Numer Algor DOI 10.1007/s11075-007-9088-0 ORIGINAL PAPER
New embedded boundary-type quadrature formulas for the simplex F. A. Costabile · F. Dell’Accio
Received: 12 December 2006 / Accepted: 26 March 2007 © Springer Science + Business Media B.V. 2007
Abstract In this paper we consider the problem of the approximation of the integral of a smooth enough function f (x, y) on the standard simplex 2 ⊂ IR2 by cubature formulas of the following kind: f (x, y) dxdy = 2
3 α=1 i, j
Aαij
∂ i+ j f (xα , yα ) + E ( f ) ∂ xi ∂ y j
where the nodes (xα , yα ) , α = 1, 2, 3 are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples. Keywords Boundary type quadrature · Simplex · Algebraic degree of exactness Mathematics Subject Classifications (2000) Primary 65D30 · 65D32 · Secondary 65D05 · 65D15
To Prof. Walter Gautschi for his 50 years of professional activity. F. A. Costabile · F. Dell’Accio (B) Dipartimento di Matematica, Università della Calabria, Via P. Bucci Cubo 30A, 87036 Rende (Cs), Italy e-mail:
[email protected] F. A. Costabile e-mail:
[email protected] Numer Algor
1 Introduction We consider the problem of the approximation of the integral of a smooth enough function f on a polygonal domain D ⊂ IR N , N ≥ 1, by quadrature formulas which use values of the integrand function and its partial derivatives up to a fixed order only at nodes xα ∈ ∂ D: |k| M ∂ f (x) dx ≈ Aα,k f (xα ) =: Q M ( f ) , (1) ∂xk α k D
where we set x = (x1 , . . . , x N ), k = (k1 , . . . , k N ), |k| = k1 + · · · + k N and ∂xk = ∂ xk1 1 . . . ∂ xkNN by following standard multi-index notations. We denote also by PxK , K ∈ IN, the space of polynomials in x1 , . . . x N of total degree not greater than K. In a modern terminology formulas like (1) are called boundary type quadrature formulas (BTQFs). BTQFs are particularly useful for the cases where the values of the integrand function and its derivatives inside the domain are not given or are not easily determinable. For example, BTQFs find application to numerical solution of boundary value problems of partial differential equations (see [14, 15] for general strategies for constructing BTQFs, several examples of BTQFs over three and higher dimensional triangular and cubical domains with both remainders and algebraic precision and a quite complete list of references on this topic). The algebraic degree of exactness (ADE) attainable by a BTQF depends both on the number of sides (or faces) of the polygonal domain D and on the order of derivatives used in the formula. In fact, a BTQF for a n-faced bounded convex polygonal domain D ⊂ IR N which uses only functional evaluations of the integrand cannot have ADE greater than n − 1. In order to see this, it is sufficient to test the quadrature on the product p of the linear polynomials li (x) that define the hyperplanes bounding the domain D. By analogy, if the BTQF uses derivatives of the integrand function up to the order m, then the test of the quadrature on pm+1 shows that its ADE cannot be greater than (m + 1)n − 1. Our general strategy for constructing BTQFs with ADE for a polygonal domain D is planned as follows: (1) by assuming f (x) smooth enough on D, we must firstly seek for explicit expansions of the form: f (x) = P M [ f ](x) + R M [ f ](x) ,
x ∈ D, M ∈ IN,
(2)
where P M [ f ](x) ∈ (we emphasize the dependence of the integer K M on M) has the following features: PxK M
– –
P M [ f ](x) depends only on the values that f (x) and some of its successive derivatives assume at the relevant boundary points of D; P M [·] is the identity in PxK M ;
(2) if (1) holds, then we can integrate both sides of expansion (2) over the polygonal domain D in order to get the quadrature Q M ( f ); we obtain also an exact expression for the truncation error E M ( f ), provided that the remainder R M [ f ] in (2) is known. Well-known one-dimensional BTQFs can
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be recovered by following this strategy. Three remarkable examples are: the Euler–Maclaurin summation formula [16, p. 452, eq. 11.11-6] which can be recovered by integrating the expansion in Bernoulli polynomials proposed in [6]; the Taylor two ends quadrature [17, Cap VI, §19] which can be recovered by integrating the two point Taylor expansion [12, p. 37, eq. 2.5.22]; the Euler summation formula ([3], [16, p. 459, Ex. 9]) which can be recovered by integrating the Lidstone expansion [18]. Finally, we quote also two quadrature formulas proposed in [10] which are obtained respectively by integrating on [0, 1] the Lidstone expansion of 2-nd type and its symmetric expansion with respect to the axis x = 12 (see [9] for above and other examples). In this paper we investigate three classes of BTQFs for the standard simplex 2 ⊂ IR2 of special form. We assume that the nodes of the quadrature are only the vertices (xα , yα ) of the simplex; moreover, the weights Aαij do not depend on the order of the quadrature: Q M ( f ) :=
3 α=1 i, j
Aαij
∂ i+ j f (xα , yα ) . ∂ xi ∂ y j
(3)
These requirements allow us to organize the quadrature formulas in pairs (Q N (·) , Q M (·)) s.t. the second formula of the pair reuses all functional evaluations, including the derivatives, used in the first formula. Pairs of this kind are called embedded pairs (EP) of quadrature formulas and are particularly useful in order to build up automatic procedures for the computation of integrals with a minor amount of calculations; moreover |Q N (·) − Q M (·)| can be used as numerical estimate for the error of the less precise formula [5]. It is worthy to note that in the special case (3) we can recover Q M (·) simply by adding to Q N (·) some extra terms. This condition is stronger than that one required for embedded pair; we emphasize it by calling the pair (Q N (·) , Q M (·)) a strongly embedded pair (SEP) of quadrature formulas. The paper is organized as follows. According to the proposed strategy, in Section 2 we get formulas of type (3) by integrating some polynomial expansions for smooth enough functions on 2 : the expansion in Bernoulli polynomials [7], the Lidstone expansion [8] and the Lidstone expansion of 2nd type [11], that generalize to the simplex the corresponding above mentioned one dimensional expansions. Bounds for the remainder and sufficient conditions for convergence are explicitly given in the case of Lidstone quadrature; in the remaining cases similar conditions hold. In Section 3 we show how to organize an algorithm for the automatic computation of the quadrature with estimate of the error and provide some numerical examples that conclude the paper.
2 Strongly embedded BTQFs for the simplex with ADE In the papers [7, 8, 11] three polynomial expansions like (2) for smooth enough functions on the simplex are proposed, with the following common features:
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–
Each polynomial approximant can be written as linear combination K
P M f (x, y) =
M dim P(x,y)
λh ph (x, y)
h=1
– –
where K M is an integer depending on M, the coefficients {λh } are linear combinations of partial derivatives of f at the vertices of the simplex and KM { ph (x, y)}h is a specific basis of the polynomial space P(x,y) ; By denoting with Lh the evaluation functional which acts on f by the rule Lh ( f ) = λh then Lh ( pk ) = δhk ; KM of the interThe polynomial P M f (x, y) is the unique solution in P(x,y) polation problem L h ( f ) = λh ;
– –
The operator P M [·] is the identity if restricted to the polynomial space KM P(x,y) ; An integral expression for the exact error R M f (x, y) is given.
Here we integrate on the standard simplex the Lidstone interpolation polynomial PL M f (x, y) [8], which reproduces exactly the polynomials of 2M−1 P(x,y) . This polynomial can be rewritten in terms of a basis which involves the Lidstone polynomials n (x) [2], with scalar coefficients which are even order mixed √ derivatives involving ∂/∂ x, ∂/∂ y and the directional derivative ∂/∂ν := 1/ 2 (∂/∂ x − ∂/∂ y). The sequence of Lidstone polynomials can be defined recursively as follows: ⎧ ⎨ 0 (x) = x, (x) = n−1 (x) , n ≥ 1, ⎩ n n (0) = n (1) = 0, n ≥ 1. The Lidstone interpolation polynomial PL M f (x, y) results: PL M f (x, y)
M−1 M−1−k ∂ 2 j+2k x k 2k = f (0, 0) 2 (x + y) k j (1 − x − y) ∂ x2 j∂ν 2k x+y j=0 k=0
M−1 M−1−k ∂ 2 j+2k x k 2k f 0) 2 j (x + y) + + y) (1, (x k ∂ x2 j∂ν 2k x+y j=0 k=0
+
M−1 M−1−k k=0
+
j=0
M−1 M−1−k k=0
j=0
∂ 2 j+2k f (0, 0) 2k (x + y)2k k ∂ y2 j∂ν 2k ∂ 2 j+2k f (0, 1) 2k (x + y)2k k ∂ y2 j∂ν 2k
y j (1 − x − y) x+y
y j (x + y) . x+y (4)
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As long as the integral QL M(
1 1−x f ) := 0
PL M [ f ](x, y)dydx
(5)
0
is invariant under the axial symmetry with respect to the line x − y = 0, we can exchange in the argument of the integrand function the first variable with the second one. For this reason the integrals of the first and third addenda in the r.h.s. of (4) have the same value and therefore they are equal to their arithmetic mean 1 2
1 1−x 2k (x + y)2k k 0
0
x x+y
+ k
y x+y
j (1 − x − y) dydx.
(6)
As we shall see below, the integral (6) reduces, apart for a constant factor, to 1 x2k+1 j (1 − x) dx
(7)
0
which becomes straightforward by using the identity ([2, Eq. (1.2.9), p. 5]) j (x) =
x 22 j+1 B2 j+1 1 + (2 j + 1)! 2
and the well-known expansion of Bernoulli polynomials by means of Bernoulli numbers (http://functions.wolfram.com, Eq. 05.14.06.0011.01) Bn (x) =
n
n k=0
k
Bn−k xk .
In fact, for k ≥ 0 we note that ([2, Eq. (1.2.24), p. 9])1 E2k (1 − x) = E2k (x) = (2k)! (k (x) + k (1 − x))
(8)
therefore, by performing the change of variables x → x + y,
y → y,
1 It should be noted that the definition of Euler polynomial in [2] does not agree with the definition
in [1] and in http://functions.wolfram.com, which we adopt in the paper, by a constant factor. As a consequence of it, the Equation (8) is opportunely resized with respect to the Equation (1.2.24) in [2].
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the integral in (6) becomes 1 0
⎛ x k 2k ⎝ 2 x j (1 − x) 0
⎞
y 1 E2k dy⎠ dx (2k)! x
1 2 E2k+1 (1) − E2k+1 (0) x2k+1 j (1 − x) dx = (2k)! 2k + 1 0 ⎧ 1 ⎪ ⎪ ⎪ ⎪ k = 0, ⎪ ⎪ x j (1 − x) dx, ⎨ 0 = 1 ⎪ ⎪ 2 22k+2 − 1 2k ⎪ ⎪ ⎪ x2k+1 j (1 − x) dx, k > 0, ⎪ ⎩ (2k + 1)!(2k + 2) k
0
where we have used the equations En+1 (x) En (x) dx = n+1 and (http://functions.wolfram.com, Eq. 05.13.03.0001.01) 2 2n+1 − 1 Bn+1 , n > 0. En (1) = −En (0) = n+1 Analogous considerations hold for the integrals of the second and forth addenda in the r.h.s. of (4). The Lidstone quadrature formula of order M and ADE 2M − 1 is: 2 j+2k M−1 M−1−k ∂ ∂ 2 j+2k L αkj f (0, 0) + 2 j 2k f (0, 0) QM( f ) = ∂ x2 j∂ν 2k ∂ y ∂ν j=0 k=0
+
M−1 M−1−k k=0
j=0
βkj
∂ 2 j+2k ∂ 2 j+2k f 0) + f (0, 1) (1, ∂ x2 j∂ν 2k ∂ y2 j∂ν 2k
(9)
where
⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 12 ⎪ ⎪ ⎪ j−1 ⎪ ⎪ 2 j+2 ( j−l) (2 j+1) 22l+2 B2l+2 2 j+1 ⎪ ⎪ − 2(2 j+2)! + ⎪ (−1) 2(2 j+3)! (2 j−2l+1)! (2l+2)! ⎪ ⎪ l=0 ⎪ ⎪ ⎪ ⎨ 2k+1 −2−k−1 2k+2 ) 2 B2k+2 ( αkj = (2k+2)(2k+3) (2k+2)! ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k+1 2k+2 ⎪ 2 B 2 j+1 1 2k+2 ⎪ − 2−k−1 (2k+2)! 2 ⎪ (−1) ⎪ (2 j+1)!(2 j+2k+3) ⎪ ⎪ ⎪ ⎪ ⎪ j−1 ⎪ ⎪ 22l+2 B2l+2 1 1 ⎪ ⎪ ⎩ − (2 j)!(2 j+2k+2) + (2 j+2k−2l+1)(2 j−2l−1)! (2l+2)!
k = 0, j = 0 k = 0, j > 0 k > 0, j = 0
k > 0, j > 0
l=0
(10)
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and
⎧1 ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j−1 ⎪ ( j−l)(1−2−2l−1 ) 22l+2 B2l+2 ⎪ j+1 j ⎪ − 6(2 j+1)! − ⎪ j+3)! (2 (2 j−2l+1)! (2l+2)! ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ⎨ k+1 −k−1 −2 ) 22k+2 B2k+2 βkj = (2 (2k+3) (2k+2)! ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k+1 22k+2 B2k+2 ⎪ ⎪ 1 −k−1 ⎪ −2 2 ⎪ (2k+2)! (2k+2 j+3)(2 j+1)! ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j−1 ⎪ 2l+2 1−2−2l−1 ) ( B 2 ⎪ 1 2l+2 ⎪ − − ⎩ 6(2k+2 j+1)(2 j−1)! (2k+2 j−2l+1)(2 j−2l−1)! (2l+2)!
k = 0, j = 0 k = 0, j > 0 k > 0, j = 0
k > 0, j > 0
l=1
(11) of the quadrature QL M ( f ) can be the remainder RL M f (x, y) =
Bounds for the truncation error obtained f (x, y) − from the integral expression for f (x, y) in terms of the partial derivatives and the directional derivative PL M ∂/∂ν given in [8]; these bounds are as follows: L R [ f ](x, y) M 2
≤
1
M−1
π 2M
k=0
∂ 2M 2k+1 (1 − 2−(2k+2) ) max 2(M−k) 2k f (x, 0) x∈[0,1] ∂ x k+1 ∂ν
∂ 2M 2k+1 (1 − 2−(2k+2) ) + 2M max 2(M−k) 2k f (0, y) y∈[0,1] ∂ y π k+1 ∂ν k=0 ∂ 2M 1 2 M+1 + 2M max 2M f (x, y) . π M + 1 (x,y)∈2 ∂ν 1
M−1
(12)
From the above bounds we get the following sufficient conditions for convergence: Theorem 1 If for each i =2M0, ..., M ∂ max 2(M−i) 2i f (x, 0) = O p2M , M → ∞, x∈[0,1] ∂ x ∂ν 2M ∂ max 2(M−i) 2i f (0, y) = O p2M , M → ∞, y∈[0,1] ∂ y ∂ν 2M ∂ = O p2M , M → ∞, max 2M f (x, y) (x,y)∈2 ∂ν √ where 0 < p < π/ 2, then the sequence QL M ( f ) converges to the exact value 1 1−x f (x, y) dydx. 0
0
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In [8] a special class of two-variate functions, called completely convex ridge functions, is introduced. Each function in this class is a combination in an interval f = g ◦ φ(x, y), where g(t) is any completely convex function a, b [19] and φ(x, y) = αx + βy a linear functional on IR2 . For functions in this class the quadrature QL M ( f ) converges to the exact integral, if the following conditions are satisfied: Corollary 2 Let f (x, y) = g (αx + βy) , g ∈ C ∞ (R) a completely convex ridge function and suppose that there exists a positive number p < π such that g(M) (0) = O p M , M → ∞. If max {|α| , |β|}
0 such that for each M ≥ M0 , k = 0, ..., M − 1 ∂ 2M max 2(M−i) 2i f (x, 0) ≤ Cp2M , x∈[0,1] ∂ x ∂ν 2M ∂ max 2(M−i) 2i f (0, y) ≤ Cp2M , y∈[0,1] ∂ y ∂ν 2M ∂ max 2M f (x, y) ≤ Cp2M , (x,y)∈2 ∂ν √ where 0 < p < π/ 2. If we set εM
Cp2M = 2M π
2
M−1 k=0
2k+1 − 2−(k+1) 2 M+1 + k+1 M+1
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then we find M
2
k+1
−2
−(k+1)
k+1 ε M+1 p2 k=0 = 2 εM π M−1 2k+1 −2−(k+1) k+1
k=0
2+
M k=0
2+
2 M+2 M+2 2k+1 −2−(k+1) k+1 2 M+1 M+1
;
M−1 2k+1 −2−(k+1) k+1 k=0
straightforward computations show that 2 M+1 M+1
lim
M→∞ M−1 k=0
= lim
2k+1 −2−(k+1) k+1
M→∞ M+1 M
1 1 = 1, +O M
therefore M k=0 lim M→∞ M−1 k=0
2k+1 −2−(k+1) k+1
=2
2k+1 −2−(k+1) k+1
and consequently ε M+1 = lim M→∞ ε M
√ 2 p 2 < 1. π
According to the definition of rate of convergence of an iterative process (see for example [13, p. 215]) we deduce that QL M ( f ) converges at least linearly to the exact value of the integral. The integration of the polynomial approximant in the Lidstone expansion of 2nd type is performed by analogy; as a result, we obtain the Lidstone s quadrature of 2nd type QL M ( f ) with ADE 2M [11]: s QL M ( f) =
M
∂ 2k f (0, 0) ∂ν 2k k=0 2k+2 j−1
M M−k ∂ ∂ 2k+2 j−1 − βkj f 0) + f 0) (0, (0, ∂ x2 j−1 ∂ν 2k ∂ y2 j−1 ∂ν 2k k=0 j=1 2k+2 j−1
M M−k ∂ ∂ 2k+2 j−1 + γkj f (1, 0) + 2 j−1 2k f (0, 1) . ∂ x2 j−1 ∂ν 2k ∂y ∂ν j=1 αk
k=0
This formula requires the use of only odd order derivatives at the vertices of the simplex, apart for some even order derivatives in the direction ∂/∂ν at the origin. The weights are: ⎧ 1 ⎪ ⎪ , k = 0, ⎪ ⎨2 αk = ⎪ ⎪ 2k+1 − 2−(k+1) 22k+2 B2k+2 ⎪ ⎩ , k > 0, k+1 (2k + 2)!
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⎧ 5 ⎪ ⎪ − , ⎪ ⎪ 48 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j−1 2 j−2l ⎪ 1 2 j+1 B 2 2 j ⎪ 2 j−2l 2l+1 ⎪ − + , ⎪ ⎪ 2 (2 j+2)! (2 j+1)! l=1 (2l+2)! (2 j−2l)! ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ k+1 −(k+1) ) 22k+2 B2k+2 , βkj = − (5+2k)(2 −2 2(2k+3)(2k+4) (2k+2)! ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k+1 22k+2 B2k+2 ⎪ 1 −(k+1) ⎪ ⎪ −2 2 ⎪ (2k+2)! (2k+2 j+2)(2 j)! ⎪ ⎪ ⎪ ⎪ ⎪ j−1 ⎪ 2 j−2l ⎪ 2 B 2 j−2l 1 1 ⎪ ⎪ − (2k+2 j+1)(2 + , ⎩ j−1)! (2l+2k+2)(2l)! (2 j−2l)!
k = 0, j = 1, k = 0, j > 1, k > 0, j = 1,
k > 0, j > 1,
l=1
γkj =
⎧ 1 ⎪ ⎪ , ⎪ ⎪ 16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 j+1 ⎪ ⎪ − ⎪ (2 j+2)! ⎪ ⎪ ⎪2 ⎪ ⎪ ⎪ ⎨ −(k+1)
k = 0, j = 1,
2 j−1 6(2 j)!
−
j−3
22l+4 B2l+4 1−2−2k−3 (2 j−2l−2)(2 j−2l−4)! (2l+4)!
, k = 0, j > 1,
l=0
B2k+2 2 2 −2 , k > 0, j = 1, 2(2k+4) (2k+2)! ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2k+2 ⎪ 2 B 1 2k+2 ⎪ (2k+1 − 2−(k+1) ) (2k+2)! ⎪ ⎪ (2k+2 j+2)(2 j)! ⎪ ⎪ ⎪ k > 0, j > 1. ⎪ ⎪ j−3 ⎪ ⎪ 2l+4 −(2l+3) B 2 ⎪ 1 1−2 2l+4 ⎪ , ⎪ (2k+2 j−2l−2)(2 j−2l−4)! (2l+4)! ⎩ − 6(2k+2 j)(2 j−2)! − k+1
2k+2
l=0
The bounds for the truncation errors reflect the symmetry between the Lidstone and Lidstone second type cubature: L 2 R Ms [ f ](x, y)
M ∂ 2M+1 1 2k+1 (1 − 2−(2k+2) ) ≤ 2M f (x, 0) max x∈[0,1] ∂ x2M−2k+1 ∂ν 2k π k+1 k=0 M 1 2k+1 (1 − 2−(2k+2) ) ∂ 2M+1 + 2M f (0, y) max y∈[0,1] ∂ y2M−2k+1 ∂ν 2k π k+1 k=0 2M+2 ∂ 1 2 M+2 + 2(M+1) max 2M+2 f (x, y) . π M + 2 (x,y)∈2 ∂ν
For the case of the Bernoulli expansion on the simplex [7] the integration process simplifies by rewriting the polynomial approximant in terms of a basis
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which involves the Bernoulli polynomials Bn (x) with the scalar coefficients that are differences of mixed derivatives involving ∂/∂ x, ∂/∂ y and the directional derivative ∂/∂ν: PBM f (x, y) = f (0, 0) · 1
M k−1 ∂ k−1 x Sk (x + y) ∂ f 0) − f 0) + (1, (0, k−1 k−1 ∂x ∂x x+y k! k=1
M k−1 ∂ ∂ k−1 y Sk (x + y) f (0, 1) − k−1 f (0, 0) + k−1 ∂y ∂y x+y k! k=1
y
Sk M √ ∂ k−1 ∂ k−1 x+ y k−1 (x+ y)k + f 1) − f 2 (0, (1,0) k−1 k−1 ∂ν ∂ν k! k=2
+
M M−k+1 k=2
j=2
∂ k+ j−2 ∂ k+ j−2 f 1) − f (0, 0) (0, ∂ y j−1 ∂ν k−1 ∂ y j−1 ∂ν k−1
∂ k+ j−2 ∂ k+ j−2 f 0) + f 0) (1, (0, ∂ x j−1 ∂ν k−1 ∂ x j−1 ∂ν k−1
y S j (x + y) Sk √ x+y (14) × 2k−1 (x + y)k−1 j!k! −
where we have set Sk (x) = Bk (x) − Bk , k = 0, 1, . . . We get the following BTQFs with ADE M, which use differences of derivatives of all order at the vertices of the simplex:
QBM ( f ) =
1 ( f (0, 0) + f (1, 0) + f (0, 1)) 6 k−1 M ∂ ∂ k−1 + αk f 0) − f (0, 0) (1, ∂ xk−1 ∂ xk−1 k=2
∂ k−1 ∂ k−1 + f (0, 1) − k−1 f (0, 0) ∂ yk−1 ∂y M M−k+1 ∂ k+ j−2 ∂ k+ j−2 βkj f 1) − f (0, 0) + (0, ∂ y j−1 ∂ν k−1 ∂ y j−1 ∂ν k−1 j=1 k=2
∂ k+ j−2 ∂ k+ j−2 − j−1 k−1 f (1, 0) + j−1 k−1 f (0, 0) ∂ x ∂ν ∂ x ∂ν
(15)
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were we have set: 1 1 Bl , 2 (k − l + 2)(k − l)! l! k−1
αk =
l=0
j−1 √ Bk 1 Bl βkj = − 2k−1 . k! (k + j − l + 1)( j − l)! l! l=0
B Bounds for the truncation error of the quadrature Q M ( f ) can be obtained B from the integral expression for the remainder R M f (x, y) given in [7]; we rewrite it in terms of the partial derivatives and the directional derivative ∂/∂ν and obtain:
2
RBM [
M √ 8 2i ∂M f ](x, y) ≤ max M−i+1 i−1 f (0, y) M ∂ν (2π ) i=1 i + 1 y∈[0,1] ∂ y M √ i ∂M 8 2 + max M−i+1 i−1 f (x, 0) M x∈[0,1] ∂x ∂ν (2π ) i=1 i + 1 √ M M 8 2 ∂ f (x, y) . + max M M + 2 (x,y)∈ ∂ν M 2 (2π )
3 Algorithm and numerical results For the particular case of the Bernoulli cubature formulas we set: [c]l QB 1 ( f) =
1 ( f (0, 0) + f (1, 0) + f (0, 1)) , 6
√ 1 1−x π 36 − 24 2 π x + y dydx = Table 1 sin ≈ 0.208607601619622 4 6 π2 0
M 1 2 3 4 5 6 7 8
0
Es. err. 7.36E-03 6.36E-05 8.77E-07 1.31E-08 2.02E-10 3.14E-12 4.89E-14 7.63E-16
Ex. err. 7.42E-03 3.64E-04 2.07E-05 1.25E-06 7.65E-08 4.65E-09 1.97E-10 8.10E-11
QB2M−1 , QB2M+1
Ex. err. 7.42E-03 6.45E-05 8.91E-07 1.34E-08 2.06E-10 3.19E-12 4.97E-14 7.77E-16
QLM , QLM+1
Es. err. 7.06E-03 3.43E-04 1.95E-05 1.17E-06 7.19E-08 4.46E-09 2.78E-10 1.73E-11
Ls QLs M , Q M+1
Ex. err. 9.20E-03 4.87E-04 2.82E-05 1.70E-06 1.04E-07 6.38E-09 3.04E-10 7.43E-11
Es. err. 8.72E-03 4.59E-04 2.65E-05 1.59E-06 9.79E-08 6.07E-09 3.78E-10 2.36E-11
Numer Algor
Table 2
1 1−x
π 24 −3 sinh π6 + 2 sinh π4 π sinh ≈ 0.228049265190525 x + y dydx = 4 6 π2 0
M 1 2 3 4 5 6 7 8
0
Ex. err. 8.04E-03 6.94E-05 9.58E-07 1.44E-08 2.21E-10 3.44E-12 5.35E-14 8.88E-16
EBM (
Es. err. 8.11E-03 7.03E-05 9.73E-07 1.46E-08 2.25E-10 3.49E-12 5.44E-14 8.49E-16
Ex. err. 8.04E-03 3.93E-04 2.24E-05 1.35E-06 8.31E-08 5.02E-09 4.50E-10 1.09E-10
QB2M−1 , QB2M+1
QLM , QLM+1
Es. err. 8.43E-03 4.16E-04 2.38E-05 1.43E-06 8.82E-08 5.47E-09 3.41E-10 2.13E-11
Ls QLs M , Q M+1
Ex. err. 1.06E-02 5.76E-04 3.36E-05 2.03E-06 1.25E-07 7.65E-09 6.14E-10 9.88E-11
Es. err. 1.12E-02 6.09E-04 3.56E-05 2.16E-06 1.33E-07 8.26E-09 5.15E-10 3.21E-11
∂M ∂M ∂M ∂M f ) = α M+1 f (1, 0) − M f (0, 0) + M f (0, 1) − M f (0, 0) ∂ xM ∂x ∂y ∂y M+1 ∂M ∂M βk,M−k+2 f (0, 1) − M−k+1 k−1 f (0, 0) + M−k+1 k−1 ∂y ∂ν ∂y ∂ν k=2
∂M ∂M − M−k+1 k−1 f (1, 0)+ M−k+1 k−1 f (0, 0) , ∂x ∂ν ∂x ∂ν
QBM+1 ( f ) = QBM ( f ) + EBM ( f ).
(16)
Then the automatic procedure for the approximate calculation of the integral is then based on the following considerations: – –
B used as first approximation; Q1B( f ) is E ( f ) is used as error estimate; M
1 1−x Table 3 cos 1 + x2 + y2 dydx ≈ 0.202901824664092 0
M 1 2 3 4 5 6 7 8
0
Es. err. 5.93E-02 1.46E-03 4.91E-05 1.77E-06 6.76E-08 2.71E-09 1.13E-10 4.88E-12
Ex. err. 6.09E-02 9.24E-03 1.34E-03 1.97E-04 3.02E-05 4.83E-06 8.05E-07 1.39E-07
QB2M−1 , QB2M+1
Ex. err. 6.09E-02 1.51E-03 5.09E-05 1.84E-06 7.04E-08 2.82E-09 1.18E-10 5.11E-12
QLM , QLM+1
Es. err. 5.16E-02 1.14E-03 7.90E-03 1.67E-04 2.54E-05 4.02E-06 6.66E-07 1.14E-07
Ls QLs M , Q M+1
Ex. err. 1.50E-02 2.39E-03 3.48E-04 4.98E-05 7.21E-06 1.08E-06 1.66E-07 2.62E-08
Es. err. 1.26E-02 2.04E-03 2.98E-04 4.26E-05 6.14E-06 9.10E-07 1.40E-07 2.24E-08
Numer Algor
– –
If EBM+1 ( f ) < EBM ( f ) then Ba better approximation is produced; E ( f ) is less than a prefixed tolerance or The calculations stop if M B B E M+1 ( f ) > E M ( f ) .
We do similar settings for the cases of Lidstone and Lidstone second type Ls L quadratures. The tests of the quadratures QB 2M−1 (·), Q M (·) and Q M (·) respectively of ADEs 2M − 1, 2M − 1 and 2M integer for first valuesof the positive M on the three functions sin π4 x + π6 y , sinh π4 x + π6 y and cos 1 + x2 + y2 give the results presented in Tables 1, 2 and 3. The estimated errors are computed by the use of the indicated embedded pairs of quadratures; only in the case of function cos 1 + x2 + y2 the exact errors are computed by assuming as exact the numerical integration performed by Mathematica. In this case, also, the convergence it is not assured by theoretical results given in previous section. It results that exact errors and corresponding estimated errors are very close, at least in these particular cases. Acknowledgements The authors are grateful to the anonymous referees both for their positive judgements and some useful suggestions for improvements in the paper.
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