Computing machine-efficient polynomial approximations

Question

1. What are the contributions mentioned in the paper "Computing machine-efficient polynomial approximations" ?

2. What are the basic floating-point operations that are implemented in hardware on modern processors?

3. What are the two kinds of polynomial approximations?

4. What is the idea2 to build a polytope?

Accepted Answer

The authors then have to consider polynomial approximations for which the degree-i coefficient has at most mi fractional bits ; in other words, it is a rational number with denominator 2mi.. The authors provide a general and efficient method for finding the best polynomial approximation under this constraint.

Accepted Answer

The basic floating-point operations that are implemented in hardware on modern processors are addition/subtraction, multiplication, and sometimes division and/or fused multiply-add, that is, an expression of the form x y + z, computed with one final rounding only.

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Two kinds of polynomial approximations are used: the approximations that minimize the average error, called least squares approximations, and the approximations that minimize the worst-case error, called least maximum approximations, or minimax approximations.

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The idea2 is to build a polytope , still containing the 2mi p i , such that ∩ Zn+1 is the smallest possible, which means that the number of candidate polynomials is the smallest possible, in order to reduce as much as possible the final step of computation of supremum norms.

Accepted Answer

(2) When evaluating two polynomials whose coefficients are very close on variables that belong to the same input interval, the largest roundoff errors will be very close, too.

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Once the authors get this polytope, the authors need an efficient way of producing these candidates, that is, an efficient way of scanning the integer points (i.e., to points with integer coordinates) of the rational polytope the authors built.

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—If the authors restrict their search to odd truncated polynomials (in this case, the authors put n = 2k + 1 and the authors must have d ≥ k), it suffices to replace inequalities (4) withli ≤ k∑j=0 p j x2 j+1 i ≤ ui, i = 0, . . . , dto create a polytope of Rk+1 whose points with integer coordinates the authors scan.

Accepted Answer

—If the authors restrict their search to truncated polynomials some of whose coefficientshave fixed values, (e.g., if the authors assume that the truncated polynomials sought have constant term equal to 1) it suffices to replace inequalities (4) withli ≤ 1 + n∑j=1 p j x2 j+1 i ≤ ui, i = 0, . . . , dto create a polytope of Rn whose points the authors scan with integer coordinates (we must have d ≥ n − 1).

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It is important to notice that their polynomial approximations will be computed once only and will be used very frequently (indeed, several billion times for an elementary function program in a widely distributed library).

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In the last column of Table II, the notation < ̂ means that the authors chose a value slightly smaller than ̂, namely ̂ rounded down to four fractional digits.

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T2 denotes the time in seconds for computing the norms (5) which, for the moment (cf. footnote 1), is done with Maple 9 in which the value of Digits was fixed equal to 20.

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To do this, consider a polynomial q whose degree-i coefficient is pi + δi. From Proposition 1, the authors have‖q − p‖[0,a] ≥ |δi||αi| ,where αi is the nonzero degree-i coefficient of T ∗n (x/a).

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These observations tend to indicate that, for all candidate polynomials, the roundoff errors will be very close, and the total error will be close to the sum of the approximation and roundoff errors.

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But the authors have no guarantee that p̂ is the best minimax approximation to f among the polynomials whose degreei coefficient is a multiple of 2−mi .

Accepted Answer

In the second one, the authors prove a lemma used in Section 3.2 that implies, in particular, the existence of a best truncated polynomial approximation.

Accepted Answer

In such cases, to minimize multiplier sizes (which increases speed and saves silicon area), the values of mi, for i ≥ 1, should be very small.

Computing machine-efficient polynomial approximations

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Computing machine-efficient polynomial approximations" ?

2. What are the basic floating-point operations that are implemented in hardware on modern processors?

3. What are the two kinds of polynomial approximations?

4. What is the idea2 to build a polytope?

5. What is the way to evaluate a polynomial?

6. What is the way to get rid of the candidate polytope?

7. what is the way to search for a truncated polynomial?

8. what is the way to search for truncated polynomials?

9. How often will the authors use their polynomial approximations?

10. What is the value of in the last column of Table II?

11. What is the time in seconds for computing the norms?

12. What is the degree-i coefficient of a polynomial?

13. What is the polynomial for the approximation error?

14. What is the minimax approximation to f?

15. What is the first proof of the lemma used in Section 3.2?

16. What is the way to minimize the size of the multiplier?

Figures

Citations

Faster CryptoNets: Leveraging Sparsity for Real-World Encrypted Inference

Elementary Functions Hardware Implementation Using Constrained Piecewise-Polynomial Approximations

Rigorous Polynomial Approximations and Applications

Elementary Functions and Approximate Computing

One Method for Proving Inequalities by Computer

References

Combinatorial optimization. Polyhedra and efficiency.

Combinatorial Optimization

The Chebyshev polynomials

Polynomials and Polynomial Inequalities

Elementary Functions: Algorithms and Implementation

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