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[EQUATION] where the Newton coefficient [MATH] would need to be re-computed from the divided difference over all nodes except [MATH] We can avoid this computation by noting that [MATH] has to be of degree [MATH] and therefore the highest coefficient of [MATH] |
[MATH] , must disappear. This is the case when [EQUATION] and therefore [EQUATION] Using this [MATH] , we can the compute the coefficients of [MATH] |
as [EQUATION] The whole process is shown in Algorithm The downdate of an [MATH] -node interpolation requires [MATH] operations. Algorithm 2 |
Remove a function value [MATH] at the node [MATH] from the interpolation given by the coefficients [MATH] 1: [MATH] compute [MATH] from [MATH] using back-substitution |
2: [MATH] 3: for [MATH] do 4: [MATH] 5: end for 6: [MATH] compute the coefficient [MATH] 7: [MATH] compute the new coefficients [MATH] |
A New Algorithm for the Construction of Interpolations Returning to the representation in Equation ( ), we can try to solve the Vandermonde-like system of linear equations directly. The matrix has some special characteristics which we can exploit to achieve better performance and stability than when using Gaussian elim... |
We start by de-composing the [MATH] Vandermonde-like matrix [MATH] as follows: [EQUATION] The sub-matrix [MATH] is a Vandermonde-like matrix analogous to [MATH] The column [MATH] contains the [MATH] th polynomial evaluated at the nodes [MATH] [MATH] |
[EQUATION] and the vector [MATH] contains the values of the first [MATH] polynomials at the node [MATH] [EQUATION] Inserting this into the product in Equation ( ), we obtain |
[EQUATION] which, when effected, results in the pair of equations [EQUATION] where the vectors [MATH] and [MATH] contain the first [MATH] coefficients or function values respectively. |
Before trying to solve Equation ( 23 ), we note that the columns of the matrix [MATH] contain the first [MATH] polynomials evaluated at the same [MATH] nodes each. Similarly, [MATH] contains the same polynomials evaluated at the node [MATH] Since the polynomials in the columns are of degree [MATH] and they are evaluate... |
[EQUATION] where the [EQUATION] are the Lagrange polynomials over the first [MATH] nodes [MATH] [MATH] We can write Equation ( 24 ) as |
[EQUATION] where the entries of the [MATH] vector [MATH] are [EQUATION] The entries of [MATH] can be computed recursively. Using the definition in Equation ( 25 ), we define |
[EQUATION] and re-write [MATH] as [EQUATION] Using the previous [MATH] and [MATH] , we can re-write Equation ( 27 as [EQUATION] which, re-inserted into Equation ( 27 ), gives |
[EQUATION] for all [MATH] We then compute the last entry [MATH] using [EQUATION] Therefore, starting with [MATH] , we can construct all the [MATH] [MATH] successively. Since, given [MATH] [MATH] the construction of each additional [MATH] |
requires [MATH] operations, the construction of all the [MATH] [MATH] requires a total of [MATH] operations. Returning to the Vandermonde-like matrix, inserting Equation ( 26 ) into Equation ( 23 ), we obtain |
[EQUATION] Multiplying the first line in Equation ( ) with [MATH] from the left and subtracting the bottom equation from the top one we obtain |
[EQUATION] from which we can finally isolate the coefficient [MATH] [EQUATION] Having computed [MATH] , we can now re-insert it into the first Equation in Equation ( ), resulting in the new system |
[EQUATION] in the remaining coefficients [MATH] Applying this computation recursively to [MATH] [MATH] [MATH] [MATH] we can compute the interpolation coefficients |
[MATH] The final coefficient [MATH] can be computed as [EQUATION] The complete algorithm is shown in Algorithm Since the construction of the [MATH] |
[MATH] requires [MATH] operations (for each [MATH] [MATH] operations are required to compute [MATH] and [MATH] are required to compute the new entries) and the evaluation of Equation ( 31 ) requires [MATH] operations for each [MATH] [MATH] the total cost of the algorithm is in [MATH] operations. |
Algorithm 3 Direct construction of [MATH] 1: [MATH] [MATH] init [MATH] 2: for [MATH] do 3: [MATH] fill [MATH] 4: end for 5: [MATH] [MATH] |
init [MATH] and [MATH] 6: for [MATH] do 7: [MATH] construct [MATH] 8: for [MATH] do 9: [MATH] 10: end for 11: for [MATH] do 12: [MATH] |
compute the [MATH] , Equation ( 28 13: end for 14: [MATH] compute [MATH] , Equation ( 29 15: end for 16: for [MATH] do 17: [MATH] |
compute coefficient [MATH] , Equation ( 31 18: [MATH] update the right-hand side [MATH] 19: end for 20: [MATH] compute the final [MATH] |
Note that, as opposed to the algorithm presented in Section this algorithm can not be extended to update or downdate an interpolation. It has an advantage, however, when multiple right-hand sides, i.e. interpolations over the same set of nodes, are to be computed. In such a case, the vectors [MATH] [MATH] |
need to be computed only once (Lines to 15 of Algorithm ). For any new vector [MATH] , only the Lines 16 to 20 need to be re-evaluated. |
Results To assess the stability of the two new interpolation routines described herein, we will follow the methodology used by \citeasnoun ref:Higham1988. Higham defines a set of interpolations consisting of all combinations of the nodes |
[EQUATION] with the right-hand sides [EQUATION] for [MATH] To avoid instabilities due to unfortunate orderings of the nodes [MATH] the nodes and corresponding function values were re-ordered according to the same permutation that would be produced by Gaussian elimination with partial pivoting applied to the Vandermonde... |
For each combination of nodes and right-hand sides, we compute, following Higham, the coefficients [MATH] for the Chebyshev base (see Equation ( )) for [MATH] , 10, 20 and 30 and compute the quantities |
[EQUATION] where [MATH] is the exact solution and [MATH] is the unit roundoff as defined by \citeasnoun [Section 2.4.2]ref:Golub1996. Note that for the special case of A1 or A2 using the Chebyshev base, the coefficients can also be computed efficiently and reliably using the Fast Fourier Transform ref:Battles2004 |
Results were computed using Gaussian elimination ( GE ), Higham’s extension of the algorithm of Björck and Pereyra ( BP/H ), the incremental Algorithm INCR ) and the direct Algorithm DIRECT ). The exact values were computed in Maple ref:Maple with 50 decimal digits of precision using the interp function therein. |
Results were also computed for the interpolation downdate ( DEL described in Algorithm Starting from [MATH] and [MATH] , the exact coefficients for the interpolation [MATH] and the Newton polynomial |
[MATH] respectively, we compute the coefficients [MATH] and [MATH] for [MATH] and [MATH] , resulting from the removal of the rightmost function value |
[MATH] at [MATH] [MATH] The exact coefficients [MATH] after deletion were computed and used to compute the quantities ERR and RES |
The results are shown in Tables to 12 For each [MATH] , the largest values for ERR and RES are highlighted. For the problem sets over the nodes A1 and A2 (Tables to ), the condition of the Vandermonde-like matrix is always [MATH] |
ref:Gautschi1983 , resulting in very small errors for Gaussian elimination. The Björck-Pereyra/Higham algorithm generates slightly larger residuals than both the incremental and direct algorithms for both sets of nodes. The values for ERR , however, are usually within the same order of magnitude for the three algorithm... |
For the nodes A3 (Tables to ), the condition number of the Vandermonde-like matrix is [MATH] for [MATH] , resulting in the errors of approximately that magnitude when Gaussian elimination is used. In general, both the Björck-Pereyra/Higham and the direct algorithm generate smaller errors and residues than Gaussian elim... |
Finally, the condition number of the Vandermonde-like matrix for the nodes A4 is [MATH] for [MATH] , making it numerically singular and thus resulting in the complete failure of Gaussian elimination. Note that since in such cases Matlab’s backslash-operator computes the minimum norm solution, the resulting residual err... |
The errors and residuals for the downdate algorithm are shown in the rightmost columns of Tables to 12 In general the errors of the downdate are relatively small for all test cases. The larger residues, e.g. for A2/F2, are due to cancellation in the final subtraction in Algorithm , Line |
Conclusions We have presented here two new algorithms for the construction of polynomial interpolations. The first algorithm (Algorithm ) offers no substantial improvement over that of Björck-Pereyra/Higham except that it can be easily downdated. The second algorithm, which does not allow for updates nor downdates, is ... |
# Source: arxiv 1003.6036 # Title: Computational Complexity of Iterated Maps on the Interval # Sections: all # Downloaded: 2026-03-03T02:20:42.236464+00:00 |
Computational Complexity of Iterated Maps on the Interval Abstract The correct computation of orbits of discrete dynamical systems on the interval is considered. Therefore, an arbitrary-precision floating-point approach based on automatic error analysis is chosen and a general algorithm is presented. The correctness of... |
keywords: Discrete dynamical systems , Lyapunov exponent , arbitrary-precision floating-point arithmetic MSC: 37M05 , 65P20 Introduction |
Consider a discrete dynamical system [MATH] on some compact interval [MATH] called the phase space, given by a function [MATH] , a recursion relation |
[MATH] and an initial value [MATH] . The sequence [MATH] of iterates is called the orbit of the dynamical system in phase space corresponding to the initial value [MATH] . If such a dynamical system is implemented, that is a computer program is written for calculating a finite initial segment of the orbit for given [MA... |
floating-point numbers are used. However, if the dynamical system shows chaotic behavior, a problem arises. The finite and constant length of the significand of a double variable causes rounding errors which are magnified after each iteration step. Already after a few iterations, the error is so big that the computed v... |
this phenomenon is examined for the dynamical system [MATH] with [MATH] [MATH] and the initial value [MATH] . To put things right, a rigorous method for computations with real numbers has to be used. There already exist some rigorous numerical methods in the field of dynamical systems and chaos |
In the next section, a rigorous method based on arbitrary-precision floating point arithmetic is presented and used to investigate the iteration of a generalization of the above mentioned function. Correctness of the results are obtained by using a method called running error analysis. The method and the numerics are c... |
Dynamic behavior of the logistic equation and rounding error In this section, the discrete dynamical system [MATH] with [MATH] and |
[MATH] [MATH] for some control parameter [MATH] is investigated. In the literature, the recursion relation [MATH] is called the logistic equation |
. When implementing the logistic equation on a real computer and demanding to obtain true values for the orbit [MATH] , some rigorous method is needed. Since for some values of |
[MATH] the dynamics is highly chaotic, inaccuracies are magnified exponentially in time . Therefore, it is clear that using floating-point numbers with a predefined, fixed precision makes sense only if the maximum iteration time [MATH] also is a predefined, fixed number. If the algorithm should work for any [MATH] , a ... |
Second, there are methods with automatic error control, for example interval arithmetic , the Feasible Real RAM model or significance arithmetic |
. Implementations are for example MPFI , the iRRAM and Mathematica respectively. All these methods have the same theoretical background. They are all practical instances of the model of Computable Analysis |
used in computer science. While the Feasible Real RAM model directly implements the theory of Computable Analysis, the other mentioned methods all have their background in scientific computing. Looking closer at the various validated methods in use, they all have in common implementing some kind of intervals for repres... |
[MATH] together with its computational error be represented by two floating-point numbers [MATH] and [MATH] [MATH] ) with given length [MATH] of the significand, called the precision, forming an interval [MATH] . The interval is an enclosure of the real value [MATH] , that is [MATH] |
for all [MATH] . The interval length [MATH] gives a measure of the uncertainty about [MATH] and is therefore a kind of error. Interval arithmetic often models quantities which are not known exactly. But here, the true orbit |
[MATH] can be, in principle, calculated to any given accuracy. Thus, in the present setting, the true object of interest is not an interval, but an approximation [MATH] of [MATH] together with an absolute error [MATH] . The interval is only used for mathematical convenience. To transform the interval to a floating poin... |
[EQUATION] where [MATH] performs a rounding to some floating-point number of precision [MATH] nearest to [MATH] Note that rounding to nearest is not unique if [MATH] is equidistant from two floating-point numbers. The absolute error [MATH] of [MATH] can be estimated via the interval length [MATH] by |
[EQUATION] where [MATH] is an error caused by the rounding operation [MATH] in Equation ( ). An upper bound on [MATH] will be discussed later, for now it suffices to say that in general it is small compared to [MATH] |
The aim now is to calculate, for given initial value [MATH] [MATH] and [MATH] the orbit up to time [MATH] with relative error at most [MATH] . That is, for |
[MATH] it should hold [EQUATION] Why using here and in the following the relative error and not the absolute error is discussed in some detail at the end of Subsection 3.2 The minimal [MATH] , fulfilling the precision requirement ( ) on the relative error of [MATH] , which depends on [MATH] [MATH] and [MATH] , is denot... |
[EQUATION] The loss of significance rate [MATH] , which may depend on the initial value [MATH] is given by [EQUATION] This quantity describes the limiting amount of significant precision being lost at each iteration step in the limit of infinite output precision. Significant means here the part of the digits being corr... |
[MATH] is the precision for any floating-point number needed in an algorithm doing the iteration starting with [MATH] and calculating to [MATH] , if the output should be precise to at least [MATH] decimal places. Here, [MATH] is the logarithm to base [MATH] |
2.1 Dynamic behavior of the logistic equation Before analyzing the different numerical behavior, it is worth having an analytical look at the dynamical behavior of the system. Despite the fact that these results are well known |
, they are reviewed here for the sake of self containment. First have a look at the fixed points of the logistic equation and their stability. In the range [MATH] , the equation possesses exactly one fixed point [MATH] if [MATH] and exactly two fixed points [MATH] |
and [MATH] if [MATH] . Looking at the derivatives [MATH] and [MATH] gives the stability of the fixed points. Since [MATH] for [MATH] and [MATH] |
for [MATH] [MATH] is a stable fixed point, an attractor for [MATH] and an unstable fixed point, a repeller for [MATH] If [MATH] , the only fixed point [MATH] is hyperbolic, that is [MATH] At [MATH] , a bifurcation occurs. If |
[MATH] [MATH] becomes unstable and the newly occurring fixed point [MATH] is stable. At [MATH] a second bifurcation occurs and for [MATH] |
both fixed points are unstable. Second, examine the basin of attraction of the stable fixed point. If [MATH] , the contraction mapping principle directly gives |
[MATH] for all [MATH] . If [MATH] observe that [MATH] holds for all [MATH] . Hence, any sequence [MATH] [MATH] , is strictly decreasing and bounded from below. So, also [MATH] holds for all [MATH] Finally, in the case [MATH] [MATH] |
holds for all [MATH] . For a proof, the interested reader is referred to the literature: , Proposition 5.3 in Section 1.5. Finally, for [MATH] the system goes into a region showing periodic behavior with period doubling bifurcations. Finally, for some |
[MATH] , chaotic behavior is reached. This analysis shows that in the parameter range [MATH] the orbit tends to the stable fixed point for any initial value |
[MATH] . Furthermore, there exists some closed interval [MATH] which depends on [MATH] , containing the stable fixed point such that |
[MATH] holds and [MATH] is a contraction on [MATH] . Next have a look at the computational effort in the various control parameter ranges. |
2.2 Numerical analysis of the computational complexity The logistic equation is implemented in various forms using an arbitrary-precision interval library. For that purpose, the already mentioned interval library MPFI based on the arbitrary-precision floating-point number library MPFR, both written in C, is used. For e... |
for [MATH] is searched. Then, [MATH] is calculated. First, [MATH] is implemented using a natural interval extension based on the expressions [MATH] |
[MATH] and [MATH] . The natural interval extension is obtained by replacing any occurrence of the variable [MATH] in the expression by an interval [MATH] |
The results are shown in Figures and respectively. Second, the logistic equation is implemented using a centered form, actually the mean value form |
[MATH] where [MATH] is an interval and [MATH] is the midpoint of [MATH] The result is shown in Figure . In the following, these 4 calculations are referred to as 1 to 4 respectively. |
The interval computation is in agreement with the dynamical picture only in Calculation 4. While for [MATH] , the results shown in Calculations 1, 2 and 4 are in agreement with the dynamical analysis, 3 is not since it would suggest an exponential divergence of initially nearby orbits which is not true in reality. A si... |
[MATH] and hence it can be said that in the range [MATH] the Calculations 3 and 4 are in agreement with the dynamic picture, while 1 and 2 are not. How can this be explained? |
2.3 Investigating Calculation 1 This subsection deals with the explanation of the curve obtained by Calculation 1. For doing an error analysis of the logistic equation analytically, some idealizing assumptions have to be made. Generally, executing the iteration in interval arithmetic, two types of error are present. Fi... |
[EQUATION] with the interval length [MATH] given by the recursion relation [EQUATION] with the obvious solution [MATH] . The absolute error [MATH] |
of [MATH] according to Equation ( ) can be bounded from above by [EQUATION] The ideal assumptions require the somewhat unreal setting that the precision has to be set to some finite, but big enough value [MATH] |
for representing [MATH] and a virtually infinite value [MATH] for doing the iteration. To get a relation connecting [MATH] and the output precision [MATH] in ), some upper bound on [MATH] is needed. The value of [MATH] |
is given as the rounding error by representing [MATH] as a floating-point number of precision [MATH] . For that, the well known estimate |
[EQUATION] exists. Combining ( ), ( ) and ( gives as a sufficient condition [EQUATION] for [MATH] . So, the sufficient condition gives an upper bound on |
[MATH] by [EQUATION] This finally leads to an upper bound for the loss of significance rate, [EQUATION] The curve in Figure shows that [MATH] |
exceeds the estimated bound [MATH] only slightly. So, the above made ideal assumptions seem to be valid. In the logistic equation was also investigated for [MATH] using the exact real arithmetic package iRRAM. In the paper, the maximum bounding precision needed to guarantee the correctness of the first [MATH] |
decimal places are reported up to [MATH] . Relating this quantity to [MATH] shows full agreement with the simulation results performed here. So, for [MATH] , the interval length [MATH] increases exponentially in time [MATH] which is in contrast to the dynamic behavior for [MATH] . The reason is that the natural interva... |
[MATH] . However, a local Lipschitz constant [MATH] governs the real error propagation at time step [MATH] and also describes the dynamic behavior. |
2.4 Investigating Calculation 2 Calculation 2 is similar to Calculation 1. An analogous analytic approach as in Calculation 1 gives the recursion relation |
[EQUATION] and hence [EQUATION] The recursion relation fulfills therefore [MATH] and [MATH] . As a consequence the bounds [MATH] are obtained. In analogy to Calculation 1, an upper bound for [MATH] |
[EQUATION] and hence [EQUATION] is calculated. A brief look at Figure shows that this upper bound is too rough. On the other hand, |
[MATH] suggests that the bound derived in Calculation 1 is a lower bound, hence [MATH] This is actually verified by numerical evidence. |
2.5 Investigating Calculation 3 Calculation 3 is explained here in the parameter range [MATH] , where it is not in agreement with the dynamic picture. Nevertheless it should be mentioned that the natural interval extension used here seems to be in full agreement with the dynamic picture in the parameter range [MATH] as... |
To explain the observed behavior, first note that for [MATH] [MATH] follows for all [MATH] . Hence, [MATH] holds for all [MATH] and the orbit |
[MATH] tends exponentially fast to zero. A brief look at the expression of [MATH] [MATH] , shows that the value of the second term in the difference tends exponentially fast in [MATH] to the value of the first term. Hence, the big values of the loss of significance rate for small values of [MATH] can be explained by ca... |
First note that even [MATH] may differ from the initial value [MATH] since the conversion to a floating-point number may cause the very first rounding error. Next, already mentioned, in calculating the orbit [MATH] , two types of error are present. First, error propagation due to the iteration scheme and second the rou... |
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