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for some [MATH] be given. Then the true error after one iteration step is [MATH] . Since in reality not [MATH] is calculated but some erroneous approximation [MATH] the true error can be written as |
[MATH] Inserting a constructive zero gives a sum [EQUATION] of two terms. The first term describes solely the error propagation while the second term gives exactly the newly produced error due to the approximate calculation of [MATH] |
Let us fix some [MATH] and consider the absolute error of [MATH] . To get the formulas more compact, set [MATH] . Then, [EQUATION] |
follows. Let [MATH] be assumed to be the actual precision under calculation at time [MATH] . The last term in the previous inequality can be estimated the following way. As discussed in |
, the rounding error produced in calculating [MATH] can be estimated by [EQUATION] where [MATH] is the number of rounding operations performed in computing |
[MATH] . In the case considered here, [MATH] follows. It is further crucial to mention that the factor [MATH] is only valid if [MATH] holds so that the precision must not be chosen too small. Furthermore, with |
[MATH] it follows [EQUATION] where [MATH] holds. In other words, one obtains the recursion relation [MATH] . Iterating the recursion gives [MATH] |
As already mentioned, [MATH] is bounded from above by [MATH] . To come to a sufficient condition for the precision, also a lower bound is needed. First observe that [MATH] holds for all [MATH] |
[MATH] . Hence, for [MATH] [MATH] follows. This gives the sufficient condition [EQUATION] on the precision. Note that [MATH] Then, an upper bound on [MATH] is given by |
[EQUATION] with [MATH] This leads to an upper bound on the loss of significance rate given by [MATH] for all [MATH] . Since [MATH] follows for [MATH] the final result on the loss of significance rate is |
[EQUATION] The curve in Figure shows that this upper bound is in full agreement with the numeric result. 2.6 Investigating Calculation 4 |
The observation at the end of the subsection describing Calculation 1 directly leads to the already introduced mean value form. The calculation is shown in Figure This calculation is the optimum of both, Calculation 1 and 3. The curve reflects in the parameter range [MATH] well the dynamic behavior. |
Furthermore, in the range [MATH] , the curve suggests a relation between the loss of significance rate and the Lyapunov exponent [MATH] for the logistic map: |
[EQUATION] for all [MATH] . For a curve of the Lyapunov exponent of the logistic map see . This relation will be shown in the next section for general dynamical systems on the interval. Furthermore, it will be shown that the algorithm based on Calculation 4 is optimal in some sense. |
But before, some crucial reflections governing the analysis in the next section. The mean value form representation, on which the calculation is based, can also be seen from a different viewpoint. Have again a look at Equation ( ). The true error is the sum of the error propagation (first term) according to the iterati... |
[MATH] with [MATH] . This gives directly the bound [EQUATION] The second term can be estimated in a similar way as was done in ) by |
[EQUATION] where [MATH] because there are 3 arithmetic operations and the rounding of [MATH] . Using the fact that [MATH] holds and [MATH] if [MATH] , the unknown value |
[MATH] can be estimated from above. This calculation shows that there exists a recursive equation on an upper bound [MATH] on [MATH] for all [MATH] |
[EQUATION] with [MATH] and [EQUATION] This description, which is in line with the analysis of Calculation 3, is equivalent to the interval description using the mean value form. Instead of using intervals, pairs of the form value [MATH] and corresponding guaranteed error bound [MATH] is used. This approach is an automa... |
. From a technical point of view, the representation as value and error has the advantage that the rounded values [MATH] are calculated as usual in floating-point arithmetic except that arbitrary-precision floats are used. The guaranteed error bounds may be calculated using interval arithmetic according to ( ), to real... |
are reported in by using a method analog to the one presented here . However, the connection to the Lyapunov exponent is not made in |
Before continuing, three remarks. First, interval libraries are primarily divided int two types concerning their representation of an interval |
: There exist libraries using the infimum-supremum representation of intervals, like MPFI, and there exist libraries using the midpoint-radius representation of intervals. If arbitrary precision is needed, the inf-sup libraries have the disadvantage that two floating-point variables with high precision are needed to re... |
Third it should be mentioned that, executing the first three presented calculations in Mathemathica using significance arithmetic, exactly the same results are obtained. This shows that also significance arithmetic suffers from the dependency problem as interval arithmetic does. This is already noted in |
The general algorithm and its complexity Let [MATH] be a compact real interval and [MATH] a self mapping. In the following, [MATH] is assumed to be continuous on [MATH] , two times continuously differentiable on |
[MATH] and [MATH] is bounded. Furthermore, [MATH] and [MATH] are assumed to be computable in the sense of Computable Analysis. The definition of a computable real function is given below. |
In this section, a general algorithm for computing the iteration [EQUATION] is presented. To be more precise, for given [MATH] [MATH] and |
[MATH] , this algorithm computes a finite part [MATH] of length [MATH] of the true orbit [MATH] with initial value [MATH] Each computed value [MATH] of this finite trajectory has a relative error of at most [MATH] [MATH] for all [MATH] The correctness of the algorithm and its relation to Computable Analysis is shown. F... |
3.1 Computability issues and specifying the algorithm The set of all computationally accessible real numbers are the floating-point numbers of arbitrary precision and arbitrary exponent range denoted by [MATH] . A floating-point number is a real number of the form [MATH] where [MATH] is the precision |
[MATH] the scale and [MATH] where [MATH] is called the significand . To get a unique representation of [MATH] for given [MATH] [MATH] is assumed if [MATH] and [MATH] if |
[MATH] . Since actually no bound is assumed on the precision and the scale, the set [MATH] is the set of the dyadic real numbers and therefore countable infinite. Thus, [MATH] forms a natural basis for computability considerations over finite objects. Consider some floating-point number [MATH] , then the scale and the ... |
a floating-point number representing [MATH] . Then the scale of [MATH] is generally determined by [MATH] while the precision can be chosen arbitrary. Regarding [MATH] as a data structure, then [MATH] has as its essential property the precision. In object oriented notation, the precision of [MATH] can be written as [MAT... |
Any real number [MATH] is represented in an algorithm concerning numerical computation by a pair [MATH] consisting of a floating point number [MATH] of arbitrary precision |
[MATH] approximating [MATH] and a floating-point number [MATH] of fixed precision giving an upper bound on the absolute error, [MATH] . Reversely, any such pair |
[MATH] can be seen as the real interval [MATH] If [MATH] holds for some [MATH] , then [MATH] is called an approximation of [MATH] . To represent a single real number, a sequence [MATH] |
of such pairs [MATH] are needed. A sequence [MATH] is called a floating-point name of a real number [MATH] , if any [MATH] approximates [MATH] [MATH] |
[MATH] and [MATH] holds. Clearly any real number has a floating-point name. As already indicated, it is a straightforward task to define what a computable function [MATH] is by using classical computability theory over finite objects. Additionally, computability over integers, computability of functions with mixed argu... |
. Consider a function [MATH] [MATH] and a pair [MATH] of two functions [MATH] and [MATH] having the following property. For any approximation [MATH] of some real number [MATH] , the pair |
[MATH] is an approximation of [MATH] . Thus, [MATH] gives an upper bound on the absolute error of [MATH] [MATH] . Considering [MATH] as an interval function, the above property is just the fundamental property of interval arithmetic, |
Property 2.12. Then, [MATH] is called an approximation function for [MATH] . Now consider an approximation function [MATH] for [MATH] such that for all [MATH] and any floating point name [MATH] of [MATH] [MATH] |
is a floating-point name of [MATH] . Such an approximation function is called approximation-continuous . Additionally, if the two functions [MATH] and [MATH] of an approximation function |
[MATH] are computable, then [MATH] is called a computable approximation function . Finally, [MATH] is called computable , if there exists a computable approximation function |
[MATH] for [MATH] which is approximation-continuous. The algorithm with the specification described at the beginning of this section reads |
Input parameter: [MATH] [MATH] [MATH] Initialize precision [MATH] do Initialize value and error [MATH] for [MATH] to [MATH] do If |
[MATH] then If not printed print [MATH] [MATH] [MATH] else break [MATH] 10 end for 11 [MATH] 12 while [MATH] where [MATH] is an approximation-continuous approximation function for [MATH] |
specified below. To initialize [MATH] , a rounding function [MATH] is needed where [MATH] is a floating-point number of precision [MATH] being the exactly rounded value of [MATH] for some rounding convention, in the following nearest. Clearly, the value [MATH] is an upper bound on the absolute rounding error, |
[MATH] if the rounding mode is nearest. The predicate [MATH] is a test whether the relative error of [MATH] is bounded by [MATH] The semantics reads: |
[EQUATION] While the object oriented notation is convenient for a compact and instructive description of the algorithm, in the following analytical analysis an abbreviation for this notation is sometimes more handsome. As in the line of the preceding section, floating-point numbers and functions are indicated by a hat:... |
[MATH] . Hence, [MATH] is equivalent to [MATH] and [MATH] is equivalent to [MATH] Finally a remark on optimization. The algorithm is not optimized in performance. Including performance issues, in Line 11 something like [MATH] can be used where [MATH] and [MATH] are constants. Here, the aim is to find the minimal [MATH]... |
3.2 Computability and correctness It is clear that the rounding function [MATH] is computable. So let us begin with the predicate [MATH] |
Proposition 3.1 The predicate [EQUATION] is computable and satisfies ( 10 ). Proof. Let [MATH] be an approximation of [MATH] . If |
[MATH] holds, then [MATH] follows. Using [MATH] [MATH] follows. The predicate ( 11 ) only uses the approximation [MATH] basic arithmetic and finite tests. Hence, this formula is computable. |
Note that the definition of the predicate also gives [MATH] in the singular case where [MATH] and [MATH] and hence [MATH] An algorithm for computing [MATH] is possible by assumption. To derive an algorithm for computing [MATH] on the absolute error, return to Equations ( ) and ( ). |
Proposition 3.2 Let [MATH] be given and [MATH] an approximation of [MATH] with [MATH] . Assume that [MATH] computes the value [MATH] up to a correctly rounded last bit in the significand. |
Furthermore assume [MATH] Then the absolute error of [MATH] is bounded from above by [EQUATION] Here, [MATH] Proof. Equation ( ) gives [MATH] . Using the mean value theorem, [MATH] |
follows. According to the assumption on [MATH] and Theorem 2.3 of [MATH] holds where [MATH] is the precision of [MATH] Corollary 3.1 |
Let [MATH] be as specified in the beginning of this section, [MATH] and [MATH] specified as in Proposition 3.2 . Then there exists a function |
[MATH] with [MATH] for some [MATH] such that [MATH] with [MATH] is an approximation-continuous, computable approximation function of [MATH] |
[EQUATION] Proof. Let [MATH] be some computable upper bound of [MATH] [MATH] can be computed by global optimization, for example by using interval arithmetic. Since [MATH] is continuous and [MATH] compact, [MATH] is bounded. So, [MATH] for some |
[MATH] . Also, [MATH] is computable. Using Proposition 3.2 , it follows that [MATH] is also an approximation function of [MATH] . Remains to show that |
[MATH] is approximation-continuous. Let [MATH] be some floating-point name of [MATH] . Clearly [MATH] holds. Since [MATH] and the sequences |
[MATH] and [MATH] are bounded, [MATH] follows. Furthermore, by this result and the statement of Proposition 3.2 also [MATH] holds. |
To summarize, the iteration ( ) is performed in the algorithm by iterating a value [MATH] approximating [MATH] with an upper bound on its absolute error [MATH] according to |
[EQUATION] where [MATH] is a computable upper bound on [MATH] as described in the preceding corollary and [MATH] the precision of any floating-point number involved at that stage. This is Line in the inner for -loop of the algorithm which is executed with successively increasing precision [MATH] controlled by the outer... |
Proposition 3.3 Let [MATH] with [MATH] be given and [MATH] floating-point name of [MATH] obeying [MATH] Then [MATH] follows for all [MATH] |
Proof. Since [MATH] and [MATH] there exists some [MATH] such that for all [MATH] [MATH] and [MATH] holds for all [MATH] . Then [MATH] for all |
[MATH] The next proposition makes the link to Line in the algorithm. Proposition 3.4 Let [MATH] be the [MATH] -th element of the orbit of the recursion ) and [MATH] a sequence given according to the recursion equations ( 13 ) and 14 ) with increasing precision |
[MATH] . Then [MATH] is a floating-point name of [MATH] Proof. Let [MATH] according to Corollary 3.1 and [MATH] such that [MATH] holds for all [MATH] Then Equation ( 14 ) leads to |
[MATH] . Iteration gives [MATH] Hence, for [MATH] fixed, [MATH] follows and consequently also [MATH] These two propositions finish the correctness proof of the algorithm. They show that, if [MATH] for [MATH] the outer loop eventually terminates for any [MATH] |
The drawback of the algorithm is, that in the case [MATH] for some [MATH] , the computation does not terminate. This is only due to the fact that the relative error controls the outer do-while -loop. If the absolute error would be used instead, this drawback is eliminated. However, controlling the relative error is mor... |
Absolute errors are in the line with Computable Analysis. Replacing the test [MATH] by the test on [MATH] in the algorithm would give a segment [MATH] of the orbit with accuracy [MATH] . It is now straightforward to see that the function [MATH] |
with [MATH] is computable. Here, a function [MATH] is computable if there exists a computable approximation function [MATH] for [MATH] which is approximation-continuous with respect to the first argument. |
3.3 Computational complexity After having presented the preliminary work, the main issue of the paper is addressed - the computational complexity of the presented algorithm. The complexity measure of interest here is the loss of significance rate already introduced informally in the previous section. Here is the formal... |
Definition 3.1 The minimal precision, for which the described algorithm eventually halts is denoted by [MATH] , where [MATH] [MATH] |
and [MATH] are the corresponding input parameters. The growth rate of [MATH] is given by [EQUATION] Then, the loss of significance rate |
[MATH] is defined by [EQUATION] To achieve bounds on the loss of significance rate, the drawback of the preceding subsection also makes problems here. If [MATH] for some [MATH] , the loss of significance rate may be unbounded. Therefore, one more assumption in addition to the ones on the dynamical system stated in the ... |
Assumption 3.1 The dynamical system [MATH] is assumed to have the properties already mentioned in the beginning of this section and furthermore, for any orbit [MATH] under consideration, |
[MATH] holds for any [MATH] as well as [EQUATION] If only a finite range in scale is relevant, the additional assumption is no loss of generality. An example is the logistic equation where [MATH] but [MATH] has no distinguished role. Instead of considering [MATH] , consider the following dynamical system [MATH] . Choos... |
[MATH] for all [MATH] . Then [MATH] fulfills the additional assumption. Furthermore [MATH] holds and therefore there is no substantial difference in the complexity analysis of the algorithm between the original system and the modified system. |
First, the boundedness of [MATH] is shown. Proposition 3.5 Let [MATH] be as in Assumption 3.1 and [MATH] as in Definition 3.1 . Then, for given [MATH] , there exists a constant [MATH] , depending on [MATH] , such that |
[MATH] holds for all [MATH] [MATH] Proof. According to the requirements made on [MATH] , there are some constants [MATH] and [MATH] such that [MATH] |
holds for all [MATH] and all precisions [MATH] Analogous to the treatment in the proof of Proposition 3.4 , iteration gives [MATH] Let [MATH] . Then, for all [MATH] |
[MATH] follows. If now [MATH] holds, [MATH] for all [MATH] . This leads to the bound [MATH] Corollary 3.2 Let [MATH] be as in Assumption 3.1 [MATH] as in 15 ) and [MATH] the loss of significance rate. Then, for given [MATH] , there exists some constant [MATH] such that |
[MATH] holds for all [MATH] In the following, the main statements of this paper are be formulated: A lower and an upper bound for the loss of significance rate is given. Furthermore, the relation of these bounds to the Lyapunov exponent |
[MATH] is shown. Before the theorem is stated, for sake of completeness, the definition of the Lyapunov exponent and its basic properties are presented. |
Definition 3.2 Let [MATH] be a dynamical system, [MATH] compact and [MATH] continuously differentiable on [MATH] . Then the Lyapunov exponent at [MATH] is defined by |
[EQUATION] if the limit exists. The Lyapunov exponent may depend on [MATH] . However, the following properties hold: (a) If [MATH] has an invariant measure |
[MATH] , then the limit in Equation ( 17 ) exists [MATH] -almost everywhere. (b) Furthermore, if [MATH] is ergodic then [MATH] is [MATH] -almost everywhere constant and equal to |
[EQUATION] These properties are a direct consequence of the Birkhoff ergodic theorem, see , Theorem 4.1.2 and Corollary 4.1.9. Now the first theorem. |
Theorem 3.1 Let [MATH] be as in Assumption 3.1 [MATH] as in 15 ) and [MATH] the Lyapunov exponent of [MATH] . Then [MATH] holds for all |
[MATH] [MATH] if [MATH] exists. Proof. Let [MATH] be given and [MATH] a constant with [MATH] for all [MATH] . According to Equation ( 14 ) and Proposition |
3.2 [MATH] holds. Iteration gives [MATH] So, [MATH] follows. A necessary condition for the algorithm to terminate is therefore [MATH] . This gives the bound on |
[MATH] . Following the definitions of [MATH] and the Lyapunov exponent, [MATH] follows. Before a realistic upper bound on [MATH] can be presented, one more definition is needed. |
Definition 3.3 Let [MATH] then define a function [MATH] by [EQUATION] Furthermore, for any [MATH] define [EQUATION] Proposition 3.6 |
For all [MATH] there exists some constant [MATH] such that [MATH] holds for all [MATH] . Furthermore, if the Lyapunov exponent [MATH] exists, |
[MATH] holds. Proof. According to the requirements made on [MATH] [MATH] is Lipschitz with a Lipschitz constant [MATH] . Furthermore, let [MATH] |
be given. Then for all [MATH] [MATH] holds. Hence it follows the upper bound on [MATH] The second assertion follows from the fact that [MATH] holds for all [MATH] |
[MATH] Proposition 3.7 Let [MATH] be given. If [MATH] exists, then also the limit [EQUATION] exists and [MATH] Proof. Since [MATH] holds for all [MATH] [MATH] , also |
[MATH] follows. Letting [MATH] [MATH] , the assertion follows. Theorem 3.2 Let [MATH] be as in Assumption 3.1 [MATH] as in 15 ) and [MATH] as in 18 ). Let [MATH] be given, then for any |
[MATH] there is some [MATH] such that for all [MATH] [EQUATION] holds if [MATH] exists. So there is the following bound on the loss of significance rate. |
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