Half-Sweep Arithmetic Mean Method with High-Order Newton-Cotes Quadrature Schemes to Solve Linear Second Kind Fredholm Equations

The main purpose of this paper is to examine the effectiveness of the Half-Sweep Arithmetic Mean (HSAM) method in solving the dense linear systems generated from the discretization of the linear Fredholm integral equations of the second kind. In addition, the applications of the various orders of closed Newton-Cotes quadrature discretization schemes will be investigate in order to form linear systems. Furthermore, the basic formulation and implementation for the proposed method are also presented. Some illustrative examples are given to point out the efficiency of the proposed method.


Introduction
Two-stage iterative methods are one of the widely used and successful classes of numerical algorithms to solve any system of linear algebraic equations that is generated by using approximation equations.Actually, there are many two-stage iterative methods can be considered such as the Alternating Group Explicit (AGE) [1], Iterative Alternating Decomposition Explicit (IADE) [2], Reduced Iterative Alternating Decomposition Explicit (RIADE) [3], Half-Sweep Iterative Alternating Decomposition Explicit (HSIADE) [4], Quarter-Sweep Iterative Alternating Decomposition Explicit (QSIADE) [5], Block Jacobi [6] and Arithmetic Mean (AM) [7] methods.
In 2004, Sulaiman et al. [8] modified the standard AM method also named as the Full-Sweep Arithmetic Mean (FSAM) method by combining the concept of the half-sweep iteration method and then called as the Half-Sweep Arithmetic Mean (HSAM) method.The concept of the half-sweep iteration method is introduced via the Explicit Decoupled Group (EDG) iterative method for solving two-dimensional Poisson equations [9].Further studies of the HSAM method have been also conducted to solve Poisson's equation using finite element [10] and fourth-order finite difference [11] methods.Besides that, HSAM method also have been applied to solve linear systems generated from the discretization of the Fredholm integral equations, see [12][13][14].In this paper, the application of the FSAM and HSAM methods using the approximation equation based on quadrature methods for solving linear second kind Fredholm integral equations is examined.
Generally, linear second kind integral equations of Fredholm type in the standard form can be defined as follows where the parameter λ , kernel ( ) are given, and ( ) is the unknown function to be determined.The kernel function ( ) t x K , is assumed to be absolutely integrable and satisfy other properties that are sufficient to imply the Fredholm alternative theorem.
Theorem (Fredholm alternative) [15] Let χ be a Banach space and let  [15] Let χ and Υ be normed vector space and let

Definition (Compact operators)
, the sequence has a subsequence that is convergent to some point in Υ .Compact operators are also called completely continuous operators.
The remainder of this paper is organized in following way.In next section, the formulation of the full-and half-sweep quadrature approximation equations based on repeated Newton-Cotes schemes is described.The latter section of this paper will discuss the formulations of the FSAM and HSAM methods and some numerical results will be shown to indicate the effectiveness of the HSAM method.Meanwhile, conclusion and directions of the future works are drawn in final section.

Half-Sweep Quadrature Approximation Equations
In many application areas, numerical approaches were used widely to solve Fredholm integral equations of the second kind than the analytical methods.For the solution of a linear integral equation, reduction of integral equation to the solution of system of linear algebraic equations is the basic concept used to solve the problems.There are many methods can be used to discretize the linear Fredholm integral equations into linear systems such as quadrature [15][16][17][18][19] and projection [20][21][22][23] methods.
In this paper, however, discretization schemes based on quadrature method were used to discretize the linear Fredholm integral equations of the second kind in order to form a linear system.Quadrature formulas in general have the form To facilitate in formulating the full-and half-sweep approximation equations for linear Fredholm equation of the second kind, further discussion will be restricted onto Newton-Cotes quadrature method, which is based on interpolation formulas with equally spaced data.In this paper, four different schemes in Newton-Cotes method such as repeated trapezoidal (RT), repeated Simpson's 3 1 (RS1), repeated Simpson's 8 3 (RS2) and repeated Boole (RB) will be applied to discretize the problem.RT, RS1, RS2 and RB are first, second, third and fourth order schemes respectively.Meanwhile, Figure 1 shows the finite grid networks in order to form the full-and half-sweep approximation methods for problem (1).Based on the Figure 1, the full-and half-sweep iterative methods will compute approximate values onto node points of type only until the convergence criterion is reached.Then other approximate solutions at remaining points (points of the different type, ) are obtained using the direct method [9,24,25].By applying Eq. (2) into Eq.( 1) and neglecting the error, ( ) y n ε a system of linear algebraic equations can be formed for approximation values ( ) K .The following linear system generated using the Newton-Cotes quadrature method can be easily shown as Based on repeated trapezoidal, repeated Simpson's 3 1 , repeated Simpson's 8 3 and repeated Boole schemes, numerical coefficient j A satisfied following relations respectively. (5) where the constant step size, h is defined as n is the number of subintervals in the interval [ ] b a, and then consider the discrete set of points be given as ih a x i + = . The value of p, which corresponds to 1 and 2, represents the full-and half-sweep cases respectively.

Derivation of the Half-Sweep Arithmetic Mean Method
As afore-mentioned in the previous section, the HSAM method is one of two-stage iterative methods.It means that the iterative process for this method consists of solving two independent systems such as ỹ and ∧ y .To develop formulation of HSAM and FSAM methods, matrix M needs to be decomposed into where L, D and T are strictly lower triangular, diagonal and strictly upper triangular matrices respectively.Thus, for real positive acceleration parameter, r, the general scheme for both AM methods is given by In Eq. ( 10), ) (k y represents as an unknown vector at the k th iteration and ) 0 ( y is an initial vector approximation to the solution y of (9).Practically, the value of r will be determined by implementing some computer programs and then choose one value of r, where its number of iterations is the smallest.By determining values of matrices L, D and T as stated in Eq. ( 9), the general algorithm for FSAM and HSAM schemes in Eq. ( 10) would be described in Algorithm 1.The FSAM and HSAM algorithms are explicitly performed by using all equations at level (1) and ( 2) alternatively until the specified convergence criterion is satisfied.

Numerical Simulations
In this section, some numerical simulations have been conducted to assess the effectiveness of the proposed method through three parameters such as number of iterations, execution time and maximum absolute error.In comparison, the Full-Sweep Gauss-Seidel (FSGS) method act as the control of comparison of numerical results.
In the implementation of the iterative methods, the convergence test considered the tolerance error, ε of 10 10 − .As mentioned above, Newton-cotes quadrature schemes are used to discretize and to form a linear system for the following example.
Example 1 (Wang [26]) and the exact solution of problem ( 11) is given by ( ) Results of numerical experiments, which were obtained from implementations of the iterative methods for Example 1, have been recorded in Table 1.

Conclusions
In the previous section, it has shown that the quadrature approximation equations based on Newton-Cotes formulas can be easily formulated and rewritten in general form as shown in Eq. ( 3).Through numerical results obtained in Tables 1 and 2, clearly it shows that by applying the AM methods can reduce the number of iterations compared to Gauss-Seidel (GS) methods.Through the numerical results obtained for Example 1 by using RT and RS2 schemes, number of iterations for FSAM and HSAM methods have declined approximately 56.48 -56.77% and 56.70 -56.99% respectively compared to FSGS method.Number of iterations for FSAM and HSAM methods with RS1 and RB schemes for Example 1 have decreased by 56.48 -56.70% and 56.54 -56.99% respectively compared to FSGS method.Meanwhile, number of iterations for both AM iterative methods for  Example 2 using RT, RS1, RS2 and RB discretization schemes as shown in Table 2 decreased approximately 41.82 -42.86% compared with FSGS method.For HSGS method with RT, RS1, RS2 and RB schemes, number of iterations is nearly same to the FSGS method for both examples, refer Tables 1 and 2.
Through the observation in Tables 1 and 2, FSAM and HSAM iterative methods reduce the execution time compared to the FSGS method respectively.Overall, the computational time for HSGS and HSAM methods together with RT, RS1, RS2 and RB is superior compared to the corresponding full-sweep iterative methods.This is due to the computational complexity of the half-sweep iterative methods are approximately 50% less than fullsweep iterative methods respectively.
In terms of accuracy of numerical solutions obtained, repeated Simpson's 3 1 , repeated Simpson's 8 3 and repeated Boole schemes are more accurate than the repeated trapezoidal scheme.From the observation of the results obtained, it shows that application of the half-sweep iteration concept reduce the accuracy of the solution for RS1, RS2 and RB discretization schemes compared to the full-sweep cases respectively.Decrement of the accuracy for half-sweep iterative methods is due to the computational technique for calculating the remaining points using direct method as proposed by Abdullah [9].
For future works, this study will be extended to investigate the applications of half-sweep concept with different approach for calculating the remaining points.The mesh sizes which have been proposed in this paper will generate a completed group for each discretization schemes.However, this study will also be proceeded to examine for the case of ungroup.
of the partition points of the integration interval [ ] that do not depend on the function ( )

Figure 1 :
Figure 1: a) and b) show distribution of uniform node points for the full-and half-sweep cases respectively.

Table 1 :
Comparison of a number of iterations, execution time (seconds) and maximum absolute for the iterative methods using RT, RS1, RS2 and RB discretization schemes (Example 1).

Table 2 :
Comparison of a number of iterations, execution time (seconds) and maximum absolute error for the iterative methods using RT, RS1, RS2 and RB discretization schemes (Example 2).