APL Programs for the Mathematics Classroom

The idea for this book grew out of proposals at the APL86 con­ ference in Manchester which led to the initiation of the I-APL (International APL) project, and through it to the availability of an interpreter which would bring the advantages of APL within the means of vast numbers of school children and their teachers. The motivation is that once school teachers have glimpsed the possibilities, there will be a place for an "ideas" book of short programs which will enable useful algorithms to be brought rapidly into classroom use, and perhaps even to be written and developed in front of the class. A scan of the contents will show how the conciseness of APL makes it possible to address a huge range of topics in a small number of pages. There is naturally a degree of idiosyncrasy in the choice of topics - the selection I have made reflects algo­ rithms which have either proved useful in real work, or which have caught my imagination as candidates for demonstrating the value of APL as a mathematical notation. Where appropriate, notes on the programs are intended to show the naturalness with which APL deals with the mathematics concerned, and to estab­ lish that APL is not, as is often supposed, an unreadable lan­ guage written in a bizarre character set.

1. Introduction.- 1.1 Graphics.- 1.2 Idioms.- 2. Arithmetic and Numbers.- 2.1. Basic Programs with Integers.- 2.2. Square and Triangular Numbers.- 2.3. Multiplication and other Tables.- 2.3.1. Log Tables.- 2.3.1. Trig Tables.- 2.4. Isomorphisms.- 2.5. Primes and Factors.- 2.6. HCF and LCM.- 2.7. Recurring Decimals.- 2.8. Numbers in Different Bases.- 2.9. Roman Numerals.- 2.10. Encoding and Decoding.- 2.11. Problems involving Base 10 Digits.- 2.12. Computer Arithmetic.- 2.13. Counting Series Forwards and Backwards.- 2.14 Complex Numbers.- 2.14.1. Complex Roots of Unity.- 3. Algebra and Sets.- 3.1. Some Basic Algebra.- 3.2. Roots of Quadratics.- 3.3. Matrix Operations.- 3.3.1. Determinants.- 3.4. Polynomials.- 3.5. Arithmetic and Geometric Progressions.- 3.6. Sets.- 3.7. Polynomial Coefficients from Roots.- 4. Series.- 4.1. Recurrence Relations.- 4.2. Tests for Monotonicity.- 4.3. Convergence.- 4.4. Binomial Coefficients.- 4.4.1. Pascal’s Triangle.- 4.5. Successive Differences of Series.- 4.6. Fibonacci Numbers.- 4.7. Series relating to pi.- 4.8. Series for e.- 4.9. A Series for ?2.- 4.10. Trig Series.- 4.11. Continued Fractions.- 4.12. Interpolation.- 5. Formulae and Tables.- 5.1. Compound Interest.- 5.1.1. Present Values.- 5.2. Mortgage Repayments.- 5.3. Triangle Formulae.- 5.4. Longest and Shortest Journeys.- 5.5. Pythagoras’s Theorem and Norms.- 5.6. Pythagorean Triples.- 6. Geometry and Pattern.- 6.1. Parametric Plotting.- 6.1.1. Conic Sections.- 6.1.2. Hypocycloids and Epicycloids.- 6.2. Envelopes.- 6.2.1. Conic Sections.- 6.2.2. Hypocycloids and Epicycloids.- 6.3. Transformations.- 6.4 Perspective Drawing.- 6.5. Co-ordinate Geometry in Two Dimensions.- 6.6. Polar and Cartesian Coordinates.- 6.7. Patterns by Plotting Large Numbers of Points.- 7. Calculus.- 7.1. Numerical Integration.- 7.1.1. Upper and Lower Bounds for Integration.- 7.1.2. Trapezium Rule.- 7.1.3. Simpson’s Rule.- 7.1.4. Adaptive Simpson’s Rule.- 7.2. Root Finding.- 7.2.1. Bisection Method.- 7.2.2. Iteration Method.- 7.2.3. Newton-Raphson Method.- 7.3. Ordinary Differential Equations.- 7.3.1. Euler’s Method.- 7.3.2. Mid-point Method.- 7.3.3. Trapezium Method.- 8. Probability and Statistics.- 8.1. Discrete Probability Distributions.- 8.1.1. Binomial Distribution.- 8.1.2. Poisson Distribution.- 8.1.3. Hypergeometric Distribution.- 8.2. The Birthday Problem.- 8.3. Descriptive Statistics.- 8.3.1. Variance.- 8.3.2. Standard Deviation.- 8.3.3. Partition Values.- 8.3.4. Mode and Range.- 8.4. Random Numbers from Various Distributions.- 8.5. Simulations.- 8.5.1. Dice/Coins etc..- 8.5.2. Buffon’s Needle.- 8.6. Frequency Distributions.- 8.6.1. Stem-and-leaf Plot.- 8.6.2. Two-way Frequency Distribution.- 8.6.3. Scatterplots.- 8.7. Regression.- 8.8. Correlation.- 8.8.1. Covariance and Correlation Matrices.- 8.9. Non-parametric Tests.- 8.9.1. Runs Test.- 8.9.2. Rank Correlation.- 8.9.2.1. Spearman’s Coefficient.- 8.9.2.2. Kendall’s Coefficient.- 8.9.3. Sign Test.- 8.9.4. Wilcoxon Signed Rank Test (W-test).- 8.9.5. Mann-Whitney Rank Sum Test (U-test).- 8.9.6. Goodness of Fit.- 8.10. Statistical Tables.- 8.10.1. Normal probability density fn..- 8.10.2. Student t probability density fn..- 8.10.3. Gamma and Chi-squared proby. dens. fns..- 8.10.4. F probability density funtion.- 8.11. Sample Sizes.- 9. Combinatorics.- 9.1. Permutations in Lexical Order.- 9.2. Derangements.- 9.3. Combinations.- 9.4. Selections.- 9.5. Compositions and Partitions.- 9.6. Latin Squares.- 9.7. Magic Squares of Odd Order.- 10. Games and Miscellaneous.- 10.1. Deal a Hand at Whist.- 10.2. Chessboard.- 10.3. Mastermind.- 10.4. Life.- 10.5. Recursive Algorithms.- 10.5.1. Tower of Hanoi.- 10.5.2. Ackerman’s Function.- 10.6. Optical Illusions.- Appendix 1. Graphics.- Appendix 2. Idioms and Utilities.- A2.1. Rounding, Averaging, and Removing Duplicates.- A2.2. Sorting and Ranking.- A2.3. Statement Joining.- A2.4. Branching and Prompting.- A2.5. Matrix Manipulation.- A2.6. Replication.- A2.7. Without.- A2.8. Bit Manipulation.- A2.9. Some String Handling Functions.- A2.10. Testing for Numeric/Character.- A2.11. Timing Function Execution.- Appendix 3. Graphics Functions in I-APL.- Index of Topics.- Index of Programs and Variables.