This document summarizes a study on using octonions to evaluate computer performance. It introduces octonions and describes how a program was created to calculate prime numbers using octonions. The program was used to benchmark the performance of 4 different CPUs by measuring the calculation time. The results showed the octonion benchmark was able to distinguish between the CPUs and detected improvements from additional cores. Future work is proposed to generalize the prime number theorem using octonions and apply it to education.

1. A Study of Linear Algebra:
Octonion and Application To
Computer Science
s1140183 Shota Hirabayashi
Supervised by Prof. Noriaki Kamiya

2. Table of Contents
Introduction
About Octonion
Application to Performance Evaluation
Conclusions and Future Work

3. Introduction
Linear Algebra
Matrices
Linear maps
Determinants
Eigenvalues, Eigenvectors
etc
Applied to another mathematics fields, physics,
economics, mechanical engineering, etc

4. About Octonion
Pythagorean Theorem
n
(n, a, b : Natural Number)
b
a
expansion
(n, a,…, h : Natural Number)

5. About Octonion
Definition
Prime Number Theorem (Octonion)
Reference: Appendix 1 (Table of Octonion)

6. Application To Performance Evaluation
Programming for evaluating CPU performance
Determine the
number of variables
Determine the number of threads
(This supports multi-core processors)
Display Combination Number & Calculation Time(sec)
Remark) In computing, comparing calculation times is called a ”benchmark”.

7. Application To Performance Evaluation
Super π
one of the most popular benchmark software, which calculates π to
specified number of digits after the decimal point up to a maximum of 32
million introduced by Kanada Laboratory (University of Tokyo).
Determine number of digits after the decimal point
Display Combination Number & Calculation Time (sec)

8. Application To Performance Evaluation
CPU Usage
Red : Octonion (Our benchmark) Blue : Super π

9. Results
Octonion (sec)
Specification
Computer 1 : Intel Pentium4 3.0GHz (FSB 800MHz, 1 thread (HT OFF))
Computer 2 : Intel Core2 Duo L7300 1.4GHz (FSB 800MHz, 2 threads)
Computer 3 : Intel Core2 Duo T9300 2.5GHz (FSB 800MHz, 2 threads)
Computer 4 : Intel Core2Quad Q9650 3.0GHz (FSB 1333MHz, 4 threads)

10. Results
Super π (sec)
Specification
Computer 1 : Intel Pentium4 3.0GHz (FSB 800MHz, 1 thread (HT OFF))
Computer 2 : Intel Core2 Duo L7300 1.4GHz (FSB 800MHz, 2 threads)
Computer 3 : Intel Core2 Duo T9300 2.5GHz (FSB 800MHz, 2 threads)
Computer 4 : Intel Core2Quad Q9650 3.0GHz (FSB 1333MHz, 4 threads)

11. Conclusions and Future Work
Conclusions
This experiment showed a possibility that Octonion
applies to assessment of computer hardware.
Computing requires about 8 hours, although
we use the quadcore system (computer 4).
Future Work
Generalization and Formularization of the prime
number theorem of
Application to Education System

12. Thank you for listening.

13. Result(Optional)
Octonion

14. Result(Optional)
Super π