Polynomial Evaluation via Root Product Form
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Solution Overview
Problem
Current methods for evaluating polynomials with floating-point operations lack precision, particularly when dealing with root products, leading to issues like digit cancellation and exponent overflow/underflow, which are not adequately addressed by existing polynomial evaluation techniques.
Innovation Solution
A new method transforms polynomials into a root product form that ensures precise evaluation by limiting the required precision of roots to just over twice the final working precision, using a two-step differencing process and correct rounding mechanisms to eliminate cancellation errors.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional direct evaluation methods are used for polynomials, then computation is straightforward, but precision is lost due to digit cancellation when real values approach floating point inputs
Solution Approach 1:
The polynomial is segmented into a product of linear factors (x - root_i) rather than evaluated as a sum of power terms. This segmentation allows each factor to be evaluated independently, preventing catastrophic cancellation from occurring in the overall evaluation process.
Solution Approach 2:
The roots of the polynomial are pre-computed to sufficient precision (just over twice the final working precision) before evaluation. This preliminary computation of roots enables the subsequent factor evaluation to maintain high precision without requiring unlimited precision arithmetic during the actual evaluation.
2Measurement precision
If unlimited precision is used for root representation, then precise evaluation is achieved, but computational resources become excessive
Solution Approach 1:
The precision parameter for root representation is optimized to be just over twice the final working precision, rather than using unlimited precision. This parameter change achieves the desired relative precision in the evaluation result while keeping computational resources finite and manageable.
3Measurement precision
If root product form is used for polynomial evaluation, then precision can be maintained, but exponent overflow and underflow issues arise
Solution Approach 1:
Each linear factor (x - root_i) is evaluated independently with its own local precision requirements. This local quality approach allows the exponent ranges of individual factors to be managed separately, preventing global exponent overflow or underflow that would affect the entire polynomial evaluation.
4Ease of manufacture
If conventional polynomial evaluation methods are used, then implementation is simple, but cancellation errors occur when real values are close to floating point inputs
Solution Approach 1:
The root product form acts as an intermediary representation between the polynomial coefficients and the evaluation result. By evaluating the product of (x - root_i) factors rather than the direct polynomial expression, cancellation errors are prevented while maintaining implementation feasibility through the use of pre-computed roots.
Data Source
AI summary
Based on the root-product polynomial form, this method compresses essential information of a polynomial by transforming polynomials into a form which eliminates cancellation error, when evaluating polynomials, of one unknown, for real, complex, and quaternion, which are implemented with floating point numbers. Additional filtering methods simplify evaluation, including the elimination of extremely small and large root factors, which can cause out-of-range errors. The usual setup problem for root-product forms, that of needing potentially unlimited root precision and floating point range, is largely eliminated for real polynomials, and greatly mitigated for complex and quaternion, and other hypercomplex polynomials.