Analysis of Numerical Measure and Numerical Integration Based on Measure

Abstract

We present a convergence analysis for a general numerical method to estimate measure function. By combining Lagrange interpolation, we propose a specific method for approximating the measure function and analyze the convergence order. Further, we analyze the error bound of numerical measure integration and prove that the numerical measure integration can decrease the singularity for singular integrals. Numerical examples are presented to confirm the theoretical results.

1. Introduction

A numerical integration based on the definition of the Lebesgue integral was proposed in the paper “A New Approach to Numerical Integration” [1]. The method was said to be particularly useful for integrands which are highly oscillatory in character or singular. It provides an exact reduction of multidimensional integrals to one-dimensional integrals. It appeared to be a more economical way of treating certain multidimensional integrals. For concise writing, we call the new numerical integration proposed by B. L. Burrows the numerical measure integration (NMI).

Consider the integral ∫_𝐼𝑓(𝑥)𝑑𝑥, where 𝑓(𝑥)∈𝐿(𝐼) is a nonnegative function and 𝐼=[𝑎,𝑏]. Firstly, define 𝐸(𝑓(𝑥)⩾𝑦)∶={𝑥∈𝐼∶𝑓(𝑥)⩾𝑦} and 𝜇_𝑓(𝑦)∶=𝑚(𝐸(𝑓(𝑥)⩾𝑦)), 𝑦∈ℝ. It is easy to obtain that 𝜇_𝑓(𝑦) is a monotonically decreasing and bounded function with 0⩽𝜇_𝑓(𝑦)⩽𝑏−𝑎,𝑦∈ℝ. Define 𝑦₀∶=min_𝑥∈𝐼𝑓(𝑥) and 𝑦_𝑁∶=max_𝑥∈𝐼𝑓(𝑥). Denote the range of 𝑓 by 𝑅(𝑓)∶=[𝑦₀,𝑦_𝑁], where 𝑦₀⩾0 and 𝑦_𝑁 can be infinite which means that 𝑓(𝑥) is singular. The foundational equation of NMI proposed by [1] is (𝐿)^𝑏_𝑎𝑓(𝑥)𝑑𝑥=𝑦₀𝜇_𝑓𝑦₀+(𝑅)^𝑦𝑁_𝑦0𝜇_𝑓𝑦𝑑𝑦. (1)

When calculating the Lebesgue integral numerically through (1), there are three kinds of errors [1]:(i)the error in the method used to estimate ∫^𝑦𝑁_𝑦0𝜇_𝑓(𝑦)𝑑𝑦;(ii)the error in the estimate of 𝜇_𝑓(𝑦) for some 𝑦 when necessary;(iii)the error in estimates of 𝑦₀ and 𝑦_𝑁 whose values are not known.

If the values of 𝜇_𝑓(𝑦), 𝑦₀, and 𝑦_𝑁 can be obtained exactly, the accuracy of the Lebesgue integral of 𝑓(𝑥) is determined by the properties of 𝜇_𝑓(𝑦) such as continuity and differentiability. One advantage of this method is that even though 𝑓(𝑥) is oscillatory or singular in its bounded domain, its relative measure function 𝜇_𝑓(𝑦) keeps some good properties, bounded and monotonically decreasing. It offers the opportunity to have better results than conventional numerical integral methods.

However, it is often difficult, or perhaps impossible, to find explicit formula for 𝜇_𝑓(𝑦) and exact bound 𝑦₀ and 𝑦_𝑁. The required values need to be estimated which leads to new errors (ii) and (iii) in the numerical results. What we desire is to reduce the effects of the additional errors on the final error of the Lebesgue integration as possible as we can and to obtain the extent of additional errors’ effects. Then we need to develop methods to estimate the unknown values and analyze the rate of convergence.

In this paper, we will mainly discuss the second kind of error caused by the numerical measures and its effect on the NMI. The paper is organized in 6 sections. In Section 2, we recall a general estimation of the measure function. Then we present its corresponding convergence analysis. Further, by combining Lagrange interpolation, we propose a specific method for approximating the measure function and analyze the rate of convergence. A highly accurate algorithm is presented especially for strictly monotonic functions. In Section 3, we present two examples of estimating measure with one monotonic function and another oscillatory function. In Section 4, we present the error bound of NMI which is controlled by the error of numerical measure and error of integral of exact measure. Further, we prove that the singularity can be reduced by NMI for singular integrals. In Section 5, we give three examples of numerical integrals calculated by NMI based on the numerical measure. Finally, we make a conclusion in Section 6.

2. Estimation of the Measure Function and Its Convergence Analysis

To calculate the Lebesgue integrals by the method NMI, it is significant to obtain the values of measure function at some necessary points. However, the values of measure function cannot be obtained exactly through an explicit formula in most of cases. They need to be estimated numerically. And it will be found that the accuracy of the values of measure function will greatly affect the final accuracy of the Lebesgue integral.

To estimate the measure of one-dimensional function 𝑓(𝑥), we divide 𝐼 into 𝑛 subsets, {𝐼_𝑖∶=[𝑥_𝑖,𝑥_𝑖+1]∶𝑖=1,2,…,𝑛}. Define 𝑝_𝑖∶=𝑚(𝐼_𝑖)=𝑥_𝑖+1−𝑥_𝑖 and obviously ^𝑛_𝑖=1𝑝_𝑖=𝑏−𝑎. (2) On each interval 𝐼_𝑖, we choose 𝑥_𝑖⩽𝑥_𝑖,1<𝑥_𝑖,2<⋯<𝑥_𝑖,𝑘⩽𝑥_𝑖+1, 𝑘∈ℕ, as testing points. Then the values {𝑓(𝑥_𝑖𝑗)∶𝑥_𝑖𝑗∈𝐼_𝑖,𝑖=1,…,𝑛,𝑗=1,…,𝑘}, 𝑘∈ℕ, are calculated. Given any 𝑦∈𝑅(𝑓), the approximation of 𝜇_𝑓(𝑦) follows 𝜇_𝑓𝑦=^𝑛_𝑖=1𝜖_𝑖𝑝_𝑖, (3) where 0⩽𝜖_𝑖⩽1.

Different methods of choosing {𝜖_𝑖∶𝑖=1,2,…,𝑛} will determine different approximations of 𝜇_𝑓(𝑦). A general assignment for {𝜖_𝑖∶𝑖=1,2,…,𝑛} which was proposed by [1] is𝜖_𝑖=0,if𝑦>max_{1⩽𝑗⩽𝑘}𝑓𝑥_𝑖𝑗,1,if𝑦⩽max_{1⩽𝑗⩽𝑘}𝑓𝑥_𝑖𝑗,anarbitrarynumberin(0,1),otherwise. (4) Let 𝜒_𝐸(𝑥) be the characteristic function, defined by 𝜒_𝐸(𝑥)∶=1,if𝑥∈𝐸,0,if𝑥∉𝐸. (5) Set 𝑔(𝑥)∶=𝜒_{𝐸(𝑓(𝑥)⩾𝑦)}(𝑥). We have 𝜇_𝑓𝑦=_𝐼𝑔(𝑥)𝑑𝑥. (6) Define 𝜇_𝑖(𝑦)∶=∫_𝐼𝑖𝑔(𝑥)𝑑𝑥. Then 𝜇_𝑓(𝑦)=∑^𝑛_𝑖=1𝜇_𝑖(𝑦).

Set 𝑀_𝑦∶= number of the segments of 𝐸(𝑓(𝑥)⩾𝑦). For example, consider 𝑓(𝑥)=|sin2𝑥| on [0,𝜋]; then 𝑀_𝑦=2 for 𝑦∈(0,1). The following theorem gives the error analysis of measure approximation (3).

Theorem 1. Let 𝑓(𝑥)∈𝐶[𝑎,𝑏] and 𝑓(𝑥)⩾0 for 𝑥∈[𝑎,𝑏]. {𝐼_𝑖∶𝑖=1,…,𝑛}, 𝑛∈ℕ, is a partition of [𝑎,𝑏]. If 𝜇_𝑓(𝑦) and {𝜖_𝑖∶𝑖=1,2,…,𝑛} are determined by (3) and (4), respectively, then 𝜇_𝑓𝑦−𝜇_𝑓𝑦⩽2𝑀_𝑦max_𝑖∈𝕍𝑝_𝑖, (7) where 𝕍={𝑖∈ℕ∶∃𝑥^′,𝑥^′′∈𝐼_𝑖𝑠.𝑡.𝑓(𝑥^′)⩽𝑦<𝑓(𝑥^′′)}.

Proof. According to the approximate formula (3) and equation (6), we have 𝜇_𝑓𝑦−𝜇_𝑓𝑦=^𝑛_𝑖=1𝜇_𝑖𝑦−𝜖_𝑖𝑝_𝑖⩽^𝑛_𝑖=1𝜇_𝑖𝑦−𝜖_𝑖𝑝_𝑖. (8) Set 𝕍={𝑖∈ℕ∶∃𝑥^′,𝑥^′′∈𝐼_𝑖s.t.𝑓(𝑥^′)⩽𝑦<𝑓(𝑥^′′)}. According to the definition of 𝜖_𝑖, we have 𝜇_𝑖(𝑦)−𝜖_𝑖𝑝_𝑖=0 for 𝑖∉𝕍. Then the inequality (8) can be simplified as 𝜇_𝑓𝑦−𝜇_𝑓𝑦⩽_𝑖∈𝕍𝜇_𝑖𝑦−𝜖_𝑖𝑝_𝑖. (9)
Here it is apparent that 𝜇_𝑖𝑦−𝜖_𝑖𝑝_𝑖⩽𝑝_𝑖, (10) for 0⩽𝜇_𝑖(𝑦),𝜖_𝑖𝑝_𝑖⩽𝑝_𝑖.
By the intermediate value theorem, 𝐼_𝑖, 𝑖∈𝕍 contains at least one end of a segment in the domain 𝐸(𝑓(𝑥)⩾𝑦). Since 𝐸(𝑓(𝑥)⩾𝑦) has only 𝑀_𝑦 segments, the number of 𝐼_𝑖, 𝑖∈𝕍 is not more than 2𝑀_𝑦; that is, Card(𝕍)⩽2𝑀_𝑦, (11) where Card (𝕍) denotes the cardinality of the set 𝕍.
Combining inequalities (9), (10), and (11), we obtain 𝜇_𝑓𝑦−𝜇_𝑓𝑦⩽2𝑀_𝑦max_𝑖∈𝕍𝑝_𝑖. (12)

We note that when 𝑀_𝑦 is bounded and the domain 𝐼 is equally divided into 𝑛 segments, the numerical measure is uniformly convergent and the order of convergence is linear according to Theorem 1.

When the integrand functions have some better properties, the numerical measure can be approximated more correctively. Suppose that nonnegative function 𝑓(𝑥) is invertible in each subinterval 𝐼_𝑖, 𝑖=1,2,…,𝑛. Define 𝑓_i∶=𝑓|_𝐼𝑖 and 𝜇_𝑓𝑖∶=𝑚(𝐸(𝑓_𝑖(𝑥)⩾𝑦)∩𝐼_𝑖). Denote by 𝑅(𝑓_𝑖) the range of 𝑓_𝑖 in 𝐼_𝑖 and by 𝑓_𝑖⁻¹ the inverse function of 𝑓_𝑖 in 𝐼_𝑖. Then we obtain 𝜇_𝑓𝑦=^𝑛_𝑖=1𝜇_𝑓𝑖𝑦,𝜇_𝑓𝑖𝑦=𝑓_𝑖⁻¹𝑦−𝑥_𝑖,if𝑓_𝑖isdecreasingand𝑦∈𝑅𝑓_𝑖,𝑥_𝑖+1−𝑓_𝑖⁻¹𝑦,if𝑓_𝑖isincreasingand𝑦∈𝑅𝑓_𝑖,𝑝_𝑖,if𝑦<min_{𝑥∈𝐼𝑖}𝑓_𝑖(𝑥),0,if𝑦>max_{𝑥∈𝐼𝑖}𝑓_𝑖(𝑥). (13)

Construct a (𝑘−1)th Lagrange interpolation polynomial 𝑃_𝑖(𝑥) in 𝑅(𝑓_𝑖) which interpolates {(𝑓(𝑥_𝑖𝑗),𝑥_𝑖𝑗)∶𝑗=1,2,…,𝑘}. Then the approximations of 𝜇_𝑓𝑖(𝑦) and 𝜇_𝑓(𝑦) are, respectively, 𝜇_𝑓𝑖𝑦=𝑃_𝑖𝑦−𝑥_𝑖,if𝑓_𝑖isdecreasingand𝑦∈𝑅𝑓_𝑖,𝑥_𝑖+1−𝑃_𝑖𝑦,if𝑓_𝑖isincreasingand𝑦∈𝑅𝑓_𝑖,𝑝_𝑖,if𝑦<min_{𝑥∈𝐼𝑖}𝑓_𝑖(𝑥),0,if𝑦>max_{𝑥∈𝐼𝑖}𝑓_𝑖(𝑥),𝜇_𝑓𝑦=^𝑛_𝑖=1𝜇_𝑓𝑖𝑦. (14)(15)

Set 𝜖_𝑖=𝜇_𝑓𝑖(𝑦)/𝑝_𝑖 which reveals that the estimation (15) is a specific case of the general assignment (4). That means that Theorem 1 is valid for the estimation (15).

Theorem 2. Suppose that nonnegative function 𝑓(𝑥) has a partition {𝐼_𝑖∶𝑖=1,2,…,𝑛} in 𝐼 and has inverse function 𝑓_𝑖⁻¹∈𝐶^𝑘(𝑅(𝑓_𝑖)) in each subinterval. If 𝜇_𝑓(𝑦) are determined by (14) and (15), then 𝜇_𝑓𝑦−𝜇_𝑓𝑦_∞=𝒪𝑛^−𝑘, (16) where 𝕍={𝑖∈ℕ∶∃𝑥^′,𝑥^′′∈𝐼_𝑖𝑠.𝑡.𝑓(𝑥^′)⩽𝑦<𝑓(𝑥^′′)}.

Proof. According to Theorem 1 and (13), (14), and (15), we have 𝜇_𝑓𝑦−𝜇_𝑓𝑦⩽min𝑛max_𝑖𝑓_𝑖⁻¹𝑦−𝑃_𝑖𝑦_∞,2𝑀_𝑦max_𝑖∈𝕍𝑝_𝑖. (17)
The proof is finished combining the well-known error of Lagrange interpolation polynomials.

According to the proof of Theorem 2, the accuracy of measure function depends on the accuracy of Lagrange interpolation polynomials of the inverse functions in each subinterval. Numerical experiments will be done to affirm the validity of the method. At the end of this section, we would like to introduce a highly accurate algorithm to calculate the measure especially for strictly monotonic functions. It is called split-half algorithm.

Algorithm 3 (split-half algorithm). Consider the following steps.
Step  1. Given 𝑦∈𝑅(𝑓) and 𝑁∈ℕ, set 𝑖∶=0, Δ=𝑏−𝑎, 𝑥_𝑚=(𝑏−𝑎)/2, and 𝜇^𝑖_𝑓(𝑦)=0. Determine the monotonicity of the function 𝑓, increasing or decreasing.
Step  2. Evaluate the function at 𝑥_𝑚: 𝑇₁𝑥∶=1,𝑓_𝑚𝑥>𝑦,−1,𝑓_𝑚𝑥<𝑦,0,𝑓_𝑚=𝑦. (18)
Step  3. If 𝑇₁=0, we can get the accurate measure 𝜇_𝑓^𝑖+1𝑦=𝜇^𝑖_𝑓𝑦+Δ2. (19)
Otherwise, 𝜇_𝑓^𝑖+1𝑦=𝜇^𝑖_𝑓𝑦+𝑇₁+12Δ2. (20) Set 𝑖←𝑖+1, Δ←Δ/2, and 𝑥_𝑚←𝑥_𝑚−𝑇₁Δ/2 for increasing functions or 𝑥_𝑚←𝑥_𝑚+𝑇₁Δ/2 for decreasing functions. And go back to Step  2 until 𝑖=𝑁.

The error of split-half algorithm satisfies 𝜇_𝑓(𝑦)−𝜇^𝑁_𝑓(𝑦)_∞⩽2^−𝑁(𝑏−𝑎). (21)

3. Numerical Examples of Measure

Example 1. Consider 𝑓(𝑥)=ln𝑥, 𝑥∈[1,5].

Since ln𝑥 is monotonically increasing, 𝑦₀=0, 𝑦_𝑁=ln5, and 𝜇_𝑓(𝑦)=5−𝑒^𝑦, 𝑦∈𝑅(𝑓). 𝑓⁻¹(𝑥)∈𝐶^∞(𝑅(𝑓)).

Table 1 lists the respective measure errors and their convergence orders (briefly, C.O.). The error of the approximation for values 𝑛 and 𝑘 is presented by 𝑒_𝑛,𝑘∶=‖𝜇_𝑓(𝑦)−𝜇_𝑓(𝑦)‖_∞, where 𝜇_𝑓(𝑦) is determined by (14) and each convergence order is computed by ln(𝑒_𝑛/𝑒_𝑚)/ln(𝑚/𝑛).

Table 1 

Errors of 𝜇(𝑦).

Example 2. Consider an oscillatory function 𝑓(𝑥)=|sin30𝑥|, 𝑥∈[0,𝜋/3].

The minimum and maximum of 𝑓(𝑥) are 𝑦₀=0 and 𝑦_𝑁=1, respectively. For 𝑓(𝑥) is a periodic function, we can obtain the explicit formula of measure function: 𝜇𝑦=13𝜋−2arcsin𝑦,𝑦∈𝑅𝑓. (22)

Numerical results of errors are shown in Table 2. Since the measure function does not possess good differentiability when 𝑦=0, the errors do not have stable convergence order of 𝒪(𝑛^−𝑘).

Table 2 

Errors of 𝜇(𝑦).

4. Error Analysis of NMI

In this section, we will analyze the whole error of the integral ∫^𝑏_𝑎𝑓(𝑥)𝑑𝑥. As we care greatly about the influence of errors caused by the numerical measures, we assume that the maximum and minimum of the function 𝑓(𝑥) are known; that is, we will not consider the errors of 𝑦₀ and 𝑦_𝑁 here. To analyze the error of the integral ∫^𝑏_𝑎𝑓(𝑥)𝑑𝑥, it is necessary to choose an integral rule for the Riemann integral of the measure function. An appropriate rule for the one-dimensional integral is dependent on the character of 𝜇(𝑦).

Let 𝑈⊂ℝ be an interval and assume 𝑔∈𝐶(𝑈). To approximate the integral 𝑄(𝑔)∶=∫_𝑈𝑔(𝑡)𝑑𝑡, we consider numerical quadrature rule of the form 𝑄_𝑛𝑔∶=^𝑛_𝑗=0𝑤_𝑛,𝑗𝑔𝑡_𝑛,𝑗, (23) where 𝑡_𝑛,𝑗∈𝑈,𝑗=0,1,…,𝑛, are quadrature points and 𝑤_𝑛,𝑗,𝑗=0,1,…,𝑛, are real quadrature weights.

Define 𝐸_𝑄𝑛(𝑔)∶=𝑄(𝑔)−𝑄_𝑛(𝑔) the integral error of 𝑔 under the given quadrature rule.

Integrating the measure function by the numerical quadrature rule (23) in the fundamental equation (1), we have _𝐼𝑓(𝑥)𝑑𝑥=𝑦₀(𝑏−𝑎)+𝑄_𝑛𝜇_𝑓+𝐸_𝑄𝑛𝜇_𝑓=𝑦₀(𝑏−𝑎)+𝑄_𝑛𝜇_𝑓+𝑄_𝑛𝜇_𝑓−𝜇_𝑓+𝐸_𝑄𝑛𝜇_𝑓, (24) where 𝜇_𝑓 and 𝜇_𝑓 are, respectively, the measure function and numerical measure function of 𝑓(𝑥). So the numerical quadrature rule of NMI corresponding to 𝑓 is 𝐿𝑄_𝑛𝑓∶=𝑦₀(𝑏−𝑎)+𝑄_𝑛𝜇_𝑓. (25) Define 𝐸_𝐿𝑄𝑛(𝑓)∶=∫_𝐼𝑓(𝑥)𝑑𝑥−𝐿𝑄_𝑛(𝑓). We have the following fundamental theorem for the error of NMI.

Theorem 4. Let 𝑓(𝑥)∈ℂ[𝑎,𝑏] and 𝑓(𝑥)⩾0 for 𝑥∈[𝑎,𝑏]. By estimating ∫_𝐼𝑓(𝑥)𝑑𝑥 by the numerical quadrature rule of NMI (25), the error satisfies 𝐸_𝐿𝑄𝑛𝑓⩽𝑄_𝑛𝜇_𝑓−𝜇_𝑓+𝐸_𝑄𝑛𝜇_𝑓⩽𝜇_𝑓−𝜇_𝑓∞^𝑛_𝑗=0𝑤_𝑛,𝑗+𝐸_𝑄𝑛𝜇_𝑓. (26)

According to Theorem 4, the accuracy of numerical quadrature rule of NMI depends on the accuracy of numerical measure and the character of measure function 𝜇_𝑓. A necessary condition to have better accuracy by NMI is that the measure function 𝜇_𝑓 should possess better properties. For bounded oscillatory functions, the integral of the exact measure function can be approximated more accurate for its monotonic property. As to singular functions, it can be verified in the following that NMI can decrease the degree of the singularity of the integral.

When 𝑦_𝑁 is infinite, by making the variable change of 𝑧=1/𝑦 in the fundamental equation (1) of NMI, it can be rewritten as _𝐼𝑓(𝑥)𝑑𝑥=𝑦₀(𝑏−𝑎)+^1/𝑦0₀1𝑧²𝜇_𝑓1𝑧𝑑𝑧, (27) where 𝑦₀ is supposed to be 𝑦₀⩾𝜖>0.

Denote 𝑔(𝑥)∶=(1/𝑥²)𝜇_𝑓(1/𝑥), 𝑥∈(0,1/𝑦₀]. Before we prove that 𝑔(𝑥) is less singular than 𝑓(𝑥), a lemma should be stated.

Lemma 5. Let 𝑓(𝑥)∈𝐶(𝐼), ℎ(𝑡)∶=𝑓(𝑥(𝑡)), and 𝑥(𝑡)∶=𝑘𝑡+𝑐 where 𝑘>0, 𝑐∈ℝ. 𝜇_𝑓 and 𝜇_ℎ are the measure functions corresponding to 𝑓 and ℎ, respectively. Then 𝜇_𝑓(𝑦)=𝑘𝜇_ℎ(𝑦), ∀𝑦∈𝑅(𝑓), where 𝑅(𝑓) is the range of 𝑓.

Proof. Since 𝑘>0, 𝑔 has the same monotonicity as 𝑓. When 𝑓 is monotonically decreasing, 𝜇_𝑓(𝑦)=𝑓⁻¹(𝑦)−𝑎 and 𝜇_ℎ(𝑦)=ℎ⁻¹(𝑦)−(𝑎−𝑐)/𝑘, respectively. We obtain ℎ⁻¹𝑦=1𝑘𝑓⁻¹𝑦−𝑐. (28) Substituting the inverse function of ℎ into 𝜇_ℎ, we have 𝜇_ℎ𝑦=1𝑘𝑓⁻¹𝑦=𝜇−𝑎^𝑓𝑦𝑘. (29)
When 𝑓 is monotonically increasing, 𝜇_𝑓(𝑦)=𝑏−𝑓⁻¹(𝑦) and 𝜇_ℎ(𝑦)=(𝑏−𝑐)/𝑘−ℎ⁻¹(𝑦), respectively. It can be obtained similarly that 𝜇_ℎ(𝑦)=𝜇_𝑓(𝑦)/𝑘.
For any 𝑓∈𝐶(𝐼), there is a partition {𝐼_𝑖∶𝑖∈ℕ} of 𝐼 such that 𝑓 is monotonic in each subinterval. Define 𝑓_𝑖∶=𝑓|_𝐼𝑖 and 𝜇_𝑓𝑖∶=𝑚(𝐸(𝑓_𝑖⩾𝑦)∩𝐼_𝑖), 𝑖∈ℕ. Correspondingly, define ℎ_𝑖∶=𝑓_𝑖(𝑥(𝑡)) and 𝜇_ℎ𝑖∶=𝑚(𝐸(ℎ_𝑖⩾𝑦)∩𝑥⁻¹(𝐼_𝑖)), 𝑖∈ℕ.
Then it is obvious that 𝜇_𝑓𝑦=_𝑖∈ℕ𝜇_𝑓𝑖𝑦,𝜇_ℎ𝑦=_𝑖∈ℕ𝜇_ℎ𝑖𝑦. (30) According to the preceding proof, 𝜇_𝑓𝑖𝑦=𝑘𝜇_ℎ𝑖𝑦 (31) since 𝑓_𝑖, ∀𝑖∈ℕ, is monotonic in its domain. Combining (30) and (31), we finally arrive at the equation 𝜇_𝑓(𝑦)=𝑘𝜇_ℎ(𝑦).

Let 𝑆 be a set of 𝐼=[𝑎,𝑏] containing a finite number of points. Define a function associated with 𝑆 by 𝑤_𝑆∶=inf{|𝑥−𝑡|∶𝑡∈𝑆}. (32) For 0<𝛼<1, a real unbounded function 𝑓 is said to be of Type(𝛼,𝐼,𝑆), if 𝑓(𝑥)⩽𝐶𝑤_𝑆(𝑥)^−𝛼,𝑥∉𝑆,𝑓∈𝐶(𝐼⧵𝑆), (33) where 𝐶 is a positive constant. The parameter 𝛼 is called index of singularity.

Theorem 6. Let 𝑓∈𝑇𝑦𝑝𝑒(𝛼,𝐼,𝑆) and 𝑓(𝑥)⩾𝜖>0 for 𝑥∈𝐼. 𝑔(𝑥)=(1/𝑥²)𝜇_𝑓(1/𝑥), 𝑥∈(0,1/𝑦₀], where 𝜇_𝑓 is the measure function of 𝑓 and 𝑦₀=min_𝑥∈𝐼𝑓(𝑥)⩾𝜖>0. Then 𝑔(𝑥)∈𝑇𝑦𝑝𝑒((2𝛼−1)/𝛼,(0,1/𝑦₀],{0}).

Proof. Assume 𝑆={𝑎}or{𝑏}. By making the variable change 𝑥(𝑡)=(𝑏−𝑎)𝑡+𝑎 for 𝑆={𝑎} or 𝑥(𝑡)=(𝑏−𝑎)𝑡+𝑏 for 𝑆={𝑏}, we define ℎ(𝑡)∶=𝑓(𝑥(𝑡)), 𝑡∈(0,1]. Then ℎ∈ Type(𝛼,(0,1],{0}).
Since ℎ(𝑡) is singular at 𝑡=0, there exists 𝑡₀∈(0,1] such that ℎ(𝑡) is monotonic in the interval (0,𝑡₀]. That means that ∀𝑦⩾ℎ⁻¹(𝑡₀) s.t. 𝜇_ℎ(𝑦)=|ℎ⁻¹(𝑦)|. According to the property of ℎ, we can obtain 𝑦=ℎℎ⁻¹𝑦⩽1ℎ⁻¹𝑦^𝛼. (34) The resultant inequality reveals that 𝜇_ℎ(𝑦)⩽𝑦^−1/𝛼.
By Lemma 5, we have 𝜇_𝑓(𝑦)=(𝑏−𝑎)𝜇_ℎ(𝑦)⩽(𝑏−𝑎)𝑦^−1/𝛼.
Then combining the inequality of 𝜇_𝑓 and the expression of 𝑔, we can obtain 0⩽𝑔(𝑥)⩽(𝑏−𝑎)𝑥^{−(2𝛼−1)/𝛼}, ∀𝑥∈(0,𝑡₀].
Since 𝑔(𝑥) is continuous in [𝑡₀,1/𝑦₀], there exists a positive constant 𝐶 such that ∀𝑥∈(0,1/𝑦₀], 0⩽𝑔(𝑥)⩽𝐶𝑥^{−(2𝛼−1)/𝛼}. (35) According to the definition (33), we have 𝑔(𝑥)∈ Type((2𝛼−1)/𝛼,(0,1/𝑦₀],{0}).
More generally, 𝑆 contains more than one point. Assume 𝑆={𝑎=𝑠₁<𝑠₂<⋯<𝑠_𝑚=𝑏}. Set 𝑡_2𝑖=𝑠_𝑖𝑡,𝑖=1,2,…,𝑚_2𝑖−1=12𝑠_𝑖+𝑠_𝑖+1,𝑖=1,2,…,𝑚−1. (36) In each interval (𝑡_𝑖,𝑡_𝑖+1), the function 𝑓(𝑥) has only one singularity. Namely, a singularity is located at 𝑡_𝑖 if 𝑖 is even and at 𝑡_𝑖+1 if 𝑖 is odd.
Define 𝑓_𝑖(𝑥)∶=𝑓(𝑥)|_{(𝑡𝑖,𝑡𝑖+1)}, 𝜇_𝑓𝑖(𝑦)∶=𝑚(𝐸(𝑓(𝑥)⩾𝑦)∩(𝑡_𝑖,𝑡_𝑖+1)), and 𝑔_𝑖(𝑦)∶=(1/𝑦²)𝜇_𝑓𝑖(1/𝑦). Then we have 𝜇_𝑓𝑦=^2𝑚_𝑖=1𝜇_𝑓𝑖𝑦,𝑔𝑦=^2𝑚_𝑖=1𝑔_𝑖𝑦. (37)
By the preceding proof, it can be obtained that 0⩽𝑔_𝑖(𝑥)<𝐶_𝑖𝑥^{−(2𝛼−1)/𝛼}, ∀𝑥∈(0,1/𝑦_0𝑖], where 𝑦_0𝑖∶=min_{𝑥∈(𝑡𝑖,𝑡𝑖+1)}𝑓(𝑥)⩾𝑦₀ and 𝐶_𝑖 is a positive constant, 𝑖=1,2,…,2𝑚. Since 𝜇_𝑓𝑖(1/𝑥)=0 for 𝑥∈(1/𝑦_0𝑖,1/𝑦₀], 0⩽𝑔_𝑖(𝑥)<𝐶_𝑖𝑥^{−(2𝛼−1)/𝛼} is valid for 𝑥∈(0,1/𝑦₀]. Combining (37), we prove that 𝑔(𝑥)∈ Type((2𝛼−1)/𝛼,(0,1/𝑦₀],{0}).

Since (2𝛼−1)/𝛼<𝛼,∀𝛼∈(0,1), NMI can reduce the singularity for singular integrals. It supplies the possibility to improve the accuracy of integrals. Theoretically, we need not know where the singularities lie.

As to whether NMI can realize the better accuracy of integrals, it depends on the accuracy of the numerical measure. In the following part, we will present some numerical integrals results by NMI and compare them with results by conventional methods.

5. Numerical Examples of NMI

In this section, we will present three kinds of integrals, including normal, oscillatory, and singular integrals, to verify the efficiency of NMI. In the numerical experiments, the Gauss-Legendre formula or its composite one will be used to calculate the integral of measure functions.

Example 1. Consider the integration ∫⁵₁ln(𝑥)𝑑𝑥=5ln5−4.

We have known that 𝑦₀=0, 𝑦_𝑁=ln5, and 𝜇(𝑦)=5−𝑒^𝑦, 𝑦₀⩽𝑦⩽𝑦_𝑁.

In this numerical example, we adopt 5-point Gauss-Legendre formula to calculate the integral of measure function and original integral. The errors of original integral and the integral of exact measure function are 3.4407𝑒−005 and 1.6778𝑒−010, respectively. It shows clearly that the accuracy of the numerical integral can be greatly improved by making the NMI transform for this example. Numerical results of errors of the integration by NMI are shown in Table 3. The measure is estimated by (14) and (15) with values 𝑛 and 𝑘. The high accuracy of numerical results of NMI requires the accuracy of measure functions as illustrated in Table 3.

Table 3 

Errors of integration ∫⁵₁ln(𝑥)𝑑𝑥 by NMI.

Example 2. Consider the integration of an oscillatory function ∫₀^𝜋/3|sin30𝑥|𝑑𝑥=2/3.

In this example, 𝑦₀=0, 𝑦_𝑁=1, and 𝜇(𝑦)=𝜋/3−(2arcsin(𝑦)/3), 𝑦₀⩽𝑦⩽𝑦_𝑁. Since the measure function is weakly singular at 𝑦=1, we adopt the Gauss-type quadrature for weakly singular integrals proposed by [2]. According to the quadrature rule, we set the parameters 𝑘=4 and 𝑞=6. In detail, set 𝑁=𝑛/𝑘, where 𝑛 is the number of evaluations of the function showed in Table 4. Then choose (𝑁+1) points 𝑡_𝑗=1−(𝑗/𝑁)^𝑞 so that the subintervals 𝐼_𝑗∶=[𝑡_𝑗,𝑡_𝑗+1],𝑗=0,1,…,𝑁, form a partition for [0,1]. Then transform the partition into the domain of integral and use k-point Gauss-Legendre formula in each subinterval to calculate the integral. By the quadrature, the errors of the integral with exact measure function is showed in Table 4 represented by GL. To show the efficiency of the proposed method, we compare the results with those obtained by composite Simpson’s rule with 𝑛+1 evaluations of function which is denoted by CS in Table 4. It is obvious that the accuracy can be improved under the correct measure function.

Table 4 

Errors of integration ∫₀^𝜋/3|sin30𝑥|𝑑𝑥.

Errors of the integration by NMI combining GL are shown in Table 5 which is denoted by GLNMI. We set the number of evaluations of function 32 for the integral of measure function by the Gauss-type quadrature. The measure is estimated by (14) and (15) with values 𝑛 and 𝑘. The accuracy of NMI is extremely restricted by the accuracy of measure.

Table 5 

Errors of integration ∫₀^𝜋/3|sin30𝑥|𝑑𝑥 by GLNMI.

Example 3. Consider the integration of a singular function ∫¹₀(1/𝑥^0.7)𝑑𝑥=10/3.

Since the integral and the measure integral are both weakly singular, we adopt the Gauss-type quadrature used in Example 2. We set the parameters 𝑘=2 and 𝑞=50/3. Split-half algorithm is used to approximate the measure function with 𝑁=100 in this example. The numerical results are presented in Table 6 where GL represents using only the method of Gauss-type quadrature while GLNMI represents combining NMI and Gauss-type quadrature together. It is illustrated that the accuracy can be improved by NMI.

Table 6 

Errors of integration ∫¹₀(1/𝑥^0.7)𝑑𝑥.

6. Conclusion

In this paper, we mainly discuss how to estimate the measure of a function and the convergence of the numerical measure. Further, we analyze the error of numerical measure integral and verify that the singular integrals can be improved theoretically. Accordingly, some numerical examples are given to validate the theoretical results.

To calculate the integration by NMI, it is greatly important to estimate the measures accurately as possible as we can. The method presented here is very effective for functions which have piecewise differential inverse function.

The study of the NMI field is still new. Investigations should be expanded to the multidimensional measures and integrations. Maybe, the method NMI is more suitable for multidimensionality because it presents an exact reduction of multidimensional integrals to one-dimensional integrals. More researches can be done in this field.

Acknowledgment

The authors are very grateful to Professor Tian-xiao He for his kind help and valuable recommendations. This work is partially supported by National Natural Science Foundation of China under Projects 11271370 and 61101183 and CSC Scholarship.