Journal of Applied Mathematics

Volume 2014 (2014), Article ID 742538, 13 pages

http://dx.doi.org/10.1155/2014/742538

Research Article

Accurate Evaluation of Polynomials in Legendre Basis

Peibing Du,¹ Hao Jiang,² and Lizhi Cheng¹

¹Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha 410073, China

²The State Key Laboratory for High Performance Computation, College of Computer Science, National University of Defense Technology, Changsha 410073, China

Received 22 April 2014; Accepted 2 July 2014; Published 23 July 2014

Academic Editor: Roberto Barrio

Copyright © 2014 Peibing Du et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents a compensated algorithm for accurate evaluation of a polynomial in Legendre basis. Since the coefficients of the evaluated polynomial are fractions, we propose to store these coefficients in two floating point numbers, such as double-double format, to reduce the effect of the coefficients’ perturbation. The proposed algorithm is obtained by applying error-free transformation to improve the Clenshaw algorithm. It can yield a full working precision accuracy for the ill-conditioned polynomial evaluation. Forward error analysis and numerical experiments illustrate the accuracy and efficiency of the algorithm.

1. Introduction

Legendre polynomial is often used in numerical analysis [1–3], such as approximation theory and quadrature and differential equations. Legendre polynomial satisfies 3-term recurrence relation; that is, for Legendre polynomial ,

The polynomial represented in Legendre basis is , where and is Legendre polynomial.

The Clenshaw algorithm [4, 5] is usually used to evaluate a linear combination of Chebyshev polynomials, but it can apply to any class of functions that can be defined by a three-term recurrence relation. Therefore the Clenshaw algorithm can evaluate a polynomial in Legendre basis. The error analysis of the Clenshaw algorithm was considered in the literatures [6–10]. The relative accuracy bound of the computed values by the Clenshaw algorithm verifies

For ill-conditioned problems, several researches applied error-free transformations [11] to propose accurate compensated algorithms [12–15] to evaluate the polynomials in monomial, Bernstein, and Chebyshev bases with Horner, de Casteljau, and Clenshaw algorithms, respectively. Some recent applications of high-precision arithmetic were given in [16].

Motivated by them, we apply error-free transformations to analyze the effect of round-off errors and then compensate them to the original result of the Clenshaw algorithm. Since the coefficients of the Legendre polynomial are fractions, the coefficient perturbations in the evaluation may exist when the coefficients are truncated to floating point numbers. We store the coefficients which are not floating point numbers in double-double format, with the double working precision, to get the perturbation. We also compensate the approximate perturbed errors to the original result of the Clenshaw algorithm. Based on the above, we construct a compensated Clenshaw algorithm for the evaluation of a linear combination of Legendre polynomials, which can yield a full working precision accuracy and its relative accuracy bound satisfies

The paper is organized as follows. Section 2 shows some basic notations in error analysis, floating point arithmetic, error-free transformations, compensated algorithm, Clenshaw algorithm, and condition number. Section 3 presents the compensated algorithm and its error bound. Section 4 gives several numerical experiments to illustrate the efficiency and accuracy of the compensated algorithm for polynomial in Legendre basis.

2. Mathematical and Arithmetical Preliminaries

2.1. Basic Notations and Definitions

Throughout this paper, we assume to work with a floating point arithmetic adhering to IEEE-754 floating point standard in rounding to nearest and no overflow nor underflow occurs. Let represent the floating point computation; then the computation obeys the model

where and ( is the round-off unit). We also assume that the computed result of in floating point arithmetic is denoted by and the set of all floating point numbers is denoted by . The following definition will be used in error analysis (see more details in [17]).

Definition 1. One defines

where with , , and .

There are three classic properties which will also be used in error analysis:

(i);(ii);(iii).

2.2. Accurate Sum and Product

Let , and no overflow nor underflow occurs. The transformation is regarded as an error-free transformation () that causes to exist such that , .

Let us show the error-free transformations of the sum and product of two floating point numbers in Algorithms 1-3 which are the algorithm by Knuth [18] and the algorithm by Dekker [19], respectively.

function

Algorithm 1: Sum of two floating point numbers.

function

(in double precision factor )

Algorithm 2: Split of a floating point number into two parts.

function

Algorithm 3: Product of two floating point numbers.

Algorithms 1–3 satisfy the Theorem 2.

Theorem 2 (see [11]). For and , and verify

We present the compensated algorithm for the product of three floating point numbers in Algorithm 4, which refers to [20].

function

Algorithm 4: Product of three floating point numbers.

According to Theorem 2, we have and . Hence, +; then . According to Lemma 3.2 in [20], we can propose the error bound of Algorithm 4 in Theorem 3.

Theorem 3. For and , verifies

Proof. According to Lemma 3.2 in [20], we have

In Algorithm 4, ; then

2.3. The Clenshaw Algorithm and Condition Number

The standard general algorithm for the evaluation of polynomial in Legendre basis is the Clenshaw algorithm [5]. We recall it in Algorithm 5.

function

for

end

Algorithm 5: The Clenshaw algorithm for finite Legendre series.

Barrio et al. [21] proposed a general polynomial condition number for any polynomial basis defined by a linear recurrence and used this new condition number to give the error bounds for the Clenshaw algorithm. Based on this general condition number, we propose the absolute Legendre polynomial, which is similar to the absolute polynomial mentioned in [7, 21].

Definition 4. Let be Legendre polynomial. We define the absolute Legendre polynomial which is associated with and satisfies

where , .

The absolute Legendre polynomial satisfies the following property.

Lemma 5. Let be the absolute Legendre polynomial. Then we have

Proof . Let and . From Definition 4, we have

then

Thus the equivalent form of (11) is

Since is increasing with , we obtain

So we finally obtain .

Following Definition 4, we introduce the condition number for the evaluation of polynomials in Legendre basis [21].

Definition 6. Let , where is Legendre polynomial. Let be the absolute Legendre polynomial. Then the absolute condition number is

and the relative condition number is

3. Compensated Algorithm for Evaluating Polynomials

In this section, we exhibit the exact round-off errors generated by the Clenshaw algorithm with . We also analyze the perturbations generated by truncating the fractions in Algorithm 5. We propose a compensated Clenshaw algorithm to evaluate finite Legendre series and present its error bound in the following.

Firstly, in order to analyze the perturbations, we split each coefficient into three parts as follows:

where , and . Let and . Then we describe the recurrence relation at th step of Algorithm 5 for theoretical computation as

For numerical computation associated with (19), it is

Remark 7. Let . We split a coefficient like (18); then the representation of splitting is unique and , .

Let and in Algorithm 5. Since every elementary floating point operation in Algorithm 5 causes round-off errors in numerical computation, we apply and the algorithms to take notes of all round-off errors and obtain

The sum of the perturbation and the round-off errors of the recurrence relation at th step is

where , is defined in Theorem 3. Then we obtain the following theorem.

Theorem 8. Let be a Legendre series of degree , and let be a floating point value. One assumes that and in Algorithm 5 and is the numerical result; is described in (22) for . Then one obtains

Proof. The perturbation of Algorithm 5 is

Let and in Algorithm 5. From Theorems 2–3 and (21), we can easily obtain

Let for ; then

From (18), (19), (22), and (24), we have

Thus

then

To devise a compensated algorithm of Algorithm 5 and give its error bound, we need the following lemmas.

Lemma 9. Let be a Legendre series of degree , a floating point value, and the numerical result of Algorithm 5. Then

Proof. Applying the standard model of floating point arithmetic, from Algorithm 5 and Remark 7, we have

Thus we obtain

Then, from Definition 1, we get

Lemma 10. Let be a Legendre series of degree and a floating point value. We assume that , where is real numbers; then

Proof. According to (31), we get

Since , from Definition 1, we have

then

According to Lemma 5, we obtain

Among the perturbed and round-off errors in Algorithm 5, we deem that some errors do not influence the numerical result in working precision. Now we can give the perturbed error bounds and the round-off error bounds, respectively. At first we analyze the perturbed error in (24). Let . According to and Remark 7, we obtain

Then let and from Lemma 10, taking into account that , we have

is , so this coefficient perturbation may influence the accuracy; we need to consider it in our compensated algorithm.

Similarly, we let ; then

is , so this coefficient perturbation does not influence the accuracy.

Remark 11. When , the th step of Algorithm 5 is , so that we only need to consider the perturbation of coefficient .

Next we deduce the round-off error bound. Let . According to Theorems 2 and 3 and Remark 7, we obtain

then

Let and using it into Lemma 10, from , we have

is , so this round-off error may influence the accuracy, we also need to consider it in our compensated algorithm.

From Theorem 3 we have known that round-off error . According to Lemma 10 we let , from and , we have

is , so this round-off error does not influence the accuracy.

Observing the error bounds we described above, the perturbation generated by the third part of coefficients in (18) does not influence the accuracy. Thanks to the algorithm in Appendix B (see Algorithm 13), we only need to split the coefficients into two floating point numbers.

Applying , the algorithm and the algorithm, considering all errors which may influence the numerical result in working precision in Algorithm 5, we obtain the compensated algorithm in Algorithm 6.

function

for

end

Algorithm 6: Compensated Clenshaw algorithm.

Here we give the error bound of Algorithm 6.

Theorem 12. Let be a Legendre series of degree and a floating point value. The forward error bound of the compensated Clenshaw algorithm is

Proof. From Algorithm 6, we obtain

where .

According to Theorem 8, we have ; then

Next we analyze the bound of .

Let ; then

Thus we obtain

According to Lemma 9, we get

From (50), (51) and , we derive

According to and Remark 7, we obtain

From (53) we use into Lemma 10; taking into account that and , according to (52), we get

Next we let ; then

thus

According to Lemma 9, we obtain

From (56), (57), and , we derive

From (43), we let and using it into Lemma 10, taking into account that and , we have

Combining (54) and (59), we get

From Definitions 4 and 6 and Theorem 12, we easily get the following corollary.

Corollary 13. Let be a Legendre series of degree and a floating point value. The relative error bound of the compensated Clenshaw algorithm is

4. Numerical Results

All our experiments are performed using IEEE-754 double precision as working precision. Here, we consider the polynomials in Legendre basis with real coefficients and floating point entry . All the programs about accuracy measurements have been written in MATLAB R2012b and that about timing measurements have been written in C code on a 2.53-GHz Intel Core i5 laptop.

4.1. Evaluation of the Polynomial in Legendre Basis

In order to construct an ill-conditioned polynomial, we consider the evaluation of the polynomial in Legendre basis converted by the polynomials in the neighborhood of its multiple roots 0.75 and 1. We use the , algorithms and the Symbolic Toolbox to evaluate the polynomial in Legendre basis. In order to observe the perturbation of polynomial coefficients clearly, we propose the algorithm in Appendix A (see Algorithm 7) to obtain the coefficients of the Legendre series. To decrease the perturbation of the coefficients we also propose the algorithm in Appendix B (see Algorithm 16) to store the coefficients in double-double format. That is, the coefficients of the polynomial evaluated, which are obtained by the algorithm and algorithm, are in double format and double-double format, respectively. In this experiment we evaluate the polynomials for 400 equally spaced points in the intervals , , and .

function

are the vectors of monomial polynomial and orthogonal polynomial’s coefficients, respectively

,

,

for

    for

    end

end

Algorithm 7: The conversion algorithm from power basis to Legendre basis.

From Figures 1-2 we observe that the polynomial evaluated by algorithm (on the top figure) is oscillating, and the compensated algorithm is more smooth drawing. The polynomials we evaluated by Symbolic Toolbox (on the bottom of Figures 1-2) are different because the perturbations of coefficients obtained by the algorithm are smaller than those by the algorithm. We can see that the accuracy of the polynomials evaluated by the compensated algorithm is the same with evaluated by Symbolic Toolbox in Figure 1. The polynomials in Legendre basis evaluated by the compensated algorithm are much more smooth drawing and just a little oscillation in the intervals in Figure 2. In fact, if we use the Symbolic Toolbox to get the polynomial coefficients, the oscillation will be smaller than it is in Figure 2. However, this method is expensive. We just need to use the algorithm to get the coefficients; the result obtained by the algorithm is almost the same as that by using the Symbolic Toolbox in working precision.

Figure 1: Evaluation of polynomial converted from by the algorithm in Legendre basis in the neighborhood of its multiple roots, using the Clenshaw algorithm (up), the algorithm (middle), and Symbolic Toolbox (down).

Figure 2: Evaluation of polynomial converted from by the algorithm in Legendre basis in the neighborhood of its multiple roots, using the Clenshaw algorithm (up), the algorithm (middle), and Symbolic Toolbox (down).

4.2. Accuracy of the Compensated Algorithm

The closer to the root, the larger the condition number. Thus, in this experiment, the evaluation is for 120 points near the root 0.75, that is, , for 1 : 40 and for 1 : 80. We compare the compensated algorithm with multiple precision library. Since the working precision is double precision, we choose the double-double arithmetic [22] (see Appendix B) which is the most efficient way to yield a full precision accuracy of evaluating the polynomial in Legendre basis to compare with the compensated algorithm. We evaluate the polynomials by the , , and algorithms in Appendix B (see Algorithm 15) and the Symbolic Toolbox, respectively, so that the relative forward errors can be obtained by and the relative error bounds are described from Corollaries 13 and A.2 in Appendix A. Then we propose the relative forward errors of evaluation of the polynomial in Legendre basis in Figure 3. As we can see, the relative errors of the compensated algorithm and double-double arithmetic are both smaller than () when the condition number is less than . And the accuracy of both algorithms is decreasing linearly for the condition number larger than . However, the algorithm cannot yield the working precision; the accuracy of which decreases linearly since the condition number is less than . When the condition number is lager than , the Clenshaw algorithm cannot obtain even one significant bit.

Figure 3: Accuracy of evaluation of polynomial converted from by the algorithm in Legendre basis with respect to the condition number.

4.3. Time Performances

We can easily know that the , , and algorithms in Appendix B (Algorithm 8) require 6, 17, and 3 flops, respectively. Then we obtain the computational cost of the , , and algorithms:

(i): 5n-2 flops;(ii): 79n-29 flops;(iii): 110n-44 flops.

function

Algorithm 8: EFT of the sum of two floating point numbers .

function

Algorithm 9: Addition of a double-double number and a double number.

function

Algorithm 10: Addition of a double-double number and a double-double number.

function

Algorithm 11: Multiplication of a double-double number by a double number.

function

Algorithm 12: Multiplication of a double-double number by a double-double number.

function

Algorithm 13: Division of a double number by a double number.

Considering the previous comparison of the accuracy, we observe that is as accurate as in double precision, but it only needs about 71.8% of flops counting on average. We also implement and by using Microsoft Visual C++ 2008 on Windows 7. Similar to the statement in [23], we assume that the computing time of these algorithms does not depend on the coefficients of polynomial in Legendre basis nor the argument . So we generate the tested polynomials with random coefficients and arguments in the interval , whose degrees vary from 20 to 10000 by the step 50. The average measured computing time ratio of to in C code is . The reason why the measured computing time ratio is better than the theoretical flop count one can be referred to the analysis in terms of instruction level parallelism (ILP) described in [24, 25].

5. Conclusions

This paper introduces a compensated Clenshaw algorithm for accurate evaluation of the finite Legendre series. The algorithm is not precise enough for an ill-conditioned problem, especially evaluating a polynomial in the neighborhood of a multiple root. However, this new algorithm can yield a full precision accuracy in working precision, as the same as the original Clenshaw algorithm using double-double arithmetic and rounding into the working precision. Meanwhile this compensated Clenshaw algorithm is more efficient which means that it is much more useful to accurately evaluate the polynomials in Legendre basis for ill-conditioned situations.

Appendices

A. The Coefficients Conversion Algorithm from Polynomial in Power Basis to Polynomial in Legendre Basis

In order to design ill-conditioned problem of the polynomial evaluation, we evaluate a polynomial in the neighborhood of a multiple root. Thus we need an algorithm for converting a polynomial in power basis to the polynomial in Legendre basis. Motivated by [26], the conversion algorithm can be deduced as follows.

Let and be the polynomials in power basis and Legendre basis, respectively; according to

we have

Thus we get Algorithm 7.

Theorem A.1. Let and be the polynomials in power basis and Legendre basis, respectively. The forward error bound of Algorithm 7 is

Proof. According to Definition 1 and Remark 7, we have

When , we get that

Hence we can obtain

Then we have

According to the properties of Definition 1, we obtain

thus

According to Theorem A.1 and Theorem 3.5 in [21], we obtain the following corollary.

Corollary A.2. Let and be the polynomials in power basis and Legendre basis, respectively. Then the relative error bound of Clenshaw algorithm by using the algorithm is

B. Double-Double Library

The double-double arithmetic is based on Algorithms 8–14 [27]. Algorithms 15 and 16 are Algorithms 5 and 7 in double-double arithmetic, respectively.

function

Algorithm 14: Division of a double-double number by a double-double number.

function

for

end

Algorithm 15: DDClenshaw algorithm of evaluating an Legendre series in double-double arithmetic.

function

are the vectors of monomial polynomial and orthogonal polynomial’s coefficients, respectively

,

,

for

     for

             if

             if

             if

     end

end

Algorithm 16: DDConvert algorithm for the polynomial from power basis to Legendre basis.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the Science Project of National University of Defense Technology (JC120201) and the National Natural Science Foundation of Hunan Province in China (13JJ2001).

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