Numerical illustrations
To illustrate the usefulness of the Gauss-Daubechies quadrature rule derived here in comparison
2K — l
to other methods for evaluating the integrals _{Q} ip(x)f(x)dx, we have considered four examples with fi(x) = sin x , /г(а;) = cos 2(x— 2) + sin 3(x — 2), /з(ж) = cos 2x — l|+sin 3|ai — 1| and fi(x) = cos 2x — 2|+sin 3|ж — 2| available in the literature. The pseudo- or quasi-weight functions appearing in the integrals will be the interior and truncated scale functions. While the integral involving fi(x) was considered by Sweldens and Piessens (Sweldens and Piessens, 1994b) to illustrate the usefulness of their quadrature rule based on shifting trick, integrals involving /2(ж). /з(ж) and /4(0,’) have been considered by Huybrechs and Vandewalle (Huybrechs and Vandewalle, 2005) to demonstrate the effectiveness of their composite quadrature formula for evaluation of integrals involving piecewise smooth and singular functions. We have examined here the consequences due to the presence of some complex nodes and weights in the quadrature rule and its convergence in the evaluation of the above integrals. To evaluate the integrals
within [a, b] at higher resolutions we introduce the symbol
■2K-1
The use of the refinement equation
l=o
to
The summation over I will be from —К +1 to К whenever supp
[—К +1, К]. The same formula with hi replaced by gi can be used to evaluate integrals involving wavelets in place of scale functions. The number of nodes n' of the quadrature rule in the range [max{Z, 2a —2k}, min{/-(-2/v — 1,2b —2k}}
within the sum in the right side assumes higher admissible value adjacent to n, whenever quadrature rule with n nodes within this range does not exist. Such admissible nodes and weights can be efficiently obtained by calling the following program in MATHEMATICA with input data, low-pass filter h, range of integration к 1, k2 and degree nn of FOP.
The output of the program is nodes (a;,) and pseudo- or quasi-weights (ca,) for some n' > n.
In the evaluation of the four integrals mentioned above we have considered cases for which nodes and pseudo- or quasi-weights of the quadrature rule are all real or some of them are complex. As expected from the Theorem 3.7 in §3.4.1.3, appearance of complex nodes and weights in the quadrature formula do not pose any difficulty in the evaluation of the integrals. In § 3.4.1.8, it has been found that the error in the evaluation of integral by using Q_{n} [/, ip] involves derivatives of order 2n of /(ж). Here f(x) = sin ж% cos 2(ж’ — 2) + sin 3(ж — 2), cos 2|ж — l|+sin 3|ж’ — 1| and cos 2|ж’ — 2|+sin 3|ж — 2|, which are all trigonometric functions. Hence, ^_{2n}jj' is bounded on [0,5] for i = 1,2; on [0,3] — {1} for i = 3 and on [0,5] — {2} for i = 4. Thus the value of each integral obtained by Q_{n}[f-, ^p^{1}} or composite formula Qn[f,^{l}P^{L}'^{r}] + Qn'[f ,
RT]
converges smoothly to the exact value as n increases. Here, we regard the values obtained by using the quadrature rules Q24[/i,! = ОаиЩ),
5]](г = 1,2), Q_{2}o[/3,In Tables 3.1 and 3.2, we have compared the absolute value of errors in the evaluation of ^{1} ip^{1} (x)fi(x)dx,i = l,---,4 by using the quadrature rule Q„[/,p^{!}] = ) involving n real or complex nodes and weights (RCNW) and other quadrature rules based on lifting trick
(LTr) with c = ^ of Barinka et al. (Barinka et ah, 2001) and shifting trick (STr) of Sweldens and Piessens (Sweldens and Piessens, 1994b). The numbers within parenthesis in this table indicate exponent of 10. The exact values of the integrals correct up to sixteen decimal places are given in the last row. From this table it appears that accuracy of the values of the integrals for smooth functions obtained by Gauss-type quadrature rule involving RCNW seems to be better than those obtained by the quadrature rule based on STr of Sweldens and Piessens. The accuracy appears to be uniform irrespective of nodes and weights of the quadrature rule being all real or some being complex. However, the results exhibit Runge-like phenomenon for integrals involving non-smooth function /_{3 }and Dan2(). 3], slow rate of convergence for non-smooth function f_{4} and the pseudo-weight function Dau'3[0, 5).
It is interesting to note that the same trend in accuracy appears whenever the integrals have been evaluated by using the quadrature rule
based on LTr of Barinka et al. (Barinka et ah, 2001). However, it is important to note that for a given n, the above mentioned formula involves 2n distinct nodes ({a^, xf, i = 1,2, •••,«)} and their corresponding weights = 1,2, • • •, n)}. Consequently, one may regard this formula
as effectively a quadrature rule of 2n nodes with stability constant
Comparison of errors in Tables 3.1 and 3.2 exhibits the superiority of the proposed quadrature rule over the existing one for smooth functions.
To evaluate the approximate values of integrals involving non-smooth functions, e.g., _{0} ip(x)fi(x)dx, i = 3.4 (non-smooth at x_{cr} = 1 for f-s(x) and x_{cr} = 2 for f(x)) with a rapidly convergent quadrature formula and to check the effectiveness of Gauss-Daubechies quadrara- ture rule for integrals involving truncated scale functions, the above mentioned integrals have been split into £^{сг} tp^{LT}(x)fi(x)dx and ^{1} (x)fi(x)dx. Now
The number of nodes and weights used in the quadrature rules for
LT(a
:) and tp^{RT}(x) for given К are different due to the fact that equal number of nodes and weights may not always be available. For the purpose of calculating the functions f,(x) (i = 3,4) at a complex node x, we usein the quadrature rule mentioned above.
The errors in the evaluation of f_{0}
(i = 3,4) by the present method and by the method given by Huybrechs and Vandewalle (Huybrechs and Vandewalle, 2005) are displayed in Table 3.3. From this table it is obvious that the present method is superior to the method of Huybrechs and Vandewalle (Huybrechs and Vandewalle, 2005).
The stability constants for the quadrature rule Q_{n} [/, p^{l}] developed here and for the rule ip^{1}] developed by Barinka et al. (Barinka et al., 2001) have been compared in Table 3.4. Regarding the choice of c in the quadrature rule Q^{(}n. it may be noted that it is to be chosen such that the lifted function
c(x) =
+ c x_{SU}pp _{v} should be non-negative for x € supp ip. For scale functions in Daubechies family c > ^. We choose c = ^ as this choice leads to better accuracy as well as better stability constants which are evident from Tables 3.1, 3.2 and 3.4. From Table 3.4 it is found that the stability of the present method is better than that of the Barinka et al. (Barinka et al., 2001).
Thus, Gauss-Daubechies quadrature rules having complex nodes and weights for integrals involving scale functions with variable signs regarded as pseudo- or quasi-weight functions and smooth or non-smooth functions can be treated as efficient and almost in the same footing as the classical Gauss quadrature rule for integrals restricted to positive semi-definite weight functions.
Table 3.1: Comparison of errors in evaluating approximate value of integrals
/о*"^{1} ip^{I}{x)fi{x)dx, i = 1,2.
No. of Nodes | |||||||
n |
3 = o RCNW |
LTr |
STr |
3 = i RCNW |
3 = 0 RCNW |
LTr |
3 = 1 RCNW |
5 |
8.4(-8) |
1.5(—6) |
6.1(-4) |
8.8(-11) |
1 -4(—3) |
3.8(-2) |
3-4(-6) |
9/10^{SI>} |
3.3(-16) |
3.9(-15) |
9.8(-5) |
«io-^{2} |
2.2(-8) |
1.1 (-6) |
3.5(-13) |
20 |
«io-^{47} |
«io-^{45} |
4.3(-6) |
« 10-^{5} |
«io-^{29} |
«io-^{26} |
и 10-^{42} |
.741 104 421 925 904 6 |
-.644 487 735 893 018 1 |
Table 3.2: Comparison of errors in evaluating approximate value of integrals
Jo^{2}*^{-1} ifi^{I}{x)f_{i}(x)dx, i = 3,4 for j = 0.
Nodes | |||||
n |
RCNW |
LTr |
n |
RCNW |
LTi- |
4 |
3.5(—2) |
5.5(-3) |
5 |
3.5(-2) |
1.3(-1) |
7 |
4.3(-2) |
6. (-2) |
11 |
l-9( 2) |
9.4(-3) |
13 |
l-5(—2) |
l-3(—2) |
21 |
4.5(-4) |
1■1(—2) |
1.38 501 797 074 570 12 |
-.604 713 724 795 161 5 |
Table 3.3: Comparison of errors in Qnn' [/, 4>^{LT},
BT).
_{r}2K-1
Jo
n |
n’ |
7 = 0 RCNW |
n=n’ |
HV |
n |
IV |
7=0 RCNW |
n=n’ |
HV |
4 |
4 |
1.6(—5) |
5 |
7.1 ( -2) |
5 |
5 |
1.1С-в) |
7 |
l-4(—2) |
7 |
7 |
9.8(-ll) |
9 |
2-1 (-4) |
11 |
12 |
и io-^{19} |
13 |
5.4 (-6) |
13 |
13 |
« 10^{-24} |
17 |
4-3(—11) |
21 |
21 |
a io-^{37} |
25 |
9.6(—13) |
Table 3.4: Stability constants a_{n} and er£ for the quadrature rules Q_{n}[f(x). p^{1}} and Q'_{n} (/, 9?] for c = | and 1.
4 |
1.1 |
4 |
7 |
5 |
1.3 |
6 |
11 |
7 |
1.2 |
4 |
7 |
9 |
1.2 |
6 |
11 |
13 |
1.2 |
4 |
7 |
20 |
1.3 |
6 |
11 |
evaluation
of
3,4 by using
We now study the utility of Gauss-Daubechies quadrature rule
(Xi,0Ji are nodes and weights associated with pseudo-weight function ф[х)) for evaluation of the integral /[/, ф,—К + 1, K involving wavelets. As mentioned earlier, this integral can also be evaluated with the help of the formula
The errors in Q_{n}[f, Ф] and Q,,[f, ф, — К + 1, К] can be estimated by using the results in the previous section as
and
Here < /(^{2n}) >_{lf>l} is the average of /(^{2,}')(_{a;}) _{over S}upp so that
The sum in the right side of E_{n} [/. ip, —K + 1, K] is zero for f(x) = x^{s}, s = 2n,......,2n + К — 1
in addition to s = 0,......,2n — 1 due to the vanishing property of the moment of 'ф(х). Above
mentioned two formulas for errors apparently suggest that for a given n, the quadrature rule Q_{n} [/, '0, —К + 1 ,K] will provide more accurate value of the integral I[f, ip,—К + 1 ,K than the quadrature rule Q_{n}[/. ip]. However, from a close observation of the results presented in the columns j = 0 and j = 1 of Table 3.1 and the two quadrature rules mentioned above, it is found that for given n, the number of arithmetic operations in Q„[f. ip, —K + 1. K] is 2К times the arithmetic operations hi Qn[f ■ Ф]- Thus, for a given computational time, while Q_{n}[f,ip,—K + 1, K can accurately evaluate the integral of product of ip(x) and a polynomial of degree 2n + К — 1, Q_{n}[f, ip can evaluate the same for polynomials of degree up to AKn — 1. Therefore, for pre-assigned order of accuracy, computational cost for evaluating wavelet coefficients by using Gauss-Daubechies quadrature rule Qn[f, ip] will be much less than the same for quadrature rule Q_{n}[f, ip, ~K + 1, К].
Accurate computation of these types of integrals is of concern in numerical computat ions where wavelets are used for their multiresolution approximation. In that setting, one typically has to evaluate many such integrals. Many workers focus on quadrature rules for these integrals with equidistant nodes, since in that case function evaluation can be reused for other integrals resulting in a considerable reduction of overall computational cost. While studying compression techniques and optimal complexity estimates for boundary integral equations, Dahrnen et al. (Dahmen et al., 2006) concluded that for stable Galerkin scheme with optimal order of convergence, such integrals have to be computed with full accuracy in the coarser resolution while the same on the finer scale is allowed to have less accuracy. The necessary accuracy can be achieved within the allowed expenses if one employ an exponentially convergent quadrature method. As mentioned above, Gauss-Daubechies quadrature rule is much stable and has higher rate of convergence than the quadrature rules involving equidistant nodes and for such integrals on higher resolution, just a few quadrature points are generally sufficient. The additional calculations due to non-equidistant nodes of Gauss-Daubechies quadrature rule is expected to be more than balanced by the reduction of number of nodes and higher rate of convergence in the quadrature rule. An in-depth discussion on this aspect of quadrature rule involving complex nodes and weights including relevant numerical data are available in the study of Panja and Mandal (Panja and Mandal, 2011: Panja and Mandal, 2015).