A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed.The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

In the paper, the function-valued Padé-type approximation (2DFPTA) is used to solve 2DF-II.In order to compute 2DFPTA, a recursive algorithm based on Sylvester identity is proposed.The remainder of this paper is organized as follows.In Section 2, we give the definition of 2DFPTA [18].In Section 3, we apply Sylvester identity to propose a recursive algorithm to compute 2DFPTA.In Section 4, we compare our algorithm with the two algorithms in [4,18].

Function-Valued Padé-Type Approximants and Its Convergence Analysis
In this section, we will give the definition of the functionvalued Padé-type approximation [15][16][17][18].Let  () : P → C be a linear functional on the polynomial space P, and define it by where  () (  ) = 0 for  +  < 0. We can obtain from linear functional  =  (0) in (6) that Let where V  ∈ P  is a scalar polynomial of degree  and assume   ̸ = 0. Define the polynomial  with function-valued coefficient by Note that  acts on  and   (, ) is a function-valued polynomial in  of degree .Set w (, ) =     (,  −1 ) .
The polynomial V  is called the generating polynomial of the Padé-type approximation (/).It can be arbitrarily chosen and we have  degrees of freedom.
According to the relations ( 9) and (11), we can from [19] obtain the following error formula with the linear functional form: Note that the error formula ( 15) is disadvantageous to estimate error numerically, and it implies from (13) that the Padé-type approximation property is as  → 0 for Ṽ () ̸ = 0.
In order to compute conveniently, we choose max ∈ | ∑  =0   c  ()| as the maximum absolute value of the coefficient of  +1 on  in (13).We can also observe from (15) and ( 16) that if the coefficients are given, the order of the approximation is  +1 .

A Triangle Recursive Algorithm
of (/)  (,) As the definition given in Section 2, we find that the key to calculating 2DFPTA is to compute its the generating polynomial V  ().If the generator polynomial V  () is determined, we can compute the 2DFPTA of type (, ) according to (10) and (12).In the section, we will apply Sylvester's identity [20] to propose a three-term recurrence formula for computing Ṽ () and establish a complete recursive algorithm for calculating 2DFPTA.
Substitute them into the formula (18) and we conclude that Due to  ()   () −1 ̸ = 0, we have By (25) we get  (+1) −1 =  (+1) −1 V (+1) −1 and substitute it into the above formula; then that is, According to the higher-order linear functional and nature of the determinant, we have and Dividing ( 34) by ( 35), we deduce that From ( 33) and (36), we obtain that Example 6.Let us choose an example of  = 6 to illustrate the process to generate V 6 ().By the three-term recurrence relation ( 16), the calculation program of V 6 () can be arranged according to the following bottom-down triangle: When computing V ()  we mainly use the left polynomial V (+1) −1 , the polynomial V () −1 in the top left corner in the table, and their relation (29).It is not difficult to find that, in every column, the number of polynomial is in decline when k is bigger and bigger.
It is noticed that V ()  can also be arranged by another pattern as follows: When computing V ()  we mainly use the above polynomial V () −1 , the polynomial V (+1) −1 in the top-right corner in the table, and their relation (29).Similarly, the number of polynomials is in decline when  is bigger and bigger in every row.In this way, we can recursively get V 6 () = V (0) 6 .From the three-term recurrence relation (29), we now build a complete recursive algorithm for calculating 2DFPTA.The advantage of this algorithm is that in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and is gradually calculated.And its specific computational steps are as follows.
In the following we use the triangle algorithm to yield (2/2)  (, ) of the integral equation.

Solution.
The first few terms of the power series for (, ) are given by Here  =  = 1,  =  −  + 1 = 1 and we set  = .

Numerical Results
In this section, we will consider the square domain case of  = [, ] × [, ].We apply Algorithm 7 for 2DFPTA to approximate the corresponding 2DF-II [11].Compared with the methods in [4,18], numerical experiments show that our method improves the calculation accuracy.Furthermore, our method can estimate the corresponding characteristic values of (, ) from the real roots of Ṽ ().
From Figure 1, we can observe that the error is 1.05 × 10 −13 and in the process we just only employ the first + = 4 items of Neumann series (4), while in [4] the error was 8.7 × 10 −5 when the first 5 items of Chebyshev series corresponding to this equation were used and in [18] the corresponding error was 5.3 × 10 −13 .By Algorithm 7, we also get that where their eigenvalue has estimates; see Table 1.
From Figure 2, we can observe that the error is 3.7×10 −15 and in the process we just only employ the first  +  = 4 items of Neumann series (4).While in [4] the error was 8.8 × 10 −7 when the first 5 items of Chebyshev series corresponding to this equation were used and in [18] the corresponding error was 5.5 × 10 −15 .