APL with a Mathematical Accent

C.A. Reiter

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351466332
• Edition: 1
• Topic: Mathematik
• Subtopic: Mathematik Allgemein

This book should be of interest to mathematics scientists working in the areas of linear algebra, abstract algebra, number theory, numerical analysis, operations research and mathematical modelling.

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APL

With a Mathematical Accent

CHAPTER

1

First Steps with APL

Chapter One begins by discussing how to interact with APL and interpret basic APL statements. It gives several familiar mathematical functions and their APL denotations. It also introduces some of APL’s powerful vector processing capabilities and special functions. Finally, the chapter briefly discusses some facts about the APL work environment and its management.

1.1 Interacting with APL

Using APL can be somewhat like using a calculator. You enter data or commands, and APL responds. If you enter 3 + 2, for example, the response is 5. The ongoing exchange between you and APL is recorded on paper or on a video screen. The record, or session log, of a few arithmetic problems could look like this:

The APL response begins at the left margin on the line below your entry. After the response there is an automatic advance to the next line and a six-space indent; then APL is ready for further input. This format makes it easy to distinguish your input from APL’s response. This book has several small examples from a session log that are printed across the page like this:

1.2 Order of Execution

Common mathematical notation has rules governing the order in which mathematical operations are performed. For example, multiplication and division take precedence over addition and subtraction. The commonly used rules interpret 3 X 4 –5 as 12–5, not 3×(–1) and 2+6÷3 as 2+2, not 8+3. Conventions for interpreting more complicated expressions such as sin²ab also exist, but they are intricate to codify and can lead to ambiguous expressions when many functions are involved.

APL has but one rule governing the order of function execution within APL expressions. There is no hierarchy of functions with some taking precedence. The resulting simplicity is especially important because APL has many primitive (built-in) functions. Simply expressed the rule is

APL functions are executed in order from right to left.

Note that the rule is that you start at the right and work to the left. For example, the APL interpretation of the expression 3 × 4 + 5 is as follows: First add the 4 and 5 to get 9 (the rightmost function is +), then multiply 9 by 3 to get 27. Thus, the APL evaluation of 3×4 + 5 is 27. As with conventional mathematical notation, expressions may be grouped with parentheses; the right-to-left rule also applies within the parentheses.

Here are some examples; note the effect of parentheses:

The order of execution rule applies to all functions; here are examples with the power function or exponentiation, denoted*:

Roots are treated as fractional powers in APL; for example,

1.3 Negative Sign, Minus Sign

Negative numbers are indicated with a high minus sign; for example, ⁻2. Note the difference in position between the minus sign that indicates the arithmetic function subtraction, as in 3−5, and the APL negative sign that serves as part of the symbol denoting a negative number, as in ⁻2. Here are examples:

1.4 Assignment and Variable Names

APL variables are given values by assignment, which is symbolized with a left-pointing arrow as in A←5. When you enter A ← 5, 5 is stored with the name A, the automatic advancement and indentation occurs in the session log, and APL awaits your next entry. After a variable has been assigned a value, entering that variable name alone results in the display in the session log of the current value of the variable. A new assignment to a variable name replaces any value formerly stored with that name. Here is a sample session log:

APL displays the result of the last function applied in a line unless the last operation is assignment.

Multiple assignments are convenient; for example, K← J←1 would assign both K and J the value 1. The expression F←2+E←D×D←3 would assign the values 3, 9, and 11 to the variables D,E, and F, respectively.

More suggestive variable names can be helpful; for example, to find the area of a rectangle,

A letter followed by any combination of letters and numerals is a valid variable name; typically, systems allow up to 77 characters in a name. Thus, AA, B14C, X5Y05, Z3TERM, and MOON are valid variable names. Modern systems usually have uppercase and lowercase alphabets, permitting ample flexibility in constructing names.

1.5 Example: Heron’s Formula

Suppose A,B, and C are the lengths of...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• CHAPTER 1 First Steps With APL
• CHAPTER 2 Fundamentals of Arrays
• CHAPTER 3 Elementary Function Definition
• CHAPTER 4 Matrix Algebra
• CHAPTER 5 Data Comparison and Logical Functions
• CHAPTER 6 Simulation and More Mathematical Functions
• CHAPTER 7 Statistics and Graphics
• CHAPTER 8 More Array Manipulation
• CHAPTER 9 Sorting and Coding
• CHAPTER 10 More Function Definition
• APPENDIX A Help Error Messages, and Debugging
• APPENDIX B Workspace Environment
• APPENDIX C Keyboards
• APPENDIX D Answers to Selected Exercises
• Bibliography
• References
• Index