10

Absolute value of error in j-th eigenmode

0

10

0.01 error level

-5

10

-10

10

-15

10

0 5 10 15 20 25 30

mode number j

Figure 7.3: Example One: Absolute errors in the eigenvalues for 16-point (solid) and 32-

point (circles) Chebyshev pseudospectral discretization.

numerically with the corresponding exact modes. The third mode, because it oscillates

slowly with x, is well-resolved and the exact and numerical eigenfunctions are graphically

indistinguishable. The ¬fteenth mode, however, is oscillating so rapidly that it cannot be

resolved by the ¬rst sixteen Chebyshev polynomials. To be sure, T15 (x) has ¬fteen roots,

just like the exact ¬fteenth mode, but the Chebyshev oscillations are not uniform. Instead,

T15 (x) oscillates slowly in the center of the interval and very rapidly near to the endpoints,

in contrast to the uniform oscillations of the eigenmode it is vainly trying to mimic.

To resolve more eigenmodes, merely increase N as illustrated in Fig. 7.3, which com-

pares the errors for 16-point and 32-point calculations. The number of good eigenvalues

has risen from seven to sixteen. The graph is ¬‚at for small j because of roundoff error. In

multiple precision, the errors for the ¬rst three or four modes would be smaller than 10’15 ,

off the bottom of the graph!

For “nice” eigenvalue problems, this behavior is typical. To show this, consider a sec-

ond example which is posed on an in¬nite interval and is solved using not the Chebyshev

polynomials but rather the “rational Chebyshev” functions T Bn (x), which are a good basis

for an unbounded domain (Chapter 17).

EXAMPLE TWO:

uxx + (» ’ x2 ) u = 0, |u| ’ 0 as x ’ ∞ (7.5)

The exact eigenfunctions are the Hermite functions,

1

uj (x) = exp ’ x2 Hj (y) (7.6)

2

where Hj is a polynomial of degree j, the j-th Hermite polynomial. The eigenvalues are

(7.7)

»j = 2j + 1, j = 0, 1, 2, . . .

Fig. 7.4 shows that again the lowest few modes are accurately approximated. The in¬-

nite interval problem is harder than a ¬nite interval problem, so there are only four “good”

CHAPTER 7. LINEAR EIGENVALUE PROBLEMS

132

4

10

Absolute value of error in j-th eigenmode

3

10

2

10

1

10

0

10

-1

10

-2

10

0.01 error level

-3

10

-4

10

-5

10

0 5 10 15

mode number j

Figure 7.4: Example Two (Eigenfunctions of parabolic cylinder equation on an in¬nite

interval): Absolute errors in the eigenvalues as given by a 16-point Rational Chebyshev

(T Bn ) discretization with the map parameter L = 4.

eigenvalues versus seven for the 16-point discretization of Example One. Fig. 7.5 shows

that the second eigenmode is well-approximated, but the seventh eigenmode is poorly

approximated. The most charitable comment one can make about the seventh numerical

eigenfunction is that it vaguely resembles the true eigenmode by having a lot of wiggles.

Fig. 7.6 con¬rms that increasing N also increases the number of “good” eigenvalues, in this

case from four to ten.

It is important to note that for both examples, the approximation to the lowest mode

is extremely accurate with errors smaller than 10’11 for N = 32 for both examples. The

error increases exponentially fast with mode number j until ¬nally the error is comparable in

magnitude to the eigenvalue itself.

These examples suggest the following heuristic.

Rule-of-Thumb 8 (EIGENVALUE RULE-OF-THUMB)

In solving a linear eigenvalue problem by a spectral method using (N + 1) terms in the truncated

spectral series, the lowest N/2 eigenvalues are usually accurate to within a few percent while the

larger N/2 numerical eigenvalues differ from those of the differential equation by such large amounts

as to be useless.

Warning #1: the only reliable test is to repeat the calculation with different N and compare the

results.

Warning #2: the number of good eigenvalues may be smaller than (N/2) if the modes have

boundary layers, critical levels, or other areas of very rapid change, or when the interval is un-

bounded.

Although this rule-of-thumb is representative of a wide class of examples, not just the

two shown above, nasty surprises are possible. We attempt a crude classi¬cation of linear

eigenvalue problems in the next section.

7.3. EIGENVALUE RULE-OF-THUMB 133

j=1 mode

1

Circles: Exact 0.5

0

-0.5

-1

-10 -5 0 5 10

j=6 mode

1

Solid: Numerical

0.5

0

-0.5

-1

-10 -5 0 5 10

x

Figure 7.5: Example Two (Exact modes are Hermite functions): 16-point Rational Cheby-

shev pseudospectral method. Exact (circles) and numerical (solid) approximations to the

eigenmodes. Upper panel: Second Mode. Lower panel: Seventh mode.

4

10

Absolute value of error in j-th eigenmode

2

10

0

10

-2

10

0.01 error level

-4

10

-6

10

-8

10

-10

10

-12

10

0 5 10 15 20 25 30

mode number j

Figure 7.6: Example Two (In¬nite Interval): Absolute errors in the eigenvalues for 16-point

(solid disks) and 32-point (open circles) T Bn discretization.

CHAPTER 7. LINEAR EIGENVALUE PROBLEMS

134

7.4 Four Kinds of Sturm-Liouville Eigenproblems

and Continuous Spectra

Solving partial differential equations by separation-of-variables generates eigenproblems

of the form of Eq. 7.8 below. Sturm and Liouville showed in the mid-nineteenth century

that as long as the equation coef¬cients p(x) and r(x) are positive everywhere on the in-

terval x ∈ [a, b] and q(x) is free of singularities on the interval, then all eigenfunctions are

discrete and orthogonal. Such “nice” classical eigenproblems are very common in applica-

tions, but unfortunately do not exhaust all the possibilities. This motivated the following

classi¬cation scheme.

De¬nition 19 (Sturm-Liouville Eigenproblems: Four Kinds)

A Sturm-Liouville eigenproblem is

[p(x)ux ]x + {q(x) + »r(x)} u = 0, x ∈ [a, b] (7.8)

subject to various homogeneous boundary conditions. There are four varieties in the classi¬cation

scheme of Boyd(1981a):

• First Kind: p, q, r analytic everywhere on the interval

• Second Kind: p, q, r analytic everywhere on the interval except the endpoints.

• Third Kind: differential equation has singularities on the interior of the interval, but the

singularities are only “apparent”.

• Fourth Kind: differential equation and eigenfunctions are singular on the interior of the in-

terval.

SL problems of the First Kind are guaranteed to be “nice” and well-behaved. However,

problems of the Second Kind, which includes most eigenproblems on an unbounded do-

main, may be either regular, with a discrete in¬nity of eigenvalues all of the same sign and

orthogonal eigenfunctions, or they may have only a ¬nite number of discrete eigenvalues

plus a continuous eigenspectrum.

The differential equation satis¬ed by the Associated Legendre functions is a good il-

lustration of the Second Kind. The Legendre functions are the solutions on x ∈ [’1, 1]

of

D2 u + { »(1 ’ x2 ) + E}u = 0 (7.9)

where the differential operator D is

d

D ≡ (1 ’ x2 ) (7.10)

dx

The eigenfunctions are those solutions which are regular at the endpoints except perhaps

for a branchpoint:

E = ’m2

m

(7.11)

u = Pn (x), » = n(n + 1),

When derived from problems in spherical coordinates, x is the cosine of colatitude and the

periodicity of the longitudinal coordinate demands that m be an integer. Then Eq.(7.9) is

a regular Sturm-Liouville problem of the second kind and » is the eigenvalue. For each

longitudinal wavenumber m, there is a countable in¬nity of discrete eigenvalues: n must

be an integer with the further requirement n ≥ |m|.

7.4. FOUR KINDS OF STURM-LIOUVILLE PROBLEMS 135

However, the same differential equation is also an important exactly-solvable illustra-

tion of the stationary Schroedinger equation of quantum mechanics with radically different

behavior: a continuous spectrum plus a ¬nite number of discrete eigenmodes. The crucial

difference is that the roles of the parameters » and E are interchanged in quantum me-

chanics: E is now the eigenvalue (“energy”) and » measures the strength of the speci¬ed

potential.

The unimportant difference is that quantum problems are usually posed on an in¬nite

interval, which requires the change of variable

[“Mercator coordinate”] (7.12)

x = tanh(y)

The Legendre equation (7.9) becomes

uyy + { » sech2 (y) + E}u = 0, y ∈ [ ’∞, ∞] (7.13)

For all positive E, the equation has solutions which are everywhere bounded; these are