NAG Library Routine Document

D03PDF/D03PDA

1  Purpose

2  Specification

2.1  Specification for D03PDF

2.2  Specification for D03PDA

3  Description

4  References

5  Parameters

1:     NPDE – INTEGERInput

On entry: the number of PDEs in the system to be solved.

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

2:     M – INTEGERInput

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:     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:     TOUT – REAL (KIND=nag_wp)Input

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

5:     PDEDEF – SUBROUTINE, supplied by the user.External Procedure

PDEDEF must compute the values of the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$ and $U_x$ and must be evaluated at a set of points.

The specification of PDEDEF for D03PDF is:

SUBROUTINE PDEDEF ( NPDE, T, X, NPTL, U, UX, P, Q, R, IRES) INTEGER  NPDE, NPTL, IRES REAL (KIND=nag_wp)  T, X(NPTL), U(NPDE,NPTL), UX(NPDE,NPTL), P(NPDE,NPDE,NPTL), Q(NPDE,NPTL), R(NPDE,NPTL)

The specification of PDEDEF for D03PDA is:

SUBROUTINE PDEDEF ( NPDE, T, X, NPTL, U, UX, P, Q, R, IRES, IUSER, RUSER) INTEGER  NPDE, NPTL, IRES, IUSER(*) REAL (KIND=nag_wp)  T, X(NPTL), U(NPDE,NPTL), UX(NPDE,NPTL), P(NPDE,NPDE,NPTL), Q(NPDE,NPTL), R(NPDE,NPTL), RUSER(*)

1:     NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2:     T – REAL (KIND=nag_wp)Input

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

3:     X(NPTL) – REAL (KIND=nag_wp) arrayInput

On entry: contains a set of mesh points at which $P_{i,j}$, $Q_i$ and $R_i$ are to be evaluated. $\mathbf{X}\left(1\right)$ and $\mathbf{X}\left(\mathbf{NPTL}\right)$ contain successive user-supplied break points and the elements of the array will satisfy $\mathbf{X}\left(1\right)<\mathbf{X}\left(2\right)<\cdots <\mathbf{X}\left(\mathbf{NPTL}\right)$.

4:     NPTL – INTEGERInput

On entry: the number of points at which evaluations are required (the value of $\mathbf{NPOLY}+1$).

5:     U(NPDE,NPTL) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{U}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$ where $x=\mathbf{X}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{j}=1,2,\dots ,\mathbf{NPTL}$.

6:     UX(NPDE,NPTL) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{UX}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ where $x=\mathbf{X}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{j}=1,2,\dots ,\mathbf{NPTL}$.

7:     P(NPDE,NPDE,NPTL) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{P}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must be set to the value of $P_{\mathit{i},\mathit{j}}\left(x,t,U,U_x\right)$ where $x=\mathbf{X}\left(\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$, $\mathit{j}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{k}=1,2,\dots ,\mathbf{NPTL}$.

8:     Q(NPDE,NPTL) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{Q}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $Q_{\mathit{i}}\left(x,t,U,U_x\right)$ where $x=\mathbf{X}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{j}=1,2,\dots ,\mathbf{NPTL}$.

9:     R(NPDE,NPTL) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{R}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $R_{\mathit{i}}\left(x,t,U,U_x\right)$ where $x=\mathbf{X}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{j}=1,2,\dots ,\mathbf{NPTL}$.

10:   IRES – INTEGERInput/Output

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

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 $\mathbf{IRES}=3$, then D03PDF/D03PDA returns to the calling subroutine with the error indicator set to $\mathbf{IFAIL}=\mathbf{4}$.

Note: the following are additional parameters for specific use with D03PDA. Users of D03PDF therefore need not read the remainder of this description.

11:   IUSER($*$) – INTEGER arrayUser Workspace

12:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

PDEDEF is called with the parameters IUSER and RUSER as supplied to D03PDF/D03PDA. You are free to use the arrays IUSER and RUSER to supply information to PDEDEF as an alternative to using COMMON global variables.

PDEDEF must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PDF/D03PDA is called. Parameters denoted as Input must not be changed by this procedure.

6:     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 D03PDF is:

SUBROUTINE BNDARY ( NPDE, T, U, UX, IBND, BETA, GAMMA, IRES) INTEGER  NPDE, IBND, IRES REAL (KIND=nag_wp)  T, U(NPDE), UX(NPDE), BETA(NPDE), GAMMA(NPDE)

The specification of BNDARY for D03PDA is:

SUBROUTINE BNDARY ( NPDE, T, U, UX, IBND, BETA, GAMMA, IRES, IUSER, RUSER) INTEGER  NPDE, IBND, IRES, IUSER(*) REAL (KIND=nag_wp)  T, U(NPDE), UX(NPDE), BETA(NPDE), GAMMA(NPDE), RUSER(*)

1:     NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2:     T – REAL (KIND=nag_wp)Input

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

3:     U(NPDE) – REAL (KIND=nag_wp) arrayInput

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

4:     UX(NPDE) – REAL (KIND=nag_wp) arrayInput

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

5:     IBND – INTEGERInput

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, $x=b$.

6:     BETA(NPDE) – REAL (KIND=nag_wp) arrayOutput

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

7:     GAMMA(NPDE) – REAL (KIND=nag_wp) arrayOutput

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

8:     IRES – INTEGERInput/Output

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

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 $\mathbf{IRES}=3$, then D03PDF/D03PDA returns to the calling subroutine with the error indicator set to $\mathbf{IFAIL}=\mathbf{4}$.

Note: the following are additional parameters for specific use with D03PDA. Users of D03PDF therefore need not read the remainder of this description.

9:     IUSER($*$) – INTEGER arrayUser Workspace

10:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

BNDARY is called with the parameters IUSER and RUSER as supplied to D03PDF/D03PDA. You are free to use the arrays IUSER and RUSER to supply information to BNDARY as an alternative to using COMMON global variables.

BNDARY must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PDF/D03PDA is called. Parameters denoted as Input must not be changed by this procedure.

7:     U(NPDE,NPTS) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if $\mathbf{IND}=1$ the value of U must be unchanged from the previous call.

On exit: $\mathbf{U}\left(i,j\right)$ will contain the computed solution at $t=\mathbf{TS}$.

8:     NBKPTS – INTEGERInput

On entry: the number of break points in the interval $\left[a,b\right]$.

Constraint: $\mathbf{NBKPTS}\ge 2$.

9:     XBKPTS(NBKPTS) – REAL (KIND=nag_wp) arrayInput

On entry: the values of the break points in the space direction. $\mathbf{XBKPTS}\left(1\right)$ must specify the left-hand boundary, $a$, and $\mathbf{XBKPTS}\left(\mathbf{NBKPTS}\right)$ must specify the right-hand boundary, $b$.

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

10:   NPOLY – INTEGERInput

On entry: the degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of break points.

Constraint: $1\le \mathbf{NPOLY}\le 49$.

11:   NPTS – INTEGERInput

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

Constraint: $\mathbf{NPTS}=\left(\mathbf{NBKPTS}-1\right)\times \mathbf{NPOLY}+1$.

12:   X(NPTS) – REAL (KIND=nag_wp) arrayOutput

On exit: the mesh points chosen by D03PDF/D03PDA in the spatial direction. The values of X will satisfy $\mathbf{X}\left(1\right)<\mathbf{X}\left(2\right)<\cdots <\mathbf{X}\left(\mathbf{NPTS}\right)$.

13:   UINIT – SUBROUTINE, supplied by the user.External Procedure

UINIT must compute the initial values of the PDE components $U_{\mathit{i}}\left(x_{\mathit{j}},t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{j}=1,2,\dots ,\mathbf{NPTS}$.

The specification of UINIT for D03PDF is:

SUBROUTINE UINIT ( NPDE, NPTS, X, U) INTEGER  NPDE, NPTS REAL (KIND=nag_wp)  X(NPTS), U(NPDE,NPTS)

The specification of UINIT for D03PDA is:

SUBROUTINE UINIT ( NPDE, NPTS, X, U, IUSER, RUSER) INTEGER  NPDE, NPTS, IUSER(*) REAL (KIND=nag_wp)  X(NPTS), U(NPDE,NPTS), RUSER(*)

1:     NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2:     NPTS – INTEGERInput

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

3:     X(NPTS) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{X}\left(\mathit{j}\right)$, contains the values of the $\mathit{j}$th mesh point, for $\mathit{j}=1,2,\dots ,\mathbf{NPTS}$.

4:     U(NPDE,NPTS) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{U}\left(\mathit{i},\mathit{j}\right)$ must be set to the initial value $U_{\mathit{i}}\left(x_{\mathit{j}},t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPDE}$ and $\mathit{j}=1,2,\dots ,\mathbf{NPTS}$.

Note: the following are additional parameters for specific use with D03PDA. Users of D03PDF therefore need not read the remainder of this description.

5:     IUSER($*$) – INTEGER arrayUser Workspace

6:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

UINIT is called with the parameters IUSER and RUSER as supplied to D03PDF/D03PDA. You are free to use the arrays IUSER and RUSER to supply information to UINIT as an alternative to using COMMON global variables.

UINIT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PDF/D03PDA is called. Parameters denoted as Input must not be changed by this procedure.

14:   ACC – REAL (KIND=nag_wp)Input

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

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

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

15:   RSAVE(LRSAVE) – REAL (KIND=nag_wp) arrayCommunication 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.

16:   LRSAVE – INTEGERInput

On entry: the dimension of the array RSAVE as declared in the (sub)program from which D03PDF/D03PDA is called.

Constraint: $\mathbf{LRSAVE}\ge 11\times \mathbf{NPDE}\times \mathbf{NPTS}+50+\mathit{nwkres}+\mathit{lenode}$.

17:   ISAVE(LISAVE) – INTEGER arrayCommunication 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.

18:   LISAVE – INTEGERInput

On entry: the dimension of the array ISAVE as declared in the (sub)program from which D03PDF/D03PDA is called.

Constraint: $\mathbf{LISAVE}\ge \mathbf{NPDE}\times \mathbf{NPTS}+24$.

19:   ITASK – INTEGERInput

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 $3$.

20:   ITRACE – INTEGERInput

On entry: the level of trace information required from D03PDF/D03PDA and the underlying ODE solver. ITRACE may take the value $-1$, $0$, $1$, $2$ or $3$.

$\mathbf{ITRACE}=-1$

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 $\mathbf{ITRACE}<-1$, then $-1$ is assumed and similarly if $\mathbf{ITRACE}>3$, then $3$ 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.

21:   IND – INTEGERInput/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 parameters TOUT and IFAIL should be reset between calls to D03PDF/D03PDA.

Constraint: $\mathbf{IND}=0$ or $1$.

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

22:   IFAIL – INTEGERInput/Output

Note: for D03PDA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.

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).

Note: the following are additional parameters for specific use with D03PDA. Users of D03PDF therefore need not read the remainder of this description.

22:   IUSER($*$) – INTEGER arrayUser Workspace

IUSER is not used by D03PDF/D03PDA, but is passed directly to PDEDEF, BNDARY and UINIT and may be used to pass information to these routines as an alternative to using COMMON global variables.

23:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

24:   CWSAV($10$) – CHARACTER(80) arrayCommunication Array

25:   LWSAV($100$) – LOGICAL arrayCommunication Array

26:   IWSAV($505$) – INTEGER arrayCommunication Array

27:   RWSAV($1100$) – REAL (KIND=nag_wp) arrayCommunication Array

28:   IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{TOUT}\le \mathbf{TS}$,
or	$\mathbf{TOUT}-\mathbf{TS}$ is too small,
or	$\mathbf{ITASK}\ne 1$, $2$ or $3$,
or	$\mathbf{M}\ne 0$, $1$ or $2$,
or	$\mathbf{M}>0$ and $\mathbf{XBKPTS}\left(1\right)<0.0$,
or	$\mathbf{NPDE}<1$,
or	$\mathbf{NBKPTS}<2$,
or	$\mathbf{NPOLY}<1$ or $\mathbf{NPOLY}>49$,
or	$\mathbf{NPTS}\ne \left(\mathbf{NBKPTS}-1\right)\times \mathbf{NPOLY}+1$,
or	$\mathbf{ACC}\le 0.0$,
or	$\mathbf{IND}\ne 0$ or $1$,
or	break points $\mathbf{XBKPTS}\left(i\right)$ are not ordered,
or	LRSAVE is too small,
or	LISAVE is too small.

$\mathbf{IFAIL}=2$

The underlying ODE solver cannot make any further progress across the integration range from the current point $t=\mathbf{TS}$ with the supplied value of ACC. The components of U contain the computed values at the current point $t=\mathbf{TS}$.

$\mathbf{IFAIL}=3$

In the underlying ODE solver, there were repeated errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or ACC is too small for the integration to continue. Integration was successful as far as $t=\mathbf{TS}$.

$\mathbf{IFAIL}=4$

In setting up the ODE system, the internal initialization routine was unable to initialize the derivative of the ODE system. This could be due to the fact that IRES was repeatedly set to $3$ in at least PDEDEF or BNDARY, when the residual in the underlying ODE solver was being evaluated.

$\mathbf{IFAIL}=5$

In solving the ODE system, a singular Jacobian has been encountered. You should check your problem formulation.

$\mathbf{IFAIL}=6$

When evaluating the residual in solving the ODE system, IRES was set to $2$ in at least PDEDEF or BNDARY. Integration was successful as far as $t=\mathbf{TS}$.

$\mathbf{IFAIL}=7$

The value of ACC is so small that the routine is unable to start the integration in time.

$\mathbf{IFAIL}=8$

In one of PDEDEF or BNDARY, IRES was set to an invalid value.

$\mathbf{IFAIL}=9$ (D02NNF)

A serious error has occurred in an internal call to the specified routine. Check the problem specification and all parameters and array dimensions. Setting $\mathbf{ITRACE}=1$ may provide more information. If the problem persists, contact NAG.

$\mathbf{IFAIL}=10$

The required task has been completed, but it is estimated that a small change in ACC is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $\mathbf{ITASK}\ne 2$.)

$\mathbf{IFAIL}=11$

An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current error message unit).

$\mathbf{IFAIL}=12$

Not applicable.

$\mathbf{IFAIL}=13$

Not applicable.

$\mathbf{IFAIL}=14$

The flux function $R_i$ was detected as depending on time derivatives, which is not permissible.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d03pdfe.f90)

Program Text (d03pdae.f90)

9.2  Program Data

Program Data (d03pdfe.d)

Program Data (d03pdae.d)

9.3  Program Results

Program Results (d03pdfe.r)

Program Results (d03pdae.r)