NAG FL Interface
d03phf  (dim1_parab_dae_fd_old)
d03pha (dim1_parab_dae_fd)

1 Purpose

2 Specification

2.1 Specification for d03phf

2.2 Specification for d03pha

3 Description

4 References

5 Arguments

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs to be solved.

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

2: $\mathbf{m}$ – Integer Input

On entry: the coordinate system used:

$\mathbf{m}=0$

Indicates Cartesian coordinates.

$\mathbf{m}=1$

Indicates cylindrical polar coordinates.

$\mathbf{m}=2$

Indicates spherical polar coordinates.

Constraint: $\mathbf{m}=0$, $1$ or $2$.

3: $\mathbf{ts}$ – Real (Kind=nag_wp) Input/Output

On entry: the initial value of the independent variable $t$.

On exit: the value of $t$ corresponding to the solution values in u. Normally $\mathbf{ts}=\mathbf{tout}$.

Constraint: $\mathbf{ts}<\mathbf{tout}$.

4: $\mathbf{tout}$ – Real (Kind=nag_wp) Input

On entry: the final value of $t$ to which the integration is to be carried out.

5: $\mathbf{pdedef}$ – Subroutine, supplied by the user. External Procedure

pdedef must evaluate the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$, $U_x$ and $V$. $Q_i$ may depend linearly on $\stackrel{.}{V}$. pdedef is called approximately midway between each pair of mesh points in turn by d03phf/​d03pha.

The specification of pdedef for d03phf is:

Fortran Interface copy

Subroutine pdedef ( npde, t, x, u, ux, nv, v, vdot, p, q, r, ires) Integer, Intent (In) :: npde, nv Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, x, u(npde), ux(npde), v(nv), vdot(nv) Real (Kind=nag_wp), Intent (Out) :: p(npde,npde), q(npde), r(npde)

C Header Interface copy

void  pdedef (const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires)

C++ Header Interface copy

#include <nag.h> extern "C" { void  pdedef (const Integer &npde, const double &t, const double &x, const double u[], const double ux[], const Integer &nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer &ires) }

The specification of pdedef for d03pha is:

Fortran Interface copy

Subroutine pdedef ( npde, t, x, u, ux, nv, v, vdot, p, q, r, ires, iuser, ruser) Integer, Intent (In) :: npde, nv Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, x, u(npde), ux(npde), v(nv), vdot(nv) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: p(npde,npde), q(npde), r(npde)

C Header Interface copy

void  pdedef (const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires, Integer iuser[], double ruser[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  pdedef (const Integer &npde, const double &t, const double &x, const double u[], const double ux[], const Integer &nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer &ires, Integer iuser[], double ruser[]) }

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

2: $\mathbf{t}$ – Real (Kind=nag_wp) Input

On entry: the current value of the independent variable $t$.

3: $\mathbf{x}$ – Real (Kind=nag_wp) Input

On entry: the current value of the space variable $x$.

4: $\mathbf{u}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{u}\left(\mathit{i}\right)$ contains the value of the component $U_{\mathit{i}}(x,t)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5: $\mathbf{ux}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{ux}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}(x,t)}{\partial x}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

6: $\mathbf{nv}$ – Integer Input

On entry: the number of coupled ODEs in the system.

7: $\mathbf{v}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

8: $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly in $Q_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

9: $\mathbf{p}(\mathbf{npde},\mathbf{npde})$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{p}(\mathit{i},\mathit{j})$ must be set to the value of $P_{\mathit{i},\mathit{j}}(x,t,U,U_x,V)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

10: $\mathbf{q}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{q}\left(\mathit{i}\right)$ must be set to the value of $Q_{\mathit{i}}(x,t,U,U_x,V,\stackrel{.}{V})$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

11: $\mathbf{r}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{r}\left(\mathit{i}\right)$ must be set to the value of $R_{\mathit{i}}(x,t,U,U_x,V)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

12: $\mathbf{ires}$ – Integer Input/Output

On entry: set to $\mathrm{-1}$ or $1$.

On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, d03phf/​d03pha returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.

13: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

14: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

pdedef is called with the arguments iuser and ruser as supplied to d03phf/​d03pha. You should use the arrays iuser and ruser to supply information to pdedef.

pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03phf/​d03pha is called. Arguments denoted as Input must not be changed by this procedure.

Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03phf/​d03pha. If your code inadvertently does return any NaNs or infinities, d03phf/​d03pha is likely to produce unexpected results.

6: $\mathbf{bndary}$ – Subroutine, supplied by the user. External Procedure

bndary must evaluate the functions ${\beta }_i$ and ${\gamma }_i$ which describe the boundary conditions, as given in (4).

The specification of bndary for d03phf is:

Fortran Interface copy

Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, gamma, ires) Integer, Intent (In) :: npde, nv, ibnd Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv) Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)

C Header Interface copy

void  bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires)

C++ Header Interface copy

#include <nag.h> extern "C" { void  bndary (const Integer &npde, const double &t, const double u[], const double ux[], const Integer &nv, const double v[], const double vdot[], const Integer &ibnd, double beta[], double gamma[], Integer &ires) }

The specification of bndary for d03pha is:

Fortran Interface copy

Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, gamma, ires, iuser, ruser) Integer, Intent (In) :: npde, nv, ibnd Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)

C Header Interface copy

void  bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires, Integer iuser[], double ruser[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  bndary (const Integer &npde, const double &t, const double u[], const double ux[], const Integer &nv, const double v[], const double vdot[], const Integer &ibnd, double beta[], double gamma[], Integer &ires, Integer iuser[], double ruser[]) }

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

2: $\mathbf{t}$ – Real (Kind=nag_wp) Input

On entry: the current value of the independent variable $t$.

3: $\mathbf{u}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{u}\left(\mathit{i}\right)$ contains the value of the component $U_{\mathit{i}}(x,t)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

4: $\mathbf{ux}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{ux}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}(x,t)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5: $\mathbf{nv}$ – Integer Input

On entry: the number of coupled ODEs in the system.

6: $\mathbf{v}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

7: $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly in $Q_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

8: $\mathbf{ibnd}$ – Integer Input

On entry: specifies which boundary conditions are to be evaluated.

$\mathbf{ibnd}=0$

bndary must set up the coefficients of the left-hand boundary, $x=a$.

$\mathbf{ibnd}\ne 0$

bndary must set up the coefficients of the right-hand boundary, $x=b$.

9: $\mathbf{beta}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{beta}\left(\mathit{i}\right)$ must be set to the value of ${\beta }_{\mathit{i}}(x,t)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

10: $\mathbf{gamma}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{gamma}\left(\mathit{i}\right)$ must be set to the value of ${\gamma }_{\mathit{i}}(x,t,U,U_x,V,\stackrel{.}{V})$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

11: $\mathbf{ires}$ – Integer Input/Output

On entry: set to $\mathrm{-1}$ or $1$.

On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, d03phf/​d03pha returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.

12: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

13: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

bndary is called with the arguments iuser and ruser as supplied to d03phf/​d03pha. You should use the arrays iuser and ruser to supply information to bndary.

bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03phf/​d03pha is called. Arguments denoted as Input must not be changed by this procedure.

Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03phf/​d03pha. If your code inadvertently does return any NaNs or infinities, d03phf/​d03pha is likely to produce unexpected results.

7: $\mathbf{u}\left(\mathbf{neqn}\right)$ – Real (Kind=nag_wp) array Input/Output

On entry: the initial values of the dependent variables defined as follows:

• $\mathbf{u}\left(\mathbf{npde}\times (\mathit{j}-1)+\mathit{i}\right)$ contain $U_{\mathit{i}}(x_{\mathit{j}},t_0)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and
• $\mathbf{u}\left(\mathbf{npts}\times \mathbf{npde}+\mathit{i}\right)$ contain $V_{\mathit{i}}\left(t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

On exit: the computed solution $U_{\mathit{i}}(x_{\mathit{j}},t)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and $V_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,\mathbf{nv}$, evaluated at $t=\mathbf{ts}$, as follows:

• $\mathbf{u}\left(\mathbf{npde}\times (\mathit{j}-1)+\mathit{i}\right)$ contain $U_{\mathit{i}}(x_{\mathit{j}},t)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and
• $\mathbf{u}\left(\mathbf{npts}\times \mathbf{npde}+\mathit{i}\right)$ contain $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

8: $\mathbf{npts}$ – Integer Input

On entry: the number of mesh points in the interval $[a,b]$.

Constraint: $\mathbf{npts}\ge 3$.

9: $\mathbf{x}\left(\mathbf{npts}\right)$ – Real (Kind=nag_wp) array Input

On entry: the mesh points in the space direction. $\mathbf{x}\left(1\right)$ must specify the left-hand boundary, $a$, and $\mathbf{x}\left(\mathbf{npts}\right)$ must specify the right-hand boundary, $b$.

Constraint: $\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{npts}\right)$.

10: $\mathbf{nv}$ – Integer Input

On entry: the number of coupled ODE components.

Constraint: $\mathbf{nv}\ge 0$.

11: $\mathbf{odedef}$ – Subroutine, supplied by the NAG Library or the user. External Procedure

odedef must evaluate the functions $F$, which define the system of ODEs, as given in (3).

If you wish to compute the solution of a system of PDEs only ($\mathbf{nv}=0$), odedef must be the dummy routine d03pck for d03phf (or d53pck for d03pha). d03pck and d53pck are included in the NAG Library.

The specification of odedef for d03phf is:

Fortran Interface copy

Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, f, ires) Integer, Intent (In) :: npde, nv, nxi Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi) Real (Kind=nag_wp), Intent (Out) :: f(nv)

C Header Interface copy

void  odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires)

C++ Header Interface copy

#include <nag.h> extern "C" { void  odedef (const Integer &npde, const double &t, const Integer &nv, const double v[], const double vdot[], const Integer &nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer &ires) }

The specification of odedef for d03pha is:

Fortran Interface copy

Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, f, ires, iuser, ruser) Integer, Intent (In) :: npde, nv, nxi Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: f(nv)

C Header Interface copy

void  odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Integer iuser[], double ruser[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  odedef (const Integer &npde, const double &t, const Integer &nv, const double v[], const double vdot[], const Integer &nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer &ires, Integer iuser[], double ruser[]) }

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

2: $\mathbf{t}$ – Real (Kind=nag_wp) Input

On entry: the current value of the independent variable $t$.

3: $\mathbf{nv}$ – Integer Input

On entry: the number of coupled ODEs in the system.

4: $\mathbf{v}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

5: $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

6: $\mathbf{nxi}$ – Integer Input

On entry: the number of ODE/PDE coupling points.

7: $\mathbf{xi}\left(\mathbf{nxi}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left(\mathit{i}\right)$ contains the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$.

8: $\mathbf{ucp}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucp}(\mathit{i},\mathit{j})$ contains the value of $U_{\mathit{i}}(x,t)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

9: $\mathbf{ucpx}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpx}(\mathit{i},\mathit{j})$ contains the value of $\frac{\partial U_{\mathit{i}}(x,t)}{\partial x}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

10: $\mathbf{rcp}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{rcp}(\mathit{i},\mathit{j})$ contains the value of the flux $R_{\mathit{i}}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

11: $\mathbf{ucpt}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpt}(\mathit{i},\mathit{j})$ contains the value of $\frac{\partial U_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

12: $\mathbf{ucptx}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{ucptx}(\mathit{i},\mathit{j})$ contains the value of $\frac{{\partial }^2{U}_{\mathit{i}}}{\partial x\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

13: $\mathbf{f}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{f}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $F$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, where $F$ is defined as

$$F=G-A\stackrel{.}{V}-B\left(\begin{array}{c}{U}_t^{*}\\ U_{xt}^{*}\end{array}\right)\text{,}$$

(5)

or

$$F=-A\stackrel{.}{V}-B\left(\begin{array}{c}{U}_t^{*}\\ U_{xt}^{*}\end{array}\right)\text{.}$$

(6)

The definition of $F$ is determined by the input value of ires.

14: $\mathbf{ires}$ – Integer Input/Output

On entry: the form of $F$ that must be returned in the array f.

$\mathbf{ires}=1$

Equation (5) must be used.

$\mathbf{ires}=\mathrm{-1}$

Equation (6) must be used.

On exit: should usually remain unchanged. However, you may reset ires to force the integration routine to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, d03phf/​d03pha returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.

15: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

16: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

odedef is called with the arguments iuser and ruser as supplied to d03phf/​d03pha. You should use the arrays iuser and ruser to supply information to odedef.

odedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03phf/​d03pha is called. Arguments denoted as Input must not be changed by this procedure.

Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03phf/​d03pha. If your code inadvertently does return any NaNs or infinities, d03phf/​d03pha is likely to produce unexpected results.

12: $\mathbf{nxi}$ – Integer Input

On entry: the number of ODE/PDE coupling points.

Constraints:

• if $\mathbf{nv}=0$, $\mathbf{nxi}=0$;
• if $\mathbf{nv}>0$, $\mathbf{nxi}\ge 0$.

13: $\mathbf{xi}\left(\mathbf{nxi}\right)$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$, must be set to the ODE/PDE coupling points.

Constraint: $\mathbf{x}\left(1\right)\le \mathbf{xi}\left(1\right)<\mathbf{xi}\left(2\right)<\cdots <\mathbf{xi}\left(\mathbf{nxi}\right)\le \mathbf{x}\left(\mathbf{npts}\right)$.

14: $\mathbf{neqn}$ – Integer Input

On entry: the number of ODEs in the time direction.

Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

15: $\mathbf{rtol}\left(*\right)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array rtol must be at least $1$ if $\mathbf{itol}=1$ or $2$ and at least $\mathbf{neqn}$ if $\mathbf{itol}=3$ or $4$.

On entry: the relative local error tolerance.

Constraint: $\mathbf{rtol}\left(i\right)\ge 0.0$ for all relevant $i$.

16: $\mathbf{atol}\left(*\right)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array atol must be at least $1$ if $\mathbf{itol}=1$ or $3$ and at least $\mathbf{neqn}$ if $\mathbf{itol}=2$ or $4$.

On entry: the absolute local error tolerance.

Constraint: $\mathbf{atol}\left(i\right)\ge 0.0$ for all relevant $i$.

Note: corresponding elements of rtol and atol cannot both be $0.0$.

17: $\mathbf{itol}$ – Integer Input

On entry: a value to indicate the form of the local error test. itol indicates to d03phf/​d03pha whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $\Vert e_i/w_i\Vert <1.0$, where $w_i$ is defined as follows:

itol	rtol	atol	${\mathit{w}}_{\mathit{i}}$
1	scalar	scalar	$\mathbf{rtol}\left(1\right)\times \left|U_i\right|+\mathbf{atol}\left(1\right)$
2	scalar	vector	$\mathbf{rtol}\left(1\right)\times \left|U_i\right|+\mathbf{atol}\left(i\right)$
3	vector	scalar	$\mathbf{rtol}\left(i\right)\times \left|U_i\right|+\mathbf{atol}\left(1\right)$
4	vector	vector	$\mathbf{rtol}\left(i\right)\times \left|U_i\right|+\mathbf{atol}\left(i\right)$

In the above, $e_{\mathit{i}}$ denotes the estimated local error for the $\mathit{i}$th component of the coupled PDE/ODE system in time, $\mathbf{u}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{neqn}$.

The choice of norm used is defined by the argument norm.

Constraint: $1\le \mathbf{itol}\le 4$.

18: $\mathbf{norm}$ – Character(1) Input

On entry: the type of norm to be used.

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

Maximum norm.

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

Averaged $L_2$ norm.

If ${\mathbf{u}}_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged $L_2$ norm

$${\mathbf{u}}_{\mathrm{norm}}=\sqrt{\frac{1}{\mathbf{neqn}}\sum _{i=1}^{\mathbf{neqn}}{(\mathbf{u}\left(i\right)/w_i)}^2}\text{,}$$

while for the maximum norm

$${\mathbf{u}}_{\mathrm{norm}}=\underset{i}{\max }|\mathbf{u}\left(i\right)/w_i|\text{.}$$

See the description of itol for the formulation of the weight vector $w$.

Constraint: $\mathbf{norm}=\text{'M'}$ or $\text{'A'}$.

19: $\mathbf{laopt}$ – Character(1) Input

On entry: the type of matrix algebra required.

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

Full matrix methods to be used.

$\mathbf{laopt}=\text{'B'}$

Banded matrix methods to be used.

$\mathbf{laopt}=\text{'S'}$

Sparse matrix methods to be used.

Constraint: $\mathbf{laopt}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.

Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., $\mathbf{nv}=0$).

20: $\mathbf{algopt}\left(30\right)$ – Real (Kind=nag_wp) array Input

On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, $\mathbf{algopt}\left(1\right)$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:

$\mathbf{algopt}\left(1\right)$

Selects the ODE integration method to be used. If $\mathbf{algopt}\left(1\right)=1.0$, a BDF method is used and if $\mathbf{algopt}\left(1\right)=2.0$, a Theta method is used. The default value is $\mathbf{algopt}\left(1\right)=1.0$.

If $\mathbf{algopt}\left(1\right)=2.0$, $\mathbf{algopt}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,4$ are not used.

$\mathbf{algopt}\left(2\right)$

Specifies the maximum order of the BDF integration formula to be used. $\mathbf{algopt}\left(2\right)$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is $\mathbf{algopt}\left(2\right)=5.0$.

$\mathbf{algopt}\left(3\right)$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If $\mathbf{algopt}\left(3\right)=1.0$ a modified Newton iteration is used and if $\mathbf{algopt}\left(3\right)=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, there is an automatic switch to the modified Newton iteration. The default value is $\mathbf{algopt}\left(3\right)=1.0$.

$\mathbf{algopt}\left(4\right)$

Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as $P_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$, for some $i$ or when there is no ${\stackrel{.}{V}}_i\left(t\right)$ dependence in the coupled ODE system. If $\mathbf{algopt}\left(4\right)=1.0$, the Petzold test is used. If $\mathbf{algopt}\left(4\right)=2.0$, the Petzold test is not used. The default value is $\mathbf{algopt}\left(4\right)=1.0$.

If $\mathbf{algopt}\left(1\right)=1.0$, $\mathbf{algopt}\left(\mathit{i}\right)$, for $\mathit{i}=5,6,7$, are not used.

$\mathbf{algopt}\left(5\right)$

Specifies the value of Theta to be used in the Theta integration method. $0.51\le \mathbf{algopt}\left(5\right)\le 0.99$. The default value is $\mathbf{algopt}\left(5\right)=0.55$.

$\mathbf{algopt}\left(6\right)$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If $\mathbf{algopt}\left(6\right)=1.0$, a modified Newton iteration is used and if $\mathbf{algopt}\left(6\right)=2.0$, a functional iteration method is used. The default value is $\mathbf{algopt}\left(6\right)=1.0$.

$\mathbf{algopt}\left(7\right)$

Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If $\mathbf{algopt}\left(7\right)=1.0$, switching is allowed and if $\mathbf{algopt}\left(7\right)=2.0$, switching is not allowed. The default value is $\mathbf{algopt}\left(7\right)=1.0$.

$\mathbf{algopt}\left(11\right)$

Specifies a point in the time direction, $t_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of $t_{\mathrm{crit}}$ is described under the argument itask. If $\mathbf{algopt}\left(1\right)\ne 0.0$, a value of $0.0$ for $\mathbf{algopt}\left(11\right)$, say, should be specified even if itask subsequently specifies that $t_{\mathrm{crit}}$ will not be used.

$\mathbf{algopt}\left(12\right)$

Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, $\mathbf{algopt}\left(12\right)$ should be set to $0.0$.

$\mathbf{algopt}\left(13\right)$

Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, $\mathbf{algopt}\left(13\right)$ should be set to $0.0$.

$\mathbf{algopt}\left(14\right)$

Specifies the initial step size to be attempted by the integrator. If $\mathbf{algopt}\left(14\right)=0.0$, the initial step size is calculated internally.

$\mathbf{algopt}\left(15\right)$

Specifies the maximum number of steps to be attempted by the integrator in any one call. If $\mathbf{algopt}\left(15\right)=0.0$, no limit is imposed.

$\mathbf{algopt}\left(23\right)$

Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, $U_t$, $V$ and $\stackrel{.}{V}$. If $\mathbf{algopt}\left(23\right)=1.0$, a modified Newton iteration is used and if $\mathbf{algopt}\left(23\right)=2.0$, functional iteration is used. The default value is $\mathbf{algopt}\left(23\right)=1.0$.

$\mathbf{algopt}\left(29\right)$ and $\mathbf{algopt}\left(30\right)$ are used only for the sparse matrix algebra option, $\mathbf{laopt}=\text{'S'}$.

$\mathbf{algopt}\left(29\right)$

Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<\mathbf{algopt}\left(29\right)<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If $\mathbf{algopt}\left(29\right)$ lies outside this range then the default value is used. If the routines regard the Jacobian matrix as numerically singular then increasing $\mathbf{algopt}\left(29\right)$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is $\mathbf{algopt}\left(29\right)=0.1$.

$\mathbf{algopt}\left(30\right)$

Is used as a relative pivot threshold during subsequent Jacobian decompositions (see $\mathbf{algopt}\left(29\right)$) below which an internal error is invoked. If $\mathbf{algopt}\left(30\right)$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see $\mathbf{algopt}\left(29\right)$). The default value is $\mathbf{algopt}\left(30\right)=0.0001$.

21: $\mathbf{rsave}\left(\mathbf{lrsave}\right)$ – Real (Kind=nag_wp) array Communication Array

If $\mathbf{ind}=0$, rsave need not be set on entry.

If $\mathbf{ind}=1$, rsave must be unchanged from the previous call to the routine because it contains required information about the iteration.

22: $\mathbf{lrsave}$ – Integer Input

On entry: the dimension of the array rsave as declared in the (sub)program from which d03phf/​d03pha is called. Its size depends on the type of matrix algebra selected.

If $\mathbf{laopt}=\text{'F'}$, $\mathbf{lrsave}\ge \mathbf{neqn}\times \mathbf{neqn}+\mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

If $\mathbf{laopt}=\text{'B'}$, $\mathbf{lrsave}\ge (3\mathit{mlu}+1)\times \mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

If $\mathbf{laopt}=\text{'S'}$, $\mathbf{lrsave}\ge 4\mathbf{neqn}+11\mathbf{neqn}/2+1+\mathit{nwkres}+\mathit{lenode}$.

Where $\mathit{mlu}$ is the lower or upper half bandwidths such that

for PDE problems only (no coupled ODEs),

$\mathit{mlu}=3\mathbf{npde}-1\text{;}$

for coupled PDE/ODE problems,

$\mathit{mlu}=\mathbf{neqn}-1\text{.}$

Where $\mathit{nwkres}$ is defined by

if $\mathbf{nv}>0\text{ and }\mathbf{nxi}>0$,

$\mathit{nwkres}=\mathbf{npde}(2\mathbf{npts}+6\mathbf{nxi}+3\mathbf{npde}+26)+\mathbf{nxi}+\mathbf{nv}+7\mathbf{npts}+2\text{;}$

if $\mathbf{nv}>0\text{ and }\mathbf{nxi}=0$,

$\mathit{nwkres}=\mathbf{npde}(2\mathbf{npts}+3\mathbf{npde}+32)+\mathbf{nv}+7\mathbf{npts}+3\text{;}$

if $\mathbf{nv}=0$,

$\mathit{nwkres}=\mathbf{npde}(2\mathbf{npts}+3\mathbf{npde}+32)+7\mathbf{npts}+4\text{.}$

Where $\mathit{lenode}$ is defined by

the BDF method is used,

$\mathit{lenode}=(6+\mathrm{int}\left(\mathbf{algopt}\left(2\right)\right))\times \mathbf{neqn}+50\text{;}$

the Theta method is used,

$\mathit{lenode}=9\mathbf{neqn}+50\text{.}$

Note: when $\mathbf{laopt}=\text{'S'}$, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if $\mathbf{itrace}>0$ and the routine returns with $\mathbf{ifail}=\mathbf{15}$.

23: $\mathbf{isave}\left(\mathbf{lisave}\right)$ – Integer array Communication Array

If $\mathbf{ind}=0$, isave need not be set on entry.

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the routine because it contains required information about the iteration. In particular:

$\mathbf{isave}\left(1\right)$

Contains the number of steps taken in time.

$\mathbf{isave}\left(2\right)$

Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.

$\mathbf{isave}\left(3\right)$

Contains the number of Jacobian evaluations performed by the time integrator.

$\mathbf{isave}\left(4\right)$

Contains the order of the last backward differentiation formula method used.

$\mathbf{isave}\left(5\right)$

Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.

24: $\mathbf{lisave}$ – Integer Input

On entry: the dimension of the array isave as declared in the (sub)program from which d03phf/​d03pha is called. Its size depends on the type of matrix algebra selected:

• if $\mathbf{laopt}=\text{'F'}$, $\mathbf{lisave}\ge 24$;
• if $\mathbf{laopt}=\text{'B'}$, $\mathbf{lisave}\ge \mathbf{neqn}+24$;
• if $\mathbf{laopt}=\text{'S'}$, $\mathbf{lisave}\ge 25\times \mathbf{neqn}+24$.

Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed on the current error message unit if $\mathbf{itrace}>0$ and the routine returns with $\mathbf{ifail}=\mathbf{15}$.

25: $\mathbf{itask}$ – Integer Input

On entry: specifies the task to be performed by the ODE integrator.

$\mathbf{itask}=1$

Normal computation of output values u at $t=\mathbf{tout}$.

$\mathbf{itask}=2$

One step and return.

$\mathbf{itask}=3$

Stop at first internal integration point at or beyond $t=\mathbf{tout}$.

$\mathbf{itask}=4$

Normal computation of output values u at $t=\mathbf{tout}$ but without overshooting $t=t_{\mathrm{crit}}$ where $t_{\mathrm{crit}}$ is described under the argument algopt.

$\mathbf{itask}=5$

Take one step in the time direction and return, without passing $t_{\mathrm{crit}}$, where $t_{\mathrm{crit}}$ is described under the argument algopt.

Constraint: $\mathbf{itask}=1$, $2$, $3$, $4$ or $5$.

26: $\mathbf{itrace}$ – Integer Input

On entry: the level of trace information required from d03phf/​d03pha and the underlying ODE solver. itrace may take the value $\mathrm{-1}$, $0$, $1$, $2$ or $3$.

$\mathbf{itrace}=\mathrm{-1}$

No output is generated.

$\mathbf{itrace}=0$

Only warning messages from the PDE solver are printed on the current error message unit (see x04aaf).

$\mathbf{itrace}>0$

Output from the underlying ODE solver is printed on the current advisory message unit (see x04abf). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

If $\mathbf{itrace}<\mathrm{-1}$, $\mathrm{-1}$ is assumed and similarly if $\mathbf{itrace}>3$, $3$ is assumed.

The advisory messages are given in greater detail as itrace increases. You are advised to set $\mathbf{itrace}=0$, unless you are experienced with Sub-chapter D02M–N.

27: $\mathbf{ind}$ – Integer Input/Output

On entry: indicates whether this is a continuation call or a new integration.

$\mathbf{ind}=0$

Starts or restarts the integration in time.

$\mathbf{ind}=1$

Continues the integration after an earlier exit from the routine. In this case, only the arguments tout and ifail should be reset between calls to d03phf/​d03pha.

Constraint: $\mathbf{ind}=0$ or $1$.

On exit: $\mathbf{ind}=1$.

28: $\mathbf{ifail}$ – Integer Input/Output

Note: for d03pha, ifail does not occur in this position in the argument list. See the additional arguments described below.

On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.

28: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

29: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by d03phf/​d03pha, but are passed directly to pdedef, bndary and odedef and may be used to pass information to these routines.

30: $\mathbf{cwsav}\left(10\right)$ – Character(80) array Communication Array

31: $\mathbf{lwsav}\left(100\right)$ – Logical array Communication Array

32: $\mathbf{iwsav}\left(505\right)$ – Integer array Communication Array

33: $\mathbf{rwsav}\left(1100\right)$ – Real (Kind=nag_wp) array Communication Array

If $\mathbf{ind}=0$, cwsav, lwsav, iwsav and rwsav need not be set on entry.

If $\mathbf{ind}=1$, cwsav, lwsav, iwsav and rwsav must be unchanged from the previous call to d03phf/​d03pha.

34: $\mathbf{ifail}$ – Integer Input/Output

Note: see the argument description for ifail above.

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, at least one point in xi lies outside $[\mathbf{x}\left(1\right),\mathbf{x}\left(\mathbf{npts}\right)]$: $\mathbf{x}\left(1\right)=⟨\mathit{\text{value}}⟩$ and $\mathbf{x}\left(\mathbf{npts}\right)=⟨\mathit{\text{value}}⟩$.

On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$, $\mathbf{x}\left(\mathit{i}\right)=⟨\mathit{\text{value}}⟩$, $\mathit{j}=⟨\mathit{\text{value}}⟩$ and $\mathbf{x}\left(\mathit{j}\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{npts}\right)$.

On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$, $\mathbf{xi}\left(\mathit{i}+1\right)=⟨\mathit{\text{value}}⟩$ and $\mathbf{xi}\left(\mathit{i}\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{xi}\left(\mathit{i}+1\right)>\mathbf{xi}\left(\mathit{i}\right)$.

On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$ and $\mathbf{atol}\left(\mathit{i}\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{atol}\left(\mathit{i}\right)\ge 0.0$.

On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$ and $\mathit{j}=⟨\mathit{\text{value}}⟩$.
Constraint: corresponding elements $\mathbf{atol}\left(\mathit{i}\right)$ and $\mathbf{rtol}\left(\mathit{j}\right)$ cannot both be $0.0$.

On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$ and $\mathbf{rtol}\left(\mathit{i}\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{rtol}\left(\mathit{i}\right)\ge 0.0$.

On entry, $\mathbf{ind}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ind}=0$ or $1$.

On entry, $\mathbf{itask}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{itask}=1$, $2$, $3$, $4$ or $5$.

On entry, $\mathbf{itol}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{itol}=1$, $2$, $3$ or $4$.

On entry, $\mathbf{laopt}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{laopt}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.

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

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

On entry, $\mathbf{m}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{m}=0$, $1$ or $2$.

On entry, $\mathbf{m}=⟨\mathit{\text{value}}⟩$ and $\mathbf{x}\left(1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{m}\le 0$ or $\mathbf{x}\left(1\right)\ge 0.0$

On entry, $\mathbf{neqn}=⟨\mathit{\text{value}}⟩$, $\mathbf{npde}=⟨\mathit{\text{value}}⟩$, $\mathbf{npts}=⟨\mathit{\text{value}}⟩$ and $\mathbf{nv}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

On entry, $\mathbf{norm}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{norm}=\text{'A'}$ or $\text{'M'}$.

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

On entry, $\mathbf{npts}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{npts}\ge 3$.

On entry, $\mathbf{nv}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nv}\ge 0$.

On entry, $\mathbf{nv}=⟨\mathit{\text{value}}⟩$ and $\mathbf{nxi}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nxi}=0$ when $\mathbf{nv}=0$.

On entry, $\mathbf{nv}=⟨\mathit{\text{value}}⟩$ and $\mathbf{nxi}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nxi}\ge 0$ when $\mathbf{nv}>0$.

On entry, on initial entry $\mathbf{ind}=1$.
Constraint: on initial entry $\mathbf{ind}=0$.

On entry, $\mathbf{tout}=⟨\mathit{\text{value}}⟩$ and $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{tout}>\mathbf{ts}$.

On entry, $\mathbf{tout}-\mathbf{ts}$ is too small: $\mathbf{tout}=⟨\mathit{\text{value}}⟩$ and $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.