NAG Library Routine Document

D02NVF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NEQMAX – INTEGERInput

On entry: a bound on the maximum number of differential equations to be solved.

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

2:     SDYSAV – INTEGERInput

On entry: the second dimension of the array YSAV that will be supplied to the integrator, as declared in the (sub)program from which the integrator is called.

Constraint: $\mathbf{SDYSAV}\ge \mathbf{MAXORD}+1$.

3:     MAXORD – INTEGERInput

On entry: the maximum order to be used for the BDF method.

Constraint: $0<\mathbf{MAXORD}\le 5$.

4:     METHOD – CHARACTER(1)Input

On entry: specifies the method to be used to solve the system of nonlinear equations arising on each step of the BDF code.

$\mathbf{METHOD}=\text{'N'}$

A modified Newton iteration is used.

$\mathbf{METHOD}=\text{'F'}$

Functional iteration is used.

$\mathbf{METHOD}=\text{'D'}$

A modified Newton iteration is used.

Note:  a linear algebra setup routine must be called even when using functional iteration, since if difficulty is encountered a switch is made to a modified Newton method.

Only the first character of the actual parameter METHOD is passed to D02NVF; hence it is permissible for the actual argument to be more descriptive e.g., ‘Newton’, ‘Functional iteration’ or ‘Default’ in a call to D02NVF.

Constraint: $\mathbf{METHOD}=\text{'N'}$, $\text{'F'}$ or $\text{'D'}$.

5:     PETZLD – LOGICALInput

On entry: specifies whether the Petzold local error test is to be used. If PETZLD is set to .TRUE. on entry, then the Petzold local error test is used, otherwise a conventional test is used. The Petzold test results in extra overhead cost but is more stable and reliable for differential/algebraic equations.

6:     CON($6$) – REAL (KIND=nag_wp) arrayInput/Output

On entry: values to be used to control step size choice during integration. If any $\mathbf{CON}\left(i\right)=0.0$ on entry, it is replaced by its default value described below. In most cases this is the recommended setting.

$\mathbf{CON}\left(1\right)$, $\mathbf{CON}\left(2\right)$, and $\mathbf{CON}\left(3\right)$ are factors used to bound step size changes. If the current step size $h$ fails, then the modulus of the next step size is bounded by $\mathbf{CON}\left(1\right)\times \left|h\right|$. The default value of $\mathbf{CON}\left(1\right)$ is $2.0$. Note that the new step size may be used with a method of different order to the failed step. If the initial step size is $h$, then the modulus of the step size on the second step is bounded by $\mathbf{CON}\left(3\right)\times \left|h\right|$. At any other stage in the integration, if the current step size is $h$, then the modulus of the next step size is bounded by $\mathbf{CON}\left(2\right)\times \left|h\right|$. The default values are $10.0$ for $\mathbf{CON}\left(2\right)$ and $1000.0$ for $\mathbf{CON}\left(3\right)$.

$\mathbf{CON}\left(4\right)$, $\mathbf{CON}\left(5\right)$ and $\mathbf{CON}\left(6\right)$ are ‘tuning’ constants used in determining the next order and step size. They are used to scale the error estimates used in determining whether to keep the same order of the BDF method, decrease the order or increase the order respectively. The larger the value of $\mathbf{CON}\left(\mathit{i}\right)$, for $\mathit{i}=4,5,6$, the less likely the choice of the corresponding order. The default values are: $\mathbf{CON}\left(4\right)=1.2$, $\mathbf{CON}\left(5\right)=1.3$, $\mathbf{CON}\left(6\right)=1.4$.

Constraints:

These constraints must be satisfied after any zero values have been replaced by their default values.

• $0.0<\mathbf{CON}\left(1\right)\le \mathbf{CON}\left(2\right)\le \mathbf{CON}\left(3\right)$;
• $\mathbf{CON}\left(\mathit{i}\right)\ge 1.0$, for $\mathit{i}=2,3,\dots ,6$.

On exit: the values actually used by D02NVF.

7:     TCRIT – REAL (KIND=nag_wp)Input

On entry: a point beyond which integration must not be attempted. The use of TCRIT is described under the parameter ITASK in the specification for the integrator (e.g., see D02NBF). A value, $0.0$ say, must be specified even if ITASK subsequently specifies that TCRIT will not be used.

8:     HMIN – REAL (KIND=nag_wp)Input

On entry: the minimum absolute step size to be allowed. Set $\mathbf{HMIN}=0.0$ if this option is not required.

9:     HMAX – REAL (KIND=nag_wp)Input

On entry: the maximum absolute step size to be allowed. Set $\mathbf{HMAX}=0.0$ if this option is not required.

10:   H0 – REAL (KIND=nag_wp)Input

On entry: the step size to be attempted on the first step. Set $\mathbf{H0}=0.0$ if the initial step size is calculated internally.

11:   MAXSTP – INTEGERInput

On entry: the maximum number of steps to be attempted during one call to the integrator after which it will return with $\mathbf{IFAIL}=\mathbf{2}$. Set $\mathbf{MAXSTP}=0$ if no limit is to be imposed.

12:   MXHNIL – INTEGERInput

On entry: the maximum number of warnings printed (if $\mathbf{ITRACE}\ge 0$) per problem when $t+h=t$ on a step ($h=$ current step size). If $\mathbf{MXHNIL}\le 0$, a default value of $10$ is assumed.

13:   NORM – CHARACTER(1)Input

On entry: indicates the type of norm to be used.

$\mathbf{NORM}=\text{'M'}$

Maximum norm.

$\mathbf{NORM}=\text{'A'}$

Averaged L2 norm.

$\mathbf{NORM}=\text{'D'}$

Is the same as $\mathbf{NORM}=\text{'A'}$.

If $\mathit{vnorm}$ denotes the norm of the vector $v$ of length $n$, then for the averaged L2 norm

$$\mathit{vnorm}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(v_i/w_i\right)}^2}\text{,}$$

while for the maximum norm

$$\mathit{vnorm}=\underset{i}{\max }\left|v_i/w_i\right|\text{.}$$

If you wish to weight the maximum norm or the L2 norm, then RTOL and ATOL should be scaled appropriately on input to the integrator (see under ITOL in the specification of the integrator for the formulation of the weight vector $w_i$ from RTOL and ATOL, e.g., see D02NBF).

Only the first character to the actual parameter NORM is passed to D02NVF; hence it is permissible for the actual argument to be more descriptive e.g., ‘Maximum’, ‘Average L2’ or ‘Default’ in a call to D02NVF.

Constraint: $\mathbf{NORM}=\text{'M'}$, $\text{'A'}$ or $\text{'D'}$.

14:   RWORK($50+4\times \mathbf{NEQMAX}$) – REAL (KIND=nag_wp) arrayCommunication Array

This must be the same workspace array as the array RWORK supplied to the integrator. It is used to pass information from the setup routine to the integrator and therefore the contents of this array must not be changed before calling the integrator.

15:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ 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 $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

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

$\mathbf{IFAIL}=1$

On entry, an illegal input was detected.

7  Accuracy

8  Further Comments

9  Example