Integration 203 4.5 Romberg Integration 213 4.6 Adaptive Quadrature Methods 220 4.7 Gaussian Quadrature 228 4.8 Multiple Integrals 235 4.9 Improper Integrals 250 4.10 Survey of Methods and Software 256 5 Initial-Value Problems for Ordinary Differential Equations 259 5.1 The Elementary Theory of Initial-Value Problems 260 5.2 Euler’s Method 266 5.3 Higher-Order Taylor Methods 276 5.4 Runge-Kutta Methods 282 5.5 Error Control and the Runge-Kutta-Fehlberg Method 293 5.6 Multistep Methods 302 5.7 Variable Step-Size Multistep Methods 315 5.8 Extrapolation Methods 321 5.9 Higher-Order Equations and Systems of Differential Equations 328 5.10 Stability 339 5.11 Stiff Differential Equations 348 5.12 Survey of Methods and Software 355 6 Direct Methods for Solving Linear Systems 357 6.1 Linear Systems of Equations 358 6.2 Pivoting Strategies 372 6.3 Linear Algebra and Matrix Inversion 381 6.4 The Determinant of a Matrix 396 6.5 Matrix Factorization 400 6.6 Special Types of Matrices 411 6.7 Survey of Methods and Software 428 7 IterativeTechniques in Matrix Algebra 431 7.1 Norms of Vectors and Matrices 432 7.2 Eigenvalues and Eigenvectors 443 7.3 The Jacobi and Gauss-Siedel Iterative Techniques 450 7.4 Relaxation Techniques for Solving Linear Systems 462 7.5 Error Bounds and Iterative Refinement 469 7.6 The Conjugate Gradient Method 479 7.7 Survey of Methods and Software 495 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it