Gaussian Integration Methods

Topics in Integration

3.6 Gaussian Integration Methods

the linear space of all polynomials whose degree does not exceed j. In addi- tion, we define the scalar product

,b

(f, g):=' w(x)f(x)g(x) dx

'a

on the linear space L2[a, b] of all functions for which the integral

,b

(f,J)

= I

w(x)(f(x}f dx

°a

is well defined and finite, The functionsf, 9 E L2[a, b] are called orthogonal if (f, g)

=

O. The following theorem establishes the existence of a sequence of mutually orthogonal polynomials, the system of orthogonal polynomials as- sociated with the weight function w(x),

(3.6.3) Theorem. There exist polynomials Pj E llj' j

=

0, 1, 2, ... , such that (3.6.4)

These polynomials are uniquely defined by the recursions (3.6.5a)

(3.6.5b)

Po(x) == 1,

Pi+l(X) == (x - bi+dpi(X) - 'l'f+lPi-l(X) for i ~ 0, where P-l(X) == 0 andl

(3.6.6a) (3.6.6b)

PROOF. The polynomials can be constructed recursively by a technique known as Gram-Schmidt orthogonalization. Cleariy Po(x) == 1. Suppose then, as an induction hypothesis, that all orthogonal polynomials with the above properties have been constructed for j ~ i and have been shown to be unique. We proceed to show that there exists a unique polynomial Pi+ 1 E lli+ 1 with

(3.6.7)

and that this polynomial satisfies (3.6.5b). Any polynomial Pi + 1 E lli + 1 can be written uniquely in the form

Pi+l(X) == (x - bi+dpi(X)

+

Ci-lPi-l(X)

+

^Ci- 2Pi-2(X)

+ , .. +

coPo(x),

1 XPi denotes the polynomial with values XPi(X) for all x.

because its leading coefficient and those of the polynomials Pj' j ~ i, have value 1. Since (Pj' Pk)

=

0 for allj, k ~ i withj

1-

k, (3.6.7) holds if and only if (3.6.8a) (Pi+ 1, Pi)

=

(XPi' p;) - bi+ ¹ (Pi' p;)

=

0,

(3.6.8b) (Pi + 1, Pj-

d =

(XPj_ bPi)

+

Cj-l (Pj-l, Pj-

d =

0 for j ~ i.

The condition (3.6.1c)-with

Pf

^{and PJ-b} respectively, in the role of the nonnegative polynomial s-rules out (Pi' p;)

=

0 and (Pj-l' Pj_

d ⁼

^{0 for}

1 ~ j ~ i. Therefore, the equations (3.6.8) can be solved uniquely. (3.6.8a) gives (3.6.6a). By the induction hypothesis,

pAx) == (x - bj)Pj-l(X) - yJpj-2(X)

for j ~ i. From this, by solving for XPj_ 1 (X), we have (xPj_ b p;) = (Pj' Pi) for j ~ i, so that

(Pj,Pi)

J-Yf+l

^forj=i,

Cj-l

= -

(Pj-l' pj-d

= \

0 for j < i,

in view of (3.6.8). Thus (3.6.5b) has been established for i

+

^{1. D}

Every polynomial P E ilk is clearly representable as a linear combination of the orthogonal polynomials Pi' i ~ k. We thus have:

(3.6.9) Corollary. (p, Pn) = 0 for all p E iln - 1·

(3.6.10) Theorem. The roots Xi' i

=

1, ... , n, of Pn are real and simple. They all lie in the open interval (a, b).

PROOF. Consider those roots of Pn which lie in (a, b) and which are of odd multiplicity, that is, at which Pn changes sign:

a < ^Xl < ... < X, < b.

The polynomial

I

q(x):=

n

^{(x - Xj) E}

^Il,

j=l

is such that the polynomial Pn(x )q(x) does not change sign in [a, b], ^{so that}

b

(Pn, q) =

J

w(x)Pn(x)q(x) dx

1-

0

a

by (3.6.1c). Thus degree(q)

=

1= n must hold, as otherwise (Pn, q)

=

0 by

Corollary (3.6.9). D

Next we have the

(3.6.11) Theorem. The n x n matrix

is nonsingular for mutually distinct arguments ti, ⁱ

=

1, ... , n.

PROOF. Assume A is singular. Then there is a row vector c^T = (co, ... , cn-

d

^{=1= 0 with cT A} = O. The polynomial

n-l

q(x):==

L

^CiPi(X),

i=O

with degree(p) < n, has the n distinct roots t1, ••• , t"n and must vanish iden- tically. Let I be the largest index with CI =1= O. Then

1^{1- 1} PI(X)

= - - L

^CiPi(X),

CI i=O

This is a contradiction, since the polynomial to the right has a lower degree

than Pl E ill' 0

Theorem (3.6.11) shows that the interpolation problem of finding a func- tion of the form

n-l

p(X) ==

L

^CiPi(X)

i=O

with p(ti ) = J;, i = 1, ... , n is always uniquely solvable. The condition ofthe theorem is known as the Baar condition. Any sequence of functions Po, Ph ... which satisfy the Haar condition is said to form a Chebychev system. Theorem (3.6.11) states that sequences of orthogonal polynomials are Chebychev systems.

Now we arrive at the main result of this section.

(3.6.12) Theorem.

(a) Let Xh ... , Xn be the roots of the nth orthogonal polynomial Pn(x), and let

WI> ... , Wn be the solution of the (nonsingular) system of equations (3.6.13) ^~

( ) _ i

(Po, Po) if k

=

0,

~Pk x· w· - \

i=l I I 0 ifk=I,2, ... ,n-1.

Then Wi >

o

^{for i} = 1,2, ... , n, and (3.6.14)

b n

f

w(x)p(x) dx =

L

^WiP(XJ

a i= 1

holds for all polynomials p E II2n - l . The positive numbers Wi are called

" weights."

(b) Conversely, if the numbers ^{Wi' Xi'} i = 1, ... , n, are such that (3.6.14) holds for all p E II2n - 1, then the Xi are the roots of Pn and the weights Wi satisfy

(3.6.13).

(c) It is not possible to find numbers Xi> Wi> i

=

1, ... , n, such that (3.6.14) holds for all polynomials p E II2n .

PROOF. By Theorem (3.6.10), the roots ^Xi' i

=

1, ... , n, of Pn are real and mutually distinct numbers in the open interval (a, b). The matrix

(3.6.15)

is nonsingular by Theorem (3.6.11), so that the system of equations (3.6.13) has a unique solution.

Consider an arbitrary polynomial P ^E II2n _ l' It can be written in the form

(3.6.16) p(X) == Pn(x)q(x)

+

r(x),

where q, r are polynomials in IIn-b which we can express as linear combina- tions of orthogonal polynomials

n-l n-1

q(x} ==

L

<XkPk(X}, r(x} ==

L

^f3kPk(X},

k=O k=O

Since Po(x) == 1, it follows from (3.6.16) and Corollary (3.6.9) that

r

b w(x}p(x} dx

=

(Pn, q)

+

(r, Po)

=

Po(Po, Po}·

"a

On the other hand, by (3.6.16) [since Pn(xJ = 0] and by (3.6.13),

it1WiP(Xi) = ^itwir(XJ =

:t>kCt

^WiPk(X

^J)

⁼ Po(Po, Po)·

Thus (3.6.14) is satisfied.

We observe that

(3.6.17). If Wi' Xi' i = 1, ... , n, are such that (3.6.14) holds for all polynomials P E II2n - 1, then Wi >

o

^{for i = 1, ... , n.}

This is readily verified by applying (3.6.14) to the polynomials pj(x}:==

n

n ^{(x -Xh)2 E II2n - 2 ,}

h=1 h~j

j = 1, ... , n,

and noting that

.b n n

°

^<

^I

w(x)pAx) dx

= L

WiPj(X;)

=

^Wj

n

(X j - Xh)2

'a i=l h=l

h t-j

by (3.6.1c). This completes the proof of (3.6.12a).

Assume that Wi' Xi' i

=

1, ... , n, are such that (3.6.14) even holds for all polynomials P E _{Il 2n .} Then

n

p(x):==

n

^{(x - xy E Il2n}

j= 1

contradicts this claim, since by (3.6.1c)

b n

0<

f

w(x)p(x) dx

= L

WiP(Xi)

=

0.

'a i= 1

This proves (3.6.12c)

To prove (3.6.12b), suppose that Wi' Xi' i

=

1, ... , n are such that (3.6.14) holds for all P E Il 2n - 1. Note that the abscissas Xi must be mutually distinct, since otherwise we could formulate the same integration rule using only n - 1 of the abscissas Xi' contradicting (3.6.12c).

Applying (3.6.14) to the orthogonal polynomials P

=

Pk, k

=

0, ... , n - 1, themselves, we find

if k =

°

if k

=

1, ... , n - 1.

In other words, the weights Wi must satisfy (3.6.13).

Applying (3.6.14) to p(x):== Pk(X)Pn(x), k = 0, ... , n - 1, gives by (3.6.9)

°

^{= (Pk' Pn) =}

^L

n WiPn(X;)Pk(X;), ^{k = 0, ... , n - 1.}

i= 1

In other words, the vector c:= (W1Pn(Xd, ... , wnPn(xn)Y solves the homo- geneous system of equations Ac =

°

with A the matrix (3.6.15). Since the abscissas Xi' i

=

1, ... , n, are mutually distinct, the matrix A is nonsingular by Theorem (3.6.11). Therefore c =

°

^{and Wi Pn(X;) =}

°

^{for i =} 1, ... , n. Since Wi>

°

by (3.6.17), we have Pn(xi)

=

0, i

=

1, ... , n. This completes the proof

of (3.6.12b). D

For the most common weight function w(x) == 1 and the interval [-1, 1], the results of Theorem (3.6.12) are due to Gauss. The corresponding ortho- gonal polynomials are (see Exercise 16)

(3.6.18) k

=

0,1, ... ,

Indeed, Pk ^{E ilk} and integration by parts establishes (Pi' Pk) =

°

^{for i =1= k.}

Up to a factor, the polynomials (3.6.18) are the Legendre polynomials. In the

following table we give some values for Wi' Xi in this important special case.

For further values see the National Bureau of Standard's Handbook of Mathematical Functions [Abramowitz and Stegun (1964)].

n Wi Xi

WI = 2 ^XI = 0

2 WI = W2 = 1 ^X2 = -XI = 0.577 350 2692 ...

3 WI = ^W3 = ~ ^X3 = ^-XI = 0.7745966692 ...

W2 = ~ ^X2 = 0

4 WI = W4 = 0.3478548451... ^X4 = ^-XI = 0.861 1363116 ...

W2 = ^W3 = 0.652 145 1549 ... ^X3 = ^-X2 = 0.339 9810436 ...

5 ^WI = ^Ws = 0.236 926 8851... ^Xs = -XI = 0.9061798459 ...

W2 = W4 = 0.478 628 6705 ... ^X4 = ^-X2 = 0.5384693101...

W3 =

ill

⁼ 0.568 888 8889 ... X3 = 0

Other important cases which lead to Gaussian integration rules are listed in the following table:

[a, b] w(x) Orthogonal polynomials

[ -1, 1] (1 - x²

t

^1/2 T,,(x), Chebychev polynomials [0, (0) ^e~X Ln(x), Laguerre polynomials (-00,00] ^e-x2 Hn(x), Hermite polynomials

We have characterized the quantities Wi' Xi which enter the Gaussian inte- gration rules for given weight functions, but we have yet to discuss methods for their actual calculation. We will examine this problem under the assump- tion that the coefficients (ji' Yi of the recursion (3.6.5) are given. Golub and Welsch (1969) and Gautschi (1968, 1970) discuss the much harder problem of finding the coefficients (ji' Yi.

The theory of orthogonal polynomials ties in with the theory of real tridiagonal matrices

(3.6.19) J n =

and their principal submatrices b1 Y2

j'2

Such matrices will be studied in Sections 5.5, 5.6 and 6.6.1. In section 5.5 it will be seen that the characteristic polynomials Pj of Jj satisfy the recursions (3.6.5) with the matrix elements bi, Yi as the coefficients. Therefore, Pn is the characteristic polynomial of the tridiagonal matrix In . Consequently we have

(3.6.20) Theorem. The roots Xi' i

=

1, ... , n, of the nth orthogonal polynomial Pn are the eigenvalues of the tridiagonal matrix I n in (3.6.19).

The bisection method of Section 5.6, the QR method of section 6.6.6, and others are available to calculate the eigenvalues of these tridiagonal systems.

With respect to the weights Wi- we have [Szego (1959), Golub and Welsch (1969)]'

(3.6.21) Theorem. Let vii) ^:= (vyl, ... , v~)Y be an eigenvector of J n (3.6.19) for the eigenvalue Xi' J n vii) = Xi di). Suppose di) is scaled in such a way that

.b

vIi) T vii) = (Po, Po) =

I

^{w(x) dx.}

·a

Then the weights are given by

Wi = (vy))2, i = 1, ... , n.

PROOF. We verify that the vector

VIi) = (PoPo(xJ, P1P1(XJ, ... , Pn-1Pn-1(XJ) where-note that Yi

+- °

by (3.6.6b)---

~

^{1 for j = 0,}

Pj'= , 1

I

^{for j} = 1, ... , n - 1 Ih···Yj+1

is an eigenvector of J n for the eigenvalue Xi : J n vIi) = Xi vIi). By (3.6.5), for any x,

For j = 2, ... , n - 1, similarly,

and finally,

YjPj-2Pj-2{X)

+

bjPj-1 Pj-I{X)

+

Yj+IPjPj{X)

= Pj-I[Y7Pj-2{X)

+

^bjpj_I{X)

+

^pAx)]

= XPj-IPj-I{X),

Pn-I[Y;Pn-2

+

bnPn-I{X)] = XPn-IPn-I{X) - Pn-IPn(X), so that

YnPn-2Pn-2{X)

+

bnPn-IPn-I{X) = XiPn-IPn-I(Xi) holds, provided Pn{Xi)

=

O.

Since Pj

1-

0, j = 0, ... , n - 1, the system of equations (3.6.13) for Wi is equivalent to

(3.6.22)

with W = (WI' ... ' wnf, el = {1, 0, ... , Of.

Eigenvectors of symmetric matrices for distinct eigenvalues are orthogonal.

Therefore, multiplying (3.6.22) by V(ilT from the left yields (V(i)TV(i»)Wi = (Po, Po)vy).

Since Po

=

1 and Po(x) == 1, we have vyl

=

1. Thus (3.6.23)

Using again the fact that vyl = 1, we find vY)v(i) = vIi), and multiplying (3.6.23) by (vyl)2 gives

(V(ilT v(il)Wi = (vy»)2(po, Po).

Since v(ilT v(il = (Po, Po) by hypothesis, we obtain Wi = (vy»)2. D If the QR-method is employed for determining the eigenvalues of In , then the calculation of the first components vY) of the eigenvectors vIi) is readily included in that algorithm: calculating the abscissas Xi and the weights Wi can be done concurrently [Golub and Welsch (1969)].

Finally, we will estimate the error of Gaussian integration:

(3.6.24) Theorem. Iff E C2n[ a, b], then

b n p2nl(~)

fa

w{x)f{x) dx -

i~l

^WJ(Xi)

=

(2n)! (Pn, Pn) for some ~ E (a, b).

PROOF. Consider the solution h E I12n-¹ of the Hermite interpolation problem (see Section 2.1.5)

h(xJ = f(x;), h'(Xi) = f'(x;), i = 1, ... , n.

Since degree(h) < 2n,

b n n

f

w(x)h(x) dx =

L

wih(x;) =

L

wJ(x;)

'a i=l i=l

by Theorem (3.6.12). Therefore the error term has the integral representation

b n b

f

w(x)f(x) dx -

L

^WJ(Xi)

⁼ f

w(x)(J(x) - h(x)) dx.

~a i = l " ' a

By Theorem (2.1.5.10), and since the Xi are the roots of Pn(x) ^E

fin,

,[<2n)(0 2 2 ,[<2n)(0 2 f(x) - h(x) = (2n)! (X - Xd ... (X - Xn) = (2n)! Pn(X) for some' = nx) in the interval I(Xl' ... , xn , x) spanned by Xl' ... , xn , x.

Next,

,[<2n)(nx)) (2n)!

f(x) - h(x) p;(x)

is continuous on [a, b] so that the mean-value theorem of integral calculus applies:

b ,b f(2n)(~)

{ w(x)(J(x) - h(x)) dx =

t

w(x),[<2n)(((x))p;(x) dx = (2n)! (Pn, Pn)

for some ~ E (a, b).

o

Comparing the various integration rules (Newton-Cotes formulas, extra- polation methods, Gaussian integration), we find that, computational efforts being equal, Gaussian integration yields the most accurate results. If only one knew ahead of time how to chose n so as to achieve specified accuracy for any given integral, then Gaussian integration would be clearly superior to other methods. Unfortunately, it is frequently not possible to use the error formula (3.6.24) for this purpose, because the 2nth derivative is difficult to estimate and also because such an estimate may be much too pessimistic.

For these reasons, one will usually apply Gaussian integration for increasing values of n until successive approximate values agree within the specified accuracy. Since the function values which had been calculated for n cannot be used for n

+

1 (at least not in the classical case w(x) == 1), the apparent advantages of Gauss integration are soon lost. There have been attempts to remedy this situation [e.g. Kronrod (1965)]. It appears, however, that extra- polation methods are to be preferred in practice.