D02KEF : NAG Library, Mark 22

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

D02KEF has essentially the same purpose as D02KDF with minor modifications to enable values of the eigenfunction to be obtained after convergence to the eigenvalue has been achieved.

It first finds a specified eigenvalue λ~ of a Sturm–Liouville system defined by a self-adjoint differential equation of the second-order

px y′ ′ + q x;λ y=0 , a<x<b ,

together with appropriate boundary conditions at the two, finite or infinite, end points a and b. The functions p and q, which are real-valued, are defined by COEFFN. The boundary conditions must be defined by BDYVAL, and, in the case of a singularity at a or b, take the form of an asymptotic formula for the solution near the relevant end point.

When the final estimate λ=λ~ of the eigenvalue has been found, the routine integrates the differential equation once more with that value of λ, and with initial conditions chosen so that the integral

S=∫aby x 2 ∂q ∂λ x;λdx

is approximately one. When qx;λ is of the form λwx+qx, which is the most common case, S represents the square of the norm of y induced by the inner product

f,g = ∫ab fx gx wx dx ,

with respect to which the eigenfunctions are mutually orthogonal. This normalization of y is only approximate, but experience shows that S generally differs from unity by only one or two per cent.

During this final integration the REPORT is called at each integration mesh point x. Sufficient information is returned to permit you to compute yx and y′x for printing or plotting. For reasons described in Section 8.2, D02KEF passes across to REPORT, not y and y′, but the Pruefer variables β, ϕ and ρ on which the numerical method is based. Their relationship to y and y′ is given by the equations

px y′ = β expρ2 cosϕ2 , y=1β expρ2 sinϕ2 .

A specimen REPORT is given in Section 9 below.

For the theoretical basis of the numerical method to be valid, the following conditions should hold on the coefficient functions:

(a)	px must be nonzero and must not change sign throughout the interval a,b; and,
(b)	∂q ∂λ must not change sign throughout the interval a,b for all relevant values of λ, and must not be identically zero as x varies, for any λ.

Points of discontinuity in the functions p and q or their derivatives are allowed, and should be included as ‘break points’ in the array XPOINT.

A good account of the theory of Sturm–Liouville systems, with some description of Pruefer transformations, is given in Chapter X of Birkhoff and Rota (1962). An introduction to the use of Pruefer transformations for the numerical solution of eigenvalue problems arising from physics and chemistry is given in Bailey (1966).

The scaled Pruefer method is described in a short note by Pryce and Hargrave (1977) and in some detail in the technical report by Pryce (1981).

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Bailey P B (1966) Sturm–Liouville eigenvalues via a phase function SIAM J. Appl. Math. 14 242–249

Banks D O and Kurowski I (1968) Computation of eigenvalues of singular Sturm–Liouville Systems Math. Comput. 22 304–310

Birkhoff G and Rota G C (1962) Ordinary Differential Equations Ginn & Co., Boston and New York

Pryce J D (1981) Two codes for Sturm–Liouville problems Technical Report CS-81-01 Department of Computer Science, Bristol University

Pryce J D and Hargrave B A (1977) The scaled Prüfer method for one-parameter and multi-parameter eigenvalue problems in ODEs IMA Numerical Analysis Newsletter 1 (3)

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

Note: error exits with IFAIL=2, 3, 6, 7, 8 or 11 are caused by being unable to set up or solve the differential equation at some iteration and will be immediately preceded by a call of MONIT giving diagnostic information. For other errors, diagnostic information is contained in HMAX2j, for j=1,2,…,m, where appropriate.

See the discussion in Section 8.2.

The time taken by D02KEF depends on the complexity of the coefficient functions, whether they or their derivatives are rapidly changing, the tolerance demanded, and how many iterations are needed to obtain convergence. The amount of work per iteration is roughly doubled when TOL is divided by 16. To make the most economical use of the routine, one should try to obtain good initial values for ELAM and DELAM, and, where appropriate, good asymptotic formulae. Also the boundary matching points should not be set unnecessarily close to singular points. The extra time needed to compute the eigenfunction is principally the cost of one additional integration once the eigenvalue has been found.

D02KEF defines a miss-distance fλ based on the rotation about the origin of the point Px=pxy′x,yx in the Phase Plane as the solution proceeds from a to b. The boundary conditions define the ray (i.e., two-sided line through the origin) on which px should start, and the ray on which it should finish. The eigenvalue k defines the total number of half-turns it should make. Numerical solution is actually done by ‘shooting forward’ from x=a and ‘shooting backward’ from x=b to a matching point x=c. Then fλ is taken as the angle between the rays to the two resulting points Pac and Pbc. A relative scaling of the py′ and y axes, based on the behaviour of the coefficient functions p and q, is used to improve the numerical behaviour.

Figure 1

The resulting function fλ is monotonic over -∞<λ<∞, increasing if ∂q ∂λ >0 and decreasing if ∂q ∂λ <0, with a unique zero at the desired eigenvalue λ~. The routine measures fλ in units of a half-turn. This means that as λ increases, fλ varies by about 1 as each eigenvalue is passed. (This feature implies that the values of fλ at successive iterations – especially in the early stages of the iterative process – can be used with suitable extrapolation or interpolation to help the choice of initial estimates for eigenvalues near to the one currently being found.)

The routine actually computes a value for fλ with errors, arising from the local errors of the differential equation code and from the asymptotic formulae provided by you if singular points are involved. However, the error estimate output in DELAM is usually fairly realistic, in that the actual error λ~-ELAM is within an order of magnitude of DELAM.

We pass the values of β, ϕ, ρ across through REPORT rather than converting them to values of y, y′ inside D02KEF, for the following reasons. First, there may be cases where auxiliary quantities can be more accurately computed from the Pruefer variables than from y and y′. Second, in singular problems on an infinite interval y and y′ may underflow towards the end of the range, whereas the Pruefer variables remain well-behaved. Third, with high-order eigenvalues (and therefore highly oscillatory eigenfunctions) the eigenfunction may have a complete oscillation (or more than one oscillation) between two mesh points, so that values of y and y′ at mesh points give a very poor representation of the curve. The probable behaviour of the Pruefer variables in this case is that β and ρ vary slowly whilst ϕ increases quickly: for all three Pruefer variables linear interpolation between the values at adjacent mesh points is probably sufficiently accurate to yield acceptable intermediate values of β, ϕ, ρ (and hence of y,y′) for graphical purposes.

Similar considerations apply to the exponentially decaying ‘tails’ of the eigenfunctions that often occur in singular problems. Here ϕ has approximately constant value whilst ρ increases rapidly in the direction of integration, though the step length is generally fairly small over such a range.

If the solution is output through REPORT at x values which are too widely spaced, the step length can be controlled by choosing HMAX suitably, or, preferably, by reducing TOL. Both these choices will lead to more accurate eigenvalues and eigenfunctions but at some computational cost.

This point is always one of the values xi in array XPOINT. It may be specified using the parameter MATCH. The default value is chosen to be the value of that xi, 2≤i≤m-1, that lies closest to the middle of the interval x2,xm-1. If there is a tie, the rightmost candidate is chosen. In particular if there are no break points, then c=xm-1 (=x3); that is, the shooting is from left to right in this case. A break point may be inserted purely to move c to an interior point of the interval, even though the form of the equations does not require it. This often speeds up convergence especially with singular problems.

Note that the shooting method used by the code integrates first from the left-hand end xl, then from the right-hand end xr, to meet at the matching point c in the middle. This will of course be reflected in printed or graphical output. The diagram shows a possible sequence of nine mesh points τ1 through τ9 in the order in which they appear, assuming there are just two sub-intervals (so m=5).

Figure 2

Since the shooting method usually fails to match up the two ‘legs’ of the curve exactly, there is bound to be a jump in y, or in pxy′ or both, at the matching point c. The code in fact ‘shares’ the discrepancy out so that both y and pxy′ have a jump. A large jump does not imply an inaccurate eigenvalue, but implies either

(a)	a badly chosen matching point: if qx;λ has a ‘humped’ shape, c should be chosen near the maximum value of q, especially if q is negative at the ends of the interval;
(b)	an inherently ill-conditioned problem, typically one where another eigenvalue is pathologically close to the one being sought. In this case it is extremely difficult to obtain an accurate eigenfunction.

In Section 9, we find the 11th eigenvalue and corresponding eigenfunction of the equation

y′′+λ-x-2/x2y=0 on 0<x<∞,

the boundary conditions being that y should remain bounded as x tends to 0 and x tends to ∞. The coding of this problem is discussed in detail in Section 8.5.

The choice of matching point c is open. If we choose c=30.0 as in D02KDF example program we find that the exponentially increasing component of the solution dominates and we get extremely inaccurate values for the eigenfunction (though the eigenvalue is determined accurately). The values of the eigenfunction calculated with c=30.0 are given schematically in Figure 3.

Figure 3

If we choose c as the maximum of the hump in qx;λ (see item (a) above) we instead obtain the accurate results given in Figure 4

Figure 4

Coding COEFFN is straightforward except when break points are needed. The examples below show:

(a)	a simple case,
(b)	a case in which discontinuities in the coefficient functions or their derivatives necessitate break points, and
(c)	a case where break points together with the HMAX parameter are an efficient way to deal with a coefficient function that is well-behaved except over one short interval.

(Some of these cases are among the examples in Section 9.)

Example A

The modified Bessel equation

x xy′ ′ + λx2-ν2 y=0 .

Assuming the interval of solution does not contain the origin and dividing through by x, we have px=x and qx;λ=λx-ν2/x. The code could be

SUBROUTINE COEFFN (P,Q,DQDL,X,ELAM,JINT)
...
P = X
Q = ELAM*X - NU*NU/X
DQDL = X
RETURN
END

where NU (standing for ν) is a double precision variable that might be defined in a DATA statement, or might be in user-declared COMMON so that its value could be set in the main program.

Example B

The Schroedinger equation

y′′+λ+qxy=0,

where

qx = x2- 10 x≤ 4, 6x x> 4,

over some interval ‘approximating to -∞,∞’, say -20,20. Here we need break points at ±4, forming three sub-intervals i1=-20,-4, i2=-4,4, i3=4,20. The code could be

SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)
...
IF (JINT.EQ.2) THEN
  Q = ELAM + X*X - 10.0D0
ELSE
  Q = ELAM + 6.0D0/ABS(X)
ENDIF
P = 1.0D0
DQDL = 1.0D0
RETURN
END

The array XPOINT would contain the values x1, -20.0, -4.0, +4.0, +20.0, x6 and m would be 6. The choice of appropriate values for x1 and x6 depends on the form of the asymptotic formula computed by BDYVAL and the technique is discussed in Section 8.5.

Example C

y′′+λ1-2⁢e-100⁢x2y=0, -10≤x≤10.

Here qx;λ is nearly constant over the range except for a sharp inverted spike over approximately -0.1≤x≤0.1. There is a danger that the routine will build up to a large step size and ‘step over’ the spike without noticing it. By using break points – say at ±0.5 – one can restrict the step size near the spike without impairing the efficiency elsewhere.

The code for COEFFN could be

SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)
...
P = 1.0D0
DQDL = 1.0D0 - 2.0D0*EXP(-100.0D0*X*X)
Q = ELAM*DQDL
RETURN
END

XPOINT might contain -10.0, -10.0, -0.5, 0.5, 10.0, 10.0 (assuming ±10 are regular points) and m would be 6. HMAX1j, for j=1,2,3, might contain 0.0, 0.1 and 0.0.

Quoting from page 243 of Bailey (1966): ‘Usually ... the differential equation has two essentially different types of solution near a singular point, and the boundary condition there merely serves to distinguish one kind from the other. This is the case in all the standard examples of mathematical physics.’

In most cases the behaviour of the ratio pxy′/y near the point is quite different for the two types of solution. Essentially what you provide through the BDYVAL is an approximation to this ratio, valid as x tends to the singular point (SP).

You must decide (a) how accurate to make this approximation or asymptotic formula, for example how many terms of a series to use, and (b) where to place the boundary matching point (BMP) at which the numerical solution of the differential equation takes over from the asymptotic formula. Taking the BMP closer to the SP will generally improve the accuracy of the asymptotic formula, but will make the computation more expensive as the Pruefer differential equations generally become progressively more ill-behaved as the SP is approached. You are strongly recommended to experiment with placing the BMPs. In many singular problems quite crude asymptotic formulae will do. To help you avoid needlessly accurate formulae, D02KEF outputs two ‘sensitivity coefficients’ σl,σr which estimate how much the errors at the BMPs affect the computed eigenvalue. They are described in detail in Section 8.6.

Example of coding BDYVAL:

The example below illustrates typical situations:

y′′+λ-x-2x2 y=0, 0<x<∞

the boundary conditions being that y should remain bounded as x tends to 0 and x tends to ∞.

At the end x=0 there is one solution that behaves like x2 and another that behaves like x-1. For the first of these solutions pxy′/y is asymptotically 2/x while for the second it is asymptotically -1/x. Thus the desired ratio is specified by setting

YL1=x and YL2=2.0.

At the end x=∞ the equation behaves like Airy's equation shifted through λ, i.e., like y′′-ty=0 where t=x-λ, so again there are two types of solution. The solution we require behaves as

exp-23t32/t4

and the other as

exp+23t32/t4.

Hence, the desired solution has pxy′/y∼-t so that we could set YR1=1.0 and YR2=-x-λ. The complete subroutine might thus be

SUBROUTINE BDYVAL (XL,XR,ELAM,YL,YR)
double precision XL, XR, ELAM, YL(3), YR(3)
YL(1) = XL
YL(2) = 2.0D0
YR(1) = 1.0D0
YR(2) = -SQRT(XR-ELAM)
RETURN
END

Clearly for this problem it is essential that any value given by D02KEF to XR is well to the right of the value of ELAM, so that you must vary the right-hand BMP with the eigenvalue index k. One would expect λk to be near the kth zero of the Airy function Aix, so there is no problem estimating ELAM.

More accurate asymptotic formulae are easily found: near x=0 by the standard Frobenius method, and near x=∞ by using standard asymptotics for Aix, Ai′x, (see page 448 of Abramowitz and Stegun (1972)).

For example, by the Frobenius method the solution near x=0 has the expansion

y=x2c0+c1x+c2x2+⋯

with

c0= 1,c1= 0, c2=-λ10, c3=118,⋯, cn=cn- 3-λ cn- 2 nn+ 3 .

This yields

pxy′y=2-25λx2+⋯ x 1-λ10x2+⋯ .

This example finds the 11th eigenvalue and eigenfunction of the example of Section 8.5, using the simple asymptotic formulae for the boundary conditions.

Comparison of the results from this example program with the corresponding results from D02KDF example program shows that similar output is produced from MONIT, followed by the eigenfunction values from REPORT, and then a further line of information from MONIT (corresponding to the integration to find the eigenfunction). Final information is printed within the example program exactly as with D02KDF.

Note the discrepancy at the matching point c (=43, the maximum of qx;λ, in this case) between the solutions obtained by integrations from left- and right-hand end points.

SUBROUTINE D02KEF (	XPOINT, M, MATCH, COEFFN, BDYVAL, K, TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN, MONIT, REPORT, IFAIL)
INTEGER	M, MATCH, K, MAXIT, MAXFUN, IFAIL
*double precision*	XPOINT(M), TOL, ELAM, DELAM, HMAX(2,M)
EXTERNAL	COEFFN, BDYVAL, MONIT, REPORT

SUBROUTINE COEFFN (	P, Q, DQDL, X, ELAM, JINT)
INTEGER	JINT
*double precision*	P, Q, DQDL, X, ELAM

SUBROUTINE MONIT (	MAXIT, IFLAG, ELAM, FINFO)
INTEGER	MAXIT, IFLAG
*double precision*	ELAM, FINFO(15)

SUBROUTINE REPORT (	X, V, JINT)
INTEGER	JINT
*double precision*	X, V(3)

HMAX21=1:	M<4;
HMAX21=2:	K<0;
HMAX21=3:	TOL≤0.0;
HMAX21=4:	XPOINT1 to XPOINTm are not in ascending order. HMAX22 gives the position i in XPOINT where this was detected.

NAG Library Routine Document

D02KEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Timing

8.2 General Description of the Algorithm

8.3 The Position of the Shooting Matching Point c

8.4 Examples of Coding the COEFFN

8.5 Examples of Boundary Conditions at Singular Points

8.6 The Sensitivity Parameters σl and σr

8.7 ‘Missed Zeros’

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

SUBROUTINE BDYVAL (	XL, XR, ELAM, YL, YR)
*double precision*	XL, XR, ELAM, YL(3), YR(3)

(a)	the wrong eigenvalue will be computed for the given index k – in fact some λk+k′ will be found where k′≥1;
(b)	the same index k can cause convergence to any of several eigenvalues depending on the initial values of ELAM and DELAM.

NAG Library Routine DocumentD02KEF

+− Contents

NAG Library Routine Document

D02KEF