NAG Library Routine Document
D02KDF
1 Purpose
D02KDF finds a specified eigenvalue of a regular or singular second-order Sturm–Liouville system on a finite or infinite interval, using a Pruefer transformation and a shooting method. Provision is made for discontinuities in the coefficient functions or their derivatives.
2 Specification
| SUBROUTINE D02KDF ( | XPOINT, M, COEFFN, BDYVAL, K, TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN, MONIT, IFAIL) |
| INTEGER | M, K, MAXIT, MAXFUN, IFAIL |
| double precision | XPOINT(M), TOL, ELAM, DELAM, HMAX(2,M) |
| EXTERNAL | COEFFN, BDYVAL, MONIT |
3 Description
D02KDF finds a specified eigenvalue
λ~ of a Sturm–Liouville system defined by a self-adjoint differential equation of the second-order
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.
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.
The eigenvalue
λ~ is determined by a shooting method based on the Scaled Pruefer form of the differential equation as described in
Pryce (1981), with certain modifications. The Pruefer equations are integrated by a special internal routine using Merson's Runge–Kutta formula with automatic control of local error. Providing certain assumptions (see
Section 8.1) are met, the computed value of
λ~ will be correct to within a mixed absolute/relative error specified by
TOL.
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).
4 References
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) 5 Parameters
- 1: XPOINT(M) – double precision arrayInput
-
On entry: the points where the boundary conditions computed by
BDYVAL are to be imposed, and also any break points, i.e.,
XPOINT1 to
XPOINTm must contain values
x1,…,xm such that
with the following meanings:
| (a) |
x1 and xm are the left- and right-hand end points, a and b, of the domain of definition of the Sturm–Liouville system if these are finite. If either a or b is infinite, the corresponding value x1 or xm may be a more-or-less arbitrarily ‘large’ number of appropriate sign. |
| (b) |
x2 and xm-1 are the Boundary Matching Points (BMPs), that is the points at which the left and right boundary conditions computed in BDYVAL are imposed.
If the left-hand end point is a regular point then you should set x2=x1 =a, while if it is a singular point you must set x2>x1. Similarly xm-1=xm (=b) if the right-hand end point is regular, and xm-1<xm if it is singular. |
| (c) |
The remaining m-4 points x3,…,xm-2, if any, define ‘break points’ which divide the interval x2,xm-1 into m-3 sub-intervals
Numerical integration of the differential equation is stopped and restarted at each break point. In simple cases no break points are needed. However, if px or qx;λ are given by different formulae in different parts of the interval, then integration is more efficient if the range is broken up by break points in the appropriate way. Similarly points where any jumps occur in px or qx;λ, or in their derivatives up to the fifth-order, should appear as break points.
Examples are given in Sections 8 and 9. XPOINT determines the position of the Shooting Matching Point (SMP), as explained in Section 8.3. |
Constraint:
XPOINT1≤XPOINT2<⋯<XPOINTM-1≤XPOINTM.
- 2: M – INTEGERInput
-
On entry:
the number of points in the array
XPOINT.
Constraint:
M≥4.
- 3: COEFFN – SUBROUTINE, supplied by the user.External Procedure
-
COEFFN must compute the values of the coefficient functions
px and
qx;λ for given values of
x and
λ.
Section 3 states the conditions which
p and
q must satisfy.
Sections 8.4 and
9
The specification of
COEFFN is:
| SUBROUTINE COEFFN ( | P, Q, DQDL, X, ELAM, JINT) |
| INTEGER | JINT |
| double precision | P, Q, DQDL, X, ELAM |
- 1: P – double precisionOutput
-
On exit: the value of px for the current value of x.
- 2: Q – double precisionOutput
-
On exit: the value of qx;λ for the current value of x and the current trial value of λ.
- 3: DQDL – double precisionOutput
-
On exit: the value of
∂q
∂λ
x;λ
for the current value of
x and the current trial value of
λ. However
DQDL is only used in error estimation and, in the rare cases where it may be difficult to evaluate, an approximation (say to within
20%) will suffice.
- 4: X – double precisionInput
-
On entry: the current value of x.
- 5: ELAM – double precisionInput
-
On entry: the current trial value of the eigenvalue parameter λ.
- 6: JINT – INTEGERInput
-
On entry: the index
j of the sub-interval
ij (see specification of
XPOINT) in which
x lies.
COEFFN must be declared as EXTERNAL in the (sub)program from which D02KDF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 4: BDYVAL – SUBROUTINE, supplied by the user.External Procedure
-
BDYVAL must define the boundary conditions. For each end point,
BDYVAL must return (in
YL or
YR) values of
yx and
pxy′x which are consistent with the boundary conditions at the end points; only the ratio of the values matters. Here
x is a given point (
XL or
XR) equal to, or close to, the end point.
For a
regular end point (
a, say),
x=a, a boundary condition of the form
can be handled by returning constant values in
YL, e.g.,
YL1=c2 and
YL2=-c1pa.
For a
singular end point however,
YL1 and
YL2 will in general be functions of
XL and
ELAM, and
YR1 and
YR2 functions of
XR and
ELAM, usually derived analytically from a power-series or asymptotic expansion. Examples are given in
Sections 8.5 and
9.
The specification of
BDYVAL is:
| SUBROUTINE BDYVAL ( | XL, XR, ELAM, YL, YR) |
| double precision | XL, XR, ELAM, YL(3), YR(3) |
- 1: XL – double precisionInput
-
On entry: if
a is a regular end point of the system (so that
a=x1=x2), then
XL contains
a. If
a is a singular point (so that
a≤x1<x2), then
XL contains a point
x such that
x1<x≤x2.
- 2: XR – double precisionInput
-
On entry: if
b is a regular end point of the system (so that
xm-1=xm=b), then
XR contains
b. If
b is a singular point (so that
xm-1<xm≤b), then
XR contains a point
x such that
xm-1≤x<xm.
- 3: ELAM – double precisionInput
-
On entry: the current trial value of λ.
- 4: YL(3) – double precision arrayOutput
-
On exit: YL1 and YL2 should contain values of yx and pxy′x respectively (not both zero) which are consistent with the boundary condition at the left-hand end point, given by x=XL. YL3 should not be set.
- 5: YR(3) – double precision arrayOutput
-
On exit: YR1 and YR2 should contain values of yx and pxy′x respectively (not both zero) which are consistent with the boundary condition at the right-hand end point, given by x=XR. YR3 should not be set.
BDYVAL must be declared as EXTERNAL in the (sub)program from which D02KDF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 5: K – INTEGERInput
-
On entry:
k, the index of the required eigenvalue when the eigenvalues are ordered
Constraint:
K≥0.
- 6: TOL – double precisionInput
-
On entry: the tolerance parameter which determines the accuracy of the computed eigenvalue. The error estimate held in
DELAM on exit satisfies the mixed absolute/relative error test
where
ELAM is the final estimate of the eigenvalue.
DELAM is usually somewhat smaller than the right-hand side of
(1) but not several orders of magnitude smaller.
Constraint:
TOL>0.0.
- 7: ELAM – double precisionInput/Output
-
On entry: an initial estimate of the eigenvalue λ~.
On exit: the final computed estimate, whether or not an error occurred.
- 8: DELAM – double precisionInput/Output
-
On entry: an indication of the scale of the problem in the
λ-direction.
DELAM holds the initial ‘search step’ (positive or negative). Its value is not critical, but the first two trial evaluations are made at
ELAM and
ELAM+DELAM, so the routine will work most efficiently if the eigenvalue lies between these values. A reasonable choice (if a closer bound is not known) is half the distance between adjacent eigenvalues in the neighbourhood of the one sought. In practice, there will often be a problem, similar to the one in hand but with known eigenvalues, which will help one to choose initial values for
ELAM and
DELAM.
If DELAM=0.0 on entry, it is given the default value of 0.25×max1.0,ELAM.
On exit: if
IFAIL=0,
DELAM holds an estimate of the absolute error in the computed eigenvalue, that is
λ~-ELAM≃DELAM. (In
Section 8.2 we discuss the assumptions under which this is true.) The true error is rarely more than twice, or less than a tenth, of the estimated error.
If
IFAIL≠0,
DELAM may hold an estimate of the error, or its initial value, depending on the value of
IFAIL. See
Section 6 for further details.
- 9: HMAX(2,M) – double precision arrayInput/Output
-
On entry:
HMAX1j should contain a maximum step size to be used by the differential equation code in the
jth sub-interval
ij (as described in the specification of parameter
XPOINT), for
j=1,2,…,m-3. If it is zero the routine generates a maximum step size internally.
It is recommended that
HMAX1j be set to zero unless the coefficient functions
p and
q have features (such as a narrow peak) within the
jth sub-interval that could be ‘missed’ if a long step were taken. In such a case
HMAX1j should be set to about half the distance over which the feature should be observed. Too small a value will increase the computing time for the routine. See
Section 8 for further suggestions.
The rest of the array is used as workspace.
On exit:
HMAX1m-1 and
HMAX1m contain the sensitivity coefficients
σl,σr, described in
Section 8.6. Other entries contain diagnostic output in the case of an error exit (see
Section 6).
- 10: MAXIT – INTEGERInput/Output
-
On entry: a bound on
nr, the number of root-finding iterations allowed, that is the number of trial values of
λ that are used. If
MAXIT≤0, no such bound is assumed. (See also
MAXFUN.)
Suggested value:
MAXIT=0.
On exit: will have been decreased by the number of iterations actually performed, whether or not it was positive on entry.
- 11: MAXFUN – INTEGERInput
-
On entry: a bound on
nf, the number of calls to
COEFFN made in any one root-finding iteration. If
MAXFUN≤0, no such bound is assumed.
Suggested value:
MAXFUN=0.
MAXFUN and
MAXIT may be used to limit the computational cost of a call to D02KDF, which is roughly proportional to
nr×nf.
- 12: MONIT – SUBROUTINE, supplied by the NAG Library or the user.External Procedure
-
MONIT is called by D02KDF at the end of each root-finding iteration and allows you to monitor the course of the computation by printing out the parameters (see
Section 9 for an example).
If no monitoring is required, the dummy (sub)program D02KAY may be used. (D02KAY is included in the NAG Library.)
The specification of
MONIT is:
- 1: MAXIT – INTEGERInput
-
On entry: the current value of the parameter
MAXIT of D02KDF, this is decreased by one at each iteration.
- 2: IFLAG – INTEGERInput
-
On entry: describes what phase the computation is in.
- IFLAG<0
- An error occurred in the computation at this iteration; an error exit from D02KDF with IFAIL=-IFLAG will follow.
- IFLAG=1
- The routine is trying to bracket the eigenvalue λ~.
- IFLAG=2
- The routine is converging to the eigenvalue λ~ (having already bracketed it).
- 3: ELAM – double precisionInput
-
On entry: the current trial value of λ.
- 4: FINFO(15) – double precision arrayInput
-
On entry: information about the behaviour of the shooting method, and diagnostic information in the case of errors. It should not normally be printed in full if no error has occurred (that is, if
IFLAG>0), though the first few components may be of interest to you. In case of an error (
IFLAG<0) all the components of
FINFO should be printed.
The contents of
FINFO are as follows:
- FINFO1
- The current value of the ‘miss-distance’ or ‘residual’ function fλ on which the shooting method is based. (See Section 8.2 for further information.) FINFO1 is set to zero if IFLAG<0.
- FINFO2
- An estimate of the quantity ∂λ defined as follows. Consider the perturbation in the miss-distance fλ that would result if the local error in the solution of the differential equation were always positive and equal to its maximum permitted value. Then ∂λ is the perturbation in λ that would have the same effect on fλ. Thus, at the zero of fλ,∂λ is an approximate bound on the perturbation of the zero (that is the eigenvalue) caused by errors in numerical solution. If ∂λ is very large then it is possible that there has been a programming error in COEFFN such that q is independent of λ. If this is the case, an error exit with IFAIL=5 should follow. FINFO2 is set to zero if IFLAG<0.
- FINFO3
- The number of internal iterations, using the same value of λ and tighter accuracy tolerances, needed to bring the accuracy (that is, the value of ∂λ) to an acceptable value. Its value should normally be 1.0, and should almost never exceed 2.0.
- FINFO4
- The number of calls to COEFFN at this iteration.
- FINFO5
- The number of successful steps taken by the internal differential equation solver at this iteration. A step is successful if it is used to advance the integration.
- FINFO6
- The number of unsuccessful steps used by the internal integrator at this iteration.
- FINFO7
- The number of successful steps at the maximum step size taken by the internal integrator at this iteration.
- FINFO8
- Not used.
- FINFO9 to FINFO15
- Set to zero, unless IFLAG<0 in which case they hold the following values describing the point of failure:
- FINFO9
- The index of the sub-interval where failure occurred, in the range 1 to m-3. In case of an error in BDYVAL, it is set to 0 or m-2 depending on whether the left or right boundary condition caused the error.
- FINFO10
- The value of the independent variable, x, the point at which the error occurred. In case of an error in BDYVAL, it is set to the value of XL or XR as appropriate (see the specification of BDYVAL).
- FINFO11, FINFO12, FINFO13
- The current values of the Pruefer dependent variables β, ϕ and ρ respectively. These are set to zero in case of an error in BDYVAL.
D02KEF
- FINFO14
- The local-error tolerance being used by the internal integrator at the point of failure. This is set to zero in the case of an error in BDYVAL.
- FINFO15
- The last integration mesh point. This is set to zero in the case of an error in BDYVAL.
MONIT must be declared as EXTERNAL in the (sub)program from which D02KDF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 13: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
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:
- IFAIL=1
-
A parameter error. All parameters (except
IFAIL) are left unchanged. The reason for the error is shown by the value of
HMAX21 as follows:
| 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. |
- IFAIL=2
-
At some call to
BDYVAL, invalid values were returned, that is, either
YL1=YL2=0.0, or
YR1=YR2=0.0 (a programming error in
BDYVAL). See the last call of
MONIT for details.
This error exit will also occur if
px is zero at the point where the boundary condition is imposed. Probably
BDYVAL was called with
XL equal to a singular end point
a or with
XR equal to a singular end point
b.
- IFAIL=3
-
At some point between
XL and
XR the value of
px computed by
COEFFN became zero or changed sign. See the last call of
MONIT for details.
- IFAIL=4
-
MAXIT>0 on entry, and after
MAXIT iterations the eigenvalue had not been found to the required accuracy.
- IFAIL=5
-
The ‘bracketing’ phase (with parameter
IFLAG of the
MONIT equal to
1) failed to bracket the eigenvalue within ten iterations. This is caused by an error in formulating the problem (for example,
q is independent of
λ), or by very poor initial estimates of
ELAM and
DELAM.
On exit,
ELAM and
ELAM+DELAM give the end points of the interval within which no eigenvalue was located by the routine.
- IFAIL=6
-
MAXFUN>0 on entry, and the last iteration was terminated because more than
MAXFUN calls to
COEFFN were used. See the last call of
MONIT for details.
- IFAIL=7
-
To obtain the desired accuracy the local error tolerance was set so small at the start of some sub-interval that the differential equation solver could not choose an initial step size large enough to make significant progress. See the last call of
MONIT for diagnostics.
- IFAIL=8
-
At some point inside a sub-interval the step size in the differential equation solver was reduced to a value too small to make significant progress (for the same reasons as with
IFAIL=7). This could be due to pathological behaviour of
px and
qx;λ or to an unreasonable accuracy requirement or to the current value of
λ making the equations ‘stiff’. See the last call of
MONIT for details.
- IFAIL=9
-
TOL is too small for the problem being solved and the
machine precision is being used. The final value of
ELAM should be a very good approximation to the eigenvalue.
- IFAIL=10
-
C05AZF, called by D02KDF, has terminated with the error exit corresponding to a pole of the residual function
fλ. This error exit should not occur, but if it does, try solving the problem again with a smaller value for
TOL.
- IFAIL=11
- IFAIL=12
-
A serious error has occurred in an internal call. Check all (sub)program calls and array dimensions. Seek expert help.
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.
7 Accuracy
See the discussion in
Section 8.2.
8 Further Comments
8.1 Timing
The time taken by D02KDF 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.
8.2 General Description of the Algorithm
A shooting method, for differential equation problems containing unknown parameters, relies on the construction of a ‘miss-distance function’, which for given trial values of the parameters measures how far the conditions of the problem are from being met. The problem is then reduced to one of finding the values of the parameters for which the miss-distance function is zero, that is to a root-finding process. Shooting methods differ mainly in how the miss-distance is defined.
D02KDF 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.
8.3 The Position of the Shooting Matching Point c
This point is always one of the values
xi in array
XPOINT. It 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.
8.4 Examples of Coding the COEFFN
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
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
where
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
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.
8.5 Examples of Boundary Conditions at Singular Points
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, D02KDF 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:
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
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
and the other as
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 D02KDF 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
with
This yields
8.6 The Sensitivity Parameters σl and σr
The sensitivity parameters
σl,
σr (held in
HMAX1m-1 and
HMAX1m on output) estimate the effect of errors in the boundary conditions. For sufficiently small errors
Δy,
Δpy′ in
y and
py′ respectively, the relations
are satisfied, where the subscripts
l,
r denote errors committed at the left- and right-hand BMPs respectively, and
Δλ denotes the consequent error in the computed eigenvalue.
8.7 ‘Missed Zeros’
This is a pitfall to beware of at a singular point. If the BMP is chosen so far from the SP that a zero of the desired eigenfunction lies in between them, then the routine will fail to ‘notice’ this zero. Since the index of
k of an eigenvalue is the number of zeros of its eigenfunction, the result will be that
| (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. |
It is up to you to take suitable precautions – for instance by varying the position of the BMPs in the light of knowledge of the asymptotic behaviour of the eigenfunction at different eigenvalues.
9 Example
This example finds the 11th eigenvalue of the example of
Section 8.5, using the simple asymptotic formulae for the boundary conditions. The results exhibit slow convergence, mainly because
XPOINT is set so that the shooting matching point
c is at the right-hand end
x=30.0. The example results for
D02KEF show that much faster convergence is obtained if
XPOINT is set to contain an additional break point at or near the maximum of the coefficient function
qx;λ, which in this case is at
x=43.
9.1 Program Text
Program Text (d02kdfe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d02kdfe.r)
