### Fields Validated Optimization WorkshopClarify Taylor Models

Prepared by George Corliss, May 27, 2002

We have followed with interest the discussions on the reliable_computing email listserv about Taylor models

Arnold Neumaier has posted at http://www.mat.univie.ac.at/~neum/papers.html#taylor a paper Taylor Forms - Use and Limits.'' Here, that paper is referenced by page as [AN.2] for page 2, etc. In many places, [AN] invited Martin to respond or to supply additional details.

If we ask Martin to explain these matters to us, we ask questions until we are satisfied, and we take careful notes, our deliberations can serve to help

1. Us learn more about the details of Taylor models
2. Martin formulate a response to Arnold.
Ideal outcomes are
1. We understand the details, benefits, and limitations
2. A _joint_ Arnold + Martin paper?

Good places to start include

• Arnold's covering email
• [AN.28, Conclusions]
• (0) [AN.16] Specifications for Taylor models?

It is my intention to understand and to help Martin focus the discussion. It is absolutely NOT my intention to either tell Martin what he should do or to respond to Arnold ourselves.

I characterize issues as: (x) refers to the circled notes in my copy of [AN]

## ROUNDING ERRORS

(1) [AN.2] the mechanism for handling rounding errors is undocumented''

(1) [AN.5]; (1) [AN.17]; (1) [AN.17]

(1) [AN.28 Conclusion 4]

Other implementations? Ned, Stadther?

## WRAPPING EFFECT - SHRINK WRAPPING

(15) [AN.10]; (16) [AN.10]; (15.1) [AN.11]; (15) [AN.21-24]

(15) [AN.12] vs. cancellation & dependence, level of nestedness

(20) [AN.25] Example 2

(15) [AN.28, Conclusion 3 & 4]

## ORDER OF ENCLOSURE - RANGE BOUNDING: Large, medium, small boxes

(4) [AN.3] factor of d+1 for method of order d

(9) [AN.5] interval evaluations of the Taylor form are needed, and these are nontrivial''

(10) [AN.6] asymptotic results for narrow intervals are possible''

cf. convergence results of Tibor?

(11) [AN.6] independence of the location of the midpoint is essential''

(4.1) [AN.6]; (9.1) [AN.8] bicentered form?

(9) [AN.7] bounding the range of the Taylor form

(9.2) [AN.8] linear dominated range bounder

(9.3) [AN.8] f(x) = -x_1^2 - ... - x_n^2 in [x] = [0.1h, 1.1h]^n

(9.4) [AN.9] evaluate the exact range of the quadratics, cf. Cornelius

(*) [AN.9] f(x) = Sum ...

(13) [AN.9] Baker explain peeling?''

(18) [AN.15] generalizes Makino on bounding polynomials

(9) [AN.17] optimal range?

(19) [AN.18] very little accuracy beyond centered forms'' f(x) = \frac{1}{1-r} - \frac{1}{2-r}

(19) [AN.19] Taylor forms generate dependence'' \frac{1}{1-x} = 1 + x + x^2 + ... + \frac{x^{2k}}{1-x}

(19) [AN.20] Taylor forms lose their superiority when no cancellation is present'' E.g., f(x) = (1 + x)^4

(19) [AN.21] fail to give large improvements if the Taylor coefficients do not decay quickly'' f(x) = \frac{1}{n} sum_{k=1}^n \sin (kx)

(19) [AN.21] high order terms with LARGE coefficients should be purged and moved into low order terms''

## SUPPORTING THEORY

(12) [AN.8] offer a cure for the dimensionality curse''

(16) [AN.11] proof of enclosure of solutions over long times

## COMPARISONS WITH OTHER WORK

(2) [AN.2] evidence that the use of floating point numbers as coefficients is an essential improvment over using narrow interval coefficients''

(2) [AN.2] theoretical analysis or thorough comparison

(5) [AN.3]; (6) [AN.4]; (14) [AN.10] cf. Kaucher?

(7) [AN.4]; (8) [AN.5] cf. slopes and centered forms

(17) [AN.15] using constraint propagation

(21) [AN.25] COSY-VI not publically available?

## CHALLENGING EXAMPLES

(9.3) [AN.8] f(x) = -x_1^2 - ... - x_n^2 in [x] = [0.1h, 1.1h]^n

(*) [AN.9] f(x) = Sum ...

(19) [AN.18] very little accuracy beyond centered forms'' f(x) = \frac{1}{1-r} - \frac{1}{2-r}

(19) [AN.19] Taylor forms generate dependence'' \frac{1}{1-x} = 1 + x + x^2 + ... + \frac{x^{2k}}{1-x}

(19) [AN.20] Taylor forms lose their superiority when no cancellation is present'' E.g., f(x) = (1 + x)^4

(19) [AN.21] fail to give large improvements if the Taylor coefficients do not decay quickly'' f(x) = \frac{1}{n} sum_{k=1}^n \sin (kx)

(20) [AN.25] Example 1

(20) [AN.25] Example 2