NAG Library Routine Document

D02SAF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: $\mathbf{P}\left(\mathit{i}\right)$ must be set to an estimate of the $\mathit{i}$th parameter, $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.

On exit: the corrected value for the $i$th parameter, unless an error has occurred, when it contains the last calculated value of the parameter.

2:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

Constraint: $\mathbf{M}>0$.

3:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the total number of differential equations.

Constraint: $\mathbf{N}>0$.

4:     $\mathrm{N1}$ – INTEGERInput

On entry: $n_1$, the number of differential equations active in the matching process. The active equations must be placed last in the numbering in FCN and BC. The first $\mathbf{N}-\mathbf{N1}$ equations are used as the driving equations.

Constraint: $\mathbf{N1}\le \mathbf{N}$, $\mathbf{N1}\le \mathbf{M}$ and $\mathbf{N1}>0$.

5:     $\mathrm{PE}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{PE}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,m$, must be set to a positive value for use in the convergence test in the $i$th parameter $p_i$. See the description of PF for further details.

Constraint: $\mathbf{PE}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,m$.

6:     $\mathrm{PF}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: $\mathbf{PF}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,m$, should be set to a ‘floor’ value in the convergence test on the $i$th parameter $p_i$. If $\mathbf{PF}\left(i\right)\le 0.0$ on entry then it is set to the small positive value $\sqrt{\epsilon }$ (where $\epsilon$ may in most cases be considered to be machine precision); otherwise it is used unchanged.

The Newton iteration is presumed to have converged if a full Newton step is taken ($\mathbf{ISTATE}=1$ in the specification of MONIT), the singular values of the Jacobian are not being significantly perturbed (also see MONIT) and if the Newton correction $C_i$ satisfies

$$\left|C_i\right|\le \mathbf{PE}\left(i\right)\times \max \left(\left|p_i\right|,\mathbf{PF}\left(i\right)\right)\text{, \hspace{1em}}i=1,2,\dots ,m\text{,}$$

where $p_i$ is the current value of the $i$th parameter. The values $\mathbf{PF}\left(i\right)$ are also used in determining the Newton iterates as discussed in Section 3, see equation (6).

On exit: the values actually used.

7:     $\mathrm{E}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: values for use in controlling the local error in the integration of the differential equations. If ${\mathit{err}}_{\mathit{i}}$ is an estimate of the local error in $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, then

$$\left|{\mathit{err}}_i\right|\le \mathbf{E}\left(i\right)\times \max \left\{\sqrt{\epsilon },\left|y_i\right|\right\}\text{,}$$

where $\epsilon$ may in most cases be considered to be machine precision.

Suggested value: $\mathbf{E}\left(i\right)={10}^{-5}$.

Constraint: $\mathbf{E}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,\mathbf{N}$.

8:     $\mathrm{DP}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: a value to be used in perturbing the parameter $p_i$ in the numerical differentiation to estimate the Jacobian used in Newton's method. If $\mathbf{DP}\left(i\right)=0.0$ on entry, an estimate is made internally by setting

$$\mathbf{DP}\left(i\right)=\sqrt{\epsilon }\times \max \left(\mathbf{PF}\left(i\right),\left|p_i\right|\right)\text{,}$$

(7)

where $p_i$ is the initial value of the parameter supplied by you and $\epsilon$ may in most cases be considered to be machine precision. The estimate of the Jacobian, $J$, is made using forward differences, that is for each $\mathit{i}$, for $\mathit{i}=1,2,\dots ,m$, $p_{\mathit{i}}$ is perturbed to $p_{\mathit{i}}+\mathbf{DP}\left(\mathit{i}\right)$ and the $\mathit{i}$th column of $J$ is estimated as

$$\left(r\left(p_{\mathit{i}}+\mathbf{DP}\left(\mathit{i}\right)\right)-r\left(p_{\mathit{i}}\right)\right)/\mathbf{DP}\left(\mathit{i}\right)$$

where the other components of $p$ are unchanged (see (3) for the notation used). If this fails to produce a Jacobian with significant columns, backward differences are tried by perturbing $p_{\mathit{i}}$ to $p_{\mathit{i}}-\mathbf{DP}\left(\mathit{i}\right)$ and if this also fails then central differences are used with $p_{\mathit{i}}$ perturbed to $p_{\mathit{i}}+10.0\times \mathbf{DP}\left(\mathit{i}\right)$. If this also fails then the calculation of the Jacobian is abandoned. If the Jacobian has not previously been calculated then an error exit is taken. If an earlier estimate of the Jacobian is available then the current parameter set, $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$, is abandoned in favour of the last parameter set from which useful progress was made and the singular values of the Jacobian used at the point are modified before proceeding with the Newton iteration. You are recommended to use the default value $\mathbf{DP}\left(i\right)=0.0$ unless you have prior knowledge of a better choice. If any of the perturbations described are likely to lead to an unfortunate set of parameter values then you should use the LOGICAL FUNCTION CONSTR to prevent such perturbations (all changes of parameters are checked by a call to CONSTR).

On exit: the values actually used.

9:     $\mathrm{NPOINT}$ – INTEGERInput

On entry: 2 plus the number of break-points in the range of definition of the system of differential equations (1).

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

10:   $\mathrm{SWP}\left(\mathbf{LDSWP},6\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: $\mathbf{SWP}\left(i,1\right)$ must contain an estimate for an initial step size for integration across the $i$th sub-interval $\left[\mathbf{X}\left(\mathit{i}\right),\mathbf{X}\left(\mathit{i}+1\right)\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}-1$, (see RANGE). $\mathbf{SWP}\left(i,1\right)$ should have the same sign as $\mathbf{X}\left(i+1\right)-\mathbf{X}\left(i\right)$ if it is nonzero. If $\mathbf{SWP}\left(i,1\right)=0.0$, on entry, a default value for the initial step size is calculated internally. This is the recommended mode of entry.

$\mathbf{SWP}\left(i,3\right)$ must contain a lower bound for the modulus of the step size on the $\mathit{i}$th sub-interval $\left[\mathbf{X}\left(\mathit{i}\right),\mathbf{X}\left(\mathit{i}+1\right)\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}-1$. If $\mathbf{SWP}\left(i,3\right)=0.0$ on entry, a very small default value is used. By setting $\mathbf{SWP}\left(i,3\right)>0.0$ but smaller than the expected step sizes (assuming you have some insight into the likely step sizes) expensive integrations with parameters $p$ far from the solution can be avoided.

$\mathbf{SWP}\left(\mathit{i},2\right)$ must contain an upper bound on the modulus of the step size to be used in the integration on $\left[\mathbf{X}\left(\mathit{i}\right),\mathbf{X}\left(\mathit{i}+1\right)\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}-1$. If $\mathbf{SWP}\left(i,2\right)=0.0$ on entry no bound is assumed. This is the recommended mode of entry unless the solution is expected to have important features which might be ‘missed’ in the integration if the step size were permitted to be chosen freely.

On exit: $\mathbf{SWP}\left(\mathit{i},1\right)$ contains the initial step size used on the last integration on $\left[\mathbf{X}\left(\mathit{i}\right),\mathbf{X}\left(\mathit{i}+1\right)\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}-1$, (excluding integrations during the calculation of the Jacobian).

$\mathbf{SWP}\left(\mathit{i},2\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}-1$, is usually unchanged. If the maximum step size $\mathbf{SWP}\left(i,2\right)$ is so small or the length of the range $\left[\mathbf{X}\left(i\right),\mathbf{X}\left(i+1\right)\right]$ is so short that on the last integration the step size was not controlled in the main by the size of the error tolerances $\mathbf{E}\left(i\right)$ but by these other factors, then $\mathbf{SWP}\left(\mathbf{NPOINT},2\right)$ is set to the floating-point value of $i$ if the problem last occurred in $\left[\mathbf{X}\left(i\right),\mathbf{X}\left(i+1\right)\right]$. Any results obtained when this value is returned as nonzero should be viewed with caution.

$\mathbf{SWP}\left(\mathit{i},3\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}-1$, are unchanged.

If an error exit with $\mathbf{IFAIL}=\mathbf{4}$, $\mathbf{5}$ or $\mathbf{6}$ (see Section 6) occurs on the integration made from $\mathbf{X}\left(i\right)$ to $\mathbf{X}\left(i+1\right)$ the floating-point value of $i$ is returned in $\mathbf{SWP}\left(\mathbf{NPOINT},1\right)$. The actual point $x\in \left[\mathbf{X}\left(i\right),\mathbf{X}\left(i+1\right)\right]$ where the error occurred is returned in $\mathbf{SWP}\left(1,5\right)$ (see also the specification of W). The floating-point value of NPOINT is returned in $\mathbf{SWP}\left(\mathbf{NPOINT},1\right)$ if the error exit is caused by a call to BC.

If an error exit occurs when estimating the Jacobian matrix ($\mathbf{IFAIL}=\mathbf{7}$, $\mathbf{8}$, $\mathbf{9}$, $\mathbf{10}$, $\mathbf{11}$ or $\mathbf{12}$, see Section 6) and if parameter $p_i$ was the cause of the failure then on exit $\mathbf{SWP}\left(\mathbf{NPOINT},1\right)$ contains the floating-point value of $i$.

$\mathbf{SWP}\left(\mathit{i},4\right)$ contains the point $\mathbf{X}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}$, used at the solution $p$ or at the final values of $p$ if an error occurred.

SWP is also partly used as workspace.

11:   $\mathrm{LDSWP}$ – INTEGERInput

On entry: the first dimension of the array SWP as declared in the (sub)program from which D02SAF is called.

Constraint: $\mathbf{LDSWP}\ge \mathbf{NPOINT}$.

12:   $\mathrm{ICOUNT}$ – INTEGERInput

On entry: an upper bound on the number of Newton iterations. If $\mathbf{ICOUNT}=0$ on entry, no check on the number of iterations is made (this is the recommended mode of entry).

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

13:   $\mathrm{RANGE}$ – SUBROUTINE, supplied by the user.External Procedure

RANGE must specify the break-points $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}$, which may depend on the parameters $p_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$.

The specification of RANGE is:

SUBROUTINE RANGE ( X, NPOINT, P, M) INTEGER  NPOINT, M REAL (KIND=nag_wp)  X(NPOINT), P(M)

1:     $\mathrm{X}\left(\mathbf{NPOINT}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the $\mathit{i}$th break-point, for $\mathit{i}=1,2,\dots ,\mathbf{NPOINT}$. The sequence $\left(\mathbf{X}\left(i\right)\right)$ must be strictly monotonic, that is either

$$a=\mathbf{X}\left(1\right)<\mathbf{X}\left(2\right)<\cdots <\mathbf{X}\left(\mathbf{NPOINT}\right)=b$$

or

$$a=\mathbf{X}\left(1\right)>\mathbf{X}\left(2\right)>\cdots >\mathbf{X}\left(\mathbf{NPOINT}\right)=b\text{.}$$

2:     $\mathrm{NPOINT}$ – INTEGERInput

On entry: $2$ plus the number of break-points in $\left(a,b\right)$.

3:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the current estimate of the $\mathit{i}$th parameter, for $\mathit{i}=1,2,\dots ,m$.

4:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

RANGE must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

14:   $\mathrm{BC}$ – SUBROUTINE, supplied by the user.External Procedure

BC must place in G1 and G2 the boundary conditions at $a$ and $b$ respectively.

The specification of BC is:

SUBROUTINE BC ( G1, G2, P, M, N) INTEGER  M, N REAL (KIND=nag_wp)  G1(N), G2(N), P(M)

1:     $\mathrm{G1}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the value of $y_{\mathit{i}}\left(a\right)$, for $\mathit{i}=1,2,\dots ,n$, (where this may be a known value or a function of the parameters $p_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$).

2:     $\mathrm{G2}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the value of $y_{\mathit{i}}\left(b\right)$, for $\mathit{i}=1,2,\dots ,n$, (where these may be known values or functions of the parameters $p_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$). If $n>n_1$, so that there are some driving equations, then the first $n-n_1$ values of G2 need not be set since they are never used.

3:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: an estimate of the $\mathit{i}$th parameter, $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.

4:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

5:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of differential equations.

BC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

15:   $\mathrm{FCN}$ – SUBROUTINE, supplied by the user.External Procedure

FCN must evaluate the functions $f_{\mathit{i}}$ (i.e., the derivatives $y_{\mathit{i}}^{\prime }$), for $\mathit{i}=1,2,\dots ,n$.

The specification of FCN is:

SUBROUTINE FCN ( X, Y, F, N, P, M, I) INTEGER  N, M, I REAL (KIND=nag_wp)  X, Y(N), F(N), P(M)

1:     $\mathrm{X}$ – REAL (KIND=nag_wp)Input

On entry: $x$, the value of the argument.

2:     $\mathrm{Y}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.

3:     $\mathrm{F}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the derivative of $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, evaluated at $x$. $\mathbf{F}\left(i\right)$ may depend upon the parameters $p_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$. If there are any driving equations (see Section 3) then these must be numbered first in the ordering of the components of F.

4:     $\mathrm{N}$ – INTEGERInput

On entry: $\mathit{n}$, the number of equations.

5:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the current estimate of the $\mathit{i}$th parameter $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.

6:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

7:     $\mathrm{I}$ – INTEGERInput

On entry: specifies the sub-interval $\left[x_i,x_{i+1}\right]$ on which the derivatives are to be evaluated.

FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

16:   $\mathrm{EQN}$ – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

EQN is used to describe the additional algebraic equations to be solved in the determination of the parameters, $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$. If there are no additional algebraic equations (i.e., $m=n_1$) then EQN is never called and the dummy routine D02HBZ should be used as the actual argument.

The specification of EQN is:

SUBROUTINE EQN ( E, Q, P, M) INTEGER  Q, M REAL (KIND=nag_wp)  E(Q), P(M)

1:     $\mathrm{E}\left(\mathbf{Q}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the vector of residuals, $r_2\left(p\right)$, that is the amount by which the current estimates of the parameters fail to satisfy the algebraic equations.

2:     $\mathrm{Q}$ – INTEGERInput

On entry: the number of algebraic equations, $m-n_1$.

3:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the current estimate of the $\mathit{i}$th parameter $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.

4:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

EQN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

17:   $\mathrm{CONSTR}$ – LOGICAL FUNCTION, supplied by the user.External Procedure

CONSTR is used to prevent the pseudo-Newton iteration running into difficulty. CONSTR should return the value .TRUE. if the constraints are satisfied by the parameters $p_1,p_2,\dots ,p_m$. Otherwise CONSTR should return the value .FALSE.. Usually the dummy function D02HBY, which returns the value .TRUE. at all times, will suffice and in the first instance this is recommended as the actual parameter.

The specification of CONSTR is:

FUNCTION CONSTR ( P, M) LOGICAL CONSTR INTEGER  M REAL (KIND=nag_wp)  P(M)

1:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: an estimate of the $\mathit{i}$th parameter, $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.

2:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

CONSTR must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

18:   $\mathrm{YMAX}$ – REAL (KIND=nag_wp)Input

On entry: a non-negative value which is used as a bound on all values ${‖y\left(x\right)‖}_{\infty }$ where $y\left(x\right)$ is the solution at any point $x$ between $\mathbf{X}\left(1\right)$ and $\mathbf{X}\left(\mathbf{NPOINT}\right)$ for the current parameters $p_1,p_2,\dots ,p_m$. If this bound is exceeded the integration is terminated and the current parameters are rejected. Such a rejection will result in an error exit if it prevents the initial residual or Jacobian, or the final solution, being calculated. If $\mathbf{YMAX}=0$ on entry, no bound on the solution $y$ is used; that is the integrations proceed without any checking on the size of ${‖y‖}_{\infty }$.

19:   $\mathrm{MONIT}$ – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

MONIT enables you to monitor the values of various quantities during the calculation. It is called by D02SAF after every calculation of the norm ${‖{\mathbf{D}}^{-1}{\tilde{J}}^{+}r‖}_2$ which determines the strategy of the Newton method, every time there is an internal error exit leading to a change of strategy, and before an error exit when calculating the initial Jacobian. Usually the routine D02HBX will be adequate and you are advised to use this as the actual parameter for MONIT in the first instance. (In this case a call to X04ABF must be made before the call of D02SAF.) If no monitoring is required, the dummy routine D02SAS may be used.

The specification of MONIT is:

SUBROUTINE MONIT ( ISTATE, IFLAG, IFAIL1, P, M, F, PNORM, PNORM1, EPS, D) INTEGER  ISTATE, IFLAG, IFAIL1, M REAL (KIND=nag_wp)  P(M), F(M), PNORM, PNORM1, EPS, D(M)

1:     $\mathrm{ISTATE}$ – INTEGERInput

On entry: the state of the Newton iteration.

$\mathbf{ISTATE}=0$

The calculation of the residual, Jacobian and ${‖{\mathbf{D}}^{-1}{\tilde{J}}^{+}r‖}_2$ are taking place.

$\mathbf{ISTATE}=1$ to $5$

During the Newton iteration a factor of $2^{\left(-\mathbf{ISTATE}+1\right)}$ of the Newton step is being used to try to reduce the norm.

$\mathbf{ISTATE}=6$

The current Newton step has been rejected and the Jacobian is being re-calculated.

$\mathbf{ISTATE}=-6$ to $-1$

An internal error exit has caused the rejection of the current set of parameter values, $p$. $-\mathbf{ISTATE}$ is the value which ISTATE would have taken if the error had not occurred.

$\mathbf{ISTATE}=-7$

An internal error exit has occurred when calculating the initial Jacobian.

2:     $\mathrm{IFLAG}$ – INTEGERInput

On entry: whether or not the Jacobian being used has been calculated at the beginning of the current iteration. If the Jacobian has been updated then $\mathbf{IFLAG}=1$; otherwise $\mathbf{IFLAG}=2$. The Jacobian is only calculated when convergence to the current parameter values has been slow.

3:     $\mathrm{IFAIL1}$ – INTEGERInput

On entry: if $-6\le \mathbf{ISTATE}\le -1$, IFAIL1 specifies the IFAIL error number that would be produced were control returned to you. IFAIL1 is unspecified for values of ISTATE outside this range.

4:     $\mathrm{P}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the current estimate of the $\mathit{i}$th parameter $p_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.

5:     $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of parameters.

6:     $\mathrm{F}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $r$, the residual corresponding to the current parameter values, provided $1\le \mathbf{ISTATE}\le 5$ or $\mathbf{ISTATE}=-7$. F is unspecified for other values of ISTATE.

7:     $\mathrm{PNORM}$ – REAL (KIND=nag_wp)Input

On entry: a quantity against which all reductions in norm are currently measured.

8:     $\mathrm{PNORM1}$ – REAL (KIND=nag_wp)Input

On entry: $p$, the norm of the current parameters. It is set for $1\le \mathbf{ISTATE}\le 5$ and is undefined for other values of ISTATE.

9:     $\mathrm{EPS}$ – REAL (KIND=nag_wp)Input

On entry: gives some indication of the convergence rate. It is the current singular value modification factor (see Gay (1976)). It is zero initially and whenever convergence is proceeding steadily. EPS is ${\epsilon }^{3/8}$ or greater (where $\epsilon$ may in most cases be considered machine precision) when the singular values of $J$ are approximately zero or when convergence is not being achieved. The larger the value of EPS the worse the convergence rate. When EPS becomes too large the Newton iteration is terminated.

10:   $\mathrm{D}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $J$, the singular values of the current modified Jacobian matrix. If $\mathbf{D}\left(m\right)$ is small relative to $\mathbf{D}\left(1\right)$ for a number of Jacobians corresponding to different parameter values then the computed results should be viewed with suspicion. It could be that the matching equations do not depend significantly on some parameter (which could be due to a programming error in FCN, BC, RANGE or EQN). Alternatively, the system of differential equations may be very ill-conditioned when viewed as an initial value problem, in which case D02SAF is unsuitable. This may also be indicated by some singular values being very large. These values of $\mathbf{D}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,m$, should not be changed.

MONIT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

20:   $\mathrm{PRSOL}$ – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

PRSOL can be used to obtain values of the solution $y$ at a selected point $z$ by integration across the final range $\left[\mathbf{X}\left(1\right),\mathbf{X}\left(\mathbf{NPOINT}\right)\right]$. If no output is required D02HBW can be used as the actual parameter.

The specification of PRSOL is:

SUBROUTINE PRSOL ( Z, Y, N) INTEGER  N REAL (KIND=nag_wp)  Z, Y(N)

1:     $\mathrm{Z}$ – REAL (KIND=nag_wp)Input/Output

On entry: contains $x_1$ on the first call. On subsequent calls Z contains its previous output value.

On exit: the next point at which output is required. The new point must be nearer $\mathbf{X}\left(\mathbf{NPOINT}\right)$ than the old.

If Z is set to a point outside $\left[\mathbf{X}\left(1\right),\mathbf{X}\left(\mathbf{NPOINT}\right)\right]$ the process stops and control returns from D02SAF to the (sub)program from which D02SAF is called. Otherwise the next call to PRSOL is made by D02SAF at the point Z, with solution values $y_1,y_2,\dots ,y_n$ at Z contained in Y. If Z is set to $\mathbf{X}\left(\mathbf{NPOINT}\right)$ exactly, the final call to PRSOL is made with $y_1,y_2,\dots ,y_n$ as values of the solution at $\mathbf{X}\left(\mathbf{NPOINT}\right)$ produced by the integration. In general the solution values obtained at $\mathbf{X}\left(\mathbf{NPOINT}\right)$ from PRSOL will differ from the values obtained at this point by a call to BC. The difference between the two solutions is the residual $r$. You are reminded that the points $\mathbf{X}\left(1\right),\mathbf{X}\left(2\right),\dots ,\mathbf{X}\left(\mathbf{NPOINT}\right)$ are available in the locations $\mathbf{SWP}\left(1,4\right),\mathbf{SWP}\left(2,4\right),\dots ,\mathbf{SWP}\left(\mathbf{NPOINT},4\right)$ at all times.

2:     $\mathrm{Y}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the solution value $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, at $z$.

3:     $\mathrm{N}$ – INTEGERInput

On entry: $\mathit{n}$, the total number of differential equations.

PRSOL must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02SAF is called. Parameters denoted as Input must not be changed by this procedure.

21:   $\mathrm{W}\left(\mathbf{LDW},\mathbf{SDW}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: in the case of an error exit of the type where the point of failure is returned in $\mathbf{SWP}\left(1,5\right)$, the solution at this point of failure is returned in $\mathbf{W}\left(\mathit{i},1\right)$, for $\mathit{i}=1,2,\dots ,n$.

Otherwise W is used for workspace.

22:   $\mathrm{LDW}$ – INTEGERInput

On entry: the first dimension of the array W as declared in the (sub)program from which D02SAF is called.

Constraint: $\mathbf{LDW}\ge \max \left(\mathbf{N},\mathbf{M}\right)$.

23:   $\mathrm{SDW}$ – INTEGERInput

On entry: the second dimension of the array W as declared in the (sub)program from which D02SAF is called.

Constraint: $\mathbf{SDW}\ge 3\times \mathbf{M}+12+\max \left(11,\mathbf{M}\right)$.

24:   $\mathrm{IFAIL}$ – 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

$\mathbf{IFAIL}=1$

One or more of the parameters N, N1, M, LDSWP, NPOINT, ICOUNT, LDW, SDW, E, PE or YMAX is incorrectly set.

$\mathbf{IFAIL}=2$

The constraints have been violated by the initial parameters.

$\mathbf{IFAIL}=3$

The condition $\mathbf{X}\left(1\right)<\mathbf{X}\left(2\right)<\cdots <\mathbf{X}\left(\mathbf{NPOINT}\right)$ (or $\mathbf{X}\left(1\right)>\mathbf{X}\left(2\right)>\cdots >\mathbf{X}\left(\mathbf{NPOINT}\right)$) has been violated on a call to RANGE with the initial parameters.

$\mathbf{IFAIL}=4$

In the integration from $\mathbf{X}\left(1\right)$ to $\mathbf{X}\left(\mathbf{NPOINT}\right)$ with the initial or the final parameters, the step size was reduced too far for the integration to proceed. Consider reversing the order of the points $\mathbf{X}\left(1\right),\mathbf{X}\left(2\right),\dots ,\mathbf{X}\left(\mathbf{NPOINT}\right)$. If this error exit still results, it is likely that D02SAF is not a suitable method for solving the problem, or the initial choice of parameters is very poor, or the accuracy requirement specified by $\mathbf{E}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n$, is too stringent.

$\mathbf{IFAIL}=5$

In the integration from $\mathbf{X}\left(1\right)$ to $\mathbf{X}\left(\mathbf{NPOINT}\right)$ with the initial or final parameters, an initial step could not be found to start the integration on one of the intervals $\mathbf{X}\left(i\right)$ to $\mathbf{X}\left(i+1\right)$. Consider reversing the order of the points. If this error exit still results it is likely that D02SAF is not a suitable routine for solving the problem, or the initial choice of parameters is very poor, or the accuracy requirement specified by $\mathbf{E}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n$, is much too stringent.

$\mathbf{IFAIL}=6$

In the integration from $\mathbf{X}\left(1\right)$ to $\mathbf{X}\left(\mathbf{NPOINT}\right)$ with the initial or final parameters, the solution exceeded YMAX in magnitude (when $\mathbf{YMAX}>0.0$). It is likely that the initial choice of parameters was very poor or YMAX was incorrectly set.

$\mathbf{IFAIL}=7$

On calculating the initial approximation to the Jacobian, the constraints were violated.

$\mathbf{IFAIL}=8$

On perturbing the parameters when calculating the initial approximation to the Jacobian, the condition $\mathbf{X}\left(1\right)<\mathbf{X}\left(2\right)<\cdots <\mathbf{X}\left(\mathbf{NPOINT}\right)$ (or $\mathbf{X}\left(1\right)>\mathbf{X}\left(2\right)>\cdots >\mathbf{X}\left(\mathbf{NPOINT}\right)$) is violated.

$\mathbf{IFAIL}=9$

On calculating the initial approximation to the Jacobian, the integration step size was reduced too far to make further progress (see $\mathbf{IFAIL}=\mathbf{4}$).

$\mathbf{IFAIL}=10$

On calculating the initial approximation to the Jacobian, the initial integration step size on some interval was too small (see $\mathbf{IFAIL}=\mathbf{5}$).

$\mathbf{IFAIL}=11$

On calculating the initial approximation to the Jacobian, the solution of the system of differential equations exceeded YMAX in magnitude (when $\mathbf{YMAX}>0.0$).

$\mathbf{IFAIL}=12$

On calculating the initial approximation to the Jacobian, a column of the Jacobian is found to be insignificant. This could be due to an element $\mathbf{DP}\left(i\right)$ being too small (but nonzero) or the solution having no dependence on one of the parameters (a programming error).

$\mathbf{IFAIL}=13$

After calculating the initial approximation to the Jacobian, the calculation of its singular value decomposition failed. It is likely that the error will never occur as it is usually associated with the Jacobian having multiple singular values. To remedy the error it should only be necessary to change the initial parameters. If the error persists it is likely that the problem has not been correctly formulated.

$\mathbf{IFAIL}=14$

The Newton iteration has failed to converge after exercising all its options. You are strongly recommended to monitor the progress of the iteration via MONIT. There are many possible reasons for the iteration not converging. Amongst the most likely are:

(a)	there is no solution;
(b)	the initial parameters are too far away from the correct parameters;
(c)	the problem is too ill-conditioned as an initial value problem for Newton's method to choose suitable corrections;
(d)	the accuracy requirements for convergence are too restrictive, that is some of the components of PE (and maybe PF) are too small – in this case the final value of this norm output via MONIT will usually be very small; or
(e)	the initial parameters are so close to the solution parameters $p$ that the Newton iteration cannot find improved parameters. The norm output by MONIT should be very small.

$\mathbf{IFAIL}=15$

The number of iterations permitted by ICOUNT has been exceeded (in the case when $\mathbf{ICOUNT}>0$ on entry).

$\mathbf{IFAIL}=16$

$\mathbf{IFAIL}=17$

$\mathbf{IFAIL}=18$

$\mathbf{IFAIL}=19$

These indicate that there has been a serious error in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

$\mathbf{IFAIL}=-99$

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

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

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

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d02safe.f90)

10.2  Program Data

Program Data (d02safe.d)

10.3  Program Results

Program Results (d02safe.r)