error estimates for rayleigh?ritz approximations of eigenvalues and eigenfunctions of the mathieu...

DOI: 10.1007/s00365-002-0527-9

Constr. Approx. (2004) 20: 39–54

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APPROXIMATION© 2003 Springer-Verlag New York Inc.

Error Estimates for Rayleigh–Ritz Approximations ofEigenvalues and Eigenfunctions of the Mathieu and

Spheroidal Wave Equation

Hans Volkmer

Abstract. Error estimates are derived for the computation of eigenvalues and eigen-

vectors of infinite tridiagonal matrices by the Rayleigh–Ritz method. The results are

applied to the Mathieu and spheroidal wave equation.

1. Introduction

It is well-known that self-adjoint operators associated with the Mathieu and spheroidal

wave equation can be represented by infinite symmetric tridiagonal matrices. Since pow-

erful software packages are widely available that compute eigenvalues of finite symmet-

ric tridiagonal matrices with almost arbitrary precision, it is natural to approximate the

eigenvalues of infinite symmetric tridiagonal matrices by eigenvalues of suitably cho-

sen finite submatrices. This method was proposed by Green and Michaelson [8] for the

Mathieu equation and by Hodge [9] for the spheroidal wave equation. Today the method

is widely employed by programs in software libraries [5], [16], [19]. Recently it was

used by Falloon [7] in his development of a software package for spheroidal wave func-

tions for Matematica. Of course, the method is just a special case of the Rayleigh–Ritz

method; see Weinstein and Stenger [17]. The Rayleigh–Ritz method is also mentioned

by the author in his chapter on spheroidal wave functions in the Digital Library of Mathe-

matical Functions (DLMF), the new edition of the Handbook of Mathematical Functions

by Abramowitz and Stegun [1].

The Rayleigh–Ritz method yields a decreasing sequence µr that approximates the

desired eigenvalue λ, and a sequence gr of elementary functions that approximates the

desired eigenfunction f (Mathieu or spheroidal wave function). The approximations µr

and gr are computed from an r × r submatrix of an infinite matrix. When writing the

chapter on spheroidal wave functions the author and the editors of the DLMF project

noticed several gaps in our knowledge about the convergence properties of the method.

For example, the editors asked whether the sequence gr converges uniformly to f as

Date received: January 16, 2002. Date revised: May 28, 2002. Date accepted: September 12, 2002. Commu-

nicated by Mourad Ismail. Online publication: February 21, 2003.

AMS classification: 33E10, 65L60, 65L70.

Key words and phrases: Rayleigh–Ritz method, Mathieu equation, Spheroidal wave equation, Tridiagonal

matrices, Error estimates.

39

40 H. Volkmer

r → ∞. More ambitiously, one may ask for error bounds for the maximum norm

‖gr − f ‖∞. Such bounds are needed when we have to decide how to truncate the infinite

matrix to a finite matrix for a prescribed precision. Another natural question is to ask for

the rate of convergence µr → λ and gr → f as r → ∞. This paper provides answers

to these questions, answers that the author was unable to find in the literature.

We know more about corresponding problems for the Mathieu equation but our knowl-

edge is also not satisfactory. There are lower bounds for λ derived from the well-known

method of intermediate problems due to Aronszaijn and Weinstein; see Weinstein and

Stenger [17, Chapter 5]. However, it seems impossible to estimate the rate of conver-

gence of µr → λ by this method. Of course, the method of intermediate problems is

applicable to a much larger class of eigenvalue problems than that treated in this paper.

Green and Michaelson [8] and the “Numerical Analysis” group at Delft [15] provide

error estimates for the computation of eigenvalues of the Mathieu equation but give no

rate of convergence and no error estimates for the approximations of eigenfunctions.

The estimates are different from those given in this paper. The derivation in [8], [15] is

based on the following simple lemma due to Wilkinson [18]:

Let A be an n × n symmetric matrix, µ ∈ R and x ∈ Rn , with Euclidian norm ‖x‖ = 1.

Let ε := ‖Ax − µx‖. Then there is an eigenvalue λ of A such that |λ − µ| ≤ ε.

Unfortunately, we face a technical problem when we try to apply this lemma. If we order

the eigenvalues of A is ascending order as λ1 ≤ λ2 ≤ · · · ≤ λn , then the lemma tells

us that there is i ∈ {1, . . . , n} such that |λi − µ| ≤ ε but the lemma does not tell us the

value of the subscript i . But knowledge of this value is required in the derivation of error

bounds for the Rayleigh–Ritz method. Papers [8] and [15] make a natural assumption

what i should be. In order to avoid this indexing problem we base the analysis on the

minimum–maximum principle for eigenvalues of self-adjoint operators. This principle

allows identification of the i th eigenvalue λi . Our error bounds are sharper than those in

[8] and [15] as indicated in Section 4.

Blanch [6] treats error estimates for the computation of eigenvalues and eigenfunctions

of the Mathieu equation based on continued fractions. However, in [15, p. 41] it is

stated that this method in some cases will not produce the correct result. The method

requires the knowledge of a “sufficiently good approximation” for the desired eigenvalue

λ as an initial value for a recursive procedure. Such approximations are usually derived

from power series and asymptotic expansions of eigenvalues. Alhargan [2], [3] provides

an elaborate system of initial approximations whose suitability is tested by numerical

experimentation. Falloon [7] points out that, today, it is easier to find such approximations

using the Rayleigh–Ritz method. So even if we use continued fractions an error estimate

of the Rayleigh–Ritz approximation is useful. The main advantage of using the Rayleigh–

Ritz method to compute eigenvalues and eigenfunctions of the Mathieu and spheroidal

wave equation is that we can rely on sophisticated linear algebra software packages while

almost no additional programming is needed.

Shirts [13], [14] shows how to compute a Mathieu function of noninteger order using

finite submatrices of doubly infinite matrices. Errors are investigated by numerical ex-

perimentation. It would be of interest to derive rigorous error bounds along the lines of

this paper.

Error Estimates for Rayleigh–Ritz Approximations 41

2. The Rayleigh–Ritz Method for Self-Adjoint Operators

Let H be a Hilbert space with inner product (·, ·) and norm ‖ · ‖ and let A : H ⊃D(A) → H be a self-adjoint operator, bounded below with compact resolvent; see [10].

Then the spectrum of A agrees with the set of eigenvalues of A, it is bounded below,

and has no finite point of accumulation. Therefore, we may order the eigenvalues of A

in ascending order and counted according to multiplicity as

λ1 ≤ λ2 ≤ λ3 ≤ · · · .

The minimum–maximum principle [17, p. 12] states that, for all k ∈ N,

λk = minEk

max{(Ax, x) : x ∈ Ek, ‖x‖ = 1},(2.1)

where the minimum is taken over all linear subspaces Ek of D(A) with dimension k.

Let E be a linear subspace of D(A) of finite dimension r , let P : H → H be the

orthogonal projection of H onto E , and let Q := I − P . Let

µ1 ≤ µ2 ≤ · · · ≤ µr

denote the eigenvalues of the operator PA : E → E counted according to multiplicity.

Again we have the minimum–maximum principle

µk = minFk

max{(Ax, x) : x ∈ Fk, ‖x‖ = 1},(2.2)

where the minimum is taken over all linear subspaces Fk of E with dimension k.

Equations (2.1) and (2.2) imply the Poincare inequalities [17, p. 10],

λk ≤ µk, k = 1, . . . , r.(2.3)

We wish to estimate µk − λk . Let x1, x2, x3, . . . denote an orthonormal basis of H

such that Axj = λj x j . We introduce k × k Hermitian matrices L = (L ij), M = (Mij) by

L ij := (Qxi , x j ), Mij := ((PAP − A)xi , x j ), i, j = 1, . . . , k.

We assume that ‖L‖ < 1, where ‖L‖ denotes the spectral norm of L . Then the vectors

Px1, . . . , Pxk are linearly independent and, thus, Fk := span{Px1, . . . , Pxk} is a linear

subspace of E with dimension k. We use this space Fk in (2.2) in order to derive an upper

bound for µk . Let

0 �= w =k

∑

i=1

αi xi .

Since

(Aw, w) =k

∑

i=1

λi |αi |2 ≤ λk‖w‖2 = λk(‖Pw‖2 + ‖Qw‖2),

42 H. Volkmer

we obtain

(APw, Pw) − λk‖Pw‖2 ≤ ((PAP − A)w, w) + λk‖Qw‖2

≤ ‖M + λk L‖‖w‖2.

Now

‖w‖2 = ‖Pw‖2 + ‖Qw‖2 ≤ ‖Pw‖2 + ‖L‖‖w‖2

yields

(APw, Pw) − λk‖Pw‖2 ≤‖M + λk L‖

1 − ‖L‖‖Pw‖2.

This inequality combined with (2.2) implies the following lemma:

Lemma 1. If ‖L‖ < 1, then

µk − λk ≤‖M + λk L‖

1 − ‖L‖.

Let y1, . . . , yr be an orthonormal basis of E such that PAyi = µi yi and (xk, yk) ≥ 0.

We wish to estimate ‖yk − xk‖. Write

Pxk =r

∑

i=1

βi yi , βi = (xk, yi ).

Then we have

r∑

i=1

|βi |2(µi − λk)2 =

∥

∥

∥

∥

∥

r∑

i=1

βi (µi − λk)yi

∥

∥

∥

∥

∥

2

= ‖PAPxk − PAxk‖2.

Let

K := ‖PAPxk − PAxk‖, := rmaxk �=i=1

‖µi − λk |−1,

and assume that < ∞. Then

r∑

k �=i=1

|βi |2 ≤ 2 K 2.(2.4)

Since

1 − ‖Qxk‖2 = ‖Pxk‖2 =r

∑

i=1

|βi |2,

we obtain, from (2.4),

1 − ‖Qxk‖2 − 2 K 2 ≤ β2k .

Hence

0 ≤ 1 − βk ≤ 1 − β2k ≤ ‖Qxk‖2 + 2 K 2


and

‖Pxk − yk‖2 =r

∑

k �=i=1

|βi |2 + |βk − 1|2

imply the following lemma:

Lemma 2. We have

‖Pxk − yk‖2 ≤ 2 K 2 + (‖Qxk‖2 + 2 K 2)2

and

‖xk − yk‖2 ≤ ‖Qxk‖2 + 2 K 2 + (‖Qxk‖2 + 2 K 2)2.

3. The Rayleigh–Ritz Method for Infinite Tridiagonal Matrices

Let H = l2(N) with natural orthonormal basis en, n ∈ N. Let cn, dn, n ∈ N, be two real

sequences such that cn is bounded, cn �= 0 for all n, and dn converges to infinity. Let

S : H ⊃ D(S) → H be the linear operator defined by Sen = dnen with domain

D(S) =

{

∞∑

n=1

unen ∈ H :

∞∑

n=1

|dnun|2 < ∞

}

.

Let T : H → H be the linear operator defined by Ten = cn−1en−1 + cnen+1, where

c0 := 0. Then S is self-adjoint, bounded below with compact resolvent, while T is

bounded and self-adjoint. It follows from [10, p. 291] that A := S + T is self-adjoint,

bounded below with compact resolvent. Formally, A is given by the infinite symmetric

tridiagonal matrix

A =

d1 c1 0 0 0 . . .

c1 d2 c2 0 0 . . .

0 c2 d3 c3 0 . . .

......

......

...

.(3.1)

We denote the eigenvalues and eigenvectors of A as in Section 2 and we set xi,n :=(xi , en). Let Pr denote the orthogonal projection of H onto Er := spanr

j=1 ej and let

Qr := I − Pr . Let

µ1,r < µ2,r < · · · < µr,r

denote the eigenvalues of the r × r tridiagonal matrix Ar consisting of the first r rows

and columns of (3.1). It should be mentioned that other choices of principal submatrices

of A are possible which sometimes may lead to better error estimates.

Let y1,r , . . . , yr,r be an orthonormal basis of Er such that Ar yj,r = µj,r yj,r and

(yj,r , x j ) ≥ 0. In order to estimate µk,r −λk and ‖yk,r −xk‖ we assume given computable

quantities �i , �i,k, i = 1, . . . , k, such that

λi ≤ �i , λk − λi ≤ �i,k .

44 H. Volkmer

In applications we will use �i = µi,s with s ≥ i which is possible by the Poincare

inequality (2.3). For an estimate of �i,k it will be sufficient to note that

λ1 ≥∞infj=1

(dj − |cj | − |cj−1|).

Lemma 3. The matrix elements of L , M , and the quantity K (corresponding to P = Pr )

admit the estimates

|L ij| ≤ ‖Qr xi‖‖Qr x j‖,

|Mij + λk L ij| ≤ |cr ||xi,r+1||x j,r | + �i,k‖Qr xi‖‖Qr x j‖,

K ≤ |cr ||xk,r+1|.

Proof. The estimate of L ij = (Qr xi , Qr x j ) follows from the Cauchy–Schwarz in-

equality. Since

APr xi = −cr xi,r+1er + cr xi,r er+1 + λi Pr xi ,

we obtain

Pr APr xi − Axi = −cr xi,r+1er − λi Qr xi .

Thus

|Mij + λk L ij| ≤ |cr ||xi,r+1||x j,r | + (λk − λi )|L ij|

≤ |cr ||xi,r+1||x j,r | + �i,k‖Qr xi‖‖Qr x j‖.

The estimate for K follows from

Pr APr xk − Pr Axk = −cr xk,r+1er .

Let i ∈ N. Choose ℓi ≥ 2 so large that

dn > |cn−1| + |cn| + �i for n ≥ ℓi .(3.2)

For n ≥ ℓi we define f(0)

i,n := 1 and then, recursively,

f(q+1)

i,n :=|cn−1|

dn − �i − |cn| f(q)

i,n+1

.

By induction on q we see that

1 ≥ f(q)

i,n > f(q+1)

i,n > 0.

The following lemma estimates the eigenvector xi of A; see [12, p. 91] for a similar

estimate:

Lemma 4. For n ≥ ℓi and q ≥ 0, we have

|xi,n| ≤ f(q)

i,n |xi,n−1|.(3.3)


Proof. Since the sequence xi,n, n ∈ N, is in l2(N), there are arbitrarily large n ≥ ℓi

such that |xi,n+1| ≤ |xi,n|. Then

cn−1xi,n−1 + (dn − λi )xi,n + cn xi,n+1 = 0(3.4)

implies that

|xi,n| ≤|cn−1|

dn − �i − |cn||xi,n−1| ≤ |xi,n−1|.

This demonstrates that |xi,n| ≤ |xi,n−1| for all n ≥ ℓi . Hence (3.3) is true for q = 0. Now

induction on q using (3.4) proves (3.3) for all q .

We pick q > 0 and set

fi,n := f(q)

i,n .(3.5)

The following error estimates improve when we increase q although the choice q = 1

or q = 2 will usually be sufficient.

Lemma 5. For i ∈ N and n ≥ ℓi , we have

|xi,n| ≤ ai,n :=n

∏

j=ℓi

fi, j ,(3.6)

‖Qn xi‖ ≤ bi,n := (1 − f 2i,n+2)

−1/2ai,n+1.(3.7)

Proof. Since |xi,n| ≤ 1 for all n, we obtain (3.6) from (3.3). Moreover,

‖Qn xi‖ ≤ |xi,n+1|(1 + f 2i,n+2 + f 2

i,n+2 f 2i,n+3 + · · ·)1/2.

Since 0 < fi, j < 1 for all j ≥ ℓi , we can estimate the sum under the root by a geometric

series to obtain the desired bound (3.7).

In order to estimate µk,r − λk we now proceed as follows. For the choice of ℓi , i =1, . . . , k, we have to verify (3.2) which comprises infinitely many conditions. Therefore,

we assume given a computable sequence Dn → ∞ and a computable bounded sequence

Cn such that

Dn ≤ Dn+1, Cn ≥ Cn+1, dn ≥ Dn, |cn| ≤ Cn for n ∈ N.(3.8)

Then (3.2) is satisfied if

Dℓ > Cℓ + Cℓ−1 + �i , ℓ = ℓi .

Next, for r ≥ ℓk , we estimate the entries of the k × k matrices L and M + λk L by

applying Lemmas 3 and 5. Then we bound the spectral norms of these matrices by their

Schur norms. Finally, we apply Lemma 1 to estimate µk,r − λk .

46 H. Volkmer

This way we obtain the following theorem:

Theorem 1. Using the quantities ai,n and bi,n from Lemma 5 we have, for r ≥ ℓk ,

‖L‖ ≤k

∑

i=1

b2i,r ,

‖M + λk L‖ ≤ |cr |

(

k∑

i=1

a2i,r

k∑

j=1

a2j,r+1

)1/2

+

(

k−1∑

i=1

�2i,kb2

i,r

k∑

j=1

b2j,r

)1/2

.

If ‖L‖ < 1, then

µk,r − λk ≤‖M + λk L‖

1 − ‖L‖.

In order to estimate ‖yk,r − xk‖, we find an upper bound for K according to Lemmas 3

and 5. We derive an upper bound for

= rmaxk �=i=1

|µi,r − λk |−1

by using the estimate for λk obtained above. Then we apply Lemma 2.

Theorem 2. If r ≥ ℓk , we have

‖yk,r − xk‖2 ≤ b2k,r + 2c2

r a2k,r+1 + (b2

k,r + 2c2r a2

k,r+1)2.

Several obvious simplifications in these estimates lead to somewhat cruder error

bounds. For example, we may use that ai,n and bi,n are increasing in i .

We note two results on the rate of convergence that follow from these estimates.

Theorem 3. Let k ∈ N. For every ε > 0, we have

µk,r − λk = O(εr ) as r → ∞

and

‖yk,r − xk‖ = O(εr ) as r → ∞.

The O-constants may depend on k but are independent of r .

Theorem 4. Assume that dn > 0 for all n (which we may achieve by replacing dn by

dn + τ for sufficiently large τ ), and that

∞∑

n=1

1

dn

< ∞.

Let k ∈ N. Then we have

µk,r − λk = O

(

c21 · · · c2

r

d21 · · · d2

r dr+1

)

as r → ∞,


and

‖yk,r − xk‖ = O

(

|c1 · · · cr+1|d1 · · · dr+1

)

as r → ∞.

The O-constants may depend on k but are independent of r .

It should be noted that explicit expressions for the O-constants in Theorems 3 and 4

can be derived from the estimates in Lemmas 1, 2, 3, and 5.

4. Application to the Mathieu Equation

We consider the problem of finding eigenvalues λ for which the Mathieu equation

w′′ + (λ − 2h2 cos(2t))w = 0(4.1)

has nontrivial odd solutions of period π . Here h2 is a given real parameter. Other self-

adjoint eigenvalue problems for the Mathieu equation can be treated similarly. The

well-known representation of the differential operator

w �→ −w′′ + 2h2 cos(2t)w

with respect to the orthonormal basis

2√

πsin(2nt), n ∈ N,

in L2(0, π/2), is of the form (3.1) where

cn = h2, dn = 4n2, n ∈ N.

The eigenvalue λk of the corresponding operator A agrees with the Mathieu eigenvalue

b2k(h2) in the notation of Arscott [4, p. 53].

By Theorem 4, we obtain, for fixed k and h2,

µk,r − λk = O

(

h4r

42r+1r !2(r + 1)!2

)

as r → ∞.

The O-constant can be computed from Theorem 1 for given k and h2. For practical

work a lookup-table of such O-constants for various ranges of k and h2 could be created.

For example, if k = 1 and −1 ≤ h2 ≤ 1, using �1 = µ1,1 = 4, ℓ1 = 2, and q = 1 in

(3.5), we obtain

a1,n ≤ b1,n−1 ≤ 2.425

(

|h2|4

)n−11

n!2

which leads to

µ1,r − λ1 ≤94h4r

42r+1r !2(r + 1)!2for r ≥ 3.(4.2)

48 H. Volkmer

Table 1. Estimates from Theorem 1 for distance between Mathieu eigenvalue

b2(10) and approximation µ1,r .

r µ1,r µ1,r − b2(10) Error estimate

5 −2.3821550258499152723614319 0.32 · 10−5 1.16 · 10−5

6 −2.3821582248055205665354767 0.11 · 10−7 0.40 · 10−7

7 −2.3821582359350987602677340 0.22 · 10−10 0.79 · 10−10

8 −2.3821582359569297267420560 0.26 · 10−13 0.94 · 10−13

9 −2.3821582359569556955120616 0.20 · 10−16 0.71 · 10−16

10 −2.3821582359569557153243100 0.10 · 10−19 0.36 · 10−19

11 −2.3821582359569557153344461 0.36 · 10−23 1.30 · 10−23

This estimate is quite sharp. For example, if r = 10, h2 = 1, then

λ1 = 3.91702477299847118670341688529379991002933424614053 . . . ,

µ1,10 = 3.91702477299847118670341688529379991002989666177165 . . . ,

so the left-hand side of inequality (4.2) is approximately 0.56·10−39 whereas the estimate

on the right-hand side gives 1.01 · 10−39. These and the following computations were

carried out in high precision arithmetic with Maple.

Table 1 shows error estimates in the case k = 1, h2 = 10 for the computation of the

Mathieu eigenvalue

b2(10) = −2.3821582359569557153344497960307538858413 . . . .

We use �1 = µ1,5 and q = 2 in (3.5).

Table 2 shows error estimates in the case k = 6, h2 = 10 for the computation of the

Mathieu eigenvalue

b12(10) = 144.3502080084893260019912658518645935310531 . . . ,

using �i = µi,i+4 for i = 1, . . . , 6 and q = 2 in (3.5).

Surprisingly, the quality of the error bounds, when compared with the true error,

improves if k is increased. As was to be expected, the speed of convergence becomes

Table 2. Estimates from Theorem 1 for distance between Mathieu eigenvalue

b12(10) and approximation µ6,r .


7 144.3816507440327643267900414 0.31 · 10−1 0.40 · 10−1

8 144.3503676402783310493153412 0.16 · 10−3 0.19 · 10−3

9 144.3502083582352186876740700 0.35 · 10−6 0.40 · 10−6

10 144.3502080088932724039297084 0.40 · 10−9 0.46 · 10−9

11 144.3502080084896019655539877 0.28 · 10−12 0.31 · 10−12

12 144.3502080084893261223614652 0.12 · 10−15 0.14 · 10−15

13 144.3502080084893260020266843 0.35 · 10−19 0.40 · 10−19


Table 3. Estimates from Theorem 1 for distance between Mathieu eigenvalue b2(100) and µ1,r .


7 −141.26840751579981679504 0.12 · 10−1 0.82 · 10−1

8 −141.27933852281769510426 0.72 · 10−3 5.06 · 10−3

9 −141.28002555042468333569 0.31 · 10−4 2.20 · 10−4

10 −141.28005582197545941739 0.99 · 10−6 6.94 · 10−6

11 −141.28005678552116753108 0.23 · 10−7 1.62 · 10−7

12 −141.28005680821069021504 0.41 · 10−9 2.88 · 10−9

13 −141.28005680861389319213 0.56 · 10−11 3.91 · 10−11

worse when h2 is increased. Table 3 shows error bounds in the case k = 1, h2 = 100 for

the computation of the Mathieu eigenvalue

b2(100) = −141.2800568086194528253587605820418627831331 . . . ,

using �1 = µ1.5 and q = 2 in (3.5).

In [15, p. 45] it is mentioned that the computation of the Mathieu eigenvalue b2(1000)

with double precision requires a 515 × 515 matrix based on the error estimates given in

[8], compared with a 57 × 57 matrix based on the error estimates derived in [15]. Since

b2(1000) = −1811.5224151493534312803605191615791051136948 . . . ,

double precision requires µ1,r − b2(1000) < 10−13. Theorem 1 with �1 = µ1,10 and

q = 4 in (3.5) gives the estimate µ1,26 − b2(1000) < 0.59 · 10−13, so a 26 × 26 matrix is

sufficient. In fact, a 26 × 26 matrix is the smallest matrix that is needed to approximate

b2(1000) within an error less than 10−13. For 57×57 and 515×515 matrices, we obtain

µ1,57 − b2(1000) < 0.32 · 10−71 and µ1,515 − b2(1000) < 0.12 · 10−2274.

The Mathieu eigenfunction se2k(t) = se2k(t, h2)belonging to the eigenvalue b2k(h2) =

λk is given by

se2k(t) =∞

∑

n=1

xk,n sin(2nt),

where xk = (xk,1, xk,2, . . .) is the eigenvector of A belonging to the eigenvalue λk and∑∞

n=1 x2k,n = 1; see [4, p. 55]. We use the eigenvector yk,r = (u1, . . . , ur , 0, 0, . . .)

of Pr APr belonging to the eigenvalue µk,r to approximate se2k by the trigonometric

polynomial

pk,r (t) =r

∑

n=1

un sin(2nt).

The L2(0, π/2)-norm

‖se2k − pk,r‖ =√

π

2‖xk − yk,r‖

can be estimated as indicated in Section 3. In particular, by Theorem 4,

‖se2k − pk,r‖ = O

(

h2r+2

4r+1(r + 1)!2

)

as r → ∞.(4.3)

50 H. Volkmer

Table 4. Estimates from Theorem 2

for L2(0, π/2) distance between Math-

ieu function se12(·, 10) and approximation

p6,r .

r ‖se12 − p6,r ‖ Error estimate

7 0.14 · 10−1 0.19 · 10−1

8 0.74 · 10−3 1.12 · 10−3

9 0.29 · 10−4 0.46 · 10−4

10 0.86 · 10−6 1.39 · 10−6

11 0.20 · 10−7 0.33 · 10−7

12 0.37 · 10−9 0.63 · 10−9

13 0.58 · 10−11 1.00 · 10−11

Table 4 gives L2-error estimates in the case k = 6, h2 = 10 for the computation of

the Mathieu function se12. We choose q = 1 in (3.5).

The following theorem estimates the maximum norm of se2k − pk,r :

Theorem 5. For every t ∈ R, we have

|se2k(t) − pk,r (t)| ≤ ‖Pr xk − yk,r‖

(

r∑

n=1

sin2(2nt)

)1/2

+∞

∑

n=r+1

|xk,n|| sin(2nt)|,

where the quantities on the right-hand side of the inequality can be estimated according

to Lemmas 2, 3, and 5.

Proof. We write

se2k(t) − pk,r (t) =r

∑

n=1

(xk,n − un) sin(2nt) +∞

∑

n=r+1

xk,n sin(2nt)

and estimate the first sum using the Cauchy–Schwarz inequality.

Table 5 shows error estimates in the case k = 6, h2 = 10 for the computation of

se12(0.2, 10) = 0.8143833501330276753773030610857527020833 . . . .

5. Application to the Spheroidal Wave Equation

We consider the problem of finding eigenvalues λ for which the spheroidal wave equation

((1 − t2)w′)′ +(

λ + γ 2(1 − t2) −m2

1 − t2

)

w = 0(5.1)

has nontrivial bounded even solutions in (−1, 1). Here γ 2 is a given real parameter and

m = 0, 1, 2, 3, . . . . The eigenvalue problem for bounded odd solutions can be treated

similarly.


Table 5. Estimates from Theorem 5 for distance between se12(0.2, 10) and approximation p6,r (0.2).

r p6,r (0.2) se12(0.2, 10) − p6,r (0.2) Error estimate

7 0.81649100562612785984 −0.21 · 10−2 1.28 · 10−2

8 0.81400537451258646982 0.38 · 10−3 1.30 · 10−3

9 0.81440948923732886079 −0.26 · 10−4 0.71 · 10−4

10 0.81438236755585789352 0.98 · 10−6 2.53 · 10−6

11 0.81438337423093520623 −0.24 · 10−7 0.63 · 10−7

12 0.81438334972786824766 0.41 · 10−9 1.16 · 10−9

13 0.81438335013757800301 −0.46 · 10−11 1.56 · 10−11

In L2(0, 1) we work with the orthonormal basis

�mn (t) =

Pmn (t)

sn,m

, sn,m =

√

1

(2n + 1)

(n + m)!

(n − m)!, n = m, m + 2, m + 4, . . . ,

where Pmn denotes the associated Legendre function. The well-known representation of

the differential operator

w �→ −((1 − t2)w′)′ +(

m2

1 − t2− γ 2(1 − t2)

)

w

with respect to this orthonormal basis has the form (3.1) where

dj = (m + 2 j − 2)(m + 2 j − 1) − 2γ 2bj ,

bj =(m + 2 j − 2)(m + 2 j − 1) − 1 + m2

(2m + 4 j − 5)(2m + 4 j − 1),

cj = γ 2

√(2 j − 1)2 j

√2m + 2 j − 1

√2m + 2 j

(2m + 4 j − 1)√

2m + 4 j − 3√

2m + 4 j + 1.

The eigenvalue λk of the corresponding operator A agrees with the spheroidal eigenvalue

λmm+2(k−1)(γ

2) in the notation of [4].

Note that the sequence bj converges to 14

as j → ∞. The sequence bj is monotonically

decreasing for m ≥ 1, and monotonically increasing for m = 0 except for the first term

b1 = 13. Therefore, we can choose a sequence Dj satisfying (3.8) as

Dj =

(2 j − 2)(2 j − 1) − 12γ 2, if m = 0, γ 2 > 0, j ≥ 2,

dj , if m ≥ 1, γ 2 > 0,

dj , if m = 0, γ 2 < 0, j ≥ 2,

(m + 2 j − 2)(m + 2 j − 1) − 12γ 2, if m ≥ 1, γ 2 < 0.

The sequence cj converges to 14γ 2. The sequence |cj | is monotonically increasing for

m ≥ 1 and monotonically decreasing for m = 0. Hence, we choose the sequence C j

satisfying (3.8) as

C j =

{

|cj | if m = 0,

14|γ 2| if m ≥ 1.

52 H. Volkmer

Table 6. Estimates from Theorem 1 for distance between spheroidal eigenvalue

λ02(10) and approximation µ2,r .

r λ02(10) µ2,r − λ0

2(10) Error estimate

3 1.7962473200113952199510987 0.58 · 10−2 1.25 · 10−2

4 1.7904107081973358415378761 0.16 · 10−4 0.36 · 10−4

5 1.7903944464734022475175084 0.15 · 10−7 0.35 · 10−7

6 1.7903944312365846389547950 0.62 · 10−11 1.48 · 10−11

7 1.7903944312303699054175945 0.13 · 10−14 0.31 · 10−14

8 1.7903944312303686201621693 0.15 · 10−18 0.37 · 10−18

9 1.7903944312303686200127650 0.11 · 10−22 0.26 · 10−22

By Theorem 4, we obtain

µk,r − λk = O

(

γ 4r

42r+1(m + 2r − 1)!2(m + 2r + 1)!2

)

as r → ∞.

We use that cj = (γ 2/4)(1 + O( j−2)) for m = 0 to prove this statement for m = 0.

Table 6 shows error estimates in the case m = 0, k = 2, γ 2 = 10 for the computation

of the spheroidal eigenvalue

λ02(10) = 1.7903944312303686200127544999612315906840 . . . .

For the error estimates in this section we choose �1 = µ1,6, �2 = µ2,7, and q = 1

in (3.5).

The spheroidal wave eigenfunction Psmn (t) = Psm

n (t, γ 2), n = m+2(k−1), belonging

to the eigenvalue λk , is given by

Psmn (t) = sn,m

∞∑

j=1

xk, j�mm+2( j−1)(t),

where xk = (xk,1, xk,2, . . .) is the eigenvector of A belonging to λk and∑∞

n=1 x2k,n = 1;

see [4]. We use the eigenvector yk,r = (u1, . . . , ur , 0, 0, . . .) of Pr APr belonging to the

eigenvalue µk,r to approximate Psmn by the quasi-polynomial

pk,r (t) = sn,m

r∑

j=1

u j�mm+2( j−1)(t).

The L2(0, 1)-norm

‖Psmn − pk,r‖ = sn,m

(

∞∑

j=1

(xk, j − u j )2

)1/2

can be estimated as indicated in Section 3. In particular, by Theorem 4,

‖Psmn − pk,r‖ = O

(

|γ 2|r+1

4r+1(m + 2r + 1)!2

)

.

Table 7 gives L2-error estimates in the case m = 0, k = 2, γ 2 = 10 for the computation

of the spheroidal wave function Ps02(·, 10).


Table 7. Estimates from Theorem 2 for

L2(0, 1) distance between spheroidal wave

function Ps02(·, 10) and approximation p2,r .

r ‖Ps02(·, 10) − p2,r ‖ Error estimate

3 0.59 · 10−2 0.85 · 10−2

4 0.22 · 10−3 0.34 · 10−3

5 0.54 · 10−5 0.84 · 10−5

6 0.91 · 10−7 1.44 · 10−7

7 0.11 · 10−8 0.18 · 10−8

8 0.11 · 10−10 0.17 · 10−10

9 0.79 · 10−13 1.29 · 10−13

The following theorem estimates the maximum norm of Psmn − pk,r :

Theorem 6. For every t ∈ [−1, 1], k ∈ N, n = m + 2(k − 1), we have

|Psmn (t) − pk,r (t)| ≤ sn,m‖Pr xk − yk,r‖

(

r∑

j=1

(�mm+2( j−1)(t))

2

)1/2

+ sn,m

∞∑

j=r+1

|xk, j ||�mm+2( j−1)(t)|,

where the quantities on the right-hand side of the inequality can be estimated as indicated

in Sections 2 and 3 and

|�mn (t)| ≤

{√(2n + 1)/2 if m = 1, 2, . . . ,

√2n + 1 if m = 0.

Proof. We argue as in the proof of Theorem 5. The bound for �mn (t) follows from the

addition formula for Legendre functions; see [4, p. 259] or [11]. The latter reference also

contains some sharper bounds.

Table 8 shows error estimates in the case m = 0, k = 2, γ 2 = 10 for the computation of

Ps02(0.5, 10) = 0.1073214125922448426909203465692704761396 . . . .

Table 8. Estimates from Theorem 6 for distance between Ps02(0.5, 10) and approximation p2,r (0.5).

r p2,r (0.5) Ps02(0.5, 10) − p2,r (0.5) Error estimate

3 0.09952067097265232504 0.78 · 10−2 1.26 · 10−2

4 0.10723677849603458567 0.84 · 10−4 2.62 · 10−4

5 0.10732612678700087639 −0.47 · 10−5 1.11 · 10−5

6 0.10732152152220085124 −0.11 · 10−6 0.24 · 10−6

7 0.10732141295182414931 −0.36 · 10−9 1.61 · 10−9

8 0.10732141258305621825 0.92 · 10−11 2.47 · 10−11

9 0.10732141259215136846 0.93 · 10−13 2.35 · 10−13

54 H. Volkmer

Acknowledgment. The author thanks the referees for pointing out additional refer-

ences, suggesting inclusion of more tables, and other useful hints.

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H. Volkmer

Department of Mathematical Sciences

University of Wisconsin-Milwaukee

P. O. Box 413

Milwaukee, WI 53201

USA

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