NAG Library Function Document

nag_ode_ivp_bdf_gen (d02ejc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     neq – IntegerInput

On entry: the number of differential equations.

Constraint: $\mathbf{neq}\ge 1$.

2:     fcn – function, supplied by the userExternal Function

fcn must evaluate the first derivatives $y_i^{\prime }$ (i.e., the functions $f_i$) for given values of their arguments $x,y_1,y_2,\dots ,y_{\mathbf{neq}}$.

The specification of fcn is:

void  fcn (Integer neq, double x, const double y[], double f[], Nag_User *comm)

1:     neq – IntegerInput

On entry: the number of differential equations.

2:     x – doubleInput

On entry: the value of the independent variable $x$.

3:     y[neq] – const doubleInput

On entry: $y\left[i-1\right]$ holds the value of the variable $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

4:     f[neq] – doubleOutput

On exit: $f\left[\mathit{i}-1\right]$ must contain the value of $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

5:     comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

3:     pederv – function, supplied by the userExternal Function

pederv must evaluate the Jacobian of the system (that is, the partial derivatives $\frac{\partial f_i}{\partial y_j}$) for given values of the variables $x,y_1,y_2,\dots ,y_{\mathbf{neq}}$.

The specification of pederv is:

void  pederv (Integer neq, double x, const double y[], double pw[], Nag_User *comm)

1:     neq – IntegerInput

On entry: the number of differential equations.

2:     x – doubleInput

On entry: the value of the independent variable $x$.

3:     y[neq] – const doubleInput

On entry: $y\left[\mathit{i}-1\right]$ holds the value of the variable $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

4:     pw[$\mathbf{neq}\times \mathbf{neq}$] – doubleOutput

On exit: $\mathbf{pw}\left[\left(i-1\right)\times \mathbf{neq}+j-1\right]$ must contain the value of $\frac{\partial f_i}{\partial y_j}$, for $i,j=1,2,\dots ,\mathbf{neq}$.

5:     comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

If you do not wish to supply the Jacobian, the actual argument pederv must be the NAG defined null function pointer NULLFN.

4:     x – double *Input/Output

On entry: the value of the independent variable $x$.

Constraint: $\mathbf{x}\ne \mathbf{xend}$.

On exit: if $g$ is supplied, x contains the point where $g\left(x,y\right)=0.0$, unless $g\left(x,y\right)\ne 0.0$ anywhere on the range x to xend, in which case, x will contain xend. If $g$ is not supplied x contains xend, unless an error has occurred, when it contains the value of $x$ at the error.

5:     y[neq] – doubleInput/Output

On entry: $y\left[\mathit{i}-1\right]$ holds the value of the variable $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

On exit: the computed values of the solution at the final point $x=\mathbf{x}$.

6:     xend – doubleInput

On entry: the final value of the independent variable.

$\mathbf{xend}<\mathbf{x}$

Iintegration proceeds in the negative direction.

Constraint: $\mathbf{xend}\ne \mathbf{x}$.

7:     tol – doubleInput

On entry: a positive tolerance for controlling the error in the integration. Hence tol affects the determination of the position where $g\left(x,y\right)=0.0$, if $g$ is supplied.

nag_ode_ivp_bdf_gen (d02ejc) has been designed so that, for most problems, a reduction in tol leads to an approximately proportional reduction in the error in the solution. However, the actual relation between tol and the accuracy achieved cannot be guaranteed. You are strongly recommended to call nag_ode_ivp_bdf_gen (d02ejc) with more than one value for tol and to compare the results obtained to estimate their accuracy. In the absence of any prior knowledge, you might compare the results obtained by calling nag_ode_ivp_bdf_gen (d02ejc) with $\mathbf{tol}={10}^{-p}$ and $\mathbf{tol}={10}^{-p-1}$ if $p$ correct decimal digits are required in the solution.

Constraint: $\mathbf{tol}>0.0$.

8:     err_c – Nag_ErrorControlInput

On entry: the type of error control. At each step in the numerical solution an estimate of the local error, est, is made. For the current step to be accepted the following condition must be satisfied:

$$\mathit{est}=\sqrt{\frac{1}{\mathbf{neq}}\sum _{i=1}^{\mathbf{neq}}{\left(e_i/\left({\tau }_r\times \left|y_i\right|+{\tau }_a\right)\right)}^2}\le 1.0$$

where ${\tau }_r$ and ${\tau }_a$ are defined by

err_c	${\tau }_r$	${\tau }_a$
$\mathrm{Nag\_Relative}$	tol	$\epsilon$
$\mathrm{Nag\_Absolute}$	0.0	tol
$\mathrm{Nag\_Mixed}$	tol	tol

where $\epsilon$ is a small machine-dependent number and $e_i$ is an estimate of the local error at $y_i$, computed internally. If the appropriate condition is not satisfied, the step size is reduced and the solution is recomputed on the current step. If you wish to measure the error in the computed solution in terms of the number of correct decimal places, then err_c should be set to $\mathrm{Nag\_Absolute}$ on entry, whereas if the error requirement is in terms of the number of correct significant digits, then err_c should be set to $\mathrm{Nag\_Relative}$. If you prefer a mixed error test, then err_c should be set to $\mathrm{Nag\_Mixed}$. The recommended value for err_c is $\mathrm{Nag\_Relative}$.

Constraint: $\mathbf{err\_c}=\mathrm{Nag\_Absolute}$, $\mathrm{Nag\_Mixed}$ or $\mathrm{Nag\_Relative}$.

9:     output – function, supplied by the userExternal Function

output permits access to intermediate values of the computed solution (for example to print or plot them), at successive user-specified points. It is initially called by nag_ode_ivp_bdf_gen (d02ejc) with $\mathbf{xsol}=\mathbf{x}$ (the initial value of $x$). You must reset xsol to the next point (between the current xsol and xend) where output is to be called, and so on at each call to output. If, after a call to output, the reset point xsol is beyond xend, nag_ode_ivp_bdf_gen (d02ejc) will integrate to xend with no further calls to output; if a call to output is required at the point $\mathbf{xsol}=\mathbf{xend}$, then xsol must be given precisely the value xend.

The specification of output is:

void  output (Integer neq, double *xsol, const double y[], Nag_User *comm)

1:     neq – IntegerInput

On entry: the number of differential equations.

2:     xsol – double *Input/Output

On entry: the value of the independent variable $x$.

On exit: you must set xsol to the next value of $x$ at which output is to be called.

3:     y[neq] – const doubleInput

On entry: $y\left[\mathit{i}-1\right]$ holds the value of the variable $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

4:     comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

If you do not wish to access intermediate output, the actual argument output must be the NAG defined null function pointer NULLFN.

10:   g – function, supplied by the userExternal Function

g must evaluate $g\left(x,y\right)$ for specified values $x,y$. It specifies the function $g$ for which the first position $x$ where $g\left(x,y\right)=0$ is to be found.

The specification of g is:

double  g (Integer neq, double x, const double y[], Nag_User *comm)

1:     neq – IntegerInput

On entry: the number of differential equations.

2:     x – doubleInput

On entry: the value of the independent variable $x$.

3:     y[neq] – const doubleInput

On entry: $y\left[\mathit{i}-1\right]$ holds the value of the variable $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

4:     comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

If you do not require the root finding option, the actual argument g must be the NAG defined null double function pointer NULLDFN.

11:   comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$, of type Pointer, allows you to communicate information to and from fcn, pederv, output and g. An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ by means of a cast to Pointer in the calling program, e.g., comm.p = (Pointer)&s. The type pointer will be void * with a C compiler that defines void * and char * otherwise.

12:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_2_REAL_ARG_EQ

On entry, $\mathbf{x}=⟨\mathit{\text{value}}⟩$ while $\mathbf{xend}=⟨\mathit{\text{value}}⟩$. These arguments must satisfy $\mathbf{x}\ne \mathbf{xend}$.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument err_c had an illegal value.

NE_INT_ARG_LT

On entry, $\mathbf{neq}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{neq}\ge 1$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_NO_SIGN_CHANGE

No change in sign of the function $g\left(x,y\right)$ was detected in the integration range.

NE_REAL_ARG_LE

On entry, tol must not be less than or equal to 0.0: $\mathbf{tol}=⟨\mathit{\text{value}}⟩$.

NE_TOL_PROGRESS

The value of tol, $⟨\mathit{\text{value}}⟩$, is too small for the function to make any further progress across the integration range. Current value of $\mathbf{x}=⟨\mathit{\text{value}}⟩$.

NE_TOL_TOO_SMALL

The value of tol, $⟨\mathit{\text{value}}⟩$, is too small for the function to take an initial step.

NE_XSOL_INCONSIST

On call $⟨\mathit{\text{value}}⟩$ to the supplied print function xsol was set to a value behind the previous value of xsol in the direction of integration.
Previous $\mathbf{xsol}=⟨\mathit{\text{value}}⟩$, $\mathbf{xend}=⟨\mathit{\text{value}}⟩$, new $\mathbf{xsol}=⟨\mathit{\text{value}}⟩$.

NE_XSOL_NOT_RESET

On call $⟨\mathit{\text{value}}⟩$ to the supplied print function xsol was not reset.

NE_XSOL_SET_WRONG

xsol was set to a value behind x in the direction of integration by the first call to the supplied print function.
The integration range is $\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$, $\mathbf{xsol}=⟨\mathit{\text{value}}⟩$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d02ejce.c)

10.2  Program Data

10.3  Program Results

Program Results (d02ejce.r)