1. d-Fold Hermite-Gauss Quadrature and Computation of Special Functions
2001:  A Mathematics Odyssey
Brian Hagler UTPB

2. CONTENTS Overview d-Fold Hermite-Gauss Quadrature
Computation of Special Functions
Formulas and Algorithms
References

3. Overview Hermite Polynomials Weight Function Standard Basis
We can begin by considering a context of the space of real polynomials of one real variable which can be endowed with an inner product given by integration with respect to a weight function. Applying the Gram-Schmidt process to the standard basis results in an ordered orthogonal basis; here, the Hermite Polynomials.
Note that superscripts and a subscript, 0, have been introduced, augmenting the usual notation.
Weight Function
Standard Basis

4. . . .
. . .
. . .
We can then successively apply a transformation …

5. . . .
. . .
. . .
…, say d times, to obtain a larger inner product space of rational functions with ordered orthogonal basis which we here will call …

6. d-Fold Hermite Rationals of Parameters g,l>0
Weight Function
Basis
…the d-Fold Hermite Rationals of parameters g,l>0. The subscript d then refers to the number of times the transformation was applied. The letter “d” stands for “doubling”, …

7. …since the application of the transformation takes each member of the ordered orthogonal basis to two in the next space.

8. It does this by a change of variables, x goes to essentially x-1/x, for the first of the two, …

9. …and that same change of variables, followed by a multiplication by essentially 1/x, for the second. Of special note at this point is the fact that the transformation is more than a change of variables, although this change of variables is a key component.

10. In fact, the weight function transformation …

11. …is simply the change of variables, x goes to essentially x-1/x, that we saw before.

12. . . .
In the Hermite case which we’re discussing, we can write the weight function in this way, where vd is the d-fold composition of (1/l)(x-g/x).

13. The basis transformation can be described …

14. …as an enlargement to include twice as many singularities.

15. . . .
Thus, the space spanned by the basis Bd is a space of rational functions with 2d singularities, including the one at infinity.

16. d-Fold Doubling
. . .
. . .
In summary, beginning with an orthogonal polynomial sequence, we can recursively construct orthogonal rational function sequences via a kind of doubling transformation.
Among the possible directions of inquiry which might be subsequently pursued is Gaussian Quadrature.
. . .

17. (2d n)-Point d-Fold Hermite-Gauss Quadrature Formula of Parameters g,l>0
Here are two equivalent ways of writing this quadrature rule. The abscissa are the zeros of the d-fold Hermite rational Hd,2dn . The error term is zero for f(x) in a subspace of dimension 2d+1n, which is a way of saying the rule is “of highest precision”. With d=0, this is the classical Hermite-Gauss quadrature, and the abscissas and weights for higher d are given explicitly by recursive formulas.
This idea is made rigorous, generalized, and exemplified in an article titled "d-Fold Hermite-Gauss Quadrature" that is to appear in the Journal of Computational and Applied Mathematics very soon, perhaps in the November issue. or
where

18. n=1
n=2
n=3
n=4
d=0
d=1
d=2
d=3
d=4
The next few slides represent some numerical work done with this quadrature, with the parameters both equal to 1 for the sake of convenience. This first example has an integrand such that the quadrature is theoretically exact for d =1 and all n>=4. The case d=0 is the classical Hermite-Gauss, and the data here shows that it struggles.

19. n=1
n=2
n=3
n=4
d=0
d=1
d=2
d=3
d=4
This is another example of an integrand for which the classical quadrature, d=0, is foiled, but a higher choice of d, d=2, is theoretically exact for n large enough, n=3 or larger.

20. -20
-10 10
-0.2
0.2
0.4
0.6
0.8 1 20
n=1
n=2
n=3
n=4
d=0
d=1
d=2
d=3
d=4
-0.4
The final example is a choice of integrand where no choice of d ever gives theoretically exact results. The integrand is the Bessel function of order 0, and the data reflects mixed results.
Now, the focus in this talk is on a successful use of d-fold Hermite-Gauss quadrature in the computation of special functions, with d=1, the Laurent Polynomial case.

21. Computation of Special Functions
Here, we begin by considering a partial differential equation.

22. ... which by changes of variables has ...

23. Linear Hyperbolic (Wave) Equation
… a couple of alternate forms. This last equation can be recognized as a linear wave equation, which might describe the vertical location U of an internally damped vibrating string at a given time z and horizontal position y, for example.
Linear Hyperbolic (Wave) Equation

24. Initial Value Problem Find such that ’ subject to given initial data.
So, an initial value problem is this.

25. Separation of Variables
In solving that problem by separation of variables, ...

26. System of Differential Equations
... a system of differential equations like this one presents itself.

27. has a solution
... which has one solution ...

28. has a solution
for
... that can be represented explicitly as a sum given by d-fold Hermite-Gauss quadrature, the independent variable, x, appearing in the parameter gamma.

29. Special Functions Here are the first few of these special functions.
This completes the goal of the talk. We've demonstrated that this quadrature has uses in the computation of special functions and, in addition, provided an example in which the parameters, in particular, gamma, play a pivotal role.
The remainder of the slides in this prepared presentation, which is available to all on the web, give some further details.

30. Formulas and Algorithms
d-Fold Hermite-Gauss Quadrature Formula
Singularities
Special Functions

31. (2d n)-Point d-Fold Hermite-Gauss Quadrature

32. Error Term For d and n positive integers,
for some real number e, provided
This expression of the error shows, by simple calculation that if f(x) is a polynomial in x of degree 2n-1 or less, or if f(x) is a polynomial in 1/(x-s) of degree 2n or less for s any of the singularities, then the error is 0. The quadrature formula is exact on a 2^(d+1) n dimensional subspace of B_d.
has a continuous th derivative.

33. Weights are the n classical weights corresponding to the abscissas for

34. Abscissas are the n simple zeros of the nth Hermite Polynomial for for

35. Weight Function
is the classical Hermite weight function.

36. Singularities
is the singularity of the Hermite Polynomials
for
for

37. Special Functions Differential Equations Integral Formulas
Quadrature Formula
Recurrence Formula
From the integral formula, the recurrence formula can be obtained by integration by parts and the second differential equation by differentiation (the integrand and its partial derivative with respect to x have continuous extensions on RxR=>exchange integration and differentiation allowed, the proof of which uses the mean value theorem, see Bartle, page 245, Theorem 31.7).

38. References
d-Fold Hermite-Gauss Quadrature, J. Comput. Appl. Math., to appear.