Finding Roots

*Note:* While inclusion of all components in this chapter would,
in principle, be possible, selection of only a few of MACSYMA, MAPLE,
*Mathematica*, IDL, MATLAB, FORTRAN (with or without Numerical
Recipes), and C (with or without Numerical Recipes)
would probably be more common.

12.1 | Sample Problems |

12.1.1 Classical Turning Points | |

12.1.2 A Max-Min Problem: Equilibrium Points | |

12.1.3 Natural Frequencies of Oscillating Systems | |

12.1.4 Range of Projectile in Viscous Medium | |

12.1.5 Energy Levels in a Quantum Well | |

12.2 | Symbolic Approaches |

12.3 | Finding Roots Symbolically with MACSYMA |

12.3.1 Classical Turning Points | |

12.3.2 Equilibrium Points | |

12.3.3 Natural Frequencies | |

12.3.4 Range of Projectile in Viscous Medium | |

12.4 | Finding Roots Symbolically with MAPLE |

12.4.1 Classical Turning Points | |

12.4.2 Equilibrium Points | |

12.4.3 Natural Frequencies | |

12.4.4 Range of Projectile in Viscous Medium | |

12.5 | Finding Roots Symbolically with Mathematica |

12.5.1 Classical Turning Points | |

12.5.2 Equilibrium Points | |

12.5.3 Natural Frequencies | |

12.5.4 Range of Projectile in Viscous Medium | |

12.6 | Algorithms for Finding Roots Numerically |

12.6.1 The Method of Bisection | |

12.6.2 Newton's Method | |

12.6.3 Other Methods | |

12.6.4 Assessing Accuracy | |

12.7 | Finding Roots Numerically with IDL |

12.7.1 Classical Turning Points | |

12.7.2 Natural Frequencies | |

12.7.3 Range of Projectile | |

12.7.4 Energy Levels in Quantum Well | |

12.8 | Finding Roots Numerically with MATLAB |

12.8.1 Classical Turning Points | |

12.8.2 Natural Frequencies | |

12.8.3 Range of Projectile | |

12.8.4 Energy Levels in Quantum Well | |

12.9 | Finding Roots Numerically with MACSYMA |

12.9.1 Classical Turning Points | |

12.9.2 Range of Projectile | |

12.9.3 Energy Levels in Quantum Well | |

12.10 | Finding Roots Numerically with MAPLE |

12.10.1 Classical Turning Points | |

12.10.2 Range of Projectile | |

12.10.3 Energy Levels in Quantum Well | |

12.11 | Finding Roots Numerically with Mathematica |

12.11.1 Classical Turning Points | |

12.11.2 Range of Projectile | |

12.11.3 Energy Levels in Quantum Well | |

12.12 | Finding Roots Numerically with FORTRAN |

12.12.1 FORTRAN Programs from Scratch | |

12.12.2 Using Numerical Recipes | |

12.13 | Finding Roots Numerically with C |

12.13.1 C Programs from Scratch | |

12.13.2 Using Numerical Recipes | |

12.14 | Solving Simultaneous Equations |

12.14.1 Systems of Linear Equations | |

12.14.2 Systems of Nonlinear Equations | |

12.15 | Exercises |

12.15.1 ... using Symbolic Methods | |

12.15.2 ... using Numerical Methods | |

12.15.3 ... using Numerical Recipes | |

12.15.4 Finding More than One Unknown | |

12.A | Listing of bisect.f |

12.B | Listing of tt bisect.c |