Vectors
The vector concept
Mathematical operations on vectors
Vector arithmetics and vector functions
Arrays in Python programs
Using lists for collecting function data
Basics of numerical Python arrays
Computing coordinates and function values
Vectorization
Curve plotting
MATLAB-style plotting with Matplotlib
Matplotlib; pyplot prefix
SciTools and Easyviz
Making animations
Making videos
Curve plots in pure text
Plotting difficulties
Piecewisely defined functions
Rapidly varying functions
More advanced vectorization of functions
Vectorization of StringFunction objects
Vectorization of the Heaviside function
Vectorization of a hat function
More on numerical Python arrays
Copying arrays
In-place arithmetics
Allocating arrays
Generalized indexing
Testing for the array type
Compact syntax for array generation
Shape manipulation
High-performance computing with arrays
Scalar implementation
Vectorized implementation
Memory-saving implementation
Analysis of memory usage
Analysis of the CPU time
Higher-dimensional arrays
Matrices and arrays
Two-dimensional numerical Python arrays
Array computing
Matrix objects
Some common linear algebra operations
Inverse, determinant, and eigenvalues
Products
Norms
Sum and extreme values
Indexing
Transpose and upper/lower triangular parts
Solving linear systems
Matrix row and column operations
Computing the rank of a matrix
Symbolic linear algebra
Plotting of scalar and vector fields
Installation
Surface plots
Parameterized curve
Contour lines
The gradient vector field
Matplotlib
Surface plots
Contour plots
Vector field plots
Mayavi
Surface plots
Contour plots
Vector field plots
A 3D scalar field and its gradient field
Animations
Summary
Chapter topics
Example: Animating a function
Exercises
Exercise 1: Fill lists with function values
Exercise 2: Fill arrays; loop version
Exercise 3: Fill arrays; vectorized version
Exercise 4: Plot a function
Exercise 5: Apply a function to a vector
Exercise 6: Simulate by hand a vectorized expression
Exercise 7: Demonstrate array slicing
Exercise 8: Replace list operations by array computing
Exercise 9: Plot a formula
Exercise 10: Plot a formula for several parameters
Exercise 11: Specify the extent of the axes in a plot
Exercise 12: Plot exact and inexact Fahrenheit-Celsius conversion formulas
Exercise 13: Plot the trajectory of a ball
Exercise 14: Plot data in a two-column file
Exercise 15: Write function data to file
Exercise 16: Plot data from a file
Exercise 17: Write table to file
Exercise 18: Fit a polynomial to data points
Exercise 19: Fit a polynomial to experimental data
Exercise 20: Read acceleration data and find velocities
Exercise 21: Read acceleration data and plot velocities
Exercise 22: Plot a trip's path and velocity from GPS coordinates
Exercise 23: Vectorize the Midpoint rule for integration
Exercise 24: Vectorize a function for computing the area of a polygon
Exercise 25: Implement Lagrange's interpolation formula
Exercise 26: Plot Lagrange's interpolating polynomial
Exercise 27: Investigate the behavior of Lagrange's interpolating polynomials
Exercise 28: Plot a wave packet
Exercise 29: Judge a plot
Exercise 30: Plot the viscosity of water
Exercise 31: Explore a complicated function graphically
Exercise 32: Plot Taylor polynomial approximations to \( \sin x \)
Exercise 33: Animate a wave packet
Exercise 34: Animate a smoothed Heaviside function
Exercise 35: Animate two-scale temperature variations
Exercise 36: Use non-uniformly distributed coordinates for visualization
Exercise 37: Animate a sequence of approximations to \( \pi \)
Exercise 38: Animate a planet's orbit
Exercise 39: Animate the evolution of Taylor polynomials
Exercise 40: Plot the velocity profile for pipeflow
Exercise 41: Plot sum-of-sines approximations to a function
Exercise 42: Animate the evolution of a sum-of-sine approximation to a function
Exercise 43: Plot functions from the command line
Exercise 44: Improve command-line input
Exercise 45: Demonstrate energy concepts from physics
Exercise 46: Plot a w-like function
Exercise 47: Plot a piecewise constant function
Exercise 48: Vectorize a piecewise constant function
Exercise 49: Visualize approximations in the Midpoint integration rule
Exercise 50: Visualize approximations in the Trapezoidal integration rule
Exercise 51: Experience overflow in a function
Exercise 52: Apply a function to a rank 2 array
Exercise 53: Explain why array computations fail
Exercise 54: Verify linear algebra results
References
A list object is handy for storing tabular data, such as a sequence of objects or a table of objects. An array is very similar to a list, but less flexible and computationally much more efficient. When using the computer to perform mathematical calculations, we often end up with a huge amount of numbers and associated arithmetic operations. Storing numbers in lists may in such contexts lead to slow programs, while arrays can make the programs run much faster. This is crucial for many advanced applications of mathematics in industry and science, where computer programs may run for hours and days, or even weeks. Any clever idea that reduces the execution time by some factor is therefore paramount.
However, one can argue that programmers of mathematical software have traditionally paid too much attention to efficiency and "clever" program constructs. The resulting software often becomes very hard to maintain and extend. In this document we advocate a focus on clear, well-designed, and easy-to-understand programs that work correctly. Thereafter, one can start thinking about optimization for speed. Fortunately, arrays contribute to clear code, correctness and speed - all at once.
This document gives an introduction to arrays: how they are created and what they can be used for. Array computing usually ends up with a lot of numbers. It may be very hard to understand what these numbers mean by just looking at them. Since the human is a visual animal, a good way to understand numbers is to visualize them. In this document we concentrate on visualizing curves that reflect functions of one variable; i.e., curves of the form \( y=f(x) \). A synonym for curve is graph, and the image of curves on the screen is often called a plot. We will use arrays to store the information about points along the curve. In a nutshell, array computing demands visualization and visualization demands arrays.
All program examples in this document can be found as files in the folder src/plot.
This chapter is taken from the book A Primer on Scientific Programming with Python by H. P. Langtangen, 5th edition, Springer, 2016.