D02AGF solves a two-point boundary value problem for a system of ordinary differential equations, using initial value techniques and Newton iteration; it generalizes D02HAF to include the case where parameters other than boundary values are to be determined.
D02AGF solves a two-point boundary value problem by determining the unknown parameters of the problem. These parameters may be, but need not be, boundary values (as they are in D02HAF); they may include eigenvalue parameters in the coefficients of the differential equations, length of the range of integration, etc. The notation and methods used are similar to those of D02HAF and you are advised to study this first. (There the parameters correspond to the unknown boundary conditions.) It is assumed that we have a system of first-order ordinary differential equations of the form
and that derivatives are evaluated by AUX. The system, including the boundary conditions given by BCAUX, and the range of integration and matching point, , given by RAAUX, involves the unknown parameters which are to be determined, and for which initial estimates must be supplied. The number of unknown parameters must not exceed the number of equations . If , we assume that equations of the system are not involved in the matching process. These are usually referred to as ‘driving equations’; they are independent of the parameters and of the solutions of the other equations. In numbering the equations for AUX, the driving equations must be put last.
The estimated values of the parameters are corrected by a form of Newton iteration. The Newton correction on each iteration is calculated using a matrix whose th element depends on the derivative of the th component of the solution, , with respect to the th parameter, . This matrix is calculated by a simple numerical differentiation technique which requires evaluations of the differential system.
You are strongly recommended to read Sections 3 and 8 in conjunction with this section.
1: H – REAL (KIND=nag_wp)Input/Output
On entry: H must be set to an estimate of the step size, , needed for integration.
On entry: must be set to a small quantity to control the th parameter component. The element is used:
in the convergence test on the th parameter in the Newton iteration,
in perturbing the th parameter when approximating the derivatives of the components of the solution with respect to the th parameter, for use in the Newton iteration.
The elements should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.
4: PARAM(N1) – REAL (KIND=nag_wp) arrayInput/Output
On entry: must be set to an estimate for the th parameter, , for .
On exit: the corrected value for the th parameter, unless an error has occurred, when it contains the last calculated value of the parameter (possibly perturbed by if the error occurred when calculating the approximate derivatives).
On entry: , the total number of differential equations.
7: N1 – INTEGERInput
On entry: , the number of parameters.
If , the last differential equations (in AUX) are driving equations (see Section 3).
8: M1 – INTEGERInput
On entry: determines whether or not the final solution is computed as well as the parameter values.
The final solution is not calculated;
The final values of the solution at interval (length of range)/ are calculated and stored sequentially in the array C starting with the values of evaluated at the first end point (see RAAUX) stored in .
9: AUX – SUBROUTINE, supplied by the user.External Procedure
AUX must evaluate the functions (i.e., the derivatives ) for given values of its arguments, ,
In the description of the parameters of D02AGF below, denotes the numerical value of N1 in the call of D02AGF.
1: X0 – REAL (KIND=nag_wp)Output
On exit: must contain the left-hand end of the range, .
2: X1 – REAL (KIND=nag_wp)Output
On exit: must contain the right-hand end of the range .
3: R – REAL (KIND=nag_wp)Output
On exit: must contain the matching point, .
4: PARAM() – REAL (KIND=nag_wp) arrayInput
On entry: , for , the value of the parameters.
RAAUX must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02AGF is called. Parameters denoted as Input must not be changed by this procedure.
12: PRSOL – SUBROUTINE, supplied by the user.External Procedure
PRSOL is called at each iteration of the Newton method and can be used to print the current values of the parameters
, for , their errors, , and the sum of squares of the errors at the matching point, .
On entry: IFAIL must be set to , . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
This indicates that on entry, that is the number of parameters is greater than the number of differential equations.
As for except that the integration failed while calculating the matrix for use in the Newton iteration.
The current matching point does not lie between the current end points and . If the values , and depend on the parameters , this may occur at any time in the Newton iteration if care is not taken to avoid it when coding RAAUX.
The step length for integration H has halved more than times (or too many steps were needed to reach the end of the range of integration) in attempting to control the local truncation error whilst integrating to obtain the solution corresponding to the current values . If, on failure, H has the sign of then failure has occurred whilst integrating from to , otherwise it has occurred whilst integrating from to .
The matrix of the equations to be solved for corrections to the variable parameters in the Newton method is singular (as determined by F07ADF (DGETRF)).
A satisfactory correction to the parameters was not obtained on the last Newton iteration employed. A Newton iteration is deemed to be unsatisfactory if the sum of the squares of the residuals (which can be printed using PRSOL) has not been reduced after three iterations using a new Newton correction.
Convergence has not been obtained after satisfactory iterations of the Newton method.
A further discussion of these errors and the steps which might be taken to correct them is given in Section 8.
If the process converges, the accuracy to which the unknown parameters are determined is usually close to that specified by you; and the solution, if requested, is usually determined to the accuracy specified.
8 Further Comments
The time taken by D02AGF depends on the complexity of the system, and on the number of iterations required. In practice, integration of the differential equations is by far the most costly process involved.
There may be particular difficulty in integrating the differential equations in one direction (indicated by or ). The value of should be adjusted to avoid such difficulties.
If the matching point is at one of the end points or and some of the parameters are used only to determine the boundary values at this point, then good initial estimates for these parameters are not required, since they are completely determined by the routine (for example, see in EX1 of Section 9).
Wherever they occur in the procedure, the error parameters contained in the arrays E and PARERR are used in ‘mixed’ form; that is always occurs in expressions of the form , and always occurs in expressions of the form . Though not ideal for every application, it is expected that this mixture of absolute and relative error testing will be adequate for most purposes.
Note that convergenceisnotguaranteed. You are strongly advised to provide an output PRSOL, as shown in EX1 of Section 9, in order to monitor the progress of the iteration. Failure of the Newton iteration to converge (see or ) usually results from poor starting approximations to the parameters, though occasionally such failures occur because the elements of one or both of the arrays PARERR or E are too small. (It should be possible to distinguish these cases by studying the output from PRSOL.) Poor starting approximations can also result in the failure described under and in Section 6 (especially if these errors occur after some Newton iterations have been completed, that is, after two or more calls of PRSOL). More frequently, a singular matrix in the Newton method (monitored as ) occurs because the mathematical problem has been posed incorrectly. The case usually occurs because or has been poorly estimated, so these values should be checked first. If is monitored, the solution is sensitive to perturbations in the parameters . Reduce the size of one or more values to reduce the perturbations. Since only one value is perturbed at any time when forming the matrix, the perturbation which is too large can be located by studying the final output from PRSOL and the values of the parameters returned by D02AGF. If this change leads to other types of failure improve the initial values of by other means.
The computing time for integrating the differential equations can sometimes depend critically on the quality of the initial estimates for the parameters . If it seems that too much computing time is required and, in particular, if the values (available on each call of PRSOL) are much larger than the expected values of the solution at the matching point , then the coding of AUX, BCAUX and RAAUX should be checked for errors. If no errors can be found, an independent attempt should be made to improve the initial estimates for .
The subroutine can be used to solve a very wide range of problems, for example:
eigenvalue problems, including problems where the eigenvalue occurs in the boundary conditions;
problems where the differential equations depend on some parameters which are to be determined so as to satisfy certain boundary conditions (see EX1 in Section 9);
problems where one of the end points of the range of integration is to be determined as the point where a variable takes a particular value (see EX2 in Section 9);
singular problems and problems on infinite ranges of integration where the values of the solution at or or both are determined by a power series or an asymptotic expansion (or a more complicated expression) and where some of the coefficients in the expression are to be determined (see EX1 in Section 9); and
differential equations with certain terms defined by other independent (driving) differential equations.
For this routine two examples are presented. There is a single example program for D02AGF, with a main program and the code to solve the two example problems given in Example 1 (EX1) and Example 2 (EX2).
Example 1 (EX1)
This example finds the solution of the differential equation
on the range , with boundary conditions and .
We cannot use the differential equation at because it is singular, so we take the truncated series expansion
near the origin (which is correct to the number of terms given in this case). Here is one of the parameters to be determined. We choose the range as and setting , we can determine all the boundary conditions. We take the matching point to be , the end of the range, and so a good initial guess for is not necessary. We write , , and estimate , .
Example 2 (EX2)
This example finds the gravitational constant and the range over which a projectile must be fired to hit the target with a given velocity. The differential equations are
on the range with boundary conditions
We write , , , and we take the matching point . We estimate , and (though this estimate is not important).