Division by Summation of Inverse Powers of Two for Memory Mapping
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Solution Overview
Problem
Conventional computing systems face inefficiencies in dividing large numbers by numbers not divisible by two, particularly prime numbers, which leads to time-consuming and computationally expensive operations, limiting memory mapping capabilities and bandwidth in systems with large memory spaces.
Innovation Solution
The method involves approximating the reciprocal of the dividing number using a summation of inverses of powers of two, represented as a repeating bit pattern, allowing for efficient division and memory mapping even with prime numbers, enabling parallel calculations and reducing computational expense.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional division techniques are used to divide large numbers by numbers not divisible by two, then the division operation can be performed, but the operation becomes very time-consuming and computationally expensive
Solution Approach 1:
The patent transforms the division operation into a multiplication operation by using the reciprocal of the divisor. Instead of performing slow division by numbers not divisible by two, the system pre-calculates reciprocal values and uses multiplication with these reciprocals to achieve division, significantly improving operation speed and reducing computational time.
Solution Approach 2:
The patent pre-calculates and stores reciprocal values of divisors in lookup tables before the actual division operation is needed. This preliminary action allows the system to replace time-consuming division operations with faster multiplication operations using pre-computed reciprocal values, thereby improving productivity and reducing execution time.
2Power
If memory is partitioned into separate memory channels to achieve high bandwidth access, then memory access bandwidth is improved, but the need for mapping global addresses to channel addresses increases computational complexity
Solution Approach 1:
The patent changes the address mapping function from a complex division-based mapping to a simpler multiplication-based mapping using pre-computed reciprocal values. This parameter change simplifies the mapping complexity while maintaining the ability to distribute addresses across multiple memory channels for high bandwidth access.
Solution Approach 2:
The patent introduces pre-computed reciprocal values as an intermediary to simplify the address mapping process. Instead of directly performing complex division operations to map global addresses to channel addresses, the system uses multiplication with reciprocal intermediaries to achieve the same mapping goal with reduced complexity.
3Productivity
If division by prime numbers is performed using conventional techniques, then the division can be completed, but the computational expense is very high
Solution Approach 1:
The patent fundamentally changes the operation from division to multiplication by using reciprocal values. This parameter change is particularly effective for prime numbers, as it avoids the need for complex iterative division algorithms and reduces computational energy consumption while maintaining division accuracy.
Solution Approach 2:
The patent replaces the mechanical division process with a multiplication process using reciprocal values. This substitution eliminates the need for repeated subtraction and comparison operations inherent in conventional division, thereby reducing computational energy usage and improving efficiency for dividing by prime numbers.
Data Source
AI summary
Divisions by numbers that are not divisible by two (2) can be performed in a computing system based on a summation that estimates and/or approximates the reciprocal of the dividing number or denominator value. By way of example, dividing by three (3) can be calculated based on a summation that approximates or estimates one third (⅓) represented as the sum of a selected group of the inverses of the powers of two (2) in a pattern, namely the sum of: ¼, 1/16, 1/64, 1/256, . . . ). Applications of the division techniques are virtually unlimited and include memory mapping of global memory addresses to memory channel addresses by dividing a global memory address into the number of memory channels, allowing memory mapping to be performed in an efficient manner even for large memory spaces using a number of memory channels that are not divisible by two, including prime numbers.