Taylor model arithmetic

Similar to intervals, basic operators such as addition, multiplication can also be ex- tended to deal with TMs. Based on them, we are able to define a TM extension of a given function. Besides, the inclusion isotonicity as well as the Lipschitz property of TM extensions may be defined analogously.

Given a function f and its TM (p, I) over the domain D. A TM for h(f ) wherein h is a unary operator can be computed only based on (p, I). For example, the additive inverse and integral on a TM (p, I) over D are defined by

Additive inverse: −(p, I) = (−p, −I)

Integral: ∂−1_i (p, I) = (Rbi

ai(p(~x) − pe(~x))dxi , (Int(pe) + I) · [ai, bi])

wherein p_e consists of the terms of degrees > k in p for some truncation order k ≥ 0, and [ai, bi] is the range of xi in D. The operation Int(pe) denotes an interval enclosure

of pe over D. Here, the order k is not necessarily same as the degree of p. The reason of

specifying it is to limit the representation size of the resulting TM.

Given two functions f, g over the some domain D, assume that (p1, I1), (p2, I2) are the

TMs of them respectively. For a binary operator ◦, a TM for f ◦ g can be computed only based on (p₁, I1) and (p2, I2). For example, the sum of the two TMs is computed by

(p1, I1) + (p2, I2) = (p1+ p2, I1+ I2)

and for some truncation order k ≥ 0, their order k product is computed by

(p1, I1) · (p2, I2) = (p1· p2− pe , Int(p1) · I2+ I1· Int(p2) + I1· I2+ Int(pe))

2.3. TAYLOR MODELS 43 Algorithm 3 Compute an order k TM for f ((p, I))

Input: a TM (p, I), a continuous function f ∈ Cκ(Df) such that (p, I) ⊆ Df

Output: an order k TM for f ((p, I)) such that k < κ

1: Compute the order k Taylor polynomial pf(x) of f (x) at the midpoint of Int((p, I));

2: Evaluate a safe remainder interval I_f for p_f(x);

3: Compute the order k TM (p_r, Ir) for pf((p, I));

4: Ir ← Ir+ If;

5: return (pr, Ir);

An order κ TM (p, I) can be simplified by performing a k-truncation on it for some 0 ≤ k < κ. That is, we remove the terms of degrees > k in p and add their interval enclosure onto the remainder.

Truncation: Trunck((p, I)) = (p − pe , I + Int(pe))

The resulting TM is a simplification as well as an over-approximation of the original one. The TM division is more complicated. Given two TMs (p1, I1), (p2, I2) such that

0 /∈ Int((p2, I2)). An order k TM of (p1, I1)/(p2, I2) is computed in the following way.

Firstly, we compute the order k Taylor polynomial p_k for the function _x1 at x = c wherein c is the midpoint of Int((p2, I2)),

pk(x) = 1 c · 1 − x − c c +  x − c c 2 − x − c c 3 + · · · + (−1)k x − c c k!

and a remainder interval for p_k is evaluated by

Ir= Int  (−1)k+1 1 xk+2 · (x − c) k+1 

over x ∈ Int((p2, I2)). Secondly, we compute an order k TM (p3, I3) by substituting

(p2 − c, I2) in the place of x in (pk, Ir). Then an order k TM of (p1, I1)/(p2, I2) can be

computed by (p1, I1) · (p3, I3). Such an idea could also be applied to computing TMs for

continuous functions. We present a general procedure by Algorithm 3. For more TM operations, one may refer to [MB03].

By TM arithmetic, we may compute a TM for a function based on the TMs of its components. In that case, the resulting remainder interval is computed by a procedure consists of several subroutines. If the procedure is a contractible remainder evaluation, then we may still able to improve the accuracy by using a higher TM order during the computation.

Class of the TM domains. Although the TM domains under our consideration are

always intervals, we can still define a large class of non-convex sets by those TMs. For a function f with an arbitrary domain set D, we may compute a set of grids that over- approximate D, and then use the TMs over them to over-approximate the values of f .

2.3.4 Applications

TMs provide higher-order over-approximations for continuous functions. Unlike inter- vals, the overestimation in a TM can be measured only based on its remainder interval size which is often very small. They considerably relieve the dependency problem in many computation jobs. Therefore, TMs are often used as replacements of intervals in numerical computation tasks when accuracy is critical. However, TM arithmetic is more computational complex than interval arithmetic in general. It can be seen from their representations. An interval could be represented by their endpoints which are only two floating-point numbers. On the other hand, for a TM, we need not only to represent its interval remainder but also to keep a multivariate polynomial. In the worst case, a poly- nomial of n variables and k degree can have n+k_k
terms. Although the Taylor expansion of a function is not always a dense polynomial, the resulting polynomial from a complex computation routine is often of a large size. Lots of heuristics can be used to conserva- tively simplify the representation of a TM (p, I). For example, we may first remove the terms whose interval enclosures are smaller than a specified box from the polynomial p, then compute an interval enclosure of the removed parts and add it onto the remainder I. The use of TMs appears in a wide range of numerical analysis tasks. The most well known application is the validated integration techniques for non-linear ODEs [BM98, MB09, NJN06]. Besides, TMs can also be applied to global optimization and satisfiability (SAT) checking [BM09], reachability analysis of non-linear hybrid systems [CÁS12], or even proof systems [BJMD+12].

Polynomial interpolations as TM polynomial parts. As we discussed previously,

polynomial interpolations such as Chebyshev polynomials may provide better approxima- tions than Taylor expansions. Thereby if we replace the Taylor approximations in the TM arithmetic framework by Chebyshev polynomials, the overestimation could be greatly reduced. To do that, we have to carry out all computations in Chebyshev basis since a transformation of a polynomial from Chebyshev to monomial basis on a finite-precision machine may lead to a loss of precision. Such work is proved effective for univariate functions [BJ10]. However, the multiplication of two multivariate polynomials in Cheby- shev basis seems not as easy as that in monomial basis. Hence, the use of Chebyshev polynomials in the framework of TM arithmetic needs further investigation.