NOLTA, IEICE Invited Paper Preconditioning of Taylor models, implementation and test cases Florian B¨unger 1 a) 1 Institute for Reliable Computing, Hamburg University of Technology Am Schwarzenberg-Campus 3, D-21073 Hamburg, Germany a) fl[email protected] Received January 14, 2020; Revised April 3, 2020; Published January 1, 2021 Abstract: Makino and Berz introduced the Taylor model approach for validated integration of initial value problems (IVPs) for ordinary differential equations (ODEs). Especially, they invented preconditioning of Taylor models for stabilizing the integration and proposed the following different types: parallelepiped preconditioning (with and without blunting), QR pre- conditioning, and curvilinear preconditioning. We review these types of preconditioning and show how they are implemented in INTLAB’s verified ODE solver verifyode by stating explicit MATLAB code. Finally, we test our implementation with several examples. Key Words: ordinary differential equations, initial value problems, Taylor models, QR pre- conditioning, curvilinear preconditioning, parallelepiped preconditioning, blunting 1. Introduction Preconditioning is a general concept in large parts of numerical mathematics. It is also widely used in the area of reliable (verified) computing in which numerical results are calculated along with rigorous error bounds that cover all numerical as well as all rounding errors of a computer, meaning that a mathematically exact result is proved to be contained within these rigorous bounds. For verified integration of initial value problems (IVPs) for ordinary differential equations (ODEs) preconditioning goes back to Lohner’s fundamental research [20] in which he particularly introduced his famous parallelepiped and QR methods to reduce the well-known wrapping effect from which naive verified integration methods for IVPs necessarily suffer. A detailed quantitative analysis of such anti-wrapping schemes including Lohner’s QR method was given by Nedialkov and Jackson in [22]. Makino and Berz invented and used so-called Taylor models for verified integration of IVPs and implemented their Taylor model approach in their software package COSY INFINITY [1]. In analogy to Lohner’s method Makino and Berz [7] invented preconditioning of Taylor models and proposed the following three types: parallelepiped preconditioning (with and without blunting), QR precondition- ing, and curvilinear preconditioning. In addition, they also mentioned a fourth, very simple type of preconditioning named identity preconditioning. Recently, we implemented the Taylor model approach for solving IVPs in INTLAB [25]. In the for- mer release V11, parallelepiped and QR preconditioning were provided. The current, new release V12 additionally contains a variant of curvilinear preconditioning and simple identity preconditioning. Also blunted parallelepiped preconditioning has changed compared to V11. 2 Nonlinear Theory and Its Applications, IEICE, vol. 12, no. 1, pp. 2–40 c IEICE 2021 DOI: 10.1587/nolta.12.2 In this technically oriented report all those types of preconditioning are reviewed and their concrete MATLAB/INTLAB implementation in our verified ODE solver verifyode is explained in detail. The focus is completely on preconditioning, the overall Taylor model approach for solving IVPs is not repeated here again. For that we refer to [4–7, 9, 13, 21, 23] and the references therein. The paper is organized as follows. In Section 2 the definition and notation of Taylor models is shortly introduced. In Section 3 the implementation of preconditioning in INTLAB is described. Section 4 contains computational results for several IVPs solved with INTLAB’s verified ODE solver verifyode using different types of preconditioning. 2. Taylor models Let n, k, m ∈ N := {1, 2,...} be fixed. As usual, the set of closed real intervals is denoted by IR; n n more generally IR stands for the set of real interval vectors of length n. An interval vector I ∈ IR n is usually denoted in inf-sup notation I =[a, a] where a, a ∈ R satisfy I =[a1, a1] ×···×[an, an]. We will consider vectors p(x)=(p1(x),...,pn(x)) of length n where each component q(x)=pi(x)is a real multivariate polynomial in k unknowns x1,...,xk of degree less than or equal to m. That is α q(x)= qαx α, |α|m k k α k αi where α =(α1, ..., αk) ∈ N0 is a multiindex, |α| := i=1 αi, x := i=1 xi ,andtheqα ∈ R are real k k polynomial coefficients. A domain D = ×i=1[ui,vi] ∈ IR and a center or centering point c ∈ D are fixed and n p(D − c)={p(x − c)=(p1(x − c),...,pn(x − c)) | x ∈ D}⊆R is called the image of p on D centered at c. Such multidimensional, multivariate polynomial images are used for validated enclosures in the context of Taylor models. Their great advantage is that they can enclose non-convex sets with curved boundaries without much overestimation. This is not possible in interval arithmetic or affine arithmetic, where the former uses interval vectors, i.e., axis parallel boxes, and the latter convex polytopes for enclosing sets. To cover ubiquitous floating-point n rounding and degree truncation errors, a so-called remainder or error interval vector E ∈ IR is added enlarging the image p(D − c)to p(D − c)+E = {y + e | y ∈ p(D − c),e∈ E} which, for better distinction, will be called the range of the data p, D, c, E. The aim will always be to keep the size/volume of the remainder E small compared to that of the image p(D − c)sothat rendering by E does not change its shape essentially. Now, an m-th order Taylor model vector of length n with k-dimensional domain given by the data p, D, c, E is defined as p + E := {f ∈ C(D, Rn) |∀x ∈ D : f(x) ∈ p(x − c)+E} (1) where C(D, Rn) denotes the set of all continuous functions f : D → Rn. For ease of presentation we will just speak of Taylor models p + E if m, n, k, D, c are clear from the context. By (1) a Taylor model p + E is a set of continuous functions with domain D and values in Rn which at each point x ∈ D deviate from p(x − c) by some e ∈ E. For n = 1, arithmetic operations +, −, ×, ÷, standard functions like exp, log, sin, cos, etc., and also integration with respect to one of the k variables can be defined on the set of all Taylor models with common, fixed D, c, m. For details on these basic operations we refer to [9, 13, 21]. For n>1, vector operations like +, − immediately carry over from the one-dimensional case. As a simple example, a Taylor model y with polynomial part 2 3 p(x1,x2)=2+x1 − x2 +3x1x2 of degree d = 5, with domain D := [−1, 1]2, centering point c := (0, 0) ∈ R2, and error interval E := [−10−6, 10−6] can be created as follows in INTLAB: 3 1 p = [2;1; − 1;3]; % polynomial coefficients 2 M = [0 0;1 0;0 1;2 3]; % ... and exponents in the order of the corresp. coefficients 3 order = 7; % some degree bound m >=d 4 D. inf = [ −1;−1]; % lower domain bounds 5 D.sup = [1;1]; % upper domain bounds 6 c = [0;0]; % center point 7 E. inf = −1e −6; % error interval E 8 E.sup = 1e−6; 9 format shortg infsup 10 y = taylormodelinit(c,D,order ,M,p,E) Then, the Taylor model will be displayed in MATLAB’s command window as follows: dim order type iv_mid iv_rad im_inf im_sup ___ _____ ____ ______ __________ ______ ______ 2 7 0 0 1.0001e-06 -3 7 min max center ___ ___ ______ x1 -1 1 0 x2 -1 1 0 x1 x2 coeff __ __ _____ 00 2 10 1 01 -1 23 3 Here dim = 2 is the dimension of the domain D, that is the number of unknowns x1,x2, order = 7 is the quite arbitrarily chosen upper degree bound md=5, and iv_mid = 0, iv_rad = 1.0001e-06 is an interval enclosure of the remainder E in midpoint radius notation. A quite large enclosure [im_inf,im_sup] = [-3,7] of the polynomial image p(D − c)=p(D)=[−3, 5] is computed au- tomatically. The true range R := [−3, 5] + E of y is very much overestimated by [−3, 7] + E.In practice, Taylor models with thin ranges of diameter much less than 1 are typical. Only for ease of presentation we chose the stated data with wide range. Taylor models of type = 0 are the default, Taylor models of type = 1 are used for solving ODEs. As we will see in the next section, the latter have special domains and center points of the form D =[−1, 1] ×···×[−1, 1] × [t1,t2], c =(0,...,0,t1) where [t1,t2] is the time domain, i.e., the domain of the variable t with respect to which the integration is performed from t1 to t2. Technically, the components of a Taylor model object in INTLAB read as follows: 1 ys = struct (y) 2 ys dim = ys .dim % number of unknowns 3 ys center = ys. center % center point c 4 ys domain = [ ys . domain . inf , ys . domain . sup ] % domain D 5 ys order = ys. order % degree bound m 6 ys monomial = ys . monomial % polynomial exponents M 7 ys coefficient = ys. coefficient % polynomial coefficients p 8 ys interval = ys. interval % error interval E 9 ys type = ys . type %Taylormodeltype 10 ys image = ys . image % interval enclosure of the image p(D−c) This code produces the following output which clarifies the internal structure. struct with fields: dim: 2 | ys_dim = | ys_monomial = | ys_interval = center: [2x1 double] | 2 | 0 0 | struct with fields: domain: [1x1 struct] | ys_center = | 1 0 | inf: -1e-06 order: 7 | 0 | 0 1 | sup: 1.0001e-06 monomial: [4x2 double] | 0 | 2 3 | ys_type = coefficient: [4x1 double] | ys_domain = | ys_coefficient = | 0 interval: [1x1 struct] | -1 1 | 2 | ys_image = type: 0 | -1 1 | 1 | struct with fields: image: [1x1 struct] | ys_order = | -1 | inf: -3 | 7 | 3 | sup: 7 4 3.