Taylor models and computations in the complex plane Текст научной статьи по специальности «Математика»

УДК 519.6

A. D. Ovsyannikov, M. Berz

TAYLOR MODELS AND COMPUTATIONS IN THE COMPLEX PLANE

Introduction. Interval arithmetic on the field R of real numbers provides sharp and rigorous enclosures for the images of compact intervals under the elementary operations and intrinsic functions. However, while complex interval arithmetic can be seen as a straightforward extension of real-valued interval methods, it generally fails to provide sharp enclosures of the mathematically correct results since the necessary wrapping of results in new interval boxes often leads to significant overestimations [1-3].

Realistic problems often require the evaluation of complicated functions composed of multiple elementary operations and intrinsics. Evaluating such functions with conventional interval methods suffers from the dependency problem: if the computation involves several occurrences of the same variable, the result will be an overly pessimistic enclosure of the mathematically correct result. For the case of real-valued interval arithmetic, this has long been recognized as one of the main limitations of interval methods [4], and it has been shown [5] that high order Taylor model methods can avoid the dependency problem for practical purposes. In the case of interval arithmetic on the field C of complex numbers, the situation is further complicated by the fact that even the elementary operations and intrinsic functions are subject to the dependency problem. Thus, complex interval arithmetic is prone to significant overestimations and often fails to provide rigorous bounds with a sufficient sharpness even for polynomial expressions [6].

Complex numbers are commonly represented by pairs of two numbers: split into real and imaginary part or characterized by argument and absolute value

z = x + i ■ y = r exp (ip) . (1)

Both representations are equivalent but lead to significantly different results when used in the context of complex interval arithmetic. Writing complex numbers in terms of real and imaginary part is equivalent to identifying C with the real linear space R2. In this representation, we write complex interval numbers as X + i ■ Y with the real intervals X and Y. As such, complex intervals represent rectangles in the complex plane and the interval methods provide optimal enclosures for the addition and the multiplication with reals. However, enclosing the product of two such complex intervals in another interval generally leads to significant overestimations.

As an alternative to representing complex interval numbers by rectangles, the characterization by arguments and absolute values can be used to enclose sets of complex numbers. In that case we describe sets as r ехр^Ф) with real intervals r с R+ and

Ovsyannikov Alexander Dmitrievich — Associate professor, Faculty of Applied Mathematics and Control Processes, St. Petersburg State University. 67 publications. Optimization, Computational Methods, Control Theory. E-mail: [email protected].

Martin Berz — Professor Department of Physics and Astronomy, Michigan State University. 265 publications. High Order Rigorous Computing, Automatic Normal Forms. E-mail: [email protected].

© A. D. Ovsyannikov, M. Berz, 2009

$ C [0, 2n]. This description provides optimal validated enclosures for the product of sets of complex numbers. However, such a description generally overestimates the sum of these objects.

By not providing optimal enclosures for the elementary operations, neither of the two presented approaches utilizes the full power and rich structure of the complex field C. Other approaches of bringing self-validated computations to the complex plane while avoiding the excessive overestimation have been based on complex disks, represented by center and radius [3, 7]. However, in all cases the problem of overestimating the results of elementary operations is further aggravated in the case of the standard mathematical functions, where the actual overestimation may be arbitrary large. Thus, it has been recognized that conventional interval techniques are insufficient to properly deal with computations in the complex plane. In this paper we will show how the use of high order Taylor model methods can successfully overcome the limitations of conventional interval methods when it comes to rigorously enclosing the image sets of analytic functions defined on the complex plane.

Complex Taylor Models. Let us introduce Complex Taylor Models (CTM). The definition and set of its operations are very similar to Taylor Models (TM) in the real case. And one of the two variants presented of CTM constructing will be based on real TMs completely. Real TM were suggested and their properties were investigated in details in [8, 9]. TM are presented as a data type in the Cosy Infinity code [10]. All examples of section 3 were calculated with using of this code.

For convenience we start with definition of Taylor Models for real functions of two real variables.

Definition 1. Let u be a real function of two real variables x,y G R. Suppose the function is defined and has continuous partial derivatives up to order n +1 (at least) at each point of two-dimensional box

and a real interval Iau (the interval remainder bound), such that for any (x,y) G M the following inclusion is true

Then we say the pair Ta,u = (Pa,u, Ia,u) is a TM of function u (x, y). Here the parameter a = (n, (x0,y0) ,M) contains information about the order of the Taylor polynomial, reference point (x0,y0) and domain M.

It is natural to consider the Taylor polynomial as corresponding truncated Taylor series, and interval remainder bounds as interval estimation of remainder term. But it is practically useless to consider this interpretation as a path of finding of TM. Instead of it we should introduce rules of TM arithmetic with elementary operations (sum, product), intrinsic functions etc. In the following we assume that these rules have been implemented. For details see [9]. For example, evidently we can define a TM for the sum of functions u (x, y) and w (x, y) as

(2)

Assume that (x0, y0) G M and we have a pair of objects: a polynomial

Pa,u (x - xo,y - yo)= ^2 Kj,k (x - xo)j (y - yo)k

(3)

u (x, y) G Pa,u (x - xo,y - yo) + Ia,u.

(4)

(5)

Here the additions of two polynomials and two real interval numbers should be done in the usual ways.

Consider a complex function f of one complex variable 2 G C:

f (z) = u (x,y)+i • w (x,y). (6)

Here x = Re z, y = Im z. Suppose that assumptions of the definition 1 for u = Re f (z)

and w = Im f (z) are fulfilled and we have separate real TM Ta,u and Ta,w of the real

and imaginary parts of as real functions of two variables.

Definition 2. We say an ordered pair of TMT^Cf = (Ta,u, Ta,w) is a Coordinate Complex Taylor Model (CCTM) of the complex function f (z). Thus we have the following statement for any z = x + i • y (where (x, y) G M):

f (z) G TC = Ta,u + i • Ta,w. (7)

The next step is the introduction elementary operations and standard functions for CCTM. Let us start with addition and multiplication. Consider two complex functions f (z) = u (x,y) + i • w (x, y), g (z) = q (x,y) + i • r (x, y), and their CCTM. We can define the elementary operations like for complex numbers. For example:

Tcc + tcc tcc (T + T T + T ) (8)

Ta,f ' T a,g — Ta,f +g — \Ta,u \ Ta,q, Ta,w \ , (8)

Tcc tcc ____ tcc _____ (t T __________T T T T + T T ) (9)

a,f • Ta,g — Ta,f g — (Ta,u • Ta,q Ta,w • Ta,r, Ta,u • Ta,r + Ta,w ' Ta,q) . (9)

Now consider the exponential function

exp f (z) = exp u cos w + i • exp u sin w. (10)

Obviously we have

Tacexp f = (Ta,exp u • Ta,cos u,Ta,exp u • Ta,sin u) . (11)

We should remind that the rules of obtaining real TM Ta,expu, Ta,cosu, Ta,sinu were described in [9].

So the key idea of the coordinate approach is to split the complex function into the real and imaginary parts and manipulate with well-known real TM for them. It is clear that in this approach, it is possible to develop a complete formalism for complex TM arithmetic like in its real counterpart, the resulting method helps in the suppression of the dependency problem [5] and the remainder bounds have the high-order scaling property.

But sometimes it is reasonable to have “pure” CTM. Let us consider the analytic complex function f of one variable z in a circle S (r0, z0):

f (z): S (ro,zo) C C ^ C. (12)

Here z G C; zo G C is a reference point; ro G R+ is the radius of the circle:

S (ro, zo) = {z : \z - zo\ < ro, ro > 0} . (13)

Now let us introduce CTM. Further we assume that zo G D C S (ro,zo) where D = {z G C : ai ^ Re z ^ bi,a2 ^ Imz ^ b2} is a complex box. Here ai,bi,a2,b2 G R.

Definition 3. Let us assume that for the analytic complex function f (z) : D C C ^ C we have a pair of objects: a complex polynomial

n

Ps,f (z - zo)=Y^/ Kj • (z - zo)j (14)

j=o

and complex interval Ig,f, such that for any z G D

f (z) G ps,f (z - zo) + IS,f. (15)

Here the parameter 6 contains information about the order of the polynomial, reference point and domain interval: 6 = (n, zo, D). Then we say the pair T£ f = (Pg,f, Is,f) is a CTM of the function f. We call n the order of the TM, zo the reference point of the TM, D the domain interval of the TM, 6 the parameter of the TM. Also we call Pg,f the Taylor polynomial, Ig,f the interval remainder bound.

Let us introduce rules of the CTM constructing for the sum and product of two functions. We note that this is just a reformulation of the real TM case. With this aim we will consider a pair of analytical complex functions f (z) and g (z), for which TM are known, i. e. T£ f = (Pf, Is,f) and T£g = (PS,g, IS,g) accordingly. Thus, it is evident that

f (z)+g (z) G ((PS,f (z - zo)+IS,f ) + (PS,g (z - zo)+Is,g )) =

= (PS,f (z - zo)+PS,g (z - zo)) + (IS,f + IS,g) (16)

or it means that a TM T£ f+g for f + g can be obtained via

PS,f+g = PS,f + PS,g and IS,f+g = IS,f + IS,g. (17)

Thus we define

Ttf + Tig = Tf+g = (PS,f + PS,g ,IS,f + IS,g ) . (18)

In a similar way we will consider a product of the TM. Let us write down a true for any z G D relation as follows

f (z) • g (z) G ((PS,f (z - zo)+IS,f) • (PS,g (z - zo)+IS,g)) =

= (PS,f (z - zo) • PS,g (z - zo)) + PS,f (z - zo) • IS,g +

+ PS,g (z - zo) • IS,f + (IS,f • IS,g) . (19)

We need to note, that PS, f (z - zo) • PS,g (z - zo) is a polynomial of 2n-th order. Let

us divide this polynomial into the sum of two: the first one of n-th order and agrees with the Taylor polynomial PS, f ,g of f • g and an additional polynomial Pe so that

PS,f (z - zo) • PS,g (z - zo) = PS,f g (z - zo) + Pe (z - zo) . (20)

We denote bounds of polynomials by B (P) for polynomial P : D C C ^ C:

Vz G D,P (z - zo) G B (P). (21)

Remark 4. We demand that B (P) is at least as sharp as direct interval evaluation

of P (z - zo) on D. More sophisticated methods exist, but are not important for our purposes.

Here B (P) is a complex interval. Then Is,f.g can be found in following way:

Is,f g = B (pe) + B (Ps,f ) • Is,g + B (Ps,g) • Is,f + IS,f • Is,g ■ (22)

Thus it is possible to define

Tf • Tig = Tf g = (Ps,f g,Is,f g) ■ (23)

By using of definition of sum and product of the TM it is possible to calculate models for functions of the type Q (f): T£Qf) = (Ps,Q(f), IS,Q(f)), where Q is a complex polynomial of function f, for which the TM will be considered as known. With this numerical coefficients tk G C of polynomial Q (f) = to +1\ • f + ■ ■■+ tm • fm are represented as the following TM Ts,tk = (Ps,tk Js,tk), where Ps,tk = tk and Is,tk = [(0 + i • O'), (0 + i • 0)].

For practical use of the TM we need to have an algorithm of calculations of intrinsic functions of the TM. This algorithm is based on the theorem of expansion of analytical function into the Taylor series.

As it is known, a function f that is analytic on S (ro,zo) can be represented

as the following power series for any z G S (ro,zo):

ft \ _ \ " 1 dfc/ (zp) / _ .fc , ,

— kl dzk '

k=0

One can also consider the complex function as a sum of two real-valued functions of real arguments:

f (z) = u (x,y) + i • w (x,y), (25)

where x = Re z G R, y = Im z G R.

After splitting f (z) into real and imaginary parts we can employ Taylor expansions of the functions (with Lagrangian remainder term) u (x, y) and w (x, y) to represent them as finite sums

u (x, y)

*»(*,!/) = ith (A* tL + Aii-jL) »(«,!») +

k=0 ' v

1 d d\n+1 ,

+ (^Tin <27>

where Ax = ReAz; Ay = ImAz; Az = z — zo; xo = Rezo; yo = Imzo; xgj = xo + 0j • Ax, ygj = yo + 0j • Ay, 0 < 0j < 1, j = 1,2; and the partial differential operator

^ Ax ■ + Ay ■ works as

For the analytic function f the following formulae are correct

d _ d

<* *'<*> = */<*>, (29)

d d _ d _

t^/(^) = 7^«(a;,y) + *-7^w(x,gO, (30)

a*'£) /ra = ((Aa!+*'A!')l;) /ra =

d d x k

A^ifa + A^a;J /p) =

d d x k

= VAx dx + Ay dy J (M + *' w ’ (31)

where k is integer number, VS G S (ro,zo), x = Re S, y = Im S.

Thus after taking into account these formulae, the Taylor expansions of u (x, y) and w (x, y), and the representation of f (z) as the power series we can obtain the following statements:

f (z) = ~j7\——~ (z ~ z°) -R-e-^n+i №) + * ‘ Im-Rn+i ($2), (32)

k=o

where Rn+1 (0) = (^i)! 5< + )fQ^+i\z Zo)) (z - z0)(n+1) and 0h d2 G [0,1].

Or in the following form:

-f ( \ 1 ^kf (z°) ( \fc 1

k=o

, d(n+1')f(zei)/dz(n+1') + d(n+1')f(ze2)/dz(n+1') , >n+1 ,

+ 2 (n + 1)! (Z “ Zo) +

, 5(»+D/ (zSl) /a^+u - a(»+D/ (z02) /az(»+D7-----------------^+i

+ -------------------------2(^TTj!---------------------(*'“'o) ' №)

Here “ means conjugate complex value, zgi = zo + 0\ (z — zo), zg2 = zo + 02 (z — zo).

Let us emphasize that Rn+i has the same form as the well-known Lagrangian remainder

term in Taylor theorem for real functions. So we can formulate

Theorem 5 (Complex Interval analogy of Taylor’s theorem). For the analytic function f the following inclusion is true with any z G S (ro, zo),

k! dzk

+ („ + i)!3<’‘W>/^(^"))<—°)<’‘+1>- <34>

where 0 = [0,1] C R is a real interval. The last term (interval remainder term) has been considered as complex interval (a box in complex plane), that contains the set of values (curve

on complex plane) of complex function Rn+1 (0) = („^i ——>a^+i)Z Z°'>'> (z ~ zo)^1'1 with 0 G [0,1], and it can apparently be obtained by interval evaluation of Rn+i.

Let us illustrate this result for the special case n = 0:

/ (z) = / (zo) + Re ^—/ (zo + ■ (z — zo)) ■ (z — zo)^ +

+ * • Im (J^f (zo + 02 ■ (z - Z0)) • (z - z0)^j , (35)

or we can write it in the following form:

/H = + + (Z_Z0) +

df (z9l) /dz — df (zg2) /dz

-\ - - 2 (z-z0). (36)

Here “ means conjugate complex value, zg1 = zo + 0\ (z — zo), zg2 = zo + 02 (z — zo).

The statements give us a practical way to bound remainder term of intrinsic functions. Remark 6. The remainder of the Taylor series in the complex plane is usually expressed in the form

(z~z0)n+1 I f(C) (37)

”+ J (C_z)(C~zo)n+1-

Here z G intr, r = : |C — zo \ = r}. But this form is not convenient for simple realization.

Since it involves integration. The representation of the remainder term in the form suggested in this paper is similar to the real functions case, allows using almost the same formulae like in realized real TM and easier treatment of remainders.

Let consider us several examples of intrinsic functions. We start with the examination of the exponential function. As it is known, (exp z)(k) = exp z, where ()(k) is the operation of taking of derivative of k-th order. Bearing in mind that exp0 = 1, we can find following:

exp f (z) = exp(cSj) ■ exp(o + f(z)^ G

1 + / (z) + \ (/ (z)^J +••• + £! (/ (zfj

\ +(fcw (7(z))+ exp(e-/w)

G exp(cSJ)

(38)

Here and further f (z) = cgj + f (z), 0 = [0,1]. The right part of taken expression can serve as the source for constructing the TM for the function exp f (z). It consists of the polynomial part and the remainder. The possibility of the constructing of the TM for a polynomial has been discussed above. The remainder must be bounded by the complex interval and added into the interval remainder bounds of the TM of the polynomial part.

The same scheme is used for the construction of the TM for the function j V/I^O j .>

where {} • denotes one-valued branch of the multi-valued function of square root \Jf (z). Let 0 G Ps,f (z) (z — zo) + Is f ) for any z G D. We will also introduce such notations as

cg, f = r exp (ip), r = \cg, f |, p = arg cg,f,

£i = = Vr ■exp(i(p/2 + Tv(j - 1))), j = 1,2.

Then with taking into account of f-jVi} •) = ( — l)fc 1 — (2fc 3^n2fc_1; we get

\ 2fc U/ll

2fc {v^}.

cg,f + f(z) t G j

( 1 + M.

+ 2 °s,f

(ft*))2

222!(clS,/)i

+ ...

V

+ (—1)

,_uk (2fc-l)!! T I >-) 2fc+!(fc+l)!

fc_l (2fc-3)!!(/(z))fc 2 kk\(cS:f)k (/(^))fc+1

a/ cjj+e'/W]

Here (2k — 1)!! = n (2m — 1), cg,f = 0, 0 G cg,f + 0 ■ (B (PgJ(z) (z — zo)j + Is,f(z))

m_1 \ \ ’ / ’ /

t _ (z)V’ "o,J ' ~S,f(z),

for any z G D.

Let us consider one-valued branches of the function of natural logarithm Lnf (z). Let again 0 Psj(z) (z ~ zo) + Is for any z e D. With this we keep that J-j {Lnz}. = ^

\(k) ’

and ({Lnz}^ = (—l)fc+1 ('kzk'1'' at k > 2. Thus

{Lnf (z)}j = {Ln(cgJ + f(z)) ^

( {Ln (c<5,/)}7- + 777 — 5

№))2

J ' C<5,/ 2 (C5,f)2 + '

+ ( — 1) + (_l)fc+2 1

fc+1 1 (/0))

k (cs,f)k (/(*))

fc+1

fc+1

(cs,f + e-f(z))

(40)

As above cg,f =0, 0 G cgj + 0 ■ (B (Pgj(z) (z — zo)) + / f(z)^ for any z G D.

Definition 7. Let us consider a CTM Tgc f = (Pgj (z — zo) ,Igj). Let \I\ denotes

diameter of a complex interval I = [01,02] + i ■ [61,62]-' |-f| = \J(a2 — «l)2 + (62 — &i)2-

Then we say \Igj \ is sharpness of the TM.

Theorem 8 (Taylor Model Scaling Theorem). Let f (z) and g (z) are analytic

complex functions that have n-th order CTM Tgc f = (Pg,f (z — zo) ,Igj) and Tgcg = (Pg,g (z — zo) ,Ig,g) correspondingly. Let us consider circle of minimal possible radius h with center in point zo such that it includes domain D: D C S (h, zo). Let the remainder bounds Ig,f and Ig,g satisfy \Igj \ = O (hn+1) and \Ig,g \ = O (hn+1). Then the TM for the sum and products of f (z) and g (z) obtained via addition and multiplication of CTM satisfy

\Ig,f+g\ = O(hn+r) and \Igj g\ = O(hn+r) .

(41)

Furthermore, let F be any of the intrinsic functions defined above, then the CTM for F (f) obtained by the above instructions satisfies

\Ig,F(f) | = O (hn+1) .

We say the CTM have the (n + 1)st order scaling property.

(42)

j

Figure 1. Exact range of z6 over the complex interval domain Di = [3, 5] + i [1,4] the computed interval enclosure of the range exceeds the plotted range in all coordinate directions

Proof. The proof for the binary operations follows directly from the definition of the remainder bounds for the binaries. Similarly, the proof for the intrinsics follows because all intrinsics are composed of binary operations as well as an additional complex interval, the width (diameter) of which scales at least with the (n + 1)st power of a bound

of a function that scales at least linearly with h. ■

Examples. In this section we illustrate how the use of high order complex TM can indeed provide sharp and guaranteed bounds on the ranges of complicated complex functions. The main advantage of TM methods over the use of conventional interval techniques lies in the propagation of high order functional dependencies from one computational step to the next; suppressing the wrapping of intermediate results between each and every elementary operation.

As a first example of how even simple elementary operations in the complex plane can result in overestimating the function’s ranges, we consider a simple monomial function of order six:

fi (z)= z6. (43)

While linear methods cannot properly model f1, it is clear that any high order method

of at least order six can model the functional behavior up to machine precision. Figure 1 shows the the mathematically correct range of f1 over the domain D1 = [3,5] + i [1,4]. While conventional interval arithmetic in either the x/y or the r/p convention is bound to overestimate the exact range, modeling that function with a single TM over the whole domain results in a sharpness that is in the order of the machine precision.

Figure 2. The plot shows the exact range and interval enclosure of the range of f2 over the complex interval domain D2 = [—0.01, 0.11] + i [—0.01, 0.11]

-2

-4

Remainder bound size (logarithmic)

-10

-12

2 4 6 8 10 12 14 16 18

Order of Taylor model

Figure 3. The graph shows the decadic logarithm of the size of the Taylor model remainder bounds for f2 over the domain D2 with the reference point z0 = 0.01 + i • 0.01 as a function of Taylor model orders

Figure 4. The plot shows the exact range and interval enclosure of the range of f2 over the complex interval domain D3 = [0, 0.02] + i [—0.1, 0.12]

-2

-4

-6

-8

-10

-12

-14

-16

Remainder bound size (logarithmic)

10 12 14 16 18

Order of Taylor model

Figure 5. The graph shows the decadic logarithm of the size of the Taylor model remainder bounds for f2 over the domain D3 with the reference point zo = 0.01 + i • 0.01 as a function of Taylor model orders

As a second example for how TM methods can provide sharp enclosures even for complicated functions, consider the function f2 that is analytic in the whole complex plane C:

/2 (z) = z2 + cos (z) + 4*exp + sin (z + exp (0.5 + -s2))^ . (44)

The figure 2 shows the exact range of f2 over the domain D2 = [—0.01,0.11] + *[—0.01,0.11]. Additionally, a range enclosure obtained with conventional interval arithmetic is included. While the interval enclosure is rigorous, it does not provide a sharp enclosure of the mathematically exact range. Utilizing complex TM methods on the other hand, allows the computation of rigorous enclosures that are extremely sharp as shown in the figure 3. The graph also illustrates how for a given domain, the sharpness of TM enclosures does indeed scale with the (n + 1)-st order of that domain, where n is the order of the TM.

As another example of how high order TM methods can accurately enclose functional dependencies, Figure 4 shows the evaluation of f2 over the domain D3 = [0,0.02] + *[—0.1,0.12]. Like in the previous example, the plot shows the exact mathematical range and its interval enclosure and the strong non-linearity of f2 leads to a significant overestimation in the interval enclosure of the mathematically exact range. As shown in the Figure 5, CTM on the other hand succeed in finding sharp enclosures for the function f2.

Domain size

Figure 6. Dependence of the computed enclosure width on the size of the domain for conventional interval arithmetic and complex Taylor models of orders two, four, six, eight, and ten

As an illustration of how the TM approach can often significantly improve the sharpness of computed enclosures over sizable domains, consider the evaluation of the function f2 over the domain box D4 = (0.01 + *0.01) + A([—0.5, 0.5] + *[0.5, 0.5]). Figure 6 shows the widths of the enclosures computed with interval and TM methods for several different domain size parameters A. Apparently, for each of the different methods, the size of the computed

enclosure scales with a power of the domain size and is limited from below by the machine accuracy. However, while conventional intervals scales approximately linear with the domain width, the accuracy of TM methods does indeed scale with the (n +1)-st order of the domain size. Thus, once the domain size drops below a problem and order dependent threshold, TM computations can often achieve a much higher sharpness over larger domains than conventional interval techniques.

This example also shows that “in the large”, the TM approach can sometimes be worse than simple evaluation with conventional intervals. In fact, TM methods will usually succeed if the contributions of all the highest order polynomial terms drop faster with order than the number of coefficients of a given order. Based on this rule of thumb, for polynomials with coefficients of magnitude 1, the TM methods do not work well over domains with magnitudes larger than 1; on the other hand, if the domains have magnitudes of 0.1 or less, the TM methods tend to work very well. Thus, if a given problem can be treated with conventional interval techniques, there is usually nothing to be gained by using TM. However, if the problem requires domain splitting the use of TM often becomes advantageous, especially if the interval approach requires domain splitting beyond the threshold at which the Taylor approach shows significant improvements.

While TM methods are computationally more expensive than conventional interval methods, the increased sharpness of the computed enclosures usually outweighs the computational expense. This is especially true if the desired sharpness requires domain splitting. To illustrate this, we measure the computational expense of evaluating f2 over the domain D4 by defining the information count as the number of floating point numbers that have to be stored and processed in order to obtain the desired sharpness in the enclosure. Thus, the information count equals the number of domain intervals multiplied with the actual storage requirements for a single instance of the underlying data type. For complex functions, the information counts for intervals Nj and n-th order TM NT,n are given by

Nj = (#boxes) x 4, (45)

- +n + 2^j . (46)

Following table lists the maximum width of subdomains, and the corresponding information counts, necessary to enclose the range of f2 over the domain D4 with a uniform sharpness of 10~3 and 10~5, respectively:

Method RequiredDomainW idth InformationCount

Sharpness 10-3 10-5 10-3 10-5

Interval 0.418 • 10-3 0.418 • 10-5 > 10200 > 10200

T aylor, n = 2 0.304 • 10-1 0.627 • 10-2 1.3 • 1021 2•1097

Taylor, n = 4 0.104 0.040 26030646 4•1016

Taylor, n = 6 0.180 0.100 158136 71217904

Taylor, n = 8 0.214 0.145 72659 1525956

Taylor, n = 10 0.317 0.181 12447 333208

Taylor, n = 12 0.318 0.209 16439 158038

Taylor, n = 14 0.321 0.316 20303 21791

Taylor, n = 16 0.332 0.316 22121 27351

Due to a prohibitively large number of domain splittings, straightforward interval methods and low order TM fail to achieve the required sharpness. However, high order TM methods can provide the requested accuracy with a moderate information count and without

NTn = {#boxes) x 2 ( + ^

y 2 • n!

imposing excessive requirements on the computational overhead. Moreover, it is apparent that for each of the two problems there is an optimal computation order that finds the results at a minimal cost. At this point, any further increase of the computation order results in only negligible improvements in sharpness at the expense of increased information counts.

Conclusion. We have developed an approach that allows the rigorous representation of analytic functions by CTM. Compared to conventional interval methods that model functions by range intervals, the new method propagates information on high order derivatives together with rigorous remainder bounds. As an important consequence, the sharpness of the computed enclosures scales with a high order of the size of the domain.

In computational mathematics, functions are frequently modeled by either one of the following methods: numerical tables, symbolic representations, range intervals, finite approximations. The TM approach offers a novel approach to rigorous numerical analysis by combining many positive aspects of these seemingly different approaches. TM use high order approximations with symbolic polynomial operations and interval methods to achieve sharp and guaranteed enclosures of functional dependencies.

Lastly, we point out that TM methods are often transparent to the user of computational software. While conventional interval methods often require an adaptation of the underlying algorithms, TM methods can usually be used as straightforward drop-in replacements for floating point number methods and provide rigorous answers to common questions.

References

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Статья рекомендована к печати проф. Л.А. Петросянoм.

Статья принята к печати 28 мая 2009 г.