Gauss Quadrature with Preassigned Nodes

5.3.3 Gauss Quadrature with Preassigned Nodes

In many applications it is desirable to use Gauss-type quadrature where some nodes are preassigned and the rest chosen to maximize the order of accuracy. In the most common cases the preassigned nodes are at the endpoints of the interval. Consider a quadrature rule of the form

f (x)w(x) dx =

w i f (x i ) +

b j f (z j ) + R(f ), (5.3.16)

a i =1

j =1

where z j , j = 1 : m, are fixed nodes in [a, b] and the x i are determined so that the interpolatory rule is exact for polynomials of order 2n + m − 1. By a generalization of

Theorem 5.3.4 the remainder term is given by the formula

( 2n+m)

f n (ξ ) & * &

R(f ) =

a < ξ < b. (5.3.17) ( 2n)!

(x −z i )

(x −x i ) w(x) dx,

a i =1

i =1

In Gauss–Lobatto quadrature both endpoints are used as abscissae, z 1 = a, z 2 = b, and m = 2. For the standard interval [a, b] = [−1, 1] and the weight function w(x) = 1,

the quadrature formula has the form

f (x) dx =w 0 f( −1) + w n +1 f( 1) +

w i f (x i ) +E L . (5.3.18)

i =1

574 Chapter 5. Numerical Integration The abscissae a < x i <b are the zeros of the orthogonal polynomial φ n corresponding

to the weight function ˜w(x) = (1 − x 2 ) , i.e., up to a constant factor equal to the Jacobi polynomial J n (x,

1, 1) = P n ′ +1 (x) . The nodes lie symmetric with respect to the origin. The corresponding weights satisfy w i =w n +1−i , and are given by

w 0 =w n +1 =

, i = 1 : n. (5.3.19) (n + 2)(n + 1)

(P n +1 (x i )) 2

The Lobatto rule (5.3.18) is exact for polynomials of order 2n+1, and for f (x) ∈ C 2m [−1, 1] the error term is given by

(n

+ 2)(n + 1) 2 2n+3 (n !)

R(f )

∈ (−1, 1). (5.3.20) 2n + 3)[(2n + 2)!]

3 f ( 2n+2) (ξ ),

Nodes and weights for Lobatto quadrature are found in [1, Table 25.6]. In Gauss–Radau quadrature rules m = 1 and one of the endpoints is taken as the

abscissa, z 1 = a or z 1 = b. The remainder term (5.3.17) becomes

f ( 2n+1) (ξ ) b 0 & n

R(f ) =

a < ξ < b. (5.3.21) ( 2n)!

(x −z 1 )

(x −x i ) w(x) dx,

a i =1

Therefore, if the derivative f (n +1) (x) has constant sign in [a, b], then the error in the Gauss– Radau rule with z 1 = b will have opposite sign to the Gauss–Radau rule with z 1 = a. Thus, by evaluating both rules we obtain lower and upper bounds for the true integral. This has many applications; see Golub [163].

For the standard interval [−1, 1] the Gauss–Radau quadrature formula with z 1 =1 has the form

f (x) dx =w 0 f( −1) +

w i f (x i ) +E R 1 . (5.3.22)

i =1

The n free abscissae are the zeros of

P n (x) +P n +1 (x) , x −1

where P m (x) are the Legendre polynomials. The corresponding weights are given by

1−x i

, i = 1 : n. (5.3.23) (n + 1)

(n + 1) 2 (P n (x i )) 2

The Gauss–Radau quadrature rule is exact for polynomials of order 2n. Assuming that

f (x) ∈C 2m−1 [−1, 1], then the error term is given by

(n + 1)2 2n+1

1 (f ) =

3 (n !) f ( 2n+1) (ξ 1 ), ξ 1 ∈ (−1, 1). (5.3.24)

[(2n + 1)!]

Asimilar formula can be obtained with the fixed point +1 by making the substitution t = −x. By modifying the proof of Theorem 5.3.3 it can be shown that the weights in Gauss–

Radau and Gauss–Lobatto quadrature rules are positive if the weight function w(x) is nonnegative.

5.3. Quadrature Rules with Free Nodes 575

Example 5.3.3.

The simplest Gauss–Lobatto rule is Simpson’s rule with n = 1 interior node. Taking

n = 2 the interior nodes are the zeros of φ 2 (x) , where

1−x 2 )φ

2 (x)p(x) dx = 0 ∀p ∈ P 2 .

1, 1) = (x 2 − 1/5). Hence the interior nodes are ±1/

Thus, φ 2 is, up to a constant factor, the Jacobi polynomial J √ 2 (x,

5 and by symmetry the quadrature formula is

f (x) dx =w 0 (f ( −1) + f (1)) + w 1 (f ( −1/

5) + f (1/ 5)) + R(f ), (5.3.25)

where R(f ) = 0 for f ∈ P 6 . The weights are determined by exactness for f (x) = 1 and

f (x) =x 2 . This gives 2w 0 + 2w 1 = 2, 2w 0 + (2/5)w 1 = 2 3 , i.e., w 0 = 1 6 ,w 1 = 5 6 .

A serious drawback with Gaussian rules is that as we increase the order of the formula, all interior abscissae change , except that at the origin. Thus function values computed for the lower-order formula are not used in the new formula. This is in contrast to Romberg’s method and Clenshaw–Curtis quadrature rules, where all old function values are used also in the new rule when the number of points is doubled.

Let G n

be an n-point Gaussian quadrature rule

b n −1

f (x)w(x) dx ≈

a i f (x i ),

a i =0

where x i , i = 0 : n − 1, are the zeros of the nth degree orthogonal polynomial π n (x) . Kronrod [232, 233] considered extending G n by finding a new quadrature rule

n −1

K 2n+1 =

where the new n + 1 abscissae y i are chosen such that the degree of the rule K 2n+1 is equal to 3n + 1. The new nodes y i should then be selected as the zeros of a polynomial p n +1 (x) of degree n + 1, satisfying the orthogonality conditions

π n (x)p n +1 (x)w(x) dx = 0.

If the zeros are real and contained in the closed interval of integration [a, b] such a rule is called a Kronrod extension of the Gaussian rule. The two rules (G n ,K 2n+1 ) are called a

Gauss–Kronrod pair. Note that the number of new function evaluations are the same as for the Gauss rule G n +1 .

It has been proved that a Kronrod extension exists for the weight function w(x) = (

1−x 2 ) λ −1/2 , λ ∈ [0, 2], and [a, b] = [−1, 1]. For this weight function the new nodes interlace the original Gaussian nodes, i.e.,

−1 ≤ y 0 <x 0 <y 1 <x 1 <y 2 < ···<x n −1 <y n < 1.

576Chapter 5. Numerical Integration This interlacing property can be shown to imply that all weights are positive. Kronrod

considered extensions of Gauss–Legendre rules, i.e., w(x) = 1, and gives nodes and weights in [233] for n ≤ 40.

It is not always the case that all weights are positive. For example, it has been shown that Kronrod extensions of Gauss–Laguerre and Gauss–Hermite quadrature rules with positive weights do not exist when n > 0 in the Laguerre case and n = 3 and n > 4 in the Hermite case. On the other hand, the Kronrod extensions of Gauss–Legendre rules can

be shown to exist and have positive weights. Gauss–Kronrod rules are one of most effective methods for calculating integrals. Often one takes n = 7 and uses the Gauss–Kronrod pair (G 7 ,K 15 ) , together with the realistic but still conservative error estimate (200|G n −K 2n+1 |) 1.5 ; see Kahaner, Moler, and Nash [220, Sec. 5.5]. Kronrod extension of Gauss–Radau and Gauss–Lobatto rules can also be constructed. Kronrod extension of the Lobatto rule (5.3.25) is given by Gander and Gautschi [131] and used in an adaptive Lobatto quadrature algorithm. The simplest extension is the four-point Lobatto–Kronrod rule

(f ( −1) + f (1)) +

f( 0) + R(f ). (5.3.28)

√ This rule is exact for all f ∈ P 10 . Note that the Kronrod points ± 2/3 and 0 interlace the

previous nodes.