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Error analysis of the extended Filon-type method for highly oscillatory integrals
- Jing Gao^{1} and
- Arieh Iserles^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s40687-017-0110-4
© The Author(s) 2017
- Received: 2 August 2016
- Accepted: 8 May 2017
- Published: 1 November 2017
Abstract
We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay between the two varies for different frequencies. We are particularly concerned with two strategies for the choice of inner nodes: Clenshaw–Curtis points and zeros of an appropriate Jacobi polynomial. Once the frequency \(\omega \) is large, the asymptotic error dominates, but the situation is altogether different when \(\omega \ge 0\) is small. In the first regime, our optimal error bounds indicate that Clenshaw–Curtis points are always marginally better, but this is reversed for small \(\omega \); then, Jacobi points enjoy an advantage. The main tool in our analysis is the Peano Kernel theorem. While the main part of the paper addresses integrals without stationary points, we indicate how to extend this work to the case when stationary points are present. Numerical experiments are provided to illustrate theoretical analysis.
Keywords
- Filon-type method
- Error analysis
- Peano Kernel theorem
- Clenshaw–Curtis points
- Jacobi polynomial
Mathematics Subject Classification
- 65D32
- 65G20
- 41A55
1 Background
The quadrature of highly oscillatory integrals plays a vital role in many research fields, such as numerical analysis, electromagnetic, acoustic scattering and quantum chemistry. Historically it was regarded as a formidable challenge, requiring large number of function evaluations (scaling roughly like the frequency), but the subject has undergone substantial revolution in the last fifteen years. Using asymptotic expansions as a major analytic tool, significant number of effective numerical methods for highly oscillatory integrals have been developed: the asymptotic expansion and Filon-type methods [9–11], numerical steepest descent [8], the Levin [12] method [15], complex-valued Gaussian quadrature [1, 3] and their diverse combinations.
In this paper we wish to explore in detail what happens once internal points are allowed in a Filon method.
Definition
The absolute error of the three methods for \(s=3\) and different values of \(\omega \)
Method | \(\varvec{\omega }\) | |||||
---|---|---|---|---|---|---|
0 | 100 | 200 | 300 | 400 | 500 | |
Fig. 1 | \(9.21_{-02}\) | \(1.42_{-07}\) | \(9.02_{-09}\) | \(1.80_{-09}\) | \(5.67_{-10}\) | \(2.29_{-10}\) |
Fig. 2 (left) | \(8.24_{-06}\) | \(8.16_{-09}\) | \(3.25_{-10}\) | \(1.90_{-11}\) | \(1.61_{-11}\) | \(1.16_{-11}\) |
Fig. 2 (right) | \(2.44_{-04}\) | \(5.91_{-09}\) | \(2.33_{-10}\) | \(6.13_{-12}\) | \(1.08_{-11}\) | \(8.23_{-12}\) |
There is nothing new in the incorporation of internal points into a Filon method. They have been introduced in [11], a paper that introduced Filon methods in their modern guise and analysed their asymptotic behaviour. It has been suggested there that a good choice of internal points is the zeros of the Jacobi polynomial \(\mathrm {P}_\nu ^{(s,s)}\), since this maximises the (conventional) order of the Birkhoff–Hermite quadrature for \(\omega =0\). We dub this method EFJ (for “Extended Filon–Jacobi”): this is the method on the left of Fig. 2.
Another choice of internal points has been proposed in [5] for the case \(s=1\) and will be extended by us to all \(s\ge 1\), namely the Chebyshev points of the second kind \(\cos k\pi /(\nu +1)\), \(k=1,\ldots ,\nu \). Such points feature in Clenshaw–Curtis quadrature, which enjoys many favourable features once compared with traditional Gaussian quadrature [21]. We call this method, corresponding to the right-hand side of Fig. 2, EFCC, standing for “Extended Filon–Clenshaw–Curtis”.
Insofar as interior points, inclusive of Jacobi points and Chebyshev points of the second kind, are not new and have been justified partly [5, 13], in particular for the case \(\nu \gg 1\). The purpose of this paper is a rigorous and complete analysis of EFM and the establishment of realistic and tight upper bounds on its numerical error.
Theorem 1
This bound on the size of \( |\mathcal {L}[f]|\) neatly separates the role of \(\mathcal {L}\) and of the function f, and it is sharp because there always exists \(f\in \mathrm {C}^{m+1}(a,b)\) so that (1.6) is satisfied as an equality.
2 Large \(\omega \)
2.1 General error bounds for EFM
Figure 4 illustrates upper error bound (2.3) for \(f(x)=\mathrm {e}^x\) and \(g(x)=x\) for the same three methods as in Figs. 1 and 2. Note that the bounds are not “strict”, but this is only to be expected, because our function f does not maximise them. All PKT allows us to say is that for every s, \(\nu \) and specific choice of the \(c_k\)s there exists a function f for which the bound is strict.
Note further that the bounds, while evidently correct, are useless for small \(\omega \ge 0\). In that instance we need a different approach, which is described in Sect. 3.
2.2 Jacobi versus Clenshaw–Curtis
For fixed s and \(\nu \), bound (2.3) depends on the choice of the inner points \(c_k\), \(k=1,\ldots ,\nu \). In particular, we have singled out in Sect. 1 two choices: EFJ, whereby the \(c_k\)s are the zeros of the Jacobi polynomial \(\mathrm {P}_\nu ^{(s,s)}\), a choice that maximised the conventional order of the (nonoscillatory) quadrature for \(\omega =0\) [11], and EFCC, with \(c_k=\cos (k\pi /(\nu +1))\) (incidentally, these are also zeros of a Jacobi polynomial, \(\mathrm {P}_\nu ^{(\frac{1}{2},\frac{1}{2})}\)). The advantage of EFCC is that they are uniformly bounded [5].^{1}
Proposition 2
The function \(\rho _s^{[\nu ]}\) increases strictly monotonically for \(s>-1/2\) and \(\nu \ge 2\), while \(\rho _s^0,\rho _s^1\equiv 1\).
Proof
Insofar as the value of \(r(-1)\), the other ingredient we need for upper bound (2.3), is concerned, \(\mathrm {P}_\nu ^{(\alpha ,\beta )}(-1)=(-1)^\nu \mathrm {P}_\nu ^{(\beta ,\alpha )}(1)=(-1)^{\nu }(1+\beta )_\nu /\nu !\) [17, p. 257] implies that \(|r(-1)|=|r(1)|\).
Theorem 3
Given s and \(\nu \), upper bound (2.3) is always smaller for EFCC than for EFJ.
Proof
An immediate consequence of Proposition 1. \(\square \)
3 Small \(\omega \ge 0\)
The complication for \(\omega =0\) is that EFJ and EFCC are of different polynomial order: while EFJ is exact for all \(f\in \mathbb {P}_{2s+2\nu -1}\) (where \(\mathbb {P}_n\) is the set of nth-degree algebraic polynomials), EFCC is, on the face of it, exact just in \(\mathbb {P}_{2s+\nu -1}\). The latter, however, is not entirely true. Because of the symmetry of both the integral and of the quadrature formula, all odd polynomials are computed exactly: the integral is zero and so is quadrature. Therefore, while for even \(\nu \) the polynomial order of EFCC is \(2s+\nu -1\), once \(\nu \) is odd it increases to \(2s+\nu \). However, unless \(\nu =1\) (when a single Jacobi and Clenshaw–Curtis point coincides at the origin), EFJ has always higher order than EFCC.
Does it matter? According to [21], Clenshaw–Curtis is just as good as Gaussian quadrature and we can expect something similar to remain true in our setting. Except that [21] is concerned with convergence for large \(\nu \), while we are interested in relatively small (and fixed) values of \(\nu \). It is well known [2] that an order-p (i.e. exact for all \(f\in \mathbb {P}_{p-1}\)) quadrature method applied to the function f bears an error of \(cf^{(p)}(\xi )\), where \(c\ne 0\) is a constant, depending solely on the method (the error constant) and \(\xi \) is an intermediate point. Therefore, the error is bounded by \(c\Vert f^{(p)}\Vert _\infty \). The problem is that it does not allow for a comparison of EFJ and EFCC method with the same values of s and \(\nu \) which lead to the different derivative order, \(f^{(2s+2\nu )}\) for EFJ and \(f^{(2s+\nu )}\) for EFCC. Different derivatives, incompatible top bounds!
- 1.
We estimate the Peano kernel constant not for the order of each method but for the order of Clenshaw–Curtis. The outcome is an estimate which is compatible for both methods.
- 2.Instead of a classical PKT estimate, we use the work of Favati et al. [7]. Since the zeros of the polynomial on \([-1, 1]\) are symmetric, denote the nodes byIf \(\nu \) is odd, \(c_1 = 0\) and when it is even, \(c_1 > 0\). Thus, we have a symmetric Birkhoff–Hermite quadrature,$$\begin{aligned} -1< -c_{\left\lfloor \frac{\nu + 1}{2}\right\rfloor }< \cdots< -c_1< c_1< \cdots< c_{\left\lfloor \frac{\nu + 1}{2}\right\rfloor } < 1 = c_{\left\lfloor \frac{\nu + 1}{2}\right\rfloor + 1}. \end{aligned}$$where \(s_{\left\lfloor \frac{\nu +1}{2}\right\rfloor + 1} = s\), \(s_k = 1\) for \(k = 1, 2, \cdots , \left\lfloor \frac{\nu +1}{2}\right\rfloor \). If \(\nu \) is odd, \(b_{1,j}\) need be halved. Note that \(\mathcal {I}_0[f_{\mathrm {O}}]=\mathcal {Q}_0[f_{\mathrm {O}}]=0\) for every sufficiently smooth odd function \(f_{\mathrm {O}}\). This motivates a focus on the even part \(f_{\mathrm {E}}(x) = (f(x)+f(-x))/2\) of the functions f(x), whereby we can restrict ourselves to [0, 1],$$\begin{aligned} \mathcal {I}_0[f]=\int _{-1}^1 f(x)\hbox {d} x \approx \mathcal {Q}_0[f]=\sum _{k=1}^{\left\lfloor \frac{\nu +1}{2}\right\rfloor + 1} \sum _{j=0}^{s_k-1} b_{k,j} [f^{(j)}(c_k)+(-1)^j f^{(j)}(-c_k)], \end{aligned}$$where in our case \(b_{k, j} = 0\) for \(j \ge 1\) and \(k = 1, 2, \cdots , \left\lfloor \frac{\nu + 1}{2}\right\rfloor \). It is trivial that \(\mathcal {I}_1[f]=2\mathcal {I}_2[f]\).$$\begin{aligned} \mathcal {I}_1[f]=\int _0^1 f_{\mathrm {E}}(x)\hbox {d} x\approx \mathcal {Q}_1[f]=\sum _{k=1}^{\left\lfloor \frac{\nu +1}{2}\right\rfloor + 1} \sum _{j=0}^{s_k-1} b_{k,j} f_{\mathrm {E}}^{(j)}(c_k), \end{aligned}$$Suppose that the order of the method is \(p\ge 1\). It has been proved in [7] that, for \(\mathcal {I}_1\),for any \(d\in \{1,2,\ldots ,p-1\}\). Integrating \(2|K_d|\) in [0, 1] results in the \(\mathrm {L}_\infty \) PKT bound,$$\begin{aligned} K_d(\theta ) =\frac{(1-\theta )_+^{d+1}}{(d+1)!} - \frac{1}{d!} \sum _{k=1}^{\left\lfloor \frac{\nu + 1}{2}\right\rfloor } b_{k,0} (c_k-\theta )_+^{d} \end{aligned}$$(3.1)In our case we choose \(d=2s+\nu \) for odd \(\nu \) or \(d=2s+\nu -1\) for even \(\nu \), the order of the EFCC method.$$\begin{aligned} \left| \mathcal {I}_0[f]-\mathcal {Q}_0[f]\right| \le 2 \int _0^1 |K_d(\theta )|\hbox {d} \theta \Vert f_{\mathrm {E}}^{(d)}\Vert _\infty . \end{aligned}$$
Peano kernel constants for EFJ and EFCC methods with \(\omega = 0\)
\({\varvec{s}}\) | \(\varvec{\nu }\) | \({\varvec{d}}\) | EFJ | EFCC |
---|---|---|---|---|
1 | 2 | 3 | \(1.13_{-03}\) | \(2.92_{-03}\) |
3 | 5 | \(6.22_{-06}\) | \(2.67_{-05}\) | |
4 | 5 | \(1.25_{-06}\) | \(3.37_{-06}\) | |
5 | 7 | \(3.27_{-09}\) | \(1.97_{-08}\) | |
6 | 7 | \(7.95_{-10}\) | \(4.92_{-09}\) | |
2 | 2 | 5 | \(1.08_{-05}\) | \(1.59_{-04}\) |
3 | 7 | \(3.28_{-08}\) | \(6.30_{-07}\) | |
4 | 7 | \(4.55_{-09}\) | \(8.20_{-08}\) | |
5 | 9 | \(8.01_{-12}\) | \(2.42_{-10}\) | |
6 | 9 | \(1.53_{-12}\) | \(2.02_{-11}\) | |
3 | 2 | 7 | \(7.53_{-08}\) | \(3.15_{-06}\) |
3 | 9 | \(1.37_{-10}\) | \(6.36_{-09}\) | |
4 | 9 | \(1.35_{-11}\) | \(2.39_{-09}\) | |
5 | 11 | \(1.67_{-14}\) | \(4.17_{-12}\) | |
6 | 11 | \(2.52_{-15}\) | \(3.60_{-13}\) | |
4 | 2 | 9 | \(3.96_{-10}\) | \(3.56_{-08}\) |
3 | 11 | \(4.58_{-13}\) | \(4.15_{-11}\) | |
4 | 11 | \(3.36_{-14}\) | \(2.60_{-11}\) | |
5 | 13 | \(2.98_{-17}\) | \(2.85_{-14}\) | |
6 | 13 | \(3.61_{-18}\) | \(7.13_{-15}\) | |
5 | 2 | 11 | \(1.61_{-12}\) | \(2.67_{-10}\) |
3 | 13 | \(1.26_{-15}\) | \(1.96_{-13}\) | |
4 | 13 | \(7.02_{-17}\) | \(1.71_{-13}\) | |
5 | 15 | \(4.63_{-20}\) | \(1.26_{-16}\) | |
6 | 15 | \(4.55_{-21}\) | \(5.24_{-17}\) |
- 1.
EFJ always beats EFCC for \(\omega =0\)—note that, we are comparing methods of the same order, alike with alike.
- 2.
The advantage of EFJ over EFCC increases as s grows and \(\nu \) is fixed. For \(s=1\), it is quite minor but for \(s=5\) the difference is substantial.
- 3.
Likewise, the advantage of EFJ grows for fixed s and increasing \(\nu \)—note that for \(\nu =1\) the two methods coincide.
- 4.
The constants decrease strictly monotonically as a function of s or of \(\nu \). Note, of course, that for different rows the constants have different meaning, because they precede different derivatives of \(f_{\mathrm {E}}\), but this is interesting nonetheless.
Figure 6 displays graphically the information embedded in Table 2. The message is the same: at \(\omega =0\) EFJ is always better than EFCC, the constants decrease as \(\nu \) increases or as s increases.
In Fig. 7 we display three such sequences: on the top left for \(d=7\) and all such methods, on the top right for \(d=15\) and only the methods with an even (hence larger) \(\nu \), and the bottom is for \(d= 23\) with even \(\nu \). Note that monotonicity, at least for EFCC, is no longer true, but the sequence is predictable: for sufficiently large d it first goes down, then up, while the last value (which coincides with Jacobi) may take it down again. The EFJ sequence is much nicer, and its logarithm seems to increase as a smooth function.
What is interesting, though, is once the objective is to minimise the PKT constant (with either method) for given order of the EFCC method, a good policy is to use \(s=1\) and maximal \(\nu \). This is an exact opposite of the right policy for \(\omega \gg 1\), namely small \(\nu \) and large s. Yet another example how the \(\omega =0\) and \(\omega \gg 1\) regimes are polar opposites.
Peano kernel constants for EFJ and \(d=2\nu +2s-1\)
\({\varvec{s}}\) | \(\varvec{\nu }\) | \({\varvec{d}}\) | The constant |
---|---|---|---|
1 | 2 | 5 | \(8.47_{-05}\) |
3 | 7 | \(3.60_{-07}\) | |
4 | 9 | \(9.70_{-10}\) | |
5 | 11 | \(1.80_{-12}\) | |
6 | 13 | \(2.43_{-15}\) | |
2 | 2 | 7 | \(7.20_{-07}\) |
3 | 9 | \(1.70_{-09}\) | |
4 | 11 | \(2.88_{-12}\) | |
5 | 13 | \(3.65_{-15}\) | |
6 | 15 | \(3.58_{-18}\) | |
3 | 2 | 9 | \(4.52_{-09}\) |
3 | 11 | \(6.47_{-12}\) | |
4 | 13 | \(7.29_{-15}\) | |
5 | 15 | \(6.56_{-18}\) | |
6 | 17 | \(4.78_{-21}\) | |
4 | 2 | 11 | \(2.16_{-11}\) |
3 | 13 | \(2.01_{-14}\) | |
4 | 15 | \(1.57_{-17}\) | |
5 | 17 | \(1.03_{-20}\) | |
6 | 19 | \(5.73_{-24}\) | |
5 | 2 | 13 | \(8.02_{-14}\) |
3 | 15 | \(5.11_{-17}\) | |
4 | 17 | \(2.90_{-20}\) | |
5 | 19 | \(1.43_{-23}\) | |
6 | 21 | \(6.16_{-27}\) |
4 Stationary points
4.1 Filon with stationary points
Let us assume the existence of a single order-2 stationary point at \(x=-1\), i.e. that \(g^{\prime }(-1)=0\), \(g^{\prime \prime }(-1)\ne 0\) and \(g^{\prime }\ne 0\) in \((-1,1]\). Higher-order stationary point there can be dealt with in a similar manner, requiring more technical effort but no added insight, while an integral with several stationary points or with a stationary point in \((-1,1]\) can be converted through linear change of variables to possibly several integrals (1.1) of the kind addressed in this section.
4.2 The case \(\omega \gg 1\)
In Fig. 9 we have sketched \(\log _{10}|Q_\omega ^{\mathsf {F},2,4}[f]-I_\omega [f]|\) for \(f(x)=\sin (x^2)\) and \(g(x)=(x+1)^2\). The right-hand plot demonstrates asymptotic behaviour: as expected from our analysis, EFCC wins over EFJ for large \(\omega \).
4.3 Small \(\omega \ge 0\)
It is evident from the left-hand plot in Fig. 9 that for small \(\omega \ge 0\) EFJ is substantially better than EFCC. This is completely in line with the behaviour in the absence of stationary points, which we have analysed in Sect. 3 and consistent with the fact that, for \(\omega =0\), EFJ and EFCC are of conventional orders \(3s+2\nu +1\) and \(3s+\nu +1\), respectively.
Peano kernel constants for EFJ and EFCC in the case of a stationary point
\({\varvec{s}}\) | \(\varvec{\nu }\) | \({\varvec{d}}\) | EFJ | EFCC |
---|---|---|---|---|
1 | 2 | 5 | \(1.31_{-05}\) | \(6.67_{-05}\) |
3 | 6 | \(2.61_{-07}\) | \(7.59_{-06}\) | |
4 | 7 | \(5.12_{-09}\) | \(1.97_{-07}\) | |
5 | 8 | \(9.53_{-11}\) | \(4.37_{-09}\) | |
6 | 9 | \(1.66_{-12}\) | \(1.29_{-10}\) | |
2 | 2 | 8 | \(8.23_{-09}\) | \(2.12_{-07}\) |
3 | 9 | \(8.07_{-11}\) | \(2.16_{-08}\) | |
4 | 10 | \(8.86_{-13}\) | \(3.33_{-10}\) | |
5 | 11 | \(1.01_{-14}\) | \(3.64_{-12}\) | |
6 | 12 | \(1.16_{-16}\) | \(1.27_{-13}\) | |
3 | 2 | 11 | \(2.82_{-12}\) | \(2.28_{-10}\) |
3 | 12 | \(1.57_{-14}\) | \(1.69_{-11}\) | |
4 | 13 | \(1.06_{-16}\) | \(1.75_{-13}\) | |
5 | 14 | \(7.87_{-19}\) | \(4.62_{-15}\) | |
6 | 15 | \(6.19_{-21}\) | \(1.22_{-16}\) | |
4 | 2 | 14 | \(5.75_{-16}\) | \(1.04_{-13}\) |
3 | 15 | \(2.01_{-18}\) | \(6.14_{-15}\) | |
4 | 16 | \(8.90_{-21}\) | \(4.70_{-17}\) | |
5 | 17 | \(4.56_{-23}\) | \(1.69_{-18}\) | |
6 | 18 | \(2.56_{-25}\) | \(3.59_{-20}\) | |
5 | 2 | 17 | \(7.47_{-20}\) | \(2.54_{-17}\) |
3 | 18 | \(1.76_{-22}\) | \(1.25_{-18}\) | |
4 | 19 | \(5.44_{-25}\) | \(7.48_{-21}\) | |
5 | 20 | \(2.00_{-27}\) | \(3.08_{-22}\) | |
6 | 21 | \(8.28_{-30}\) | \(5.59_{-24}\) |
In Table 4 we display the Peano constants \(\Vert K\Vert _1\) for EFJ and EFCC and a range of values of s and \(\nu \) the same as in Tables 2 and 3. It is clear that the constants associated with EFJ are substantially smaller, implying significantly smaller error at \(\omega =0\)—this is completely consistent with Fig. 9.
The advantage of EFJ vis á vis EFCC grows fast with s, and this is demonstrated not just in Table 4 but also in Fig. 10, where we display the relevant constants to logarithmic scale.
Peano kernel constants for EFJ and \(d=3s+2\nu \) in the case of a stationary point
\({\varvec{s}}\) | \(\varvec{\nu }\) | \({\varvec{d}}\) | The constant |
---|---|---|---|
1 | 2 | 7 | \(9.00_{-07}\) |
3 | 9 | \(2.04_{-09}\) | |
4 | 11 | \(3.36_{-12}\) | |
5 | 13 | \(4.17_{-15}\) | |
6 | 15 | \(4.02_{-18}\) | |
2 | 2 | 10 | \(4.71_{-10}\) |
3 | 12 | \(5.11_{-13}\) | |
4 | 14 | \(4.58_{-16}\) | |
5 | 16 | \(3.40_{-19}\) | |
6 | 18 | \(2.10_{-22}\) | |
3 | 2 | 13 | \(1.38_{-13}\) |
3 | 15 | \(8.22_{-17}\) | |
4 | 17 | \(4.43_{-20}\) | |
5 | 19 | \(2.10_{-23}\) | |
6 | 21 | \(8.74_{-27}\) | |
4 | 2 | 16 | \(2.43_{-17}\) |
3 | 18 | \(8.83_{-21}\) | |
4 | 20 | \(3.06_{-24}\) | |
5 | 22 | \(9.83_{-28}\) | |
6 | 24 | \(2.87_{-31}\) | |
5 | 2 | 19 | \(2.79_{-21}\) |
3 | 21 | \(6.59_{-25}\) | |
4 | 23 | \(1.56_{-28}\) | |
5 | 25 | \(3.53_{-32}\) | |
6 | 27 | \(7.51_{-36}\) |
5 Conclusions
In this paper, we have derived such bounds using the methodology of the Peano Kernel theorem. This has led to tight bounds in two cases: for \(\omega \gg 1\) (which, after all, is the main objective of Filon-type methods!) and \(\omega =0\). In particular, we have compared two choices of internal points: Jacobi points, which maximise classical order for \(\omega =0\), and Clenshaw–Curtis points, which are cheaper when the number of internal points is large. Our conclusion is that for \(\omega \gg 1\) Clenshaw–Curtis is marginally more precise, but for \(\omega =0\) the honours go to Jacobi points.
All this leaves an important lacuna: What is the choice of internal points likely to deliver the least uniform (for all \(\omega \ge 0\)) error? In all our calculations [cf. Figs. 5 and 9 (left)] the pattern (modulo oscillations) is the same: small error for \(\omega =0\), subsequent increase (‘intermediate asymptotics’) and finally, once asymptotics take over, consistent decrease. The error, thus, is likely to be maximised in the regime of intermediate asymptotics. In all our calculations, Jacobi points are a clear winner there, yet there is neither a proof of this statement nor, indeed, a rigorous uniform bound on the error.
Acknowledgements
The work is supported by the Projects of International Cooperation and Exchanges NSFC-RS (Grant No. 11511130052) and the Key Science and Technology Program of Shaanxi Province of China (Grant No. 2016GY-080).
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