Evaluating Integrals
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.
11.1 | Sample Problems | |
11.1.1 One-Dimensional Trajectories | ||
11.1.2 Center of Mass | ||
11.1.3 Moment of Inertia | ||
11.1.4 The Large Amplitude Simple Pendulum | ||
11.1.5 Statistical Data Analysis | ||
11.1.6 The Cornu Spiral | ||
11.1.7 Electric and Magnetic Fields and Potentials | ||
11.1.8 Quantum Probabilities | ||
11.1.9 Expansion in Orthogonal Functions | ||
11.2 | Evaluating Integrals Symbolically with Macsyma | |
11.2.1 Relativistic Motion Under a Constant Force | ||
11.2.2 Center of Mass | ||
11.2.3 Moment of Inertia; Radius of Gyration | ||
11.2.4 Electrostatic Potential of a Finite Line Charge | ||
11.2.5 The Helmholtz Coil | ||
11.2.6 Period of a Pendulum with Modest Amplitude | ||
11.2.7 Fourier Coefficients for Half-Rectified Signal | ||
11.3 | Evaluating Integrals Symbolically with Maple | |
11.3.1 Relativistic Motion Under a Constant Force | ||
11.3.2 Center of Mass | ||
11.3.3 Moment of Inertia; Radius of Gyration | ||
11.3.4 Electrostatic Potential of a Finite Line Charge | ||
11.3.5 The Helmholtz Coil | ||
11.3.6 Period of a Pendulum with Modest Amplitude | ||
11.3.7 Fourier Coefficients for Half-Rectified Signal | ||
11.4 | Evaluating Integrals Symbolically with Mathematica | |
11.4.1 Relativistic Motion Under a Constant Force | ||
11.4.2 Center of Mass | ||
11.4.3 Moment of Inertia; Radius of Gyration | ||
11.4.4 Electrostatic Potential of a Finite Line Charge | ||
11.4.5 The Helmholtz Coil | ||
11.4.6 Period of a Pendulum with Modest Amplitude | ||
11.4.7 Fourier Coefficients for Half-Rectified Signal | ||
11.5 | Algorithms for Numerical Integration | |
11.5.1 Newton-Cotes Quadrature | ||
11.5.2 Rearrangements for Computational Efficiency | ||
11.5.3 Assessing Error | ||
11.5.4 Iterative and Adaptive Algorithms | ||
11.5.5 Gaussian Quadrature | ||
11.6 | Evaluating Integrals Numerically with IDL | |
11.6.1 Using Elementary Commands | ||
11.6.2 The Function luqsimp | ||
11.6.3 Moment of Inertia | ||
11.6.4 Quantum Probabilities | ||
11.6.5 Integrals as Functions of the Upper Limit | ||
11.6.6 The Error Function | ||
11.6.7 The Cornu Spiral | ||
11.6.8 Integrals as Functions of an Internal Parameter | ||
11.6.9 The Off-Axis Electrostatic Potential of Two Rings | ||
11.7 | Evaluating Integrals Numerically with MATLAB | |
11.7.1 Using Elementary Commands | ||
11.7.2 The Functions trapz, quad, quad8and quadl | ||
11.7.3 Moment of Inertia | ||
11.7.4 Quantum Probabilities | ||
11.7.5 Integrals as Functions of the Upper Limit | ||
11.7.6 The Error Function | ||
11.7.7 The Cornu Spiral | ||
11.7.8 Integrals as Functions of an Internal Parameter | ||
11.7.9 The Off-Axis Electrostatic Potential of Two Rings | ||
11.8 | Evaluating Integrals Numerically with MACSYMA | |
11.8.1 Romberg and Bromberg Integration | ||
11.8.2 Newton-Cotes Quadrature | ||
11.8.3 Quantum Probability | ||
11.8.4 The Error Function | ||
11.8.5 The Off-Axis Electrostatic Potential of Two Rings | ||
11.9 | Evaluating Integrals Numerically with MAPLE | |
11.9.1 Quantum Probability | ||
11.9.2 The Error Function | ||
11.9.3 The Off-Axis Electrostatic Potential of Two Rings | ||
11.10 | Evaluating Integrals Numerically with Mathematica | |
11.10.1 Quantum Probability | ||
11.10.2 The Error Function | ||
11.10.3 The Off-Axis Electrostatic Potential of Two Rings | ||
11.11 | Evaluating Integrals Numerically with FORTRAN | |
11.11.1 Writing Programs from Scratch | ||
11.11.2 Using Numerical Recipes | ||
11.12 | Evaluating Integrals Numerically with C | |
11.12.1 Writing Programs from Scratch | ||
11.12.2 Using Numerical Recipes | ||
11.13 | Exercises | |
11.13.1 ... using Symbolic Methods | ||
11.13.2 ... using Numerical Methods | ||
11.13.3 ... using Numerical Recipes | ||
11.A | Listing of trapezoidal.f | |
11.B | Listing of trapezoidal.c |