NAG Library Routine Document

E04XAF/E04XAA

1  Purpose

2  Specification

2.1  Specification for E04XAF

2.2  Specification for E04XAA

3  Description

4  References

5  Parameters

1:     $\mathrm{MSGLVL}$ – INTEGERInput

On entry: must indicate the amount of intermediate output desired (see Section 9.1 for a description of the printed output). All output is written on the current advisory message unit (see X04ABF).

Value	Definition
0	No printout
1	A summary is printed out for each variable plus any warning messages.
Other	Values other than $0$ and $1$ should normally be used only at the direction of NAG.

2:     $\mathrm{N}$ – INTEGERInput

On entry: the number $n$ of independent variables.

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

3:     $\mathrm{EPSRF}$ – REAL (KIND=nag_wp)Input

On entry: must define $e_R$, which is intended to be a measure of the accuracy with which the problem function $F$ can be computed. The value of $e_R$ should reflect the relative precision of $1+\left|F\left(x\right)\right|$, i.e., acts as a relative precision when $\left|F\right|$ is large, and as an absolute precision when $\left|F\right|$ is small. For example, if $F\left(x\right)$ is typically of order $1000$ and the first six significant digits are known to be correct, an appropriate value for $e_R$ would be $\text{1.0E−6}$.

A discussion of EPSRF is given in Chapter 8 of Gill et al. (1981). If EPSRF is either too small or too large on entry a warning will be printed if $\mathbf{MSGLVL}=1$, the parameter IWARN set to the appropriate value on exit and E04XAF/E04XAA will use a default value of $e_M^{0.9}$, where $e_M$ is the machine precision.

If $\mathbf{EPSRF}\le 0.0$ on entry then E04XAF/E04XAA will use the default value internally. The default value will be appropriate for most simple functions that are computed with full accuracy.

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

On entry: the point $x$ at which the derivatives are to be computed.

5:     $\mathrm{MODE}$ – INTEGERInput/Output

On entry: indicates which derivatives are required.

$\mathbf{MODE}=0$

The gradient and Hessian diagonal values having supplied the objective function via OBJFUN.

$\mathbf{MODE}=1$

The Hessian matrix having supplied both the objective function and gradients via OBJFUN.

$\mathbf{MODE}=2$

The gradient values and Hessian matrix having supplied the objective function via OBJFUN.

On exit: is changed only if you set MODE negative in OBJFUN, i.e., you have requested termination of E04XAF/E04XAA.

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

If $\mathbf{MODE}=0$ or $2$, OBJFUN must calculate the objective function; otherwise if $\mathbf{MODE}=1$, OBJFUN must calculate the objective function and the gradients.

The specification of OBJFUN is:

SUBROUTINE OBJFUN ( MODE, N, X, OBJF, OBJGRD, NSTATE, IUSER, RUSER) INTEGER  MODE, N, NSTATE, IUSER(*) REAL (KIND=nag_wp)  X(N), OBJF, OBJGRD(N), RUSER(*)

1:     $\mathrm{MODE}$ – INTEGERInput/Output

MODE indicates which parameter values within OBJFUN need to be set.

On entry: to OBJFUN, MODE is always set to the value that you set it to before the call to E04XAF/E04XAA.

On exit: its value must not be altered unless you wish to indicate a failure within OBJFUN, in which case it should be set to a negative value. If MODE is negative on exit from OBJFUN, the execution of E04XAF/E04XAA is terminated with IFAIL set to MODE.

2:     $\mathrm{N}$ – INTEGERInput

On entry: the number $n$ of variables as input to E04XAF/E04XAA.

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

On entry: the point $x$ at which the objective function (and gradients if $\mathbf{MODE}=1$) is to be evaluated.

4:     $\mathrm{OBJF}$ – REAL (KIND=nag_wp)Output

On exit: must be set to the value of the objective function.

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

On exit: if $\mathbf{MODE}=1$, $\mathbf{OBJGRD}\left(j\right)$ must contain the value of the first derivative with respect to $x$.

If $\mathbf{MODE}\ne 1$, OBJGRD need not be set.

6:     $\mathrm{NSTATE}$ – INTEGERInput

On entry: will be set to $1$ on the first call of OBJFUN by E04XAF/E04XAA, and is $0$ for all subsequent calls. Thus, if you wish, NSTATE may be tested within OBJFUN in order to perform certain calculations once only. For example you may read data.

7:     $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

8:     $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

OBJFUN is called with the parameters IUSER and RUSER as supplied to E04XAF/E04XAA. You are free to use the arrays IUSER and RUSER to supply information to OBJFUN as an alternative to using COMMON global variables.

OBJFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04XAF/E04XAA is called. Parameters denoted as Input must not be changed by this procedure.

7:     $\mathrm{LDH}$ – INTEGERInput

On entry: the first dimension of the array H as declared in the (sub)program from which E04XAF/E04XAA is called.

Constraint: $\mathbf{LDH}\ge \mathbf{N}$.

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

On entry: the initial trial interval for computing the appropriate partial derivative to the $j$th variable.

If $\mathbf{HFORW}\left(j\right)\le 0.0$, then the initial trial interval is computed by E04XAF/E04XAA (see Section 3).

On exit: $\mathbf{HFORW}\left(j\right)$ is the best interval found for computing a forward-difference approximation to the appropriate partial derivative for the $j$th variable.

9:     $\mathrm{OBJF}$ – REAL (KIND=nag_wp)Output

On exit: the value of the objective function evaluated at the input vector in X.

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

On exit: if $\mathbf{MODE}=0$ or $2$, $\mathbf{OBJGRD}\left(j\right)$ contains the best estimate of the first partial derivative for the $j$th variable.

If $\mathbf{MODE}=1$, $\mathbf{OBJGRD}\left(j\right)$ contains the first partial derivative for the $j$th variable evaluated at the input vector in X.

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

On exit: $\mathbf{HCNTRL}\left(j\right)$ is the best interval found for computing a central-difference approximation to the appropriate partial derivative for the $j$th variable.

12:   $\mathrm{H}\left(\mathbf{LDH},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array H must be at least $1$ if $\mathbf{MODE}=0$ and at least $\mathbf{N}$ if $\mathbf{MODE}=1$ or $2$.

On exit: if $\mathbf{MODE}=0$, the estimated Hessian diagonal elements are contained in the first column of this array.

If $\mathbf{MODE}=1$ or $2$, the estimated Hessian matrix is contained in the leading $n$ by $n$ part of this array.

13:   $\mathrm{IWARN}$ – INTEGEROutput

On exit: $\mathbf{IWARN}=0$ on successful exit.

If the value of EPSRF on entry is too small or too large then IWARN is set to $1$ or $2$ respectively on exit and the default value for EPSRF is used within E04XAF/E04XAA.

If $\mathbf{MSGLVL}>0$ then warnings will be printed if EPSRF is too small or too large.

14:   $\mathrm{WORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array WORK must be at least $\mathbf{N}$ if $\mathbf{MODE}=0$ and at least $\mathbf{N}\times \left(\mathbf{N}+1\right)$ if $\mathbf{MODE}=1$ or $2$.

15:   $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

16:   $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by E04XAF/E04XAA, but are passed directly to OBJFUN and may be used to pass information to this routine as an alternative to using COMMON global variables.

17:   $\mathrm{INFO}\left(\mathbf{N}\right)$ – INTEGER arrayOutput

On exit: $\mathbf{INFO}\left(j\right)$ represents diagnostic information on variable $j$. (See Section 6 for more details.)

18:   $\mathrm{IFAIL}$ – INTEGERInput/Output

Note: for E04XAA, 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 E04XAA. Users of E04XAF therefore need not read the remainder of this description.

18:   $\mathrm{LWSAV}\left(1\right)$ – LOGICAL arrayCommunication Array

19:   $\mathrm{IWSAV}\left(1\right)$ – INTEGER arrayCommunication Array

20:   $\mathrm{RWSAV}\left(1\right)$ – REAL (KIND=nag_wp) arrayCommunication Array

These parameters are no longer required by E04XAF/E04XAA.

21:   $\mathrm{IFAIL}$ – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6  Error Indicators and Warnings

$\mathbf{IFAIL}<0$

A negative value of IFAIL indicates an exit from E04XAF/E04XAA because you set MODE negative in OBJFUN. The value of IFAIL will be the same as your setting of MODE.

$\mathbf{IFAIL}=1$

On entry, one or more of the following conditions are satisfied: $\mathbf{N}<1$, $\mathbf{LDH}<\mathbf{N}\text{ or }\mathbf{MODE}$ is invalid.

$\mathbf{IFAIL}=2$

One or more variables have a nonzero INFO value. This may not necessarily represent an unsuccessful exit – see diagnostic information on INFO.

$\mathbf{IFAIL}=-99$

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.

$\mathbf{INFO}=1$

The appropriate function appears to be constant. $\mathbf{HFORW}\left(i\right)$ is set to the initial trial interval value (see Section 3) corresponding to a well-scaled problem and Error est. in the printed output is set to zero. This value occurs when the estimated relative condition error in the first derivative approximation is unacceptably large for every value of the finite difference interval. If this happens when the function is not constant the initial interval may be too small; in this case, it may be worthwhile to rerun E04XAF/E04XAA with larger initial trial interval values supplied in HFORW (see Section 3). This error may also occur if the function evaluation includes an inordinately large constant term or if EPSRF is too large.

$\mathbf{INFO}=2$

The appropriate function appears to be linear or odd. $\mathbf{HFORW}\left(i\right)$ is set to the smallest interval with acceptable bounds on the relative condition error in the forward- and backward-difference estimates. In this case, the estimated relative condition error in the second derivative approximation remained large for every trial interval, but the estimated error in the first derivative approximation was acceptable for at least one interval. If the function is not linear or odd the relative condition error in the second derivative may be decreasing very slowly, it may be worthwhile to rerun E04XAF/E04XAA with larger initial trial interval values supplied in HFORW (see Section 3).

$\mathbf{INFO}=3$

The second derivative of the appropriate function appears to be so large that it cannot be reliably estimated (i.e., near a singularity). $\mathbf{HFORW}\left(i\right)$ is set to the smallest trial interval.

This value occurs when the relative condition error estimate in the second derivative remained very small for every trial interval.

If the second derivative is not large the relative condition error in the second derivative may be increasing very slowly. It may be worthwhile to rerun E04XAF/E04XAA with smaller initial trial interval values supplied in HFORW (see Section 3). This error may also occur when the given value of EPSRF is not a good estimate of a bound on the absolute error in the appropriate function (i.e., EPSRF is too small).

$\mathbf{INFO}=4$

The algorithm terminated with an apparently acceptable estimate of the second derivative. However the forward-difference estimates of the appropriate first derivatives (computed with the final estimate of the ‘optimal’ forward-difference interval) and the central difference estimates (computed with the interval used to compute the final estimate of the second derivative) do not agree to half a decimal place. The usual reason that the forward- and central-difference estimates fail to agree is that the first derivative is small.

If the first derivative is not small, it may be helpful to execute the procedure at a different point.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

9.1  Description of the Printed Output

10  Example

10.1  Program Text

Program Text (e04xafe.f90)

Program Text (e04xaae.f90)

10.2  Program Data

10.3  Program Results

Program Results (e04xafe.r)

Program Results (e04xaae.r)