Floating Point Degree of Precision in Numerical Quadrature

1. Floating Point Degree of Precision in Numerical Quadrature Sanda Adam & Gheorghe Adam LIT-JINR Dubna & IFIN-HH Bucharest Ro-LCG Workshop Bucharest, RomaniaNovember 29-30, 2011

2. Overview • Bayesian automatic adaptive quadrature • Interpolatory quadrature sums • Floating point degree of precision • Features of the floating point degree of precision • Conclusions

3. Mathematical problem Given the proper (or improper) Riemann integral we seek a globally adaptive numerical solution of it within input accuracy specifications i.e.,

4. Automatic adaptive quadrature The derivation of the quantities Q and E is done within an approach based on two pillars: (i) A subrange subdivision strategy of the integration domain [a,b] implementing the ordering of the already generated subranges into a priority queue. (ii) A convenient local quadrature rule which yields, over each subrange a local pair {q, e} for the quadrature sum (q) and its associated local error estimate (e > 0) respectively. Then Q & E are obtained as sums of q & e over subranges.

5. Questions within Bayesian analysis The Bayesian analysis checks the absence of the following kinds of unacceptable integrand features: • Catastrophic cancellation by subtraction • Range of variation of the integrand beyond the maximally allowed polynomial threshold • Occurrence of integrand oscillations beyond the resolving power of the current integrand profile • Occurrence of inner isolated integrand discontinuities • Occurrence of unsolvable irregular behaviour of the integrand

6. CC 15-31 GK 10-21 GK 7-15 Lateral close proximity neighbourhoods Consist of six quadrature knots which produce information over distances equating the inter-knot distances at subrange centre.

7. Central close proximity neighbourhoods CC 15-31 GK 10-21 GK 7-15 Consist of nine quadrature knots which produce information over distances equating the inter-knot distances at subrange centre.

8. Overview • Bayesian automatic adaptive quadrature • Interpolatory quadrature sums • Floating point degree of precision • Features of the floating point degree of precision • Conclusions

9. An interpolatory quadrature sum approximates a proper or improper one-dimensional Riemann integral, by means of an interpolatory algebraic polynomial, the values of which equate those of the integrand function at a specific set of quadrature knots , pn(xk) = f (xk), k = 0, 1, ..., n. Interpolatory quadrature sums

10. The quadrature sum solves exactly the polynomial integrals over the fundamental power set, The maximum degree , at which these identities hold, defines the algebraic degree of precision of the quadrature sum . In the literature, the algebraic degree of precision, d, is considered to be a specificuniversal parameter of a given interpolatory quadrature sum, irrespective of the extent and localization of the integration domain on the real axis. Algebraic Degree of Precision

11. Overview • Bayesian automatic adaptive quadrature • Interpolatory quadrature sums • Floating point degree of precision • Features of the floating point degree of precision • Conclusions

12. In the calculation over of the set of probe integrals each monomial entering the integrand brings a distinct, non-negligible, contribution to . In floating point computations, the above property of the monomials of bringing distinct, non-negligible contributions to σm may get infringed both at integration limits β << 1 and β >> 1 . The maximum degree at which the identity of the individual monomial contributions is preserved in floating point computations defines the floating point degree of precision of the quadrature sum. Its definition is formalized in the next two slides. Floating Point Degree of Precision (1)

13. 1. Let denote the integration range of interest. 2. Let , a quadrature sum of algebraic degree of precision d , be computed over a set of t-bit floating point machine numbers (t = 52 in double precision). 3. Let ξ > 0, let fl(a) denote the floating point approximate of , and let [a] denote the ceiling of fl(a). 4. Let where Floating Point Degree of Precision (2a)

14. 5. For the integration range [α, β] we define , The quantities dXand dρ are computed from 4. 6. Then the floating point degree of precision, associated to is the positive integer Floating Point Degree of Precision (2b)

15. Overview • Bayesian automatic adaptive quadrature • Interpolatory quadrature sums • Floating point degree of precision • Features of the floating point degree of precision • Conclusions

16. Features of the Floating Point Degree of Precision • First, there is a manifold of integration ranges [α, β] over which all the terms of the polynomials πm(x) are significant and contribute distinctly to the computed output. • Then the floating point degree of precision, dfp equates the algebraic degree of precision, d. • Second, in the case of arbitrarily placed on the real axis narrow integration intervals(characterized by the property that 0 <ρ << 1), dfp << d, in agreement with the fact that the integrand properties inside [α, β] are sufficiently well described by a low degree Taylor series expansion of f(x) around one of the subrange ends.

17. Features of the Floating Point Degree of Precision • Third, the occurrence of extremalX values (0 < X << 1 for integration intervals around the origin, or X >> 1 for very large integration intervals) result in dfp << d irrespective of the value of the parameter ρ. • The case 0 < X << 1 is consistent with the observation that a low degree power series expansion around the origin approximate sufficiently well the integrand properties everywhere inside [α, β]. • The case X >> 1 corresponds to a sparse integrand discretization by the widely spaced from each other quadrature knots. Then the actual integrand behaviour inside[α, β]is poorly approximated by this sparse integrand sampling except for the case when the integrand can be well approximated by a low degree polynomial. • The discussion at X >> 1 also points to the fact that the occur-rence ofasymptotictails of the integrand cannot be tackled sufficiently accurately by general purpose quadrature rules.

18. Features of the Floating Point Degree of Precision • The above general remarks are strengthened by thefloating point degree of precision outputs obtained for specific quadrature sums. In what follows, results are reported for the GK 10-21 local quadrature rule, which is among the most attractive candidates for the implementation of the Bayesian automatic adaptive quadrature.The 21 Gauss-Kronrod abscissas result in an algebraic degree of precision d = 31. • We have considered three illustrative cases: (i) Gliding integration range [0, 1] on the real axis. A family of 1024 integration ranges was defined. (ii) Inflating integration range [α, β] over the real axis. A family of 1023 integration ranges illustrating this case has been obtained from the sampling. (iii) Non-equivalence of the siblings in the binary subrange trees. We assume the case study integration domain [0, 2n].

19. Features of the Floating Point Degree of Precision • Gliding integration rangeson the real axis. Gauss-Kronrod 10-21 local quadrature rule Floating point degrees of precision of six families of gliding ranges of lengths 1, 1/2, 1/4, 1/8, 1/16,1/32, respectively, versus their distances j from the origin Variation of the floating point degree of precision of the GK 10-21 local quadrature rule over the gliding range [0, 1] versus its distance j from the origin.It is shown thatdfp = d = 31 at low j values (j = 0, 1, 2), then dfp abruptly decreases at larger but small enough j, to show slower decreasing rates under the displacement of [0,1] far away from the origin, reaching a bottom value dfp = 5 at 701 ≤ j ≤ 1023.

20. Features of the Floating Point Degree of Precision • Inflating range[0,j]and gliding range [j-1,j]on the real axis. Gauss-Kronrod 10-21 local quadrature rule The following plot gives outputs for the families of 1023 integration ranges {j = 1, 2, ..., 1023} Variation of the floating point degree of precision of the GK 10-21 local quadrature rule over the inflating range [0, j] versus its width j.The plot of the computed values of dfp points to a behaviour of dfp which is similar to that reported in the previous case dfp = d = 31 atj = 1, 2, 3;abrupt and then milder decreasing rate down to dfp = 5 at 702 ≤ j ≤ 1023.

21. Features of the Floating Point Degree of Precision • Non-equivalence of the siblings in the binary subrange tree. Gauss-Kronrod 10-21 local quadrature rule Case study of the root domain [0, 220] A binary subrange tree is built up to n-th depth level by bisection ofthe parent ranges.Comparison of the dependencies of the floating point degrees ofprecision of the GK10-21 local quadrature rule on the depth level in the binary subrange tree generated by the root range [0, 2n], for theleftmost and the rightmost siblings are plotted for n=20. While the floating point degree of precision of the rightmost siblings in the binary subrange tree keeps the minimaldfp value of the root range [0, 2n], the values of the floating point degree of precision of GK 10-21 for the leftmost siblings increases from the initial minimal dfp value up to the maximally possible value dfp=d=31 at the last depth levels.

22. Overview • Bayesian automatic adaptive quadrature • Interpolatory quadrature sums • Floating point degree of precision • Features of the floating point degree of precision • Conclusions

23. Allowed range of variation of the Integrand • Definition: Given the subrange [α, β], its centre γ = (β+α) / 2, the • inherited integrand values fα = f(α)and fβ = f(β), together with • the newly computed fγ = f(γ), if{fα, fγ, fβ} define a strictly • monotonic sequence, thenM-1 <(fγ - fα) / (fβ - fγ) < M, where the • threshold M follows from the floating point degree of precision dfp • Bayesian inference: • proceed to subrange subdivision by bisection.

24. Conclusions The concept of the floating point degree of precision, dfp , of an interpolatory quadrature sum has been defined in an attempt to improve the integrand conditioning diagnostics over subranges in the Bayesian approach to the automatic adaptive quadrature. The study evidenced the need to consider two parameters for the definition of the dfp , namely,the maximum absolute magnitude X among the two ends of the current integration range and the ratio ρbetween the range length and X. A number of three families of case study integrals was considered: (i) glide integration range [0,1] on the real axis, (ii) inflating integration range [0, j] over the real axis, and (iii) the leftmost and rightmost siblings of a binary subrange tree. Both general arguments and data collected for the three case studies have evidenced the occurrence of a significant variation of dfp from the standard algebraic degree of precision d of an interpolatory quadrature sum. The content of information which can be extracted by means of a same local quadrature rule significantly varies with the subrange location inside the original integration domain.

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