Gaussian quadrature for products of exponential functions

li NOgrH-Bi(ilAND

Gaussian Quadrature for Products of Exponential Functions M. Cordero, C. Martin,* and J. Miller

Department of Mathematics Texas Tech University Lubbock, Texas 79409

ABSTRACT Necessary and sufficient conditions are derived for the existence of Gaussian quadrature rules on spaces spanned by products of real exponential functions.

1.

INTRODUCTION

In this paper we solve the problem of determining all sets of real numbers {Ai: i = 1 , . . . , n} for which there exist n real numbers wi and n real numbers x~ so that

b

e(~+~')t dt =

f a

for all choices of k and j. Let the cardinality of S must be those sets of A's for which this solve the resulting quadrature

wi" e(~j+ak)x'

(1.1)

i=1

S = {Ak + )tj: k, j ~< n}. We first show that either 2 n - 1 or 2 n. We then characterize condition holds; then for those sets of A's we problem.

*Supported in part by NSF Grant #DMS 8905334, NASA Grant #NAG2-89 and NSA Grant #MDA904-90-H-4009.

APPLIED MATHEMATICS AND COMPUTATION 79:189-202 (1996) © Elsevier ScienceInc., 1996 0096-3003/96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(95)00254-F

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This problem arises in reconstructing the initial data for a linear parabolic differential equation on a closed interval, [a, b], when the solution is sampled at fixed time and a series of spatial points. In this problem, it is necessary to invert the matrix

¢0(xl)

" ' "

M= ~bo( xn)

Cn(x~)

"'"

where the ¢ are the eigenfunctions of the spatial operator. It can be shown that the matrix M has the property that there exists a positive diagonal matrix S such that M t S M --- I ( ( )t denotes transpose) if and only if there exists a Gaussian quadrature rule on the set of all products of the Ck; for a complete development see [1]. It is easy to see that polynomials have this property with respect to any positive measure. It can also be shown that Ck(x) -- sin kx or = cos kx have this property [2]. In this paper we show that exponential functions seldom satisfy this condition. 2.

CARDINALITY

In this section we show that the set S defined in Section 2 must have cardinality ~< 2 n.

THEOREM 2.1. L e t {~: i = 1 , . . . , N} be a set o f distinct real numbers. I f there exist real n u m b e r s wi, i = 1 , . . . , n and real n u m b e r s xi, i = 1 , . . . , n such that f o r k = 1 . . . . , N b

f

e ~# dt = a

wke ~'xk k~l

then N <~ 2 n.

We begin by noting that by evaluating the integral we have the following system of equations e b')q _ e a'~,

~.~

Ti

k= 1

Wk e Tizk,

Gaussian Quadrature for Products of Exponentials

191

provided that Ti is nonzero. We can view this as the statement that the %'s are solutions of the following equation e bt __ e at

p(t)

~ k=l

t

W k e txk ~

O.

The goal is to determine the maximal number of solutions this equation can have. Let 9(t) = tp(t). We note that g(t) is a solution of the differential equation n

( D - a)( D - b) i-I ( D -

2

x,) u( t) = 0 .

i~1

This differential equation is linear of degree 2 n + 2 with constant coefficients and with real roots for its auxiliary equation. Hence, it follows that g(t) can have at most 2 n + 1 real zeros. Now 9(0) = 0 and by passing to the limit, we have that p(0) = 0 if and only if

b-

a=

~

wk.

(2.2)

k=l

If condition (2.2) is satisfied, then by differentiating 9, we have that 9'(0) = 0. From the fact that g(t) = tp(t) and the Leibnitz differentiation formula, we have that 9 ("+ 1)(0) = 0 if and only if p(n)(0) = 0. Because p(t)

=

E

bJ+l-

j=0

(J+

aj+l

1

1) v"

J! k=lWkX~ tj

we have that p(n)(0) = 0 if and only if bn+ 1 __ a n + 1

( n + 1)!

1 n! ~'~ wkx~" k=l

Thus, either 9 has one more zero than p or 9(2"+1)(0) = 0. If it is the latter case, then 9(t) is identically zero because it would be the solution of a

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differential equation of order 2 n + 2 with 2 n + 2 zero initial conditions. Because g(t) cannot be identically zero and have the form given, we must conclude t h a t p(t) = 0 has at most 2 n solutions. Thus, the theorem follows. 3.

C A R D I N A L I T Y O F SUMS W e prove the following theorem.

THEOREM 3.1. Let {hi: i = 1 , . . . , n} be a set of distinct real numbers with h 1 < h 2 < ... < h,. Let S = {h i + h j : i, j <~ n}. The cardinality of S is less than or equal to 2 n if and only if there exist two real numbers, a and h, so that one of the following three conditions is satisfied. 1. h k = a + ( k 1)h, k = 1 , . . . , n ; 2. A 1 = a and h k = a + kh, k = 2 , . . . , n; 3. Ak = a + ( k 1)h, k = 1 , . . . , n 1 a n d A n = a + nh. Let L = { A i + h j : i or j is 1 or n}. T h e n the elements of L form a well-ordered set which is of length 2 n - 1 and is contained in S; hence, the cardinality of S is at least 2 n - 1. W e now assume t h a t the cardinality of S is 2 n - 1, so S = L. Because h2+h._l ~Land A 1 + An_ 1 • A 2 + ALn_ 1 < h 2 + A n ,

we have h 2 -[- An_ 1 = h 1 -[- An,

and hence, t h a t A2 - h I = An -

An_ 1.

Now A3 + A._ 1 e L, and we then have A1 + A n = A2 -[- A n _ 1 < A 3 -[- An_ 1 < A 3 "Jr-An, and hence, t h a t

Gaussian Quadrature for Products of Exponentials

193

Similarly, because Ak + An_ 1 E L, for k = 1 , . . . , n t h e n Ak +

An_ 1 = Ak_ 1 +

An.

a n d hence, Ak -

Ak_ 1 = An -

An_ 1

for k = 2 , . . . , n which shows t h a t C o n d i t i o n 1 is satisfied. Obviously, if C o n d i t i o n 1 is satisfied, t h e n t h e c a r d i n a l i t y of S is 2 n - 1. F r o m t h e proof of t h e previous case we notice t h a t if B 1 = {A~ + A , _ I :

k = 2 , . . . , n - 1}

is c o n t a i n e d in L, t h e n t h e c a r d i n a l i t y of S is 2 n - 1. One can show w i t h v i r t u a l l y identical techniques t h a t if t h e set B 2 = {A 2 + A~:

k---2,...,n-

1}

is c o n t a i n e d in L, t h e n t h e c a r d i n a i i t y of S is 2 n - 1. W e now assume t h a t t h e c a r d i n a l i t y of S is 2n. Then, obviously, A2 + An- 1 is c o n t a i n e d in L if a n d only if one of B1 or B 2 is c o n t a i n e d in L. Thus, because b y a s s u m p t i o n t h e c a r d i n a i i t y of S is 2 n, b y t h e a b o v e r e m a r k s we have t h a t h 2 + An_ 1 is n o t in L. W e now show t h a t in this case, C o n d i t i o n 2 or C o n d i t i o n 3 of t h e t h e o r e m holds. W e begin b y using t h e fact t h a t if A2 + An_ ~ ~ L t h e n A3 + An_ 1 a n d A2 + An_ 2 are in L. W e n o t e t h a t

A1 "{- An-i < A3 + An-1 < A3 -l- An and AI -{- An_ 2 < A2 + An_ 2 < A2 + An.

Therefore, we can conclude t h a t

A3 + A._, ~ {A~ + A., ~ + A.}

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M. C O R D E R O E T AL.

and

)t2 @ '~n-2 • { )[1 "[- An-1, )tl -[- '~n}" W e c o n s i d e r f o u r cases.

Case 1. Case 2. Case 3.

Case 4. Case 1.

A3 -[" An_ 1 = A3 -+- An_ 1 = A3 + An_ 1 = A3 -{- An_ 1 = Wehavethat

A1 4- A n = A2 + An_ 2 A1 -{- An, A2 + An_ 2 --= A1 + An_ 1 A2 + An, A2 + An_ 2 -= A1 + A n A2 + An, A2 + An_ 2 = A1 + An_ 1 A3 - A1 = A. - An_ 1 a n d A2 - A1 = A n - An_ 2.

Now A n - An_ 2 = A2 - A1 < A3 - A1 = A n - A n _ l , a n d hence, t h i s case is self c o n t r a d i c t o r y . Case 2. W e h a v e t h a t A , - A n_l = A 3 An-2. O u r o b j e c t is t o s h o w t h a t

A1 a n d

A2

-

A I =

A._I

-

A2 + An-k = A1 + A ~ - k + l for k = 2 , . . . , n - 1. W e p r o c e e d b y i n d u c t i o n . N o w k = 2 is p a r t of t h e h y p o t h e s i s so we a s s u m e t h e s t a t e m e n t h o l d s u p t o k = m. T h e n A1 + A n - m - 1 < A2 -{- A n - m - 1 < A2 + A n - m = A1 + A n - r e + l , a n d hence, A2 + An-m-i = AI + A.-m" So w e h a v e A2 -{- An_k_ 1 = A1 -t- An_k, for k = 2 , . . . , n - 1. T h u s , w i t h h = A2 - A1, C o n d i t i o n 3 of t h e t h e o r e m holds. Case 3. T h e p r o o f for t h i s c a s e is v i r t u a l l y t h e s a m e as t h e p r o o f in C a s e 2 a n d will be left t o t h e reader. B y u s i n g t h e s a m e a r g u m e n t s , it is c o n c l u d e d t h a t C o n d i t i o n 2 of t h e t h e o r e m is satisfied. Case 4. W e h a v e f r o m t h e h y p o t h e s i s t h a t A3 - A2 = A, - A n_ 1 a n d An-1 - An-2 = A2 - At- W e will b e g i n b y s h o w i n g t h a t Ak+l + A . - 1

= A k + A.

Gaussian Quadraturefor Products of Exponentials

195

for k = 2, . . . , n - 1. Now k = 2 is p a r t of t h e hypothesis, so we assume t h e result for k = m - 1 a n d n o t e A n + Am_ 1 = Am + An_ i < Am+ i + An_ i < Am+ 1 + A n , a n d hence, Am+i + An-1 = Am + An. Hence, we have t h a t An_ 1 -- An_ 2 ~--- A n -- A n _ l ,

a n d from t h e h y p o t h e s i s we have t h a t

A k + l -- Ak = An -- A n - l ,

for k = 1 . . . . , n - 1. Thus, C o n d i t i o n 1 of t h e t h e o r e m holds, which gives a c o n t r a d i c t i o n because t h e n t h e c a r d i n a l i t y of S m u s t be 2 n - 1. It is trivial t o show t h a t if C o n d i t i o n 2 or 3 holds, t h e n t h e c a r d i n a l i t y of S i s 2n. 4.

QUADRATURE

In Sections 2 a n d 3, we have shown t h a t in order for t h e r e l a t i o n given b y 1.1 to hold, t h e e x p o n e n t s of t h e e x p o n e n t i a l families {eaot: c = 1 , . . . , n} m u s t satisfy t h e conditions of T h e o r e m 3.1. In this section, we give necessary a n d sufficient conditions for t h e i d e n t i t y 1.1 to hold. W e begin b y n o t i n g t h a t t h e q u a d r a t u r e condition can be r e d u c e d t o a question of G a u s s i a n q u a d r a t u r e over s y s t e m s of p o l y n o m i a l s by

b

/a))

(bh) Xklf X --n -- 1

W e consider t h e t h r e e cases of T h e o r e m 3.1 s e p a r a t e l y .

~x k

(4.1)

196 4.1.

M. CORDERO ET AL. Condition 1

In this case, the problem simply reduces to standard Gaussian quadrature by the above identity. This has been noted in the literature explicitly by Newberry [3]. 4.2.

Condition 2

Condition 2 of Theorem 3.1 makes it necessary (in light of 4.1) to consider the following:

PROBLEM 4.1. D o t h e r e e x i s t r e a l n u m b e r s i = 1. . . . , n such that

wi,

i =

1 , . . . , n a n d x i,

n

f ~ dg(x) = Z w,x~ a

f o r k --

i=l

0,2,3,...,2n?

First we define

m~ = f ~ xk dg( x) and proceed by using a variant of Prony's method, [4]. We are trying to solve the system of equations

g. m 0 ..~ ~

wi

i=l

m2n

=

WiX i i=l

•

Gaussian Quadrature for Products of Exponentials

197

W e establish some useful n o t a t i o n s . Let mk = ( m k , . . . ,

i n k + , - 1 ) t, k = 2 , 3 , . . . ,

n + 1,

$

1

...

X1

1

Xn

• . .

V= X~- 1

D = Diag(

~ nn - 1

''"

Xl,...

, Xn)

and

= (w~,...,

w,)'.

W e can rewrite t h e above s y s t e m of equations as a s y s t e m of m a t r i x equations

m2 = VD2~v

-= ?nn+ 1

VDn+I~

F r o m this, we establish t h e s y s t e m of recurrences

m3-- V D V - l m 2

m,+l = VDV-lm, • Note t h a t 0 0 VDV -1

0 1

""

ai's.

0/

0

=

•

0 a0 for some real n u m b e r s

1 0 0 a~

0 %

"" ...

i an-1

°

198

M. CORDERO ET AL.

By examining the last row of each of the matrix recurrences, we have the following relations

mn+ 2

a o m 2 A- a i m 3 + "" + a n _ l m n + l

ran+k-1

a o m k _ 1 -t- a i m k + " " . - b a n _ l i n k + n _ 2

m2 n

ao m n + a l t o n + 1 -4- "'" ...k an_ l m 2 n _ l.

Let p,~( x )

= x~ -

the characteristic polynomial of in terms of integrals as

a,_ l x ~-1 .....

VD

ao ,

V-1, and rewrite the moment equations

/3 =

0

=

o.

t3

fopn( )x -2(

2

Thus, the polynomial Pn(X) must be orthogonal to all polynomials of degree less than or equal to n - 2 with respect to the positive measure (x 2 dtt(x)). Let {~bk}~=0 be a sequence of polynomials orthogonal with respect to the measure ( x 2 d/t(x)) with the properties, that the degree of ~b~ is k and Ck is monic. We then have that p.(x)

=

+

1(x),

where c is an arbitrary real parameter. Now any set of nodes for the original problem will be zeros of p , ( x ) for some choice of the parameter c. Now the zeros of ¢ , and ~b,_ 1 interlace, and the zeros of p~ approach the zeros of ~b,_ 1 as n approaches + oo. We also know from classical root-locus theory, (See any elementary control theory text, for example, [5].), that the roots of p~ interlace with the roots of ¢~_ 1. Thus, for every choice of c such that the

Gaussian Quadrature for Products of Exponentials

199

roots of pn are positive we have a unique choice for the vector of weights ~. The question then reduces to whether the weights satisfy m o = 2_, wi.

For a given p~, we can calculate the weights by constructing the Lagrange interpolation polynomials for the zeros of Pn. Let the Lagrange polynomials be denoted by l~(x). We then have that B

~ = f li( ~)(~2 ~(~)), and we can perform the following calculation using the fact that n

1 = E li(~). i=1

mo -- f e~(~) ~ ~

Wi

i=1

= f~ ~ l,( x)(x2 d~(~)) ai=l t~

= fo(x2 d,(x)) m 2.

We have proven the following theorem.

THEOREM 4.1. There exist real n u m b e r s w~, i = 1 , . . . , n and positive real n u m b e r s x~, i = 1 , . . . , n such that the equations

f~ x k ~ ( ~ ) = E w~x~

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M. CORDERO ET AL.

are satisfied for k = 0, 2 , . . . , 2 n if and only if

d.( x) = Q x~ d~( x).

4.3. Condition 3 Again, from condition 3 of Theorem 3.1, we consider the problem:

PROBLEM 4.2. Do there exist real numbers wi, i = 1 , . . . , n and xi, i = 1 , . . . , n such that 0

f~ ~ d~(~)-- ~2 ~,,~,~ i=l

f o r k = 0, 1 , 2 , . . . , 2 n -

2,2n?

We will reduce this problem to Problem 4.1. We begin by relabeling the exponents as x 2"-k, k = 0, 2 , . . . , 2 n and now consider the following calculation. Let u = x-x. Then

f~

/3-1 u~(u-2° d~(u-1)) x2.- k d~(x) = fo_ or-1

Ot-I = fo,

u k dm(u)

where din(u) is the positive measure,

( - u -2" d/z( u-1)). So we ask if the equations

f

u ~ dm(~) =

i=l

~o,u,~

201

Gaussian Quadrature for Products of Exponentials

have solutions for k = 0,2, . . . , 2 n . This was answered by T h e o r e m 4.1. After a few simple calculations, we have the following theorem.

THEOREM 4.2. There exist real n u m b e r s w~, i = 1 . . . . , n and positive real n u m b e r s x,, i = 1, . . . , n such that the equations

f ~ d~(x) = a

~,x~ i=1

are satisfied f o r k = O, 1, 2 , . . . , 2 n - 2, 2 n i f and only i f 13

t3

.f X~n d~.(x) = .f x ~n-~ d~(x).

5.

CONCLUSION In s u m m a r y , we have proven the following theorem.

THEOREM 5.1. L e t {Ak: k = 1 , . . . , n} be a set of n distinct real numbers. There exist n real n u m b e r s a < x 1 < x 2 < ... < x n < b and n real n u m b e r s w k such that f o r all choices o f k and j b

f

Wi • e(aJ+ak)x`

e (a¢+'~k)t dt = a

i=1

i f and only i f there exist real n u m b e r s a and h such that 1. Ak = a + ( k - 1 ) h

, k= l,...,

n

2. A1 = a , A k = a +

kh, k = 2 , . . . , n and

or

b

b

f aeat dt = f ae(a+2h)t dt or 3. Ak = a + ( k -

1)h, k = 1 . . . . , n -

1, A n = a +

b b fae(a+(n-2)h)tdt.=fae(a+nh)tdt.

nhand

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M. CORDERO ET AL.

It would seem that the theorem should be true if dt is replaced by a more general measure dry(t). However, the proof of Section 2 fails to generalize in this more general case, the problem being that if a more general measure is used, the function b

f(t) = foist dp,(s) need not satisfy any linear differential equation. However, there is a possibility of using the representation of f as

f( t) =

E

rant n n]

n=O

where m n is the nth moment of the measure dte. This is currently under investigation. REFERENCES 1 D. Gilliam, J. Lund, and C. Martin, Inverse Parabolic Problems and Discrete Observability, submitted. 2 D. Gilliam, J. Lund, and C. Martin, Gaussian Quadrature for Trigonometric Polynomials: An Example from Observability. Preprint, Texas Tech University. 3 A. C. R. Newberry, Some extensions of Legendre quadrature, Math. Comp. 23:173-176 (1969). 4 F.B. Hilderbrand, Introduction to Numerical Analysis, Dover Publishing Inc., New York, 1974. 5 N. Sinha, Control Systems, Holt-Reinhart-Winston, New York, 1986.