HP Prime: Interval Arithmetic
Interval Arithmetic
In computing, we are working with approximate calculation. Often a number can be stated as a small interval to account for round off error.
An interval is a range of two numbers from a to b: [a, b]. The variable a represents the lower bound, with the variable b represents the upper bound. There is no restriction on how large or small the interval is.
For example, the interval [ 81.8, 82.2 ] represents the range from 81.8 to 82.2. This could stand for a calculated value of 82.0 with a plus-or-minus error of 0.2.
Applying a general function of one function, f(x) to interval, results in this:
f([a, b]) = [ min(f(a), f(b)), max(f(a), f(b)) ]
Applying this to the square root function to intervals:
√([a,b]) = [ min(√a, √b), max(√a, √b) ]
The arithmetic functions for intervals are defined as:
[a, b] + [c, d] = [a + c, b + d]
[a, b] - [c, d] = [a - d, b - c]
[a, b] × [c, d] = [min(a × c, a × d, b × c, b × d), max(a × c, a × d, b × c, b × d)]
[a, b] ÷ [c, d] = [min(a/c, a/d, b/c, b/d), max(a/c, a/d, b/c, b/d)]
Scalar multiplication, given the scalar is positive:
s × [a, b] = [ s × a, s × b ]
HP Prime Interval Programs
INTERVALHELP: help screen for interval arithmetic functions
INTERVALADD: adding two intervals
INTERVALSUB: subtracting two intervals
INTERVALMUL: multiply two intervals
INTERVALDIV: divide two intervals
In the Home mode, use the curly brackets (used for lists { }), to type the intervals. For example, the interval [ 81.8, 82.2 ] would be typed {81.8,82.2}.
EXPORT INTERVALHELP()
BEGIN
// 2022-12-22 EWS
PRINT();
PRINT("Interval Arithmetic");
PRINT("Use curly brackets {}");
PRINT("");
PRINT("INTERVALADD({a,b},{c,d})");
PRINT("add intervals");
PRINT("");
PRINT("INTERVALSUB({a,b},{c,d}");
PRINT("subtract intervals");
PRINT("");
PRINT("INTERVALMUL({a,b},{c,d})");
PRINT("multiply intervals");
PRINT("");
PRINT("INTERVALDIV({a,b},{c,d})");
PRINT("divide intervals");
END;
EXPORT INTERVALADD(v1,v2)
BEGIN
RETURN v1+v2;
END;
EXPORT INTERVALSUB(v1,v2)
BEGIN
RETURN v1-REVERSE(v2);
END;
EXPORT INTERVALMUL(v1,v2)
BEGIN
LOCAL v3;
v3:=CONCAT(v1*v2,v1*REVERSE(v2));
RETURN {MIN(v3),MAX(v3)};
END;
EXPORT INTERVALDIV(v1,v2)
BEGIN
INTERVALMUL(v1,1/REVERSE(v2));
END;
Download a zip file here: https://drive.google.com/file/d/1L8mmnm3xx58PVjavDROCiZAafBOCsH9o/view?usp=share_link
Example
Interval 1 = {13, 29}
Interval 2 = {10, 14}
Add: {23, 43}
Subtract: {-1, 19}
Multiply: {130, 406}
Divide: {0.928571428572, 2.9}
Source
Moore, Ramon E. Mathematical Elements of Scientific Computing Holt, Rinehart, and Winston, Inc: New York. 1975. ISBN 0-03-088125-0
Until next time and wish you all well,
Eddie
All original content copyright, © 2011-2023. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
