NAG Library Routine Document
d02hbf (bvp_shoot_genpar)
1
Purpose
d02hbf solves a twopoint boundary value problem for a system of ordinary differential equations, using initial value techniques and Newton iteration; it generalizes subroutine
d02haf to include the case where parameters other than boundary values are to be determined.
2
Specification
Fortran Interface
Subroutine d02hbf ( 
p, n1, pe, e, n, soln, m1, fcn, bc, range, w, sdw, ifail) 
Integer, Intent (In)  ::  n1, n, m1, sdw  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  pe(n1), e(n)  Real (Kind=nag_wp), Intent (Inout)  ::  p(n1)  Real (Kind=nag_wp), Intent (Out)  ::  soln(n,m1), w(n,sdw)  External  ::  fcn, bc, range 

C Header Interface
#include <nagmk26.h>
void 
d02hbf_ (double p[], const Integer *n1, const double pe[], const double e[], const Integer *n, double soln[], const Integer *m1, void (NAG_CALL *fcn)(const double *x, const double y[], double f[], const double p[]), void (NAG_CALL *bc)(double g1[], double g2[], const double p[]), void (NAG_CALL *range)(double *a, double *b, const double p[]), double w[], const Integer *sdw, Integer *ifail) 

3
Description
d02hbf solves a twopoint boundary value problem by determining the unknown parameters
${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$ of the problem. These parameters may be, but need not be, boundary values; 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. (The parameters
${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$ correspond precisely to the unknown boundary conditions in
d02haf.) It is assumed that we have a system of
$\mathit{n}$ firstorder ordinary differential equations of the form:
and that the derivatives
${f}_{i}$ are evaluated by
fcn. The system, including the boundary conditions given by
bc and the range of integration given by
range, involves the
${\mathit{n}}_{1}$ unknown parameters
${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$ which are to be determined, and for which initial estimates must be supplied. The number of unknown parameters
${\mathit{n}}_{1}$ must not exceed the number of equations
$\mathit{n}$. If
${\mathit{n}}_{1}<\mathit{n}$, we assume that
$\left(\mathit{n}{\mathit{n}}_{1}\right)$ 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
${\mathit{n}}_{1}$ equations. In numbering the equations for
fcn, the driving equations must be put
first.
The estimated values of the parameters are corrected by a form of Newton iteration. The Newton correction on each iteration is calculated using a Jacobian matrix whose $\left(i,j\right)$th element depends on the derivative of the $i$th component of the solution, ${y}_{i}$, with respect to the $j$th parameter, ${p}_{j}$. This matrix is calculated by a simple numerical differentiation technique which requires ${\mathit{n}}_{1}$ evaluations of the differential system.
If the argument
ifail is set appropriately, the routine automatically prints messages to inform you of the flow of the calculation. These messages are discussed in detail in
Section 9.
d02hbf is a simplified version of
d02saf which is described in detail in
Gladwell (1979).
4
References
Gladwell I (1979) The development of the boundary value codes in the ordinary differential equations chapter of the NAG Library Codes for Boundary Value Problems in Ordinary Differential Equations. Lecture Notes in Computer Science (eds B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76 Springer–Verlag
5
Arguments
You are strongly recommended to read
Sections 3 and
9 in conjunction with this section.
 1: $\mathbf{p}\left({\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: an estimate for the
$\mathit{i}$th argument, ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$.
On exit: the corrected value for the $i$th argument, unless an error has occurred, when it contains the last calculated value of the argument.
 2: $\mathbf{n1}$ – IntegerInput

On entry: ${\mathit{n}}_{1}$, the number of arguments.
Constraint:
$1\le {\mathbf{n1}}\le {\mathbf{n}}$.
 3: $\mathbf{pe}\left({\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the elements of
pe must be given small positive values. The element
${\mathbf{pe}}\left(i\right)$ is used
(i) 
in the convergence test on the $i$th argument in the Newton iteration, and 
(ii) 
in perturbing the $i$th argument when approximating the derivatives of the components of the solution with respect to this argument for use in the Newton iteration. 
The elements ${\mathbf{pe}}\left(i\right)$ should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.
Constraint:
${\mathbf{pe}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n1}}$.
 4: $\mathbf{e}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the elements of
e must be given positive values. The element
${\mathbf{e}}\left(i\right)$ is used in the bound on the local error in the
$i$th component of the solution
${y}_{i}$ during integration.
The elements ${\mathbf{e}}\left(i\right)$ should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.
Constraint:
${\mathbf{e}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
 5: $\mathbf{n}$ – IntegerInput

On entry: $\mathit{n}$, the total number of differential equations.
Constraint:
${\mathbf{n}}\ge {\mathbf{n1}}$.
 6: $\mathbf{soln}\left({\mathbf{n}},{\mathbf{m1}}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the solution when ${\mathbf{m1}}>1$.
 7: $\mathbf{m1}$ – IntegerInput

On entry: a value which controls exit values.
 ${\mathbf{m1}}=1$
 The final solution is not calculated.
 ${\mathbf{m1}}>1$
 The final values of the solution at interval (length of range)/$\left({\mathbf{m1}}1\right)$ are calculated and stored sequentially in the array soln starting with the values of the solutions evaluated at the first end point (see range) stored in the first column of soln.
Constraint:
${\mathbf{m1}}\ge 1$.
 8: $\mathbf{fcn}$ – Subroutine, supplied by the user.External Procedure

fcn must evaluate the functions
${f}_{\mathit{i}}$ (i.e., the derivatives
${y}_{\mathit{i}}^{\prime}$), for
$\mathit{i}=1,2,\dots ,\mathit{n}$, at a general point
$x$.
The specification of
fcn is:
Fortran Interface
Subroutine fcn ( 
x, y, f, p) 
Real (Kind=nag_wp), Intent (In)  ::  x, y(*), p(*)  Real (Kind=nag_wp), Intent (Out)  ::  f(*) 

C Header Interface
#include <nagmk26.h>
void 
fcn (const double *x, const double y[], double f[], const double p[]) 

In the description of the arguments of
d02hbf below,
$\mathit{n}$ and
$\mathit{n1}$ denote the numerical values of
n and
n1 in the call of
d02hbf.
 1: $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: $x$, the value of the argument.
 2: $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.
 3: $\mathbf{f}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the value of
${f}_{\mathit{i}}$, for
$\mathit{i}=1,2,\dots ,\mathit{n}$. The
${f}_{i}$ may depend upon the parameters
${p}_{\mathit{j}}$, for
$\mathit{j}=1,2,\dots ,{\mathit{n}}_{1}$. If there are any driving equations (see
Section 3) then these must be numbered first in the ordering of the components of
f in
fcn.
 4: $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the current estimate of the argument
${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$.
fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
d02hbf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: fcn should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d02hbf. If your code inadvertently
does return any NaNs or infinities,
d02hbf is likely to produce unexpected results.
 9: $\mathbf{bc}$ – Subroutine, supplied by the user.External Procedure

bc must place in
g1 and
g2 the boundary conditions at
$a$ and
$b$ respectively (see
range).
The specification of
bc is:
Fortran Interface
Subroutine bc ( 
g1, g2, p) 
Real (Kind=nag_wp), Intent (In)  ::  p(*)  Real (Kind=nag_wp), Intent (Out)  ::  g1(*), g2(*) 

C Header Interface
#include <nagmk26.h>
void 
bc (double g1[], double g2[], const double p[]) 

In the description of the arguments of
d02hbf below,
$\mathit{n}$ and
$\mathit{n1}$ denote the numerical values of
n and
n1 in the call of
d02hbf.
 1: $\mathbf{g1}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the value of
${y}_{\mathit{i}}\left(a\right)$, (where this may be a known value or a function of the parameters ${p}_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=1,2,\dots ,{\mathit{n}}_{1}$).
 2: $\mathbf{g2}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the value of
${y}_{\mathit{i}}\left(b\right)$, for
$\mathit{i}=1,2,\dots ,\mathit{n}$, (where these may be known values or functions of the parameters
${p}_{\mathit{j}}$, for
$\mathit{j}=1,2,\dots ,{\mathit{n}}_{1}$). If
$\mathit{n}>{\mathit{n}}_{1}$, so that there are some driving equations, the first
$\mathit{n}{\mathit{n}}_{1}$ values of
g2 need not be set since they are never used.
 3: $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

On entry: an estimate of the argument
${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$.
bc must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
d02hbf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: bc should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d02hbf. If your code inadvertently
does return any NaNs or infinities,
d02hbf is likely to produce unexpected results.
 10: $\mathbf{range}$ – Subroutine, supplied by the user.External Procedure

range must evaluate the boundary points
$a$ and
$b$, each of which may depend on the arguments
${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$. The integrations in the shooting method are always from
$a$ to
$b$.
The specification of
range is:
Fortran Interface
Subroutine range ( 
a, b, p) 
Real (Kind=nag_wp), Intent (In)  ::  p(*)  Real (Kind=nag_wp), Intent (Out)  ::  a, b 

C Header Interface
#include <nagmk26.h>
void 
range (double *a, double *b, const double p[]) 

In the description of the arguments of
d02hbf below,
$\mathit{n1}$ denotes the actual value of
n1 in the call of
d02hbf.
 1: $\mathbf{a}$ – Real (Kind=nag_wp)Output

On exit: $a$, one of the boundary points.
 2: $\mathbf{b}$ – Real (Kind=nag_wp)Output

On exit: the second boundary point,
$b$. Note that
${\mathbf{b}}>{\mathbf{a}}$ forces the direction of integration to be that of increasing
$x$. If
a and
b are interchanged the direction of integration is reversed.
 3: $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the current estimate of the
$\mathit{i}$th argument, ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$.
range must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
d02hbf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: range should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d02hbf. If your code inadvertently
does return any NaNs or infinities,
d02hbf is likely to produce unexpected results.
 11: $\mathbf{w}\left({\mathbf{n}},{\mathbf{sdw}}\right)$ – Real (Kind=nag_wp) arrayOutput

Used mainly for workspace.
On exit: with
${\mathbf{ifail}}={\mathbf{2}}$,
${\mathbf{3}}$,
${\mathbf{4}}$ or
${\mathbf{5}}$ (see
Section 6),
${\mathbf{w}}\left(\mathit{i},1\right)$, for
$\mathit{i}=1,2,\dots ,\mathit{n}$, contains the solution at the point
$x$ when the error occurred.
${\mathbf{w}}\left(1,2\right)$ contains
$x$.
 12: $\mathbf{sdw}$ – IntegerInput

On entry: the second dimension of the array
w as declared in the (sub)program from which
d02hbf is called.
Constraint:
${\mathbf{sdw}}\ge 3{\mathbf{n}}+14+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(11,{\mathbf{n}}\right)$.
 13: $\mathbf{ifail}$ – IntegerInput/Output

For this routine, the normal use of
ifail is extended to control the printing of error and warning messages as well as specifying hard or soft failure (see
Section 3.4 in How to Use the NAG Library and its Documentation).
On entry:
ifail must be set to a value with the decimal expansion
$\mathit{cba}$, where each of the decimal digits
$c$,
$b$ and
$a$ must have a value of
$0$ or
$1$.
$a=0$ 
specifies hard failure, otherwise soft failure; 
$b=0$ 
suppresses error messages, otherwise error messages will be printed (see Section 6); 
$c=0$ 
suppresses warning messages, otherwise warning messages will be printed (see Section 6). 
The recommended value for inexperienced users is $110$ (i.e., hard failure with all messages printed).
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{m1}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m1}}\ge 1$.
On entry, ${\mathbf{n1}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n1}}\ge 1$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n1}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge {\mathbf{n1}}$.
On entry, ${\mathbf{sdw}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{sdw}}\ge 3\times {\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(25,{\mathbf{n}}+14\right)$; that is, $\u2329\mathit{\text{value}}\u232a$.
On entry a negative or zero convergence test tolerance has been set.
On entry a negative or zero local error tolerance has been set.
 ${\mathbf{ifail}}=2$

In the integration with initial or final parameters, the step size was reduced too far for the integration to proceed. Either this routine is not a suitable method for solving the problem, or the initial choice of parameters is very poor.
The step length for the integration became too short to proceed when calculating the residual.
 ${\mathbf{ifail}}=3$

An initial steplength could be found for integration to proceed with the current parameters.
In the integration with initial or final parameters, a suitable initial step could not be found. Either this routine is not suitable for solving the problem, or the initial choice of parameters is very poor.
 ${\mathbf{ifail}}=4$

The steplength required to calculate the Jacobian to sufficient accuracy is too small
 ${\mathbf{ifail}}=5$

An initial steplength could be found for Jacobian calculation to proceed with the current parameters.
 ${\mathbf{ifail}}=6$

The Jacobian has an insignificant column. Make sure that the solution vector depends on all the parameters.
 ${\mathbf{ifail}}=7$

An internal singular value decomposition has failed.
This error can be avoided by changing the initial parameter estimates.
 ${\mathbf{ifail}}=8$

The Newton iteration has failed to converge.
This can indicate a poor initial choice of parameters or a very difficult problem.
Consider varying elements of the parameter convergence control if the residuals are small; otherwise vary initial parameter estimates.
 ${\mathbf{ifail}}=9$

Internal error in Newton method. Please contact
NAG.
 ${\mathbf{ifail}}=10$

Internal error in calculating Jacobian. Please contact
NAG.
 ${\mathbf{ifail}}=11$

Internal error in calculating residual. Please contact
NAG.
 ${\mathbf{ifail}}=12$

Internal error in calculating residual. Please contact
NAG.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
If the process converges, the accuracy to which the unknown parameters are determined is usually close to that specified by you; the solution, if requested, may be determined to a required accuracy by varying
e.
8
Parallelism and Performance
d02hbf is not thread safe and should not be called from a multithreaded user program. Please see
Section 3.12.1 in How to Use the NAG Library and its Documentation for more information on thread safety.
d02hbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by d02hbf 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.
Wherever they occur in the routine, the error arguments contained in the arrays
e and
pe are used in ‘mixed’ form; that is
${\mathbf{e}}\left(i\right)$ always occurs in expressions of the form
and
${\mathbf{pe}}\left(i\right)$ 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.
You may determine a suitable direction of integration
$a$ to
$b$ and suitable values for
${\mathbf{e}}\left(i\right)$ by integrations with
d02pef. The best direction of integration is usually the direction of decreasing solutions.
You are strongly recommended to set
ifail to obtain selfexplanatory error messages, and also monitoring information about the course of the computation. You may select the unit numbers on which this output is to appear by calls of
x04aaf (for error messages) or
x04abf (for monitoring information) – see
Section 10 for an example. Otherwise the default unit numbers will be used, as specified in the
Users' Note. The monitoring information produced at each iteration includes the current parameter values, the residuals and
$2$norms: a basic norm and a current norm. At each iteration the aim is to find parameter values which make the current norm less than the basic norm. Both these norms should tend to zero as should the residuals. (They would all be zero if the exact parameters were used as input.) For more details, in particular about the other monitoring information printed, you are advised to consult the specification of
d02saf, and especially the description of the argument
monit there.
The computing time for integrating the differential equations can sometimes depend critically on the quality of the initial estimates for the parameters
${p}_{i}$. If it seems that too much computing time is required and, in particular, if the values of the residuals printed by the monitoring routine are much larger than the expected values of the solution at
$b$, then the coding of
fcn,
bc and
range should be checked for errors. If no errors can be found, an independent attempt should be made to improve the initial estimates for
${p}_{i}$.
The subroutine can be used to solve a very wide range of problems, for example:
(a) 
eigenvalue problems, including problems where the eigenvalue occurs in the boundary conditions; 
(b) 
problems where the differential equations depend on some parameters which are to be determined so as to satisfy certain boundary conditions (see Example 2 in Section 10); 
(c) 
problems where one of the end points of the range of integration is to be determined as the point where a variable ${y}_{i}$ takes a particular value (see Example 2 in Section 10); 
(d) 
singular problems and problems on infinite ranges of integration where the values of the solution at $a$ or $b$ 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 Example 1 in Section 10); and 
(e) 
differential equations with certain terms defined by other independent (driving) differential equations. 
10
Example
For this routine two examples are presented. There is a single example program for d02hbf, 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
$0\le x\le 16$, with boundary conditions
$y\left(0\right)=0.1$ and
$y\left(16\right)=1/6$. We cannot use the differential equation at
$x=0$ because it is singular, so we take a truncated power series expansion
near the origin where
${p}_{1}$ is one of the parameters to be determined. We choose the interval as
$\left[0.1,16\right]$ and setting
${p}_{2}={y}^{\prime}\left(16\right)$, we can determine all the boundary conditions. We take
$\mathrm{X1}=16$. We write
$y={\mathbf{y}}\left(1\right)$,
${y}^{\prime}={\mathbf{y}}\left(2\right)$, and estimate
$\mathrm{PARAM}\left(1\right)=0.2$,
$\mathrm{PARAM}\left(2\right)=0.0$. Note the call to
x04abf before the call to
d02hbf.
Example 2 (EX2)
This example finds the gravitational constant ${p}_{1}$ and the range ${p}_{2}$ over which a projectile must be fired to hit the target with a given velocity.
The differential equations are
on the range
$0<x<{p}_{2}$, with boundary conditions
We write
$y={\mathbf{y}}\left(1\right)$,
$v={\mathbf{y}}\left(2\right)$,
$\varphi ={\mathbf{y}}\left(3\right)$. We estimate
${p}_{1}=\mathrm{PARAM}\left(1\right)=32$,
${p}_{2}=\mathrm{PARAM}\left(2\right)=6000$ and
${p}_{3}=\mathrm{PARAM}\left(3\right)=0.54$ (though this last estimate is not important).
10.1
Program Text
Program Text (d02hbfe.f90)
10.2
Program Data
Program Data (d02hbfe.d)
10.3
Program Results
Program Results (d02hbfe.r)