τr−2 | τr−1 | τr | τr+1 | τr+2 | else | |
Kr(τ) | 1120 | 26120 | 66120 | 26120 | 1120 | 0 |
K'r(τ) | 124Λ | 1024Λ | 0 | −1024Λ | −124Λ | 0 |
K''r(τ) | 16Λ2 | 26Λ2 | −66Λ2 | 26Λ2 | 16Λ2 | 0 |
K'''r(τ) | 12Λ3 | −22Λ3 | 0 | 22Λ3 | −12Λ3 | 0 |
Citation: Ghazanfar Latif, Jaafar Alghazo, Majid Ali Khan, Ghassen Ben Brahim, Khaled Fawagreh, Nazeeruddin Mohammad. Deep convolutional neural network (CNN) model optimization techniques—Review for medical imaging[J]. AIMS Mathematics, 2024, 9(8): 20539-20571. doi: 10.3934/math.2024998
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The Lane-Emden equation represents a dimensionless form of Poisson's equation that arises in astrophysics for the spherically symmetric, polytrophic fluids, and the gravitational potential of Newtonian self-gravitating [1,2,3]. Modeling diverse phenomena in astrophysics, physical, and mathematical physics, such as the stellar structure theory, isothermal gas spheres, the thermal dynamic of a spherical gas cloud, the thermionic current theory, chemical reactions, population evolution, and pattern formation, results in scalar and systems of Lane-Emden equations, see [4,5] and the references therein. While there has been little research on Lane-Emden equation systems, recent attention to studying this type of systems has increased considerably [5].
In this study, we consider the following Lane-Emden system of the form
d2ωi(τ)dτ2+δiτdωi(τ)dτ+ℏi(τ,ω1(τ),ω2(τ))=ℵi(τ),i=1,2, | (1) |
subject to
ωi(0)=εi,ω'i(0)=0, | (2) |
where δ1,δ2,ε1,andε2 are real constants, and ℏi and ℵi(τ),i=1,2 are given continuous functions.
Numerous approaches have been established for solving scalar and systems of Lane-Emden equations, including the Haar wavelet collocation method [6], Laplace transform and residual error function [7], Bernoulli wavelets functional matrix technique [5], B-spline methods [8,9,10], Adomian decomposition method [9], Chebyshev operational matrix method [4], variational iteration method [11,12], discontinuous finite element method [13], Bernstein collocation method [14], Bessel-collocation procedure [15], and Legendre Polynomials [16,17,18].
The literature survey reveals that collocation methods are an important tool in obtaining approximate solutions for different types of differential equations, including different classes of initial and boundary value problems, Singular differential equations, partial and fractional partial differential equations, system of partial differential equations, fractional Volterra integro-differential equations, and Abel's integral equations, [19,20,21,22,23,24,25,26,27,28,29,30,31,32], among others. One well-known established method among collocation methods is the so-called B-spline method, where the letter "B" represents "basis". This method was originally introduced by Schoenberg in 1946. The primary motivation for introducing the B-splines is the creation of a stable interpolating function across finite number of points, which maintain the smoothness and the shape of the data [33,34]. Recently, B-spline methods have been demonstrated to be useful in approximation theory, image processing, and numerical computation due to their valuable properties such as numerical computation stability, local effects of coefficient changes, and built-in smoothness between adjacent polynomial pieces.
The spline methods, as is known, provide inaccurate solutions with the presence of singularity. To defeat the drawback of these methods, we, in this work, develop an effective method based on quintic B-spline functions, known as the quintic B-spline method (QBSM), to approximate the solution of (1). To construct the QBSM, the approximate solution is forced to fulfill the considered system at the grid points, converting it into a set of algebraic equations with unknown coefficients. Solving the set of algebraic equations determines the values of these coefficients. Note that the considered problem has a singularity at τ=0. When addressing the singularity of (1) numerically, it is important to efficiently deal with the singularity via certain means. In our case, we employ the L'Hôpital rule to its second term. To the best of our knowledge, the results presented in this work are new and have not been previously presented in the literature. The method is illustrated with several test problems. It is demonstrated that the accuracy of the method is of fourth-order convergence, superior to the convergence of the cubic B-spline method, which is proven to be of second-order convergence, derived in our prior work [8]. Outcomes are compared with some other numerical solutions to demonstrate the advantage of the method.
The structure of this paper is as follows: Section 2 provides the preliminaries of the quintic B-spine functions and their properties. Section 3 is dedicated to the construction of QBSM for obtaining the solution of the considered system. Section 4 discusses the convergence of the method. Section 5 provides the numerical illustration, and, finally, Section 6 summarizes and concludes our work.
In this section, we define quintic B-spline functions and their main properties to be utilized in constructing QBSM. We construct this method upon a uniform mesh. To do this, we partition the solution domain Γ=[α,β] into k subintervals Γi=[τi,τi+1] by the grid points τi=α+iΛ (i=0,1,...,k), where Λ=(β−α)/k. Let Ω be the set of these grid points of the solution domain Γ, referred to as the partition of Γ, and is defined as Ω={τ0,τ1,…,τk}. To provide proper support for the quintic B-spline functions, it is essential to introduce an additional five grid points on each side of the solution domain Γ. Consequently, the solution domain Γ is extended to −Γ=[α−5Λ,β+5Λ] with τi=α+iΛ (i=−5,...,k+5). The linear space of quintic splines over this defined partition is expressed as
M5(Γ)={μ(τ)∈C4(Γ):μ(τ)|Γi∈P5,i=0,...,k−1}, |
where μ(τ)|Γi indicates the restriction of μ(τ) over Γi and P5 designates the set of one-variable quintic polynomials. The dimension of the linear space M5(Γ) is (k+5). According to [30], the quintic B-spline Kr(τ)(r=−2,...,k+2) is defined as
Kr(τ)=1120Λ5{(τ−τr−3)5,(τ−τr−3)5−6(τ−τr−2)5,(τ−τr−3)5−6(τ−τr−2)5+15(τ−τr−1)5,(−τ+τr+3)5−6(−τ+τr−2)5+15(−τ+τr+1)5,(−τ+τr+3)5−6(τ+τr+2)5,(τ−τr−3)5,0,τ∈[τr−3,τr−2]τ∈[τr−2,τr−1]τ∈[τr−1,τr]τ∈[τr,τr+1]τ∈[τr+1,τr+2]τ∈[τr+2,τr+3]else. |
The basis functions Kr, r=−2,...,k+2, are nonnegative and linearly independent on the domain [α,β]. The values of Kr(τ) and their derivatives up to the third order at the grid points are recorded in Table 1.
τr−2 | τr−1 | τr | τr+1 | τr+2 | else | |
Kr(τ) | 1120 | 26120 | 66120 | 26120 | 1120 | 0 |
K'r(τ) | 124Λ | 1024Λ | 0 | −1024Λ | −124Λ | 0 |
K''r(τ) | 16Λ2 | 26Λ2 | −66Λ2 | 26Λ2 | 16Λ2 | 0 |
K'''r(τ) | 12Λ3 | −22Λ3 | 0 | 22Λ3 | −12Λ3 | 0 |
For an appropriately smooth function ω(τ), one can uniquely define a quintic spline
μ(τ)=k+2∑r=−2λrKr(τ)∈M5(I) |
that fulfill the interpolation conditions μ(τi)=ω(τi),i=0,...,k, and μ'(α)=ω'(α). From Table 1, for the discretization knots τj(j=0,...,k), we get
μ(τj)=k+2∑r=−2λrKr(τj)=λj−2+26λj−1+66λj+26λj+1+λj+2120, | (3) |
μ'(τj)=k+2∑r=−2λrK'r(τj)=−λj−2−10λj−1+10λj+1+λj+224Λ, | (4) |
μ''(τj)=k+2∑r=−2λrK''r(τj)=λj−2+2λj−1−6λj+2λj+1+λj+26Λ2, | (5) |
μ'''(τj)=k+2∑r=−2λrK'''r(τj)=−λj−2+2λj−1−2λj+1+λj+22Λ3. | (6) |
Equations (3)–(6) serve as the fundamental relations in the construction of the QBSM.
This part of the study discusses the method and the convergence analysis.
In this section, we present the development of a collocation method based on quintic B-spline functions for (1) and (2). Let μi(τ)=∑k+2r=−2λi,rKr(τ),i=1,2, represents the quintic B-spline approximate solution of the exact solution ωi(τ) to (1). To overcome the singularity behavior of (1), we employ the L'Hôpital rule on the second term at τ=0, to obtain
(1+δi)d2ωi(τ)dτ2+ℏi(τ,ω1(τ),ω2(τ))=ℵi(τ),forτ=0, |
d2ωi(τ)dτ2+δiτdωi(τ)dτ+ℏi(τ,ω1(τ),ω2(τ))=ℵi(τ),forτ≠0,i=1,2. | (7) |
Discretizing (7), we get
(1+δi)d2ωi(τ0)dτ2+ℏi(τ0,ω1(τ0),ω2(τ0))=ℵi(τ0), |
d2ωi(τj)dτ2+δiτjdωi(τj)dτ+ℏi(τj,ω1(τj),ω2(τj))=ℵi(τj), | (8) |
where j=1,⋯,k. Using (3)–(5), we have
(1+δi)(λi,−2+2λi,−1−6λi,0+2λi,1+λi,26Λ2)+ℏi(τ0,ε1,ϑ1)=ℵi(τ0),(λi,j−2+2λi,j−1−6λi,j+2λi,j+1+λi,j+26Λ2)+δiτj(−λi,j−2−10λi,j−1+10λi,j+1+λi,j+224Λ)+ϱi(λ1,j−2,λ1,j−1,λ1,j,λ1,j+1,λ1,j+2,λ2,j−2,λ2,j−1,λ2,j,λ2,j+1,λ2,j+2)=ℵi(τj), | (9) |
where i=1,2 and j=1,⋯,k. Additionally, from (2), we derive the following four equations
ωi(0)=εi=λi,−2+26λi,−1+66λi,0+26λi,1+λi,2120, | (10) |
ω'i(0)=0=−λi,−2−10λi,−1+10λi,1+λi,224Λ, | (11) |
where i=1,2. Four equations are still needed. Therefore, by differentiating (1), we obtain:
d3ωi(τ)dτ2+δiτω''i(τ)−ω'i(τ)τ2+ddτℏi(τ,ω1(τ),ω2(τ))+dℏi(τ,ω1(τ),ω2(τ))dω1(τ)ω'1(τ)+dℏi(τ,ω1(τ),ω2(τ))dω2(τ)ω'2(τ)=ℵ'i(τ). | (12) |
Applying the L'Hôpital rule and using (2)–(4), (6) and (12) becomes
(1+δi2)(−λi,−2+2λi,−1−2λi,1+λi,22Λ3)+ddτℏi(τ0,ε1,ε2)=ℵ'i(0). | (13) |
Similarly, at τ=1, we obtain
−λi,k−2+2λi,k−1−2λi,k+1+λi,k+22Λ3+ϖi(λ1,k−2,λ1,k−1,λ1,k,λ1,k+1,λ1,k+2,λ2,k−2,λ2,k−1,λ2,k,λ2,k+1,λ2,k+2)=ℵ'i(1), | (14) |
where i=1,2.
Expressing (9)–(11), (13), and (14) in matrix form as
AΦ=Ψ, | (15) |
where A represents a coefficient matrix of dimension 2(k+5)×2(k+5), Φ is a column vector defined as
Φ=[λ1,−2,⋯,λ1,k+2,λ2,−2,⋯,λ2,k+2]T, |
and Ψ is a column vector with 2(k+5) entries. Solving this system yields the coefficients of the approximate solution μi(τ) for (1).
In this section, we demonstrate the convergence analysis of QBSM. To facilitate this analysis, we assume that ωi(τ)∈C5[0,1],i=1,2. From (3)–(6), we have [35,36]
μ'i(τj−2)+26μ'i(τj−1)+66μ'i(τj)+26μ'i(τj+1)+μ'i(τj+2)=−5ωi(τj−2)−50ωi(τj−1)+50ωi(τj+1)+5ωi(τj+2)Λ,μ''i(τj−2)+26μ''i(τj−1)+66μ''i(τj)+26μ''i(τj+1)+μ''i(τj+2)=20ωi(τj−2)+40ωi(τj−1)−120ωi(τj)+40ωi(τj+1)+20ωi(τj+2)Λ2,μ'''i(τj−2)+26μ'''i(τj−1)+66μ'''i(τj)+26μ'''i(τj+1)+μ'''i(τj+2)=−60ωi(τj−2)+120ωi(τj−1)−120ωi(τj+1)+60ωi(τj+2)Λ3. | (16) |
With the operator notations, Ξωi(τj)=ωi(τj+1),Dωi(τj)=ω'i(τj), and Iωi(τj)=ωi(τj), Eq (16) can be expressed as
μ'i(τj)=1Λ(−5Ξ−2−50Ξ−1+50Ξ+5Ξ2Ξ−2+26Ξ−1+66I+26Ξ+Ξ2)ωi(τj), |
μ''i(τj)=1Λ2(20Ξ−2+40Ξ−1−120I+40Ξ+20Ξ2Ξ−2+26Ξ−1+66I+26Ξ+Ξ2)ωi(τj), |
μ'''i(τj)=1Λ3(−60Ξ−2+120Ξ−1−120Ξ+60Ξ2Ξ−2+26Ξ−1+66I+26Ξ+Ξ2)ωi(τj), | (17) |
i=1,2. Setting Ξ=eΛD in (17) gives
μ'i(τj)=1Λ(−5e−2ΛD−50e−ΛD+50eΛD+5e2ΛDe−2ΛD+26e−ΛD+66+26eΛD+e2ΛD)ωi(τj), |
μ''i(τj)=1Λ2(20e−2ΛD+40e−ΛD−120+40eΛD+20e2ΛDe−2ΛD+26e−ΛD+66+26eΛD+e2ΛD)ωi(τj), |
μ'''i(τj)=1Λ3(−60e−2ΛD+120e−ΛD−120eΛD+60e2ΛDe−2ΛD+26e−ΛD+66+26eΛD+e2ΛD)ωi(τj), | (18) |
Expanding the exponential functions in (18) in powers of ΛD, we obtain
μ'i(τj)=ω'i(τj)+15040Λ6ω(7)i(τj)+O(Λ8), |
μ''i(τj)=ω''i(τj)+1720Λ4ω(6)i(τj)+O(Λ6), |
μ'''i(τj)=ω'''i(τj)−1240Λ4ω(7)i(τj)+O(Λ6), | (19) |
i=1,2. Next, let's define truncation error as follows
ei(τj)=ℵi(τj)−d2ωi(τj)dτ2−δiτjdω1(τj)dτ−ℏi(ω1(τj),ω2(τj))=[d2μi(τj)dτ2+δiτjdμ1(τj)dτ+ℏi(μ1(τj),μ2(τj))]−d2ωi(τj)dτ2−δiτjdω1(τj)dτ−ℏi(ω1(τj),ω2(τj)). | (20) |
As μi(τj)=ωi(τj),i=1,2 and j=1,...,k, Eq (20) can be simplified as
ei(τj)=[d2μi(τj)dτ2−d2ωi(τj)dτ2]+δiτj[dμi(τj)dτ−dωi(τj)dτ], | (21) |
Hence, by using (19) in (21), we can conclude that
‖ei(τj)‖∞=O(Λ4), | (22) |
and for j = 0, we have
ei(τ0)=ℵi(τ0)−(1+δi)d2ωi(τ0)dτ2−ℏi(ω1(τ0),ω2(τ0))=(1+δi)d2μi(τ0)dτ2+ℏi(μ1(τ0),μ2(τ0))−(1+δi)d2ωi(τ0)dτ2−ℏi(ω1(τ0),ω2(τ0). | (23) |
As μi(τ0)=ωi(τ0),i=1,2, Eq (23) can be simplified as
ei(τj)=(1+δi)[d2μi(τ0)dτ2−d2ωi(τ0)dτ2]. | (24) |
Hence, by using (19) in (24), we find
‖ei(τ0)‖∞=O(Λ4). | (25) |
In light of (22) and (25), it can be deduced that the truncation error for the Lane-Emden system is of the order O(Λ4).
In this section, five test problems are considered to demonstrate the accuracy and applicability of QBSM. Additionally, the obtained numerical results corresponding to the considered system have been compared with those achieved previously [4,8,37]. Note that, in our calculations, "E−n" means 10−n.
The absolute error (Absi) and L∞ error are defined by
Absi=|ωi(τj)−μi(τj)|,i=1,2, |
Li∞(k)=max0≤j≤k|ωi(τj)−μi(τj)|,i=1,2, |
where ωi(τ) and μi(τ) represent the exact and QBSM solutions at the grid point τj, respectively. Moreover, the order of convergence (OC) of the method is computed by applying the following formula:
OCi=log2(Li∞(k)Li∞(2k)),i=1,2. |
Problem 1. Consider the following system
d2ω1(τ)dτ2+3τdω1(τ)dτ−4(ω1(τ)+ω2(τ))=0, | (26) |
subject to
ω1(0)=1,ω'1(0)=0, |
ω2(0)=1,ω'2(0)=0. | (27) |
The exact solution for this system is ω1(τ)=1+τ2,ω2(τ)=1−τ2.
We apply the proposed QBSM to solve this problem for Λ=0.1. Table 2 presents the exact and approximate solutions at the grid points. It is worth mentioning that, for this problem, the outcomes are exact and the errors are only incurred caused by round-off errors in computational processes.
τ | ω1(τ) | μ1(τ)(Λ=0.1) | Abs1 | ω2(τ) | μ2(τ)(Λ=0.1) | Abs2 |
0.0 | 1 | 1 | 1.11E−16 | 1 | 1 | 0 |
0.1 | 1.01 | 1.01 | 2.22E−16 | 0.99 | 0.99 | 0 |
0.2 | 1.04 | 1.04 | 2.22E−16 | 0.96 | 0.96 | 1.11E−16 |
0.3 | 1.09 | 1.09 | 2.22E−16 | 0.91 | 0.91 | 1.11E−16 |
0.4 | 1.16 | 1.16 | 2.22E−16 | 0.84 | 0.84 | 1.11E−16 |
0.5 | 1.25 | 1.25 | 0 | 0.75 | 0.75 | 0 |
0.6 | 1.36 | 1.36 | 2.22E−16 | 0.64 | 0.64 | 1.11E−16 |
0.7 | 1.49 | 1.49 | 0 | 0.51 | 0.51 | 2.22E−16 |
0.8 | 1.61 | 1.61 | 2.22E−16 | 0.36 | 0.36 | 2.77E−16 |
0.9 | 1.81 | 1.81 | 2.22E−16 | 0.19 | 0.19 | 3.33E−16 |
1.0 | 2 | 2 | 0 | 0 | 5.64E−16 | 3.84E−16 |
Problem 2. Consider the following system
ω''1(τ)+2τω'1(τ)−(4τ2+6)ω1(τ)+ω2(τ)=τ4−τ3, |
ω''2(τ)+8τω'2(τ)+ω1(τ)+τω2(τ)=eτ2+τ5−τ4+44τ2−30τ, | (28) |
subject to
ω1(0)=1,ω'1(0)=0, |
ω2(0)=0,ω'2(0)=0. | (29) |
The exact solution of this system is
ω1(τ)=eτ2,ω2(τ)=τ4−τ3. |
We apply the proposed QBSM to solve this problem for Λ=0.1,0.01. The logarithmic plots of absolute errors for various values of k are depicted in Figure 1, which exhibits that if the value of k is increased, the error decreases. The absolute errors obtained by QBSM are given in Tables 3 and 4 along with those obtained by CBSM [8] and Chebyshev operational matrix method (COMM) [4]. Comparison reveals that QBSM yields more accurate solutions than the methods in [4,8]. The outcomes of Li∞(k) errors are listed using k=16,32,64, and 128. In addition, the OCi,i=1,2, are computed and the results are tabulated in Table 5. It can be observed that the achieved OCi,i=1,2, is four. The method's computational time (CPU time) is reported in the same Table, which confirms that the QBSM is computationally effective.
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=5 | n=6 | n=8 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | 8.00E−9 | 5.00E−10 | 0 | 0 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | ---- | ---- | ---- | 5.98E−8 | 2.88E−12 |
0.2 | 9.52E−5 | 7.17E−7 | 2.38E−5 | 1.35E−7 | 1.02E−7 | 1.58E−7 | 1.23E−11 |
0.3 | 2.02E−4 | 1.76E−6 | ---- | ---- | ---- | 3.45E−7 | 3.09E−11 |
0.4 | 3.81E−4 | 3.51E−6 | 1.26E−4 | 6.90E−6 | 2.61E−7 | 6.78E−7 | 6.39E−11 |
0.5 | 6.72E−4 | 6.35E−6 | ---- | ---- | ---- | 1.24E−6 | 1.21E−10 |
0.6 | 1.14E−3 | 1.09E−5 | 2.09E−4 | 3.05E−5 | 4.71E−7 | 2.20E−6 | 2.17E−10 |
0.7 | 1.87E−3 | 1.81E−5 | ---- | ---- | ---- | 3.78E−6 | 3.77E−10 |
0.8 | 3.04E−3 | 2.96E−5 | 6.88E−3 | 1.02E−4 | 9.09E−7 | 6.44E−6 | 6.47E−10 |
0.9 | 4.90E−3 | 4.78E−5 | ---- | ---- | ---- | 1.08E−5 | 1.10E−9 |
1.0 | 7.89E−3 | 7.71E−5 | 3.14E−2 | 6.11E−4 | 1.97E−4 | 1.84E−5 | 1.86E−9 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=5 | n=6 | n=8 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | 0 | 0 | 0 | 7.23E−21 | 2.82E−23 |
0.1 | 4.67E−5 | 1.13E−7 | ---- | ---- | ---- | 1.53E−11 | 6.59E−16 |
0.2 | 6.21E−5 | 4.46E−7 | 4.36E−8 | 1.89E−10 | 1.22E−10 | 1.76E−10 | 1.07E−14 |
0.3 | 1.20E−4 | 9.98E−7 | ---- | ---- | ---- | 7.45E−10 | 5.78E−14 |
0.4 | 1.99E−4 | 1.77E−6 | 3.43E−7 | 3.58E−8 | 3.99E−10 | 2.31E−9 | 2.00E−13 |
0.5 | 2.95E−4 | 2.74E−6 | ---- | ---- | ---- | 5.97E−9 | 5.47E−13 |
0.6 | 4.12E−4 | 3.92E−6 | 7.70E−6 | 1.02E−7 | 1.38E−9 | 1.37E−8 | 1.30E−12 |
0.7 | 5.47E−4 | 5.27E−6 | ---- | ---- | ---- | 2.91E−8 | 2.82E−12 |
0.8 | 6.96E−4 | 6.77E−6 | 6.20E−6 | 2.59E−7 | 4.55E−9 | 5.85E−8 | 5.75E−12 |
0.9 | 8.53E−4 | 8.36E−6 | ---- | ---- | ---- | 1.13E−7 | 1.12E−11 |
1.0 | 1.01E−3 | 9.96E−6 | 4.19E−5 | 7.22E−6 | 1.68E−7 | 2.12E−7 | 2.12E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 4.493×10−5 | − | 5.149×10−7 | − | 0.0156 |
16 | 2.825×10−6 | 3.991 | 3.234×10−8 | 3.993 | 0.0312 |
32 | 1.771×10−7 | 3.996 | 2.022×10−9 | 3.999 | 0.0312 |
64 | 1.108×10−8 | 3.999 | 1.263×10−10 | 3.999 | 0.0625 |
128 | 6.927×10−10 | 3.999 | 7.899×10−12 | 3.999 | 0.1406 |
Problem 3. Consider the following system
ω''1(τ)+5τω'1(τ)+8(eω1(τ)+2e−ω2(τ)2)=0,ω''2(τ)+3τω'2(τ)−8(eω1(τ)2+e−ω2(τ))=0, | (30) |
subject to
ω1(0)=1−2ln(2),ω'1(0)=0,ω2(0)=1+2ln(2),ω'2(0)=0, | (31) |
where the exact solution is
ω1(τ)=1−2ln(τ2+2),ω2(τ)=1+2ln(τ2+2). |
We apply the proposed QBSM to obtain the approximate solutions to this problem for Λ=0.1,0.01. Absolute errors of QBSM for Λ=0.1,0.01 are listed in Tables 5 and 6, respectively, along with those obtained by the CBSM [8]. From these tables, it can be observed that QBSM provides lesser error than CBSM. The logarithmic plots of absolute errors for various values of k are depicted in Figure 2. The outcomes of Li∞(k) errors are listed using k=16,32,64, and 128. In addition, the OCi,i=1,2, are computed and the results are tabulated in Tables 7 and 8. The table show that the achieved OCi,i=1,2, is four. The method's CPU time is reported in the same table, which confirms that the QBSM is computationally effective.
τ | CBSM [8] | QBSM | ||
Λ=0.1 | Λ=0.01 | Λ=0.1 | Λ=0.01 | |
0.0 | 2.22E−16 | 2.22E−16 | 2.22E−16 | 2.22E−16 |
0.1 | 1.29E−5 | 4.14E−8 | 2.47E−8 | 6.75E−13 |
0.2 | 2.12E−5 | 1.53E−7 | 2.99E−8 | 2.34E−12 |
0.3 | 3.50E−5 | 3.08E−7 | 4.56E−8 | 4.19E−12 |
0.4 | 5.03E−5 | 4.67E−7 | 5.18E−8 | 5.24E−12 |
0.5 | 6.18E−5 | 5.92E−7 | 4.21E−8 | 4.80E−12 |
0.6 | 6.68E−5 | 6.51E−7 | 1.58E−8 | 2.59E−12 |
0.7 | 6.36E−5 | 6.26E−7 | 2.47E−8 | 1.13E−12 |
0.8 | 5.16E−5 | 5.13E−7 | 7.33E−8 | 5.82E−12 |
0.9 | 3.16E−5 | 3.18E−7 | 1.24E−7 | 1.08E−11 |
1.0 | 5.10E−6 | 5.70E−8 | 1.68E−7 | 1.54E−11 |
τ | CBSM [8] | QBSM | ||
Λ=0.1 | Λ=0.01 | Λ=0.1 | Λ=0.01 | |
0.0 | 2.22E−16 | 2.22E−16 | 2.22E−16 | 2.22E−16 |
0.1 | 1.48E−5 | 6.21E−8 | 2.79E−8 | 1.02E−12 |
0.2 | 3.18E−5 | 2.33E−7 | 4.86E−8 | 3.60E−12 |
0.3 | 5.42E−5 | 4.75E−7 | 7.42E−8 | 6.64E−12 |
0.4 | 7.94E−5 | 7.39E−7 | 9.19E−8 | 8.83E−12 |
0.5 | 1.02E−4 | 9.71E−7 | 8.89E−8 | 9.10E−12 |
0.6 | 1.16E−4 | 1.12E−6 | 6.21E−8 | 6.95E−12 |
0.7 | 1.20E−4 | 1.17E−6 | 1.28E−8 | 2.51E−12 |
0.8 | 1.12E−4 | 1.10E−6 | 5.23E−8 | 3.64E−12 |
0.9 | 9.26E−5 | 9.13E−7 | 1.26E−7 | 1.07E−11 |
1.0 | 6.27E−5 | 6.21E−7 | 1.97E−7 | 1.79E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 4.299×10−7 | − | 5.072×10−7 | − | 0.0156 |
16 | 2.439×10−8 | 4.139 | 2.842×10−8 | 4.157 | 0.0312 |
32 | 1.484×10−9 | 4.038 | 1.724×10−9 | 4.042 | 0.0312 |
64 | 9.215×10−11 | 4.009 | 1.069×10−10 | 4.011 | 0.0625 |
128 | 5.746×10−12 | 4.003 | 6.643×10−12 | 4.009 | 0.1406 |
Problem 4. Consider the following system of LEE
ω''1(τ)+1τω'1(τ)−ω32(τ)(ω21+1)=0,ω''2(τ)+3τω'2(τ)+ω52(τ)(ω21+3)=0, | (32) |
subject to
ω1(0)=1,ω'1(0)=0, |
ω2(0)=1,ω'2(0)=0, | (33) |
where the exact solution is given by ω1(τ)=√1+τ2,ω2(τ)=1√1+τ2. We solve this system using the proposed QBSM for Λ=0.1,0.01. Absolute errors obtained by QBSM for Λ=0.1,0.01 are given in Tables 9 and 10, along with the errors obtained by the CBSM [8] and COMM [4]. From these tables, it seems that the errors of QBSM are less than the errors of CBSM and COMM. The logarithmic graphs of absolute errors for different values of n are presented in Figure 3. The outcomes of Li∞(k) errors are listed using k=16,32,64, and 128. In addition, the OCi,i=1,2, are computed and the results are tabulated in Table 11. The table show that the achieved OCi,i=1,2, is four. The method's CPU time is reported in the same table, which confirms that the QBSM is computationally effective.
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1=0.01 | Λ=0.1=0.01 | n=4 | n=5 | n=6 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | ---- | ---- | ---- | 2.61E−8 | 1.51E−12 |
0.2 | 9.52E−5 | 7.17E−7 | 5.09E−4 | 5.65E−5 | 7.56E−6 | 6.61E−8 | 5.11E−12 |
0.3 | 2.02E−4 | 1.76E−6 | ---- | ---- | ---- | 1.02E−7 | 8.86E−12 |
0.4 | 3.81E−4 | 3.51E−6 | 6.28E−4 | 2.16E−5 | 8.65E−6 | 1.20E−7 | 1.09E−11 |
0.5 | 6.72E−4 | 6.35E−6 | ---- | ---- | ---- | 1.13E−7 | 1.06E−11 |
0.6 | 1.14E−3 | 1.09E−5 | 2.77E−4 | 5.57E−6 | 4.56E−6 | 8.93E−8 | 8.35E−12 |
0.7 | 1.87E−3 | 1.81E−5 | ---- | ---- | ---- | 6.00E−8 | 5.27E−12 |
0.8 | 3.04E−3 | 2.96E−5 | 2.72E−4 | 7.38E−5 | 7.71E−6 | 3.81E−8 | 2.61E−12 |
0.9 | 4.90E−3 | 4.78E−5 | ---- | ---- | ---- | 3.24E−8 | 1.32E−12 |
1.0 | 7.89E−3 | 7.71E−5 | 6.44E−4 | 7.46E−5 | 6.56E−6 | 4.80E−8 | 1.96E−12 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=4 | n=5 | n=6 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | 0 | 0 | 0 | 0 | 2.22E−16 |
0.1 | 4.67E−5 | 1.13E−7 | ---- | ---- | ---- | 1.06E−7 | 3.61E−12 |
0.2 | 6.21E−5 | 4.46E−7 | 1.03E−4 | 1.65E−5 | 6.59E−6 | 1.42E−7 | 1.08E−11 |
0.3 | 1.20E−4 | 9.98E−7 | ---- | ---- | ---- | 1.39E−7 | 1.42E−11 |
0.4 | 1.99E−4 | 1.77E−6 | 2.27E−4 | 1.51E−5 | 7.65E−6 | 5.25E−8 | 8.72E−12 |
0.5 | 2.95E−4 | 2.74E−6 | ---- | ---- | ---- | 1.13E−7 | 5.17E−12 |
0.6 | 4.12E−4 | 3.92E−6 | 1.00E−4 | 1.66E−5 | 9.63E−6 | 3.03E−7 | 2.31E−11 |
0.7 | 5.47E−4 | 5.27E−6 | ---- | ---- | ---- | 4.70E−7 | 4.00E−11 |
0.8 | 6.96E−4 | 6.77E−6 | 6.92E−4 | 1.67E−5 | 8.65E−6 | 5.81E−7 | 5.22E−11 |
0.9 | 8.53E−4 | 8.36E−6 | ---- | ---- | ---- | 6.27E−7 | 5.85E−11 |
1.0 | 1.01E−3 | 9.96E−6 | 2.62E−4 | 6.08E−5 | 6.53E−7 | 6.16E−7 | 5.91E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 2.961×10−7 | − | 1.583×10−6 | − | 0.0156 |
16 | 1.766×10−8 | 4.067 | 9.263×10−8 | 4.095 | 0.0312 |
32 | 1.073×10−9 | 4.040 | 5.697×10−9 | 4.023 | 0.0312 |
64 | 6.638×10−11 | 4.015 | 3.547×10−10 | 4.005 | 0.0468 |
128 | 4.132×10−12 | 4.005 | 2.216×10−11 | 4.000 | 0.1250 |
Problem 5. Consider the following system of LEE
ω''1(τ)+8τω'1(τ)+(18ω1(τ)−4ω1(τ)lnω2(τ))=0,ω''2(τ)+4τω'2(τ)+(4ω2(τ)lnω1(τ)−10ω2(τ))=0 | (34) |
subject to
ω1(0)=1,ω'1(0)=0,ω2(0)=1,ω'2(0)=0. | (35) |
The exact solution for this system is
ω1(τ)=e−τ2,ω2(τ)=eτ2. |
We solve this system using the proposed QBSM for Λ=0.1,0.01. Absolute errors obtained by QBSM for Λ=0.1,0.01 are tabulated in Tables 12 and 13, along with the errors reported in CBSM [8] and Dickson operational matrix (DOM) [37]. We note that QBSM yields results more accurate than those obtained in [8,37]. The logarithmic graphs of absolute errors for different values of k are displayed in Figure 4. The outcomes of Li∞(k) errors are listed using k=16,32,64, and 128. In addition, the OCi,i=1,2, are computed and the results are tabulated in Table 14. The table show that the achieved OCi,i=1,2, is four. The method's CPU time is reported in the same Table, which confirms that the QBSM is computationally effective.
τ | CBSM [8] | DOM [37] | QBSM | |||
Λ=0.1 | Λ=0.01 | n=8 | n=10 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | ---- | ---- | 1.11E−16 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | 5.84E−8 | 1.17E−9 | 4.43E−8 | 9.12E−13 |
0.2 | 9.52E−5 | 7.17E−7 | 1.04E−7 | 1.67E−9 | 3.81E−8 | 3.23E−12 |
0.3 | 2.02E−4 | 1.76E−6 | 2.21E−7 | 2.17E−10 | 7.01E−8 | 5.97E−12 |
0.4 | 3.81E−4 | 3.51E−6 | 1.37E−7 | 2.91E−9 | 7.71E−8 | 7.78E−12 |
0.5 | 6.72E−4 | 6.35E−6 | 3.18E−7 | 2.17E−9 | 6.38E−8 | 7.30E−12 |
0.6 | 1.14E−3 | 1.09E−5 | 2.59E−7 | 1.59E−9 | 1.82E−8 | 3.54E−12 |
0.7 | 1.87E−3 | 1.81E−5 | 3.00E−7 | 3.33E−9 | 6.32E−8 | 3.91E−12 |
0.8 | 3.04E−3 | 2.96E−5 | 2.41E−7 | 1.16E−9 | 1.75E−7 | 1.47E−11 |
0.9 | 4.90E−3 | 4.78E−5 | 2.77E−7 | 3.21E−11 | 3.10E−7 | 2.79E−11 |
1.0 | 7.89E−3 | 7.71E−5 | 5.95E−7 | 2.19E−10 | 4.47E−7 | 4.22E−11 |
τ | CBSM [8] | DOM [37] | QBSM | |||
Λ=0.1 | Λ=0.01 | n=8 | n=10 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | ---- | ---- | 0 | 0 |
0.1 | 4.67E−5 | 1.13E−7 | 3.30E−7 | 1.27E−8 | 5.14E−8 | 1.74E−12 |
0.2 | 6.21E−5 | 4.46E−7 | 1.18E−6 | 1.28E−8 | 1.01E−7 | 7.50E−12 |
0.3 | 1.20E−4 | 9.98E−7 | 1.14E−6 | 6.24E−9 | 2.24E−7 | 1.93E−11 |
0.4 | 1.99E−4 | 1.77E−6 | 1.69E−6 | 3.15E−8 | 4.50E−7 | 4.11E−11 |
0.5 | 2.95E−4 | 2.74E−6 | 1.44E−6 | 1.35E−8 | 8.44E−7 | 7.97E−11 |
0.6 | 4.12E−4 | 3.92E−6 | 2.46E−6 | 2.31E−8 | 1.53E−6 | 1.47E−10 |
0.7 | 5.47E−4 | 5.27E−6 | 1.17E−6 | 3.51E−8 | 2.69E−6 | 2.63E−10 |
0.8 | 6.96E−4 | 6.77E−6 | 1.87E−6 | 9.14E−9 | 4.68E−6 | 4.61E−10 |
0.9 | 8.53E−4 | 8.36E−6 | 2.20E−6 | 7.33E−10 | 7.99E−6 | 7.99E−10 |
1.0 | 1.01E−3 | 9.96E−6 | 3.36E−7 | 2.57E−9 | 1.39E−5 | 1.38E−9 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 1.120×10−6 | − | 3.416×10−5 | − | 0.0156 |
16 | 6.588×10−8 | 4.088 | 2.106×10−6 | 4.019 | 0.0312 |
32 | 4.044×10−9 | 4.025 | 1.314×10−7 | 4.002 | 0.0312 |
64 | 2.515×10−10 | 4.006 | 8.213×10−9 | 4.000 | 0.0468 |
128 | 1.570×10−11 | 4.001 | 5.133×10−10 | 4.000 | 0.1250 |
As can be observed from the above tables, the proposed QBSM is fourth-order accurate and the practical convergence order aligns consistently with the theoretical convergence order obtained in the previous section.
In this study, we have established a numerical method for solving systems of Lane-Emden equations. The QBSM has been constructed using quintic B-spline functions on the uniform mesh. We investigate the convergence analysis of the QBSM and found it exhibited fourth-order convergence. To strengthen the significance of the QBSM method and validate theoretical results, we examined five test problems. We have presented tabular and graphical exhibitions to confirm the effectiveness of QBSM. Notably, the numerical solutions of QBSM are in good agreement with the exact ones, and their accuracy improves as the step sizes decrease. Moreover, we compared the QBSM with other numerical methods such as CBSM, DOM, and COMM, and the comparison exposed that the QBSM produces more accurate numerical results than the other methods. In conclusion, the method is computationally efficient, accurate, robust, easy to address the singularity, and, therefore, it can be employed to solve different classes of nonlinear singular differential equations.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare no conflicts of interest.
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τr−2 | τr−1 | τr | τr+1 | τr+2 | else | |
Kr(τ) | 1120 | 26120 | 66120 | 26120 | 1120 | 0 |
K'r(τ) | 124Λ | 1024Λ | 0 | −1024Λ | −124Λ | 0 |
K''r(τ) | 16Λ2 | 26Λ2 | −66Λ2 | 26Λ2 | 16Λ2 | 0 |
K'''r(τ) | 12Λ3 | −22Λ3 | 0 | 22Λ3 | −12Λ3 | 0 |
τ | ω1(τ) | μ1(τ)(Λ=0.1) | Abs1 | ω2(τ) | μ2(τ)(Λ=0.1) | Abs2 |
0.0 | 1 | 1 | 1.11E−16 | 1 | 1 | 0 |
0.1 | 1.01 | 1.01 | 2.22E−16 | 0.99 | 0.99 | 0 |
0.2 | 1.04 | 1.04 | 2.22E−16 | 0.96 | 0.96 | 1.11E−16 |
0.3 | 1.09 | 1.09 | 2.22E−16 | 0.91 | 0.91 | 1.11E−16 |
0.4 | 1.16 | 1.16 | 2.22E−16 | 0.84 | 0.84 | 1.11E−16 |
0.5 | 1.25 | 1.25 | 0 | 0.75 | 0.75 | 0 |
0.6 | 1.36 | 1.36 | 2.22E−16 | 0.64 | 0.64 | 1.11E−16 |
0.7 | 1.49 | 1.49 | 0 | 0.51 | 0.51 | 2.22E−16 |
0.8 | 1.61 | 1.61 | 2.22E−16 | 0.36 | 0.36 | 2.77E−16 |
0.9 | 1.81 | 1.81 | 2.22E−16 | 0.19 | 0.19 | 3.33E−16 |
1.0 | 2 | 2 | 0 | 0 | 5.64E−16 | 3.84E−16 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=5 | n=6 | n=8 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | 8.00E−9 | 5.00E−10 | 0 | 0 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | ---- | ---- | ---- | 5.98E−8 | 2.88E−12 |
0.2 | 9.52E−5 | 7.17E−7 | 2.38E−5 | 1.35E−7 | 1.02E−7 | 1.58E−7 | 1.23E−11 |
0.3 | 2.02E−4 | 1.76E−6 | ---- | ---- | ---- | 3.45E−7 | 3.09E−11 |
0.4 | 3.81E−4 | 3.51E−6 | 1.26E−4 | 6.90E−6 | 2.61E−7 | 6.78E−7 | 6.39E−11 |
0.5 | 6.72E−4 | 6.35E−6 | ---- | ---- | ---- | 1.24E−6 | 1.21E−10 |
0.6 | 1.14E−3 | 1.09E−5 | 2.09E−4 | 3.05E−5 | 4.71E−7 | 2.20E−6 | 2.17E−10 |
0.7 | 1.87E−3 | 1.81E−5 | ---- | ---- | ---- | 3.78E−6 | 3.77E−10 |
0.8 | 3.04E−3 | 2.96E−5 | 6.88E−3 | 1.02E−4 | 9.09E−7 | 6.44E−6 | 6.47E−10 |
0.9 | 4.90E−3 | 4.78E−5 | ---- | ---- | ---- | 1.08E−5 | 1.10E−9 |
1.0 | 7.89E−3 | 7.71E−5 | 3.14E−2 | 6.11E−4 | 1.97E−4 | 1.84E−5 | 1.86E−9 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=5 | n=6 | n=8 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | 0 | 0 | 0 | 7.23E−21 | 2.82E−23 |
0.1 | 4.67E−5 | 1.13E−7 | ---- | ---- | ---- | 1.53E−11 | 6.59E−16 |
0.2 | 6.21E−5 | 4.46E−7 | 4.36E−8 | 1.89E−10 | 1.22E−10 | 1.76E−10 | 1.07E−14 |
0.3 | 1.20E−4 | 9.98E−7 | ---- | ---- | ---- | 7.45E−10 | 5.78E−14 |
0.4 | 1.99E−4 | 1.77E−6 | 3.43E−7 | 3.58E−8 | 3.99E−10 | 2.31E−9 | 2.00E−13 |
0.5 | 2.95E−4 | 2.74E−6 | ---- | ---- | ---- | 5.97E−9 | 5.47E−13 |
0.6 | 4.12E−4 | 3.92E−6 | 7.70E−6 | 1.02E−7 | 1.38E−9 | 1.37E−8 | 1.30E−12 |
0.7 | 5.47E−4 | 5.27E−6 | ---- | ---- | ---- | 2.91E−8 | 2.82E−12 |
0.8 | 6.96E−4 | 6.77E−6 | 6.20E−6 | 2.59E−7 | 4.55E−9 | 5.85E−8 | 5.75E−12 |
0.9 | 8.53E−4 | 8.36E−6 | ---- | ---- | ---- | 1.13E−7 | 1.12E−11 |
1.0 | 1.01E−3 | 9.96E−6 | 4.19E−5 | 7.22E−6 | 1.68E−7 | 2.12E−7 | 2.12E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 4.493×10−5 | − | 5.149×10−7 | − | 0.0156 |
16 | 2.825×10−6 | 3.991 | 3.234×10−8 | 3.993 | 0.0312 |
32 | 1.771×10−7 | 3.996 | 2.022×10−9 | 3.999 | 0.0312 |
64 | 1.108×10−8 | 3.999 | 1.263×10−10 | 3.999 | 0.0625 |
128 | 6.927×10−10 | 3.999 | 7.899×10−12 | 3.999 | 0.1406 |
τ | CBSM [8] | QBSM | ||
Λ=0.1 | Λ=0.01 | Λ=0.1 | Λ=0.01 | |
0.0 | 2.22E−16 | 2.22E−16 | 2.22E−16 | 2.22E−16 |
0.1 | 1.29E−5 | 4.14E−8 | 2.47E−8 | 6.75E−13 |
0.2 | 2.12E−5 | 1.53E−7 | 2.99E−8 | 2.34E−12 |
0.3 | 3.50E−5 | 3.08E−7 | 4.56E−8 | 4.19E−12 |
0.4 | 5.03E−5 | 4.67E−7 | 5.18E−8 | 5.24E−12 |
0.5 | 6.18E−5 | 5.92E−7 | 4.21E−8 | 4.80E−12 |
0.6 | 6.68E−5 | 6.51E−7 | 1.58E−8 | 2.59E−12 |
0.7 | 6.36E−5 | 6.26E−7 | 2.47E−8 | 1.13E−12 |
0.8 | 5.16E−5 | 5.13E−7 | 7.33E−8 | 5.82E−12 |
0.9 | 3.16E−5 | 3.18E−7 | 1.24E−7 | 1.08E−11 |
1.0 | 5.10E−6 | 5.70E−8 | 1.68E−7 | 1.54E−11 |
τ | CBSM [8] | QBSM | ||
Λ=0.1 | Λ=0.01 | Λ=0.1 | Λ=0.01 | |
0.0 | 2.22E−16 | 2.22E−16 | 2.22E−16 | 2.22E−16 |
0.1 | 1.48E−5 | 6.21E−8 | 2.79E−8 | 1.02E−12 |
0.2 | 3.18E−5 | 2.33E−7 | 4.86E−8 | 3.60E−12 |
0.3 | 5.42E−5 | 4.75E−7 | 7.42E−8 | 6.64E−12 |
0.4 | 7.94E−5 | 7.39E−7 | 9.19E−8 | 8.83E−12 |
0.5 | 1.02E−4 | 9.71E−7 | 8.89E−8 | 9.10E−12 |
0.6 | 1.16E−4 | 1.12E−6 | 6.21E−8 | 6.95E−12 |
0.7 | 1.20E−4 | 1.17E−6 | 1.28E−8 | 2.51E−12 |
0.8 | 1.12E−4 | 1.10E−6 | 5.23E−8 | 3.64E−12 |
0.9 | 9.26E−5 | 9.13E−7 | 1.26E−7 | 1.07E−11 |
1.0 | 6.27E−5 | 6.21E−7 | 1.97E−7 | 1.79E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 4.299×10−7 | − | 5.072×10−7 | − | 0.0156 |
16 | 2.439×10−8 | 4.139 | 2.842×10−8 | 4.157 | 0.0312 |
32 | 1.484×10−9 | 4.038 | 1.724×10−9 | 4.042 | 0.0312 |
64 | 9.215×10−11 | 4.009 | 1.069×10−10 | 4.011 | 0.0625 |
128 | 5.746×10−12 | 4.003 | 6.643×10−12 | 4.009 | 0.1406 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1=0.01 | Λ=0.1=0.01 | n=4 | n=5 | n=6 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | ---- | ---- | ---- | 2.61E−8 | 1.51E−12 |
0.2 | 9.52E−5 | 7.17E−7 | 5.09E−4 | 5.65E−5 | 7.56E−6 | 6.61E−8 | 5.11E−12 |
0.3 | 2.02E−4 | 1.76E−6 | ---- | ---- | ---- | 1.02E−7 | 8.86E−12 |
0.4 | 3.81E−4 | 3.51E−6 | 6.28E−4 | 2.16E−5 | 8.65E−6 | 1.20E−7 | 1.09E−11 |
0.5 | 6.72E−4 | 6.35E−6 | ---- | ---- | ---- | 1.13E−7 | 1.06E−11 |
0.6 | 1.14E−3 | 1.09E−5 | 2.77E−4 | 5.57E−6 | 4.56E−6 | 8.93E−8 | 8.35E−12 |
0.7 | 1.87E−3 | 1.81E−5 | ---- | ---- | ---- | 6.00E−8 | 5.27E−12 |
0.8 | 3.04E−3 | 2.96E−5 | 2.72E−4 | 7.38E−5 | 7.71E−6 | 3.81E−8 | 2.61E−12 |
0.9 | 4.90E−3 | 4.78E−5 | ---- | ---- | ---- | 3.24E−8 | 1.32E−12 |
1.0 | 7.89E−3 | 7.71E−5 | 6.44E−4 | 7.46E−5 | 6.56E−6 | 4.80E−8 | 1.96E−12 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=4 | n=5 | n=6 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | 0 | 0 | 0 | 0 | 2.22E−16 |
0.1 | 4.67E−5 | 1.13E−7 | ---- | ---- | ---- | 1.06E−7 | 3.61E−12 |
0.2 | 6.21E−5 | 4.46E−7 | 1.03E−4 | 1.65E−5 | 6.59E−6 | 1.42E−7 | 1.08E−11 |
0.3 | 1.20E−4 | 9.98E−7 | ---- | ---- | ---- | 1.39E−7 | 1.42E−11 |
0.4 | 1.99E−4 | 1.77E−6 | 2.27E−4 | 1.51E−5 | 7.65E−6 | 5.25E−8 | 8.72E−12 |
0.5 | 2.95E−4 | 2.74E−6 | ---- | ---- | ---- | 1.13E−7 | 5.17E−12 |
0.6 | 4.12E−4 | 3.92E−6 | 1.00E−4 | 1.66E−5 | 9.63E−6 | 3.03E−7 | 2.31E−11 |
0.7 | 5.47E−4 | 5.27E−6 | ---- | ---- | ---- | 4.70E−7 | 4.00E−11 |
0.8 | 6.96E−4 | 6.77E−6 | 6.92E−4 | 1.67E−5 | 8.65E−6 | 5.81E−7 | 5.22E−11 |
0.9 | 8.53E−4 | 8.36E−6 | ---- | ---- | ---- | 6.27E−7 | 5.85E−11 |
1.0 | 1.01E−3 | 9.96E−6 | 2.62E−4 | 6.08E−5 | 6.53E−7 | 6.16E−7 | 5.91E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 2.961×10−7 | − | 1.583×10−6 | − | 0.0156 |
16 | 1.766×10−8 | 4.067 | 9.263×10−8 | 4.095 | 0.0312 |
32 | 1.073×10−9 | 4.040 | 5.697×10−9 | 4.023 | 0.0312 |
64 | 6.638×10−11 | 4.015 | 3.547×10−10 | 4.005 | 0.0468 |
128 | 4.132×10−12 | 4.005 | 2.216×10−11 | 4.000 | 0.1250 |
τ | CBSM [8] | DOM [37] | QBSM | |||
Λ=0.1 | Λ=0.01 | n=8 | n=10 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | ---- | ---- | 1.11E−16 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | 5.84E−8 | 1.17E−9 | 4.43E−8 | 9.12E−13 |
0.2 | 9.52E−5 | 7.17E−7 | 1.04E−7 | 1.67E−9 | 3.81E−8 | 3.23E−12 |
0.3 | 2.02E−4 | 1.76E−6 | 2.21E−7 | 2.17E−10 | 7.01E−8 | 5.97E−12 |
0.4 | 3.81E−4 | 3.51E−6 | 1.37E−7 | 2.91E−9 | 7.71E−8 | 7.78E−12 |
0.5 | 6.72E−4 | 6.35E−6 | 3.18E−7 | 2.17E−9 | 6.38E−8 | 7.30E−12 |
0.6 | 1.14E−3 | 1.09E−5 | 2.59E−7 | 1.59E−9 | 1.82E−8 | 3.54E−12 |
0.7 | 1.87E−3 | 1.81E−5 | 3.00E−7 | 3.33E−9 | 6.32E−8 | 3.91E−12 |
0.8 | 3.04E−3 | 2.96E−5 | 2.41E−7 | 1.16E−9 | 1.75E−7 | 1.47E−11 |
0.9 | 4.90E−3 | 4.78E−5 | 2.77E−7 | 3.21E−11 | 3.10E−7 | 2.79E−11 |
1.0 | 7.89E−3 | 7.71E−5 | 5.95E−7 | 2.19E−10 | 4.47E−7 | 4.22E−11 |
τ | CBSM [8] | DOM [37] | QBSM | |||
Λ=0.1 | Λ=0.01 | n=8 | n=10 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | ---- | ---- | 0 | 0 |
0.1 | 4.67E−5 | 1.13E−7 | 3.30E−7 | 1.27E−8 | 5.14E−8 | 1.74E−12 |
0.2 | 6.21E−5 | 4.46E−7 | 1.18E−6 | 1.28E−8 | 1.01E−7 | 7.50E−12 |
0.3 | 1.20E−4 | 9.98E−7 | 1.14E−6 | 6.24E−9 | 2.24E−7 | 1.93E−11 |
0.4 | 1.99E−4 | 1.77E−6 | 1.69E−6 | 3.15E−8 | 4.50E−7 | 4.11E−11 |
0.5 | 2.95E−4 | 2.74E−6 | 1.44E−6 | 1.35E−8 | 8.44E−7 | 7.97E−11 |
0.6 | 4.12E−4 | 3.92E−6 | 2.46E−6 | 2.31E−8 | 1.53E−6 | 1.47E−10 |
0.7 | 5.47E−4 | 5.27E−6 | 1.17E−6 | 3.51E−8 | 2.69E−6 | 2.63E−10 |
0.8 | 6.96E−4 | 6.77E−6 | 1.87E−6 | 9.14E−9 | 4.68E−6 | 4.61E−10 |
0.9 | 8.53E−4 | 8.36E−6 | 2.20E−6 | 7.33E−10 | 7.99E−6 | 7.99E−10 |
1.0 | 1.01E−3 | 9.96E−6 | 3.36E−7 | 2.57E−9 | 1.39E−5 | 1.38E−9 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 1.120×10−6 | − | 3.416×10−5 | − | 0.0156 |
16 | 6.588×10−8 | 4.088 | 2.106×10−6 | 4.019 | 0.0312 |
32 | 4.044×10−9 | 4.025 | 1.314×10−7 | 4.002 | 0.0312 |
64 | 2.515×10−10 | 4.006 | 8.213×10−9 | 4.000 | 0.0468 |
128 | 1.570×10−11 | 4.001 | 5.133×10−10 | 4.000 | 0.1250 |
τr−2 | τr−1 | τr | τr+1 | τr+2 | else | |
Kr(τ) | 1120 | 26120 | 66120 | 26120 | 1120 | 0 |
K'r(τ) | 124Λ | 1024Λ | 0 | −1024Λ | −124Λ | 0 |
K''r(τ) | 16Λ2 | 26Λ2 | −66Λ2 | 26Λ2 | 16Λ2 | 0 |
K'''r(τ) | 12Λ3 | −22Λ3 | 0 | 22Λ3 | −12Λ3 | 0 |
τ | ω1(τ) | μ1(τ)(Λ=0.1) | Abs1 | ω2(τ) | μ2(τ)(Λ=0.1) | Abs2 |
0.0 | 1 | 1 | 1.11E−16 | 1 | 1 | 0 |
0.1 | 1.01 | 1.01 | 2.22E−16 | 0.99 | 0.99 | 0 |
0.2 | 1.04 | 1.04 | 2.22E−16 | 0.96 | 0.96 | 1.11E−16 |
0.3 | 1.09 | 1.09 | 2.22E−16 | 0.91 | 0.91 | 1.11E−16 |
0.4 | 1.16 | 1.16 | 2.22E−16 | 0.84 | 0.84 | 1.11E−16 |
0.5 | 1.25 | 1.25 | 0 | 0.75 | 0.75 | 0 |
0.6 | 1.36 | 1.36 | 2.22E−16 | 0.64 | 0.64 | 1.11E−16 |
0.7 | 1.49 | 1.49 | 0 | 0.51 | 0.51 | 2.22E−16 |
0.8 | 1.61 | 1.61 | 2.22E−16 | 0.36 | 0.36 | 2.77E−16 |
0.9 | 1.81 | 1.81 | 2.22E−16 | 0.19 | 0.19 | 3.33E−16 |
1.0 | 2 | 2 | 0 | 0 | 5.64E−16 | 3.84E−16 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=5 | n=6 | n=8 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | 8.00E−9 | 5.00E−10 | 0 | 0 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | ---- | ---- | ---- | 5.98E−8 | 2.88E−12 |
0.2 | 9.52E−5 | 7.17E−7 | 2.38E−5 | 1.35E−7 | 1.02E−7 | 1.58E−7 | 1.23E−11 |
0.3 | 2.02E−4 | 1.76E−6 | ---- | ---- | ---- | 3.45E−7 | 3.09E−11 |
0.4 | 3.81E−4 | 3.51E−6 | 1.26E−4 | 6.90E−6 | 2.61E−7 | 6.78E−7 | 6.39E−11 |
0.5 | 6.72E−4 | 6.35E−6 | ---- | ---- | ---- | 1.24E−6 | 1.21E−10 |
0.6 | 1.14E−3 | 1.09E−5 | 2.09E−4 | 3.05E−5 | 4.71E−7 | 2.20E−6 | 2.17E−10 |
0.7 | 1.87E−3 | 1.81E−5 | ---- | ---- | ---- | 3.78E−6 | 3.77E−10 |
0.8 | 3.04E−3 | 2.96E−5 | 6.88E−3 | 1.02E−4 | 9.09E−7 | 6.44E−6 | 6.47E−10 |
0.9 | 4.90E−3 | 4.78E−5 | ---- | ---- | ---- | 1.08E−5 | 1.10E−9 |
1.0 | 7.89E−3 | 7.71E−5 | 3.14E−2 | 6.11E−4 | 1.97E−4 | 1.84E−5 | 1.86E−9 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=5 | n=6 | n=8 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | 0 | 0 | 0 | 7.23E−21 | 2.82E−23 |
0.1 | 4.67E−5 | 1.13E−7 | ---- | ---- | ---- | 1.53E−11 | 6.59E−16 |
0.2 | 6.21E−5 | 4.46E−7 | 4.36E−8 | 1.89E−10 | 1.22E−10 | 1.76E−10 | 1.07E−14 |
0.3 | 1.20E−4 | 9.98E−7 | ---- | ---- | ---- | 7.45E−10 | 5.78E−14 |
0.4 | 1.99E−4 | 1.77E−6 | 3.43E−7 | 3.58E−8 | 3.99E−10 | 2.31E−9 | 2.00E−13 |
0.5 | 2.95E−4 | 2.74E−6 | ---- | ---- | ---- | 5.97E−9 | 5.47E−13 |
0.6 | 4.12E−4 | 3.92E−6 | 7.70E−6 | 1.02E−7 | 1.38E−9 | 1.37E−8 | 1.30E−12 |
0.7 | 5.47E−4 | 5.27E−6 | ---- | ---- | ---- | 2.91E−8 | 2.82E−12 |
0.8 | 6.96E−4 | 6.77E−6 | 6.20E−6 | 2.59E−7 | 4.55E−9 | 5.85E−8 | 5.75E−12 |
0.9 | 8.53E−4 | 8.36E−6 | ---- | ---- | ---- | 1.13E−7 | 1.12E−11 |
1.0 | 1.01E−3 | 9.96E−6 | 4.19E−5 | 7.22E−6 | 1.68E−7 | 2.12E−7 | 2.12E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 4.493×10−5 | − | 5.149×10−7 | − | 0.0156 |
16 | 2.825×10−6 | 3.991 | 3.234×10−8 | 3.993 | 0.0312 |
32 | 1.771×10−7 | 3.996 | 2.022×10−9 | 3.999 | 0.0312 |
64 | 1.108×10−8 | 3.999 | 1.263×10−10 | 3.999 | 0.0625 |
128 | 6.927×10−10 | 3.999 | 7.899×10−12 | 3.999 | 0.1406 |
τ | CBSM [8] | QBSM | ||
Λ=0.1 | Λ=0.01 | Λ=0.1 | Λ=0.01 | |
0.0 | 2.22E−16 | 2.22E−16 | 2.22E−16 | 2.22E−16 |
0.1 | 1.29E−5 | 4.14E−8 | 2.47E−8 | 6.75E−13 |
0.2 | 2.12E−5 | 1.53E−7 | 2.99E−8 | 2.34E−12 |
0.3 | 3.50E−5 | 3.08E−7 | 4.56E−8 | 4.19E−12 |
0.4 | 5.03E−5 | 4.67E−7 | 5.18E−8 | 5.24E−12 |
0.5 | 6.18E−5 | 5.92E−7 | 4.21E−8 | 4.80E−12 |
0.6 | 6.68E−5 | 6.51E−7 | 1.58E−8 | 2.59E−12 |
0.7 | 6.36E−5 | 6.26E−7 | 2.47E−8 | 1.13E−12 |
0.8 | 5.16E−5 | 5.13E−7 | 7.33E−8 | 5.82E−12 |
0.9 | 3.16E−5 | 3.18E−7 | 1.24E−7 | 1.08E−11 |
1.0 | 5.10E−6 | 5.70E−8 | 1.68E−7 | 1.54E−11 |
τ | CBSM [8] | QBSM | ||
Λ=0.1 | Λ=0.01 | Λ=0.1 | Λ=0.01 | |
0.0 | 2.22E−16 | 2.22E−16 | 2.22E−16 | 2.22E−16 |
0.1 | 1.48E−5 | 6.21E−8 | 2.79E−8 | 1.02E−12 |
0.2 | 3.18E−5 | 2.33E−7 | 4.86E−8 | 3.60E−12 |
0.3 | 5.42E−5 | 4.75E−7 | 7.42E−8 | 6.64E−12 |
0.4 | 7.94E−5 | 7.39E−7 | 9.19E−8 | 8.83E−12 |
0.5 | 1.02E−4 | 9.71E−7 | 8.89E−8 | 9.10E−12 |
0.6 | 1.16E−4 | 1.12E−6 | 6.21E−8 | 6.95E−12 |
0.7 | 1.20E−4 | 1.17E−6 | 1.28E−8 | 2.51E−12 |
0.8 | 1.12E−4 | 1.10E−6 | 5.23E−8 | 3.64E−12 |
0.9 | 9.26E−5 | 9.13E−7 | 1.26E−7 | 1.07E−11 |
1.0 | 6.27E−5 | 6.21E−7 | 1.97E−7 | 1.79E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 4.299×10−7 | − | 5.072×10−7 | − | 0.0156 |
16 | 2.439×10−8 | 4.139 | 2.842×10−8 | 4.157 | 0.0312 |
32 | 1.484×10−9 | 4.038 | 1.724×10−9 | 4.042 | 0.0312 |
64 | 9.215×10−11 | 4.009 | 1.069×10−10 | 4.011 | 0.0625 |
128 | 5.746×10−12 | 4.003 | 6.643×10−12 | 4.009 | 0.1406 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1=0.01 | Λ=0.1=0.01 | n=4 | n=5 | n=6 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | ---- | ---- | ---- | 2.61E−8 | 1.51E−12 |
0.2 | 9.52E−5 | 7.17E−7 | 5.09E−4 | 5.65E−5 | 7.56E−6 | 6.61E−8 | 5.11E−12 |
0.3 | 2.02E−4 | 1.76E−6 | ---- | ---- | ---- | 1.02E−7 | 8.86E−12 |
0.4 | 3.81E−4 | 3.51E−6 | 6.28E−4 | 2.16E−5 | 8.65E−6 | 1.20E−7 | 1.09E−11 |
0.5 | 6.72E−4 | 6.35E−6 | ---- | ---- | ---- | 1.13E−7 | 1.06E−11 |
0.6 | 1.14E−3 | 1.09E−5 | 2.77E−4 | 5.57E−6 | 4.56E−6 | 8.93E−8 | 8.35E−12 |
0.7 | 1.87E−3 | 1.81E−5 | ---- | ---- | ---- | 6.00E−8 | 5.27E−12 |
0.8 | 3.04E−3 | 2.96E−5 | 2.72E−4 | 7.38E−5 | 7.71E−6 | 3.81E−8 | 2.61E−12 |
0.9 | 4.90E−3 | 4.78E−5 | ---- | ---- | ---- | 3.24E−8 | 1.32E−12 |
1.0 | 7.89E−3 | 7.71E−5 | 6.44E−4 | 7.46E−5 | 6.56E−6 | 4.80E−8 | 1.96E−12 |
τ | CBSM [8] | COMM [4] | QBSM | ||||
Λ=0.1 | Λ=0.01 | n=4 | n=5 | n=6 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | 0 | 0 | 0 | 0 | 2.22E−16 |
0.1 | 4.67E−5 | 1.13E−7 | ---- | ---- | ---- | 1.06E−7 | 3.61E−12 |
0.2 | 6.21E−5 | 4.46E−7 | 1.03E−4 | 1.65E−5 | 6.59E−6 | 1.42E−7 | 1.08E−11 |
0.3 | 1.20E−4 | 9.98E−7 | ---- | ---- | ---- | 1.39E−7 | 1.42E−11 |
0.4 | 1.99E−4 | 1.77E−6 | 2.27E−4 | 1.51E−5 | 7.65E−6 | 5.25E−8 | 8.72E−12 |
0.5 | 2.95E−4 | 2.74E−6 | ---- | ---- | ---- | 1.13E−7 | 5.17E−12 |
0.6 | 4.12E−4 | 3.92E−6 | 1.00E−4 | 1.66E−5 | 9.63E−6 | 3.03E−7 | 2.31E−11 |
0.7 | 5.47E−4 | 5.27E−6 | ---- | ---- | ---- | 4.70E−7 | 4.00E−11 |
0.8 | 6.96E−4 | 6.77E−6 | 6.92E−4 | 1.67E−5 | 8.65E−6 | 5.81E−7 | 5.22E−11 |
0.9 | 8.53E−4 | 8.36E−6 | ---- | ---- | ---- | 6.27E−7 | 5.85E−11 |
1.0 | 1.01E−3 | 9.96E−6 | 2.62E−4 | 6.08E−5 | 6.53E−7 | 6.16E−7 | 5.91E−11 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 2.961×10−7 | − | 1.583×10−6 | − | 0.0156 |
16 | 1.766×10−8 | 4.067 | 9.263×10−8 | 4.095 | 0.0312 |
32 | 1.073×10−9 | 4.040 | 5.697×10−9 | 4.023 | 0.0312 |
64 | 6.638×10−11 | 4.015 | 3.547×10−10 | 4.005 | 0.0468 |
128 | 4.132×10−12 | 4.005 | 2.216×10−11 | 4.000 | 0.1250 |
τ | CBSM [8] | DOM [37] | QBSM | |||
Λ=0.1 | Λ=0.01 | n=8 | n=10 | Λ=0.1 | Λ=0.01 | |
0.0 | 0 | 0 | ---- | ---- | 1.11E−16 | 0 |
0.1 | 3.39E−5 | 1.72E−7 | 5.84E−8 | 1.17E−9 | 4.43E−8 | 9.12E−13 |
0.2 | 9.52E−5 | 7.17E−7 | 1.04E−7 | 1.67E−9 | 3.81E−8 | 3.23E−12 |
0.3 | 2.02E−4 | 1.76E−6 | 2.21E−7 | 2.17E−10 | 7.01E−8 | 5.97E−12 |
0.4 | 3.81E−4 | 3.51E−6 | 1.37E−7 | 2.91E−9 | 7.71E−8 | 7.78E−12 |
0.5 | 6.72E−4 | 6.35E−6 | 3.18E−7 | 2.17E−9 | 6.38E−8 | 7.30E−12 |
0.6 | 1.14E−3 | 1.09E−5 | 2.59E−7 | 1.59E−9 | 1.82E−8 | 3.54E−12 |
0.7 | 1.87E−3 | 1.81E−5 | 3.00E−7 | 3.33E−9 | 6.32E−8 | 3.91E−12 |
0.8 | 3.04E−3 | 2.96E−5 | 2.41E−7 | 1.16E−9 | 1.75E−7 | 1.47E−11 |
0.9 | 4.90E−3 | 4.78E−5 | 2.77E−7 | 3.21E−11 | 3.10E−7 | 2.79E−11 |
1.0 | 7.89E−3 | 7.71E−5 | 5.95E−7 | 2.19E−10 | 4.47E−7 | 4.22E−11 |
τ | CBSM [8] | DOM [37] | QBSM | |||
Λ=0.1 | Λ=0.01 | n=8 | n=10 | Λ=0.1 | Λ=0.01 | |
0.0 | 6.00E−31 | 0 | ---- | ---- | 0 | 0 |
0.1 | 4.67E−5 | 1.13E−7 | 3.30E−7 | 1.27E−8 | 5.14E−8 | 1.74E−12 |
0.2 | 6.21E−5 | 4.46E−7 | 1.18E−6 | 1.28E−8 | 1.01E−7 | 7.50E−12 |
0.3 | 1.20E−4 | 9.98E−7 | 1.14E−6 | 6.24E−9 | 2.24E−7 | 1.93E−11 |
0.4 | 1.99E−4 | 1.77E−6 | 1.69E−6 | 3.15E−8 | 4.50E−7 | 4.11E−11 |
0.5 | 2.95E−4 | 2.74E−6 | 1.44E−6 | 1.35E−8 | 8.44E−7 | 7.97E−11 |
0.6 | 4.12E−4 | 3.92E−6 | 2.46E−6 | 2.31E−8 | 1.53E−6 | 1.47E−10 |
0.7 | 5.47E−4 | 5.27E−6 | 1.17E−6 | 3.51E−8 | 2.69E−6 | 2.63E−10 |
0.8 | 6.96E−4 | 6.77E−6 | 1.87E−6 | 9.14E−9 | 4.68E−6 | 4.61E−10 |
0.9 | 8.53E−4 | 8.36E−6 | 2.20E−6 | 7.33E−10 | 7.99E−6 | 7.99E−10 |
1.0 | 1.01E−3 | 9.96E−6 | 3.36E−7 | 2.57E−9 | 1.39E−5 | 1.38E−9 |
k | L1∞(k) | OC1 | L2∞(k) | OC2 | CPU (s) |
8 | 1.120×10−6 | − | 3.416×10−5 | − | 0.0156 |
16 | 6.588×10−8 | 4.088 | 2.106×10−6 | 4.019 | 0.0312 |
32 | 4.044×10−9 | 4.025 | 1.314×10−7 | 4.002 | 0.0312 |
64 | 2.515×10−10 | 4.006 | 8.213×10−9 | 4.000 | 0.0468 |
128 | 1.570×10−11 | 4.001 | 5.133×10−10 | 4.000 | 0.1250 |