Validated Numeric Processing with Definite-Uncertain Number Arithmetic
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Solution Overview
Problem
Existing numerical arithmetic operations, particularly those based on the IEEE 754 standard, suffer from issues such as overflow/underflow, digit loss, information loss, and round-off errors, especially when handling higher-order polynomials, leading to potential inaccuracies at the nanometer level.
Innovation Solution
The use of high-precision computer numbers, defined by definite and uncertain components, allows for validated numerics by separately expressing numbers as a subset of real numbers using n-ary numbers with a radix n, ensuring accurate arithmetic operations through extended absolute effective digits.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Speed
If IEEE 754 floating-point arithmetic is used for numerical operations, then computation speed and hardware compatibility are improved, but precision and reliability deteriorate due to overflow/underflow, digit loss, and round-off errors
Solution Approach 1:
The patent segments the numerical value into two distinct components: a definite number (representing the certain part) and an uncertain number (representing the error bound). This segmentation allows the system to track and control precision explicitly while maintaining efficient computation, resolving the contradiction between speed and precision by handling only the definite part with standard arithmetic while separately managing the uncertain part.
Solution Approach 2:
The patent changes the parameter representation from standard floating-point format to a custom format with explicit precision control. By introducing the uncertain number component and modifying how arithmetic operations are performed (particularly in determining the new uncertain number after operations), the system achieves higher reliability without sacrificing computation speed.
2Reliability
If interval arithmetic is used to ensure mathematical correctness, then precision and reliability are improved, but computational complexity and operation overhead increase significantly
Solution Approach 1:
The patent extracts the error bound information into a separate uncertain number component, distinct from the main definite value. This extraction allows the system to perform simple arithmetic on the definite part while separately tracking the uncertain part through simplified rules, avoiding the full complexity of interval arithmetic while maintaining mathematical correctness.
Solution Approach 2:
The patent performs preliminary determination of the uncertain number after each arithmetic operation using pre-established rules. Rather than performing complex interval arithmetic operations, the system applies predetermined formulas to calculate the new uncertain number based on the operation type and input uncertain numbers, significantly reducing computational complexity while ensuring reliability.
3Manufacturing precision
If higher precision arithmetic is implemented to handle nanometer-level calculations, then manufacturing precision is improved, but computational overhead and resource requirements increase
Solution Approach 1:
The patent applies partial precision by maintaining only the necessary definite and uncertain components required for nanometer-level precision, rather than using full high-precision arithmetic. This partial action approach achieves the required manufacturing precision while consuming fewer computational resources, as the system processes only the essential components needed for the task.
Data Source
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AI summary
Provided are an information processing apparatus, a program, and an information processing method capable of easily performing validated numerics. Arithmetic operation of definite numbers a1 and b1 of the computer numbers in which real numbers A and B are respectively defined by the following formulas (1) and (2) is performed to determine an absolute effective digit γ satisfying the following formula (3), A=a1+a2,a1≤Cha,−Cea≤a2<Cea B=b1+b2,a1≤Chb,−Ceb≤b2<Ceb A,B=a,b1+a,b2,−Cγ≤a,b2<Cγ wherein, a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha and hb respectively denote extended high order maxes that are minimum extended digits satisfying | a1 | ≤ Cha and | b1 | ≤ Chb, and ha and hb respectively denote high order maxes that are integers; and ea, eb and γ denote the absolute effective digits that are integers.