e04dgc minimizes an unconstrained nonlinear function of several variables using a pre-conditioned, limited memory quasi-Newton conjugate gradient method. The function is intended for use on large scale problems.

2 Specification

#include <nag.h>

void

e04dgc (Integer n,

void	(objfun)(Integer n, const double x[], double objf, double g[], Nag_Comm *comm),

double x[], double *objf, double g[], Nag_E04_Opt *options, Nag_Comm *comm, NagError *fail)

The function may be called by the names: e04dgc, nag_opt_uncon_conjgrd_comp or nag_opt_conj_grad.

3 Description

e04dgc uses a pre-conditioned conjugate gradient method and is based upon algorithm PLMA as described in Gill and Murray (1979) and Section 4.8.3 of Gill et al. (1981).

The algorithm proceeds as follows:

Let

x_{0}

be a given starting point and let

k

denote the current iteration, starting with

k = 0

. The iteration requires

g_{k}

, the gradient vector evaluated at

x_{k}

, the

k

th estimate of the minimum. At each iteration a vector

p_{k}

(known as the direction of search) is computed and the new estimate

x_{k + 1}

is given by

x_{k} + α_{k} p_{k}

where

α_{k}

(the step length) minimizes the function

F (x_{k} + α_{k} p_{k})

with respect to the scalar

α_{k}

. At the start of each line search an initial approximation

α_{0}

to the step

α_{k}

is taken as:

α_{0} = \min {1, 2 | F_{k} - F_{est} | / g_{k}^{T} g_{k}}

where

F_{est}

is a user-supplied estimate of the function value at the solution. If

F_{est}

is not specified, the software always chooses the unit step length for

α_{0}

. Subsequent step length estimates are computed using cubic interpolation with safeguards.

A quasi-Newton method computes the search direction,

p_{k}

, by updating the inverse of the approximate Hessian

(H_{k})

and computing

p_{k + 1} = - H_{k + 1} g_{k + 1} .

(1)

The updating formula for the approximate inverse is given by

H_{k + 1} = H_{k} - \frac{1}{y_{k}^{T} s_{k}} (H_{k} y_{k} s_{k}^{T} + s_{k} y_{k}^{T} H_{k}) + \frac{1}{y_{k}^{T} s_{k}} (1 + \frac{y_{k}^{T} H_{k} y_{k}}{y_{k}^{T} s_{k}}) s_{k} s_{k}^{T}

(2)

where

y_{k} = g_{k - 1} - g_{k}

and

s_{k} = x_{k + 1} - x_{k} = α_{k} p_{k}

The method used by e04dgc to obtain the search direction is based upon computing

p_{k + 1}

- H_{k + 1} g_{k + 1}

where

H_{k + 1}

is a matrix obtained by updating the identity matrix with a limited number of quasi-Newton corrections. The storage of an

n \times n

matrix is avoided by storing only the vectors that define the rank two corrections – hence the term limited-memory quasi-Newton method. The precise method depends upon the number of updating vectors stored. For example, the direction obtained with the ‘one-step’ limited memory update is given by (1) using (2) with

H_{k}

equal to the identity matrix, viz.

p_{k + 1} = - g_{k + 1} + \frac{1}{y_{k}^{T} s_{k}} (s_{k}^{T} g_{k + 1} y_{k} + y_{k}^{T} g_{k + 1} s_{k}) - \frac{s_{k}^{T} g_{k + 1}}{y_{k}^{T} s_{k}} (1 + \frac{y_{k}^{T} y_{k}}{y_{k}^{T} s_{k}}) s_{k}

e04dgc uses a two-step method described in detail in Gill and Murray (1979) in which restarts and pre-conditioning are incorporated. Using a limited-memory quasi-Newton formula, such as the one above, guarantees

p_{k + 1}

to be a descent direction if all the inner products

y_{k}^{T} s_{k}

are positive for all vectors

y_{k}

and

s_{k}

used in the updating formula.

The termination criteria of e04dgc are as follows:

Let

τ_{F}

specify an argument that indicates the number of correct figures desired in

F_{k}

(

τ_{F}

is equivalent to

options . optim_tol

in the optional parameter list, see Section 11). If the following three conditions are satisfied:

(i) $F_{k - 1} - F_{k} < τ_{F} (1 + | F_{k} |)$
(ii) $‖ x_{k - 1} - x_{k} ‖ < \sqrt{τ_{F}} (1 + ‖ x_{k} ‖)$
(iii) $‖ g_{k} ‖ \leq τ_{F}^{1 / 3} (1 + | F_{k} |)$ or $‖ g_{k} ‖ < ε_{A}$ , where $ε_{A}$ is the absolute error associated with computing the objective function

then the algorithm is considered to have converged. For a full discussion on termination criteria see Chapter 8 of Gill et al. (1981).

4 References

Gill P E and Murray W (1979) Conjugate-gradient methods for large-scale nonlinear optimization Technical Report SOL 79-15 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1983) Computing forward-difference intervals for numerical optimization SIAM J. Sci. Statist. Comput. 4 310–321

Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press

5 Arguments

1: $n$ – Integer Input

On entry: the number

n

of variables.

Constraint:

n \geq 1

2: $objfun$ – function, supplied by the user External Function

objfun must calculate the objective function

F (x)

and its gradient at a specified point

x

The specification of objfun is:

void	objfun (Integer n, const double x[], double objf, double g[], Nag_Comm comm)

1: $n$ – Integer Input

On entry: the number

n

of variables.

2: $x [n]$ – const double Input

On entry: the point

x

at which the objective function is required.

3: $objf$ – double * Output

On exit: the value of the objective function

F

at the current point

x

4: $g [n]$ – double Output

On exit:

g [i - 1]

must contain the value of

\frac{\partial F}{\partial x_{i}}

at the point

x

, for

i = 1, 2, \dots, n

5: $comm$ – Nag_Comm *

Pointer to structure of type

Nag_Comm

; the following members are relevant to objfun.

flag – IntegerInput/Output: On entry: $comm \to flag$ is always non-negative.

On exit: if objfun resets $comm \to flag$ to some negative number then e04dgc will terminate immediately with the error indicator NE_USER_STOP. If fail is supplied to e04dgc $fail . errnum$ will be set to your setting of $comm \to flag$ .

first – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the first call to objfun and Nag_FALSE for all subsequent calls.

nf – IntegerInput: On entry: the number of calculations of the objective function; this value will be equal to the number of calls made to objfun including the current one.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void * with a C compiler that defines void * and char * otherwise. Before calling e04dgc these pointers may be allocated memory and initialized with various quantities for use by objfun when called from e04dgc.

Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04dgc. If your code inadvertently does return any NaNs or infinities, e04dgc is likely to produce unexpected results.

Note: objfun should be tested separately before being used in conjunction with e04dgc. The array x must not be changed by objfun.

3: $x [n]$ – double Input/Output

On entry:

x_{0}

, an estimate of the solution point

x^{*}

On exit: the final estimate of the solution.

4: $objf$ – double * Output

On exit: the value of the objective function

F (x)

at the final iterate.

5: $g [n]$ – double Output

On exit: the objective gradient at the final iterate.

6: $options$ – Nag_E04_Opt * Input/Output

On entry/exit: a pointer to a structure of type

Nag_E04_Opt

whose members are optional parameters for e04dgc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of options is given below in Section 11.

If any of these optional parameters are required then the structure options should be declared and initialized by a call to e04xxc and supplied as an argument to e04dgc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.

7: $comm$ – Nag_Comm * Input/Output

Note: comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

On entry/exit: structure containing pointers for communication with user-supplied functions; see the above description of objfun for details. If you do not need to make use of this communication feature the null pointer NAGCOMM_NULL may be used in the call to e04dgc; comm will then be declared internally for use in calls to user-supplied functions.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $options . print_level$ had an illegal value.

On entry, argument $options . verify_grad$ had an illegal value.
NE_DERIV_ERRORS: Large errors were found in the derivatives of the objective function.
This value of fail will occur if the verification process indicated that at least one gradient component had no correct figures. You should refer to the printed output to determine which elements are suspected to be in error.
As a first step, you should check that the code for the objective values is correct – for example, by computing the function at a point where the correct value is known. However, care should be taken that the chosen point fully tests the evaluation of the function. It is remarkable how often the values $x = 0$ or $x = 1$ are used to test function evaluation procedures, and how often the special properties of these numbers make the test meaningless.
Errors in programming the function may be quite subtle in that the function value is ‘almost’ correct. For example, the function may not be accurate to full precision because of the inaccurate calculation of a subsidiary quantity, or the limited accuracy of data upon which the function depends.
NE_GRAD_TOO_SMALL: The gradient at the starting point is too small, rerun the problem at a different starting point.
The value of $g {(x_{0})}^{T} g (x_{0})$ is less than $ε | F (x_{o}) |$ , where $ε$ is the machine precision.
NE_INT_ARG_LT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_INVALID_INT_RANGE_1: Value $⟨ value ⟩$ given to $options . max_iter$ not valid. Correct range is $options . max_iter \geq 0$ .
NE_INVALID_REAL_RANGE_EF: Value $⟨ value ⟩$ given to $options . f_prec$ not valid. Correct range is $ε \leq options . f_prec < 1.0$ .

Value $⟨ value ⟩$ given to $options . optim_tol$ not valid. Correct range is $⟨ value ⟩$ $\leq options . optim_tol < 1.0$ .
NE_INVALID_REAL_RANGE_F: Value $⟨ value ⟩$ given to $options . max_line_step$ not valid. Correct range is $options . max_line_step > 0.0$ .
NE_INVALID_REAL_RANGE_FF: Value $⟨ value ⟩$ given to $options . linesearch_tol$ not valid. Correct range is $0.0 \leq options . linesearch_tol < 1.0$ .
NE_NOT_APPEND_FILE: Cannot open file $⟨ string ⟩$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $⟨ string ⟩$ .
NE_OPT_NOT_INIT: Options structure not initialized.
NE_USER_STOP: User requested termination, user flag value $= ⟨ value ⟩$ .
This exit occurs if you set $comm \to flag$ to a negative value in objfun. If fail is supplied the value of $fail . errnum$ will be the same as your setting of $comm \to flag$ .
NE_WRITE_ERROR: Error occurred when writing to file $⟨ string ⟩$ .
NW_NO_IMPROVEMENT: A sufficient decrease in the function value could not be attained during the final linesearch. Current point cannot be improved upon.
If objfun computes the function and gradients correctly, then this warning may occur because an overly stringent accuracy has been requested, i.e., $options . optim_tol$ is too small or if the minimum lies close to a step length of zero. In this case you should apply the tests described in Section 3 to determine whether or not the final solution is acceptable. For a discussion of attainable accuracy see Gill et al. (1981).
If many iterations have occurred in which essentially no progress has been made or e04dgc has failed to move from the initial point, then the function objfun may be incorrect. You should refer to the comments below under NE_DERIV_ERRORS and check the gradients using the $options . verify_grad$ argument. Unfortunately, there may be small errors in the objective gradients that cannot be detected by the verification process. Finite difference approximations to first derivatives are catastrophically affected by even small inaccuracies.
NW_STEP_BOUND_TOO_SMALL: Computed upper-bound on step length was too small
The computed upper bound on the step length taken during the linesearch was too small. A rerun with an increased value of $options . max_line_step$ ( $ρ$ say) may be successful unless $ρ \geq 10^{10}$ (the default value), in which case the current point cannot be improved upon.
NW_TOO_MANY_ITER: The maximum number of iterations, $⟨ value ⟩$ , have been performed.
If the algorithm appears to be making progress the value of $options . max_iter$ value may be too small (see Section 11), you should increase its value and rerun e04dgc. If the algorithm seems to be ‘bogged down’, you should check for incorrect gradients or ill-conditioning as described below under NW_NO_IMPROVEMENT.

7 Accuracy

On successful exit the accuracy of the solution will be as defined by the optional parameter

options . optim_tol

8 Parallelism and Performance

e04dgc is not threaded in any implementation.

9 Further Comments

9.1 Timing

Problems whose Hessian matrices at the solution contain sets of clustered eigenvalues are likely to be minimized in significantly fewer than

n

iterations. Problems without this property may require anything between

n

and

5 n

iterations, with approximately

2 n

iterations being a common figure for moderately difficult problems.

10 Example

This example minimizes the function

F = e^{x_{1}} ({4 x_{1}}^{2} + {2 x_{2}}^{2} + 4 x_{1} x_{2} + 2 x_{2} + 1) .

The data includes a set of user-defined column and row names, and data for the Hessian in a sparse storage format (see Section 10.2 for further details). The options structure is declared and initialized by e04xxc. Five option values are read from a data file by use of e04xyc.

11 Optional Parameters

A number of optional input and output arguments to e04dgc are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling e04dgc; the default settings will then be used for all arguments.

Before assigning values to options directly the structure must be initialized by a call to the function e04xxc. Values may then be assigned to the structure members in the normal C manner.

After return from e04dgc, the options structure may only be re-used for future calls of e04dgc if the dimensions of the new problem are the same. Otherwise, the structure must be cleared by a call of e04xzc) and re-initialized by a call of e04xxc before future calls. Failure to do this will result in unpredictable behaviour.

Option settings may also be read from a text file using the function e04xyc in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.

If assignment of functions and memory to pointers in the options structure is required, then this must be done directly in the calling program, they cannot be assigned using e04xyc.

11.1 Optional Parameter Checklist and Default Values

For easy reference, the following list shows the members of options which are valid for e04dgc together with their default values where relevant. The number

ε

is a generic notation for machine precision (see X02AJC).

Boolean list	Nag_TRUE
Nag_PrintType print_level	$Nag_Soln_Iter$
char outfile[512]	stdout
void (*print_fun)()	NULL
Nag_GradChk verify_grad	$Nag_SimpleCheck$
Boolean print_gcheck	Nag_TRUE
Integer obj_check_start	1
Integer obj_check_stop	n
Integer max_iter	$\max (50, 5 n)$
double f_prec	$ε^{0.9}$
double optim_tol	${options . f_prec}^{0.8}$
double linesearch_tol	0.9
double max_line_step	$10^{10}$
double f_est
Integer iter
Integer nf

11.2 Description of the Optional Parameters

list – Nag_Boolean

Default

= Nag_TRUE

On entry: if

options . list = Nag_TRUE

the argument settings in the call to e04dgc will be printed.

print_level – Nag_PrintType

Default

= Nag_Soln_Iter

On entry: the level of results printout produced by e04dgc. The following values are available:

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution.
$Nag_Iter$	One line of output for each iteration.
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.

Constraint:

options . print_level = Nag_NoPrint

Nag_Soln

Nag_Iter

Nag_Soln_Iter

outfile – const char[512]

Default

= stdout

On entry: the name of the file to which results should be printed. If

options . outfile [0] =' \0'

then the stdout stream is used.

print_fun – pointer to function

Default

=

NULL

On entry: printing function defined by you; the prototype of

options . print_fun

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

See Section 11.3.1 below for further details.

verify_grad – Nag_GradChk

Default

= Nag_SimpleCheck

On entry: specifies the level of derivative checking to be performed by e04dgc on the gradient elements defined in objfun.

options . verify_grad

may have the following values:

$Nag_NoCheck$	No derivative check is performed.
$Nag_SimpleCheck$	Perform a simple check of the gradient.
$Nag_CheckObj$	Perform a component check of the gradient elements.

options . verify_grad = Nag_SimpleCheck

then a simple ‘cheap’ test is performed, which requires only one call to objfun. If

options . verify_grad = Nag_CheckObj

then a more reliable (but more expensive) test will be made on individual gradient components. This component check will be made in the range specified by

options . obj_check_start

and

options . obj_check_stop

, default values being

1

and n respectively. The procedure for the derivative check is based on finding an interval that produces an acceptable estimate of the second derivative, and then using that estimate to compute an interval that should produce a reasonable forward-difference approximation. The gradient element is then compared with the difference approximation. (The method of finite difference interval estimation is based on Gill et al. (1983)). The result of the test is printed out by e04dgc if

options . print_gcheck = Nag_TRUE

Constraint:

options . verify_grad = Nag_NoCheck

Nag_SimpleCheck

Nag_CheckObj

print_gcheck – Nag_Boolean

Default

= Nag_TRUE

On entry: if Nag_TRUE the result of any derivative check (see

options . verify_grad

) will be printed.

obj_check_start – Integer

Default

= 1

obj_check_stop – Integer

Default

= n

On entry: these options take effect only when

options . verify_grad = Nag_CheckObj

. They may be used to control the verification of gradient elements computed by the function objfun. For example, if the first 30 variables appear linearly in the objective, so that the corresponding gradient elements are constant, then it is reasonable for

options . obj_check_start

to be set to 31.

Constraint:

1 \leq options . obj_check_start \leq options . obj_check_stop \leq n

max_iter – Integer

Default

= \max (50, 5 n)

On entry: the limit on the number of iterations allowed before termination.

Constraint:

options . max_iter \geq 0

f_prec – double

Default

= ε^{0.9}

On entry: this argument defines

ε_{r}

, which is intended to be a measure of the accuracy with which the problem function

F

can be computed. The value of

ε_{r}

should reflect the relative precision of

1 + | F (x) |

; i.e.,

ε_{r}

acts as a relative precision when

| F |

is large, and as an absolute precision when

| F |

is small. For example, if

F (x)

is typically of order 1000 and the first six significant digits are known to be correct, an appropriate value for

ε_{r}

would be

1.0e−6

. In contrast, if

F (x)

is typically of order

10^{−4}

and the first six significant digits are known to be correct, an appropriate value for

ε_{r}

would be

1.0e−10

. The choice of

ε_{r}

can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981), for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy. However when the accuracy of the computed function values is known to be significantly worse than full precision, the value of

ε_{r}

should be large enough so that e04dgc will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.

Constraint:

ε \leq options . f_prec < 1.0

optim_tol – double

Default

= {options . f_prec}^{0.8}

On entry: specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking,

options . optim_tol

indicates the number of correct figures desired in the objective function at the solution. For example, if

options . optim_tol

10^{−6}

and e04dgc terminates successfully, the final value of

F

should have approximately six correct figures. e04dgc will terminate successfully if the iterative sequence of

x

-values is judged to have converged and the final point satisfies the termination criteria (see Section 3, where

τ_{F}

represents

options . optim_tol

Constraint:

options . f_prec \leq options . optim_tol < 1.0

linesearch_tol – double

Default

= 0.9

On entry: controls the accuracy with which the step

α

taken during each iteration approximates a minimum of the function along the search direction (the smaller the value of

options . linesearch_tol

, the more accurate the linesearch). The default value requests an inaccurate search, and is appropriate for most problems. A more accurate search may be appropriate when it is desirable to reduce the number of iterations – for example, if the objective function is cheap to evaluate.

Constraint:

0.0 \leq options . linesearch_tol < 1.0

max_line_step – double

Default

= 10^{10}

On entry: defines the maximum allowable step length for the line search.

Constraint:

options . max_line_step > 0.0

f_est – double

On entry: specifies the user-supplied guess of the optimum objective function value. This value is used by e04dgc to calculate an initial step length (see Section 3). If no value is supplied then an initial step length of

1.0

will be used but it should be noted that for badly scaled functions a unit step along the steepest descent direction will often compute the function at very large values of

x

iter – Integer

On exit: the number of iterations which have been performed in e04dgc.

nf – Integer

On exit: the number of times the objective function has been evaluated (i.e., number of calls of objfun). The total excludes the calls made to objfun for purposes of derivative checking.

11.3 Description of Printed Output

The level of printed output can be controlled with the structure members

options . list

options . print_gcheck

and

options . print_level

(see Section 11.2). If

options . list = Nag_TRUE

then the argument values to e04dgc are listed, followed by the result of any derivative check if

options . print_gcheck = Nag_TRUE

. The printout of the optimization results is governed by the value of

options . print_level

. The default of

options . print_level = Nag_Soln_Iter

provides a single line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from e04dgc.

If a simple derivative check,

options . verify_grad = Nag_SimpleCheck

, is requested then the directional derivative,

g {(x)}^{T} p

, of the objective gradient and its finite difference approximation are printed out, where

p

is a random vector of unit length.

When a component derivative check,

options . verify_grad = Nag_CheckObj

, is requested then the following results are supplied for each component:

x[i]	the element of $x$ .
dx[i]	the optimal finite difference interval.
g[i]	the gradient element.
Difference approxn.	the finite difference approximation.
Itns	the number of trials performed to find a suitable difference interval.

The indicator, OK or BAD?, states whether the gradient element and finite difference approximation are in agreement.

If the gradient is believed to be in error e04dgc will exit with fail set to NE_DERIV_ERRORS.

When

options . print_level = Nag_Iter

Nag_Soln_Iter

a single line of output is produced on completion of each iteration, this gives the following values:

Itn	the current iteration number $k$ .
Nfun	the cumulative number of calls to objfun. The evaluations needed for the estimation of the gradients by finite differences are not included in the total Nfun. The value of Nfun is a guide to the amount of work required for the linesearch. e04dgc will perform at most 16 function evaluations per iteration.
Objective	the current value of the objective function, $F (x_{k})$ .
Norm g	the Euclidean norm of the gradient vector, $‖ g (x_{k}) ‖$ .
Norm x	the Euclidean norm of $x_{k}$ .
Norm(x(k-1)-x(k))	the Euclidean norm of $x_{k - 1} - x_{k}$ .
Step	the step $α$ taken along the computed search direction $p_{k}$ .

options . print_level = Nag_Soln

Nag_Soln_Iter

, the final result is printed out. This consists of:

x	the final point, $x^{*}$ .
g	the final gradient vector, $g (x^{*})$ .

options . print_level = Nag_NoPrint

then printout will be suppressed; you can print the final solution when e04dgc returns to the calling program.

11.3.1 Output of results via a user-defined printing function

You may also specify your own print function for output of the results of any gradient check, the optimization results at each iteration and the final solution. The user-defined print function should be assigned to the

options . print_fun

function pointer, which has prototype

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

The rest of this section can be skipped if the default printing facilities provide the required functionality.

When a user-defined function is assigned to

options . print_fun

this will be called in preference to the internal print function of e04dgc. Calls to the user-defined function are again controlled by means of the

options . print_gcheck

and

options . print_level

members. Information is provided through st and comm the two structure arguments to

options . print_fun

comm \to it_prt = Nag_TRUE

then the results from the last iteration of e04dgc are in the following members of st:

n – Integer: The number of variables.

x – double *: Points to the $st \to n$ memory locations holding the current point $x_{k}$ .

f – double: The value of the current objective function.

g – double *: Points to the $st \to n$ memory locations holding the first derivatives of $F$ at the current point $x_{k}$ .

step – double: The step $α$ taken along the search direction $p_{k}$ .

xk_norm – double: The Euclidean norm of $x_{k - 1} - x_{k}$ .

iter – Integer: The number of iterations performed by e04dgc.

nf – Integer: The cumulative number of calls made to objfun.

comm \to g_prt = Nag_TRUE

then the following members are set:

n – Integer: The number of variables.

x – double *: Points to the $st \to n$ memory locations holding the initial point $x_{0}$ .

g – double *: Points to the $st \to n$ memory locations holding the first derivatives of $F$ at the initial point $x_{0}$ .

Details of any derivative check performed by e04dgc are held in the following substructure of st:

gprint – Nag_GPrintSt *

Which in turn contains two substructures

gprint \to g_chk

gprint \to f_sim

and a pointer to an array of substructures,

* gprint \to f_comp

g_chk – Nag_Grad_Chk_St *

This substructure contains the members:

type – Nag_GradChk: The type of derivative check performed by e04dgc. This will be the same value as in $options . verify_grad$ .

g_error – Integer: This member will be equal to one of the error codes NE_NOERROR or NE_DERIV_ERRORS according to whether the derivatives were found to be correct or not.

obj_start – Integer: Specifies the gradient element at which any component check started. This value will be equal to $options . obj_check_start$ .

obj_stop – Integer: Specifies the gradient element at which any component check ended. This value will be equal to $options . obj_check_stop$ .

f_sim – Nag_SimSt *

The result of a simple derivative check,

g_chk \to type = Nag_SimpleCheck

, will be held in this substructure which has members:

correct – Nag_Boolean: If Nag_TRUE then the objective gradient is consistent with the finite difference approximation according to a simple check.

dir_deriv – double *: The directional derivative $g {(x)}^{T} p$ where $p$ is a random vector of unit length with elements of approximately equal magnitude.

fd_approx – double *: The finite difference approximation, $(F (x + h p) - F (x)) / h$ , to the directional derivative.

f_comp – Nag_CompSt *

The results of a component derivative check,

g_chk \to type = Nag_CheckObj

, will be held in the array of

st \to n

substructures of type

Nag_CompSt

pointed to by

gprint \to f_comp

. The procedure for the derivative check is based on finding an interval that produces an acceptable estimate of the second derivative, and then using that estimate to compute an interval that should produce a reasonable forward-difference approximation. The gradient element is then compared with the difference approximation. (The method of finite difference interval estimation is based on Gill et al. (1983)).

correct – Nag_Boolean: If Nag_TRUE then this objective gradient component is consistent with its finite difference approximation.

hopt – double *: The optimal finite difference interval. This is dx[i] in the NAG default printout.

gdiff – double *: The finite difference approximation for this gradient component.

iter – Integer: The number of trials performed to find a suitable difference interval.

comment – char: A character string which describes the possible nature of the reason for which an estimation of the finite difference interval failed to produce a satisfactory relative condition error of the second-order difference. Possible strings are: "Constant?", "Linear or odd?", "Too nonlinear?" and "Small derivative?".

The relevant members of the structure comm are:

g_prt – Nag_Boolean: Will be Nag_TRUE only when the print function is called with the result of the derivative check of objfun.

it_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the result of the current iteration.

sol_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the final result.

user – double *
iuser – Integer *
p – Pointer: Pointers for communication of user information. If used they must be allocated memory either before entry to e04dgc or during a call to objfun or $options . print_fun$ . The type Pointer will be void * with a C compiler that defines void * and char * otherwise.