NAG FL Interface
d03pcf  (dim1_parab_fd_old)
d03pca (dim1_parab_fd)

1 Purpose

2 Specification

2.1 Specification for d03pcf

2.2 Specification for d03pca

3 Description

4 References

5 Arguments

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system 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 compute the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by d03pcf/​d03pca.

The specification of pdedef for d03pcf is:

Fortran Interface copy

Subroutine pdedef ( npde, t, x, u, ux, p, q, r, ires) Integer, Intent (In) :: npde Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, x, u(npde), ux(npde) 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[], 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[], double p[], double q[], double r[], Integer &ires) }

The specification of pdedef for d03pca is:

Fortran Interface copy

Subroutine pdedef ( npde, t, x, u, ux, p, q, r, ires, iuser, ruser) Integer, Intent (In) :: npde Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, x, u(npde), ux(npde) 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[], 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[], 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: $\mathbf{p}(\mathbf{npde},\mathbf{npde})$ – Real (Kind=nag_wp) array Output

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

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

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

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

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

9: $\mathbf{ires}$ – 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 $\mathbf{ires}=3$, d03pcf/​d03pca 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 d03pca. Users of d03pcf therefore need not read the remainder of this description.

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

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

pdedef is called with the arguments iuser and ruser as supplied to d03pcf/​d03pca. 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 d03pcf/​d03pca 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 d03pcf/​d03pca. If your code inadvertently does return any NaNs or infinities, d03pcf/​d03pca is likely to produce unexpected results.

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

bndary must compute the functions ${\beta }_i$ and ${\gamma }_i$ which define the boundary conditions as in equation (3).

The specification of bndary for d03pcf is:

Fortran Interface copy

Subroutine bndary ( npde, t, u, ux, ibnd, beta, gamma, ires) Integer, Intent (In) :: npde, ibnd Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde) 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 *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 &ibnd, double beta[], double gamma[], Integer &ires) }

The specification of bndary for d03pca is:

Fortran Interface copy

Subroutine bndary ( npde, t, u, ux, ibnd, beta, gamma, ires, iuser, ruser) Integer, Intent (In) :: npde, ibnd Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde) 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 *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 &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: $\mathbf{ibnd}$ – Integer Input

On entry: determines the position of the boundary conditions.

$\mathbf{ibnd}=0$

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

$\mathbf{ibnd}\ne 0$

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

6: $\mathbf{beta}\left(\mathbf{npde}\right)$ – 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 $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

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

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

8: $\mathbf{ires}$ – 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 $\mathbf{ires}=3$, d03pcf/​d03pca 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 d03pca. Users of d03pcf therefore need not read the remainder of this description.

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

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

bndary is called with the arguments iuser and ruser as supplied to d03pcf/​d03pca. 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 d03pcf/​d03pca 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 d03pcf/​d03pca. If your code inadvertently does return any NaNs or infinities, d03pcf/​d03pca is likely to produce unexpected results.

7: $\mathbf{u}(\mathbf{npde},\mathbf{npts})$ – Real (Kind=nag_wp) array Input/Output

On entry: the initial values of $U(x,t)$ at $t=\mathbf{ts}$ and the mesh points $\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

On exit: $\mathbf{u}(\mathit{i},\mathit{j})$ will contain the computed solution at $t=\mathbf{ts}$.

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 spatial 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: $\mathbf{acc}$ – Real (Kind=nag_wp) Input

On entry: a positive quantity for controlling the local error estimate in the time integration. If $E(i,j)$ is the estimated error for $U_i$ at the $j$th mesh point, the error test is:

$$\left|E(i,j)\right|=\mathbf{acc}\times (1.0+\left|\mathbf{u}(i,j)\right|)\text{.}$$

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

11: $\mathbf{rsave}\left(\mathbf{lrsave}\right)$ – 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.

12: $\mathbf{lrsave}$ – Integer Input

On entry: the dimension of the array rsave as declared in the (sub)program from which d03pcf/​d03pca is called.

Constraint: $$\begin{gathered} \mathbf{lrsave}\ge (6\times \mathbf{npde}+10)\times \mathbf{npde}\times \mathbf{npts}+(3\times \mathbf{npde}+21)\times \mathbf{npde}+ \\ 7\times \mathbf{npts}+54 \end{gathered}$$.

13: $\mathbf{isave}\left(\mathbf{lisave}\right)$ – 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.

14: $\mathbf{lisave}$ – Integer Input

On entry: the dimension of the array isave as declared in the (sub)program from which d03pcf/​d03pca is called.

Constraint: $\mathbf{lisave}\ge \mathbf{npde}\times \mathbf{npts}+24$.

15: $\mathbf{itask}$ – 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 $t=\mathbf{tout}$.

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

16: $\mathbf{itrace}$ – Integer Input

On entry: the level of trace information required from d03pcf/​d03pca 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.

17: $\mathbf{ind}$ – 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 d03pcf/​d03pca.

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

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

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

Note: for d03pca, 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 d03pca. Users of d03pcf therefore need not read the remainder of this description.

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

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

iuser and ruser are not used by d03pcf/​d03pca, but are passed directly to pdedef and bndary and may be used to pass information to these routines.

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

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

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

23: $\mathbf{rwsav}\left(1100\right)$ – 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 d03pcf/​d03pca.

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

Note: see the argument description for ifail above.

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{acc}>0.0$.

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

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

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

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

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 $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=2$

Underlying ODE solver cannot make further progress from the point ts with the supplied value of acc. $\mathbf{ts}=⟨\mathit{\text{value}}⟩$, $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=3$

Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts: $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=4$

In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting $\mathbf{ires}=3$ in pdedef or bndary.

$\mathbf{ifail}=5$

Singular Jacobian of ODE system. Check problem formulation.

$\mathbf{ifail}=6$

In evaluating residual of ODE system, $\mathbf{ires}=2$ has been set in pdedef or bndary. Integration is successful as far as ts: $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=7$

acc was too small to start integration: $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=8$

ires set to an invalid value in call to pdedef or bndary.

$\mathbf{ifail}=9$

Serious error in internal call to an auxiliary. Increase itrace for further details.

$\mathbf{ifail}=10$

Integration completed, but a small change in acc is unlikely to result in a changed solution. $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=11$

Error during Jacobian formulation for ODE system. Increase itrace for further details.

$\mathbf{ifail}=14$

Flux function appears to depend on time derivatives.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results