NAG CL Interface
e04gbc (lsq_​uncon_​quasi_​deriv_​comp)

Note: this function is deprecated. Replaced by e04ggc.
e04ggc is part of the new NAG optimization modelling suite (see Section 4.1 in the E04 Chapter Introduction), therefore the definition of the nonlinear residual function values and gradients need to be split into two separate subroutines. e04ggc offers a significant improvement in performance over e04gbc as well as additional functionality, such as the addition of variable bounds and user-evaluation recovery, amongst many others.
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1 Purpose

e04gbc is a comprehensive algorithm for finding an unconstrained minimum of a sum of squares of m nonlinear functions in n variables (mn) . First derivatives are required.
e04gbc is intended for objective functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

2 Specification

#include <nag.h>
void  e04gbc (Integer m, Integer n,
void (*lsqfun)(Integer m, Integer n, const double x[], double fvec[], double fjac[], Integer tdfjac, Nag_Comm *comm),
double x[], double *fsumsq, double fvec[], double fjac[], Integer tdfjac, Nag_E04_Opt *options, Nag_Comm *comm, NagError *fail)
The function may be called by the names: e04gbc, nag_opt_lsq_uncon_quasi_deriv_comp or nag_opt_lsq_deriv.

3 Description

e04gbc is applicable to problems of the form:
Minimize ​ F (x) = i=1 m [ f i (x)] 2  
where x = ( x 1 , x 2 ,, x n ) T and mn . (The functions f i (x) are often referred to as ‘residuals’.) You must supply a function to calculate the values of the f i (x) and their first derivatives fi xj at any point x .
From a starting point x (1) e04gbc generates a sequence of points x (2) , x (3) , , which is intended to converge to a local minimum of F (x) . The sequence of points is given by
x (k+1) = x (k) + α (k) p (k)  
where the vector p (k) is a direction of search, and α (k) is chosen such that F ( x (k) + α (k) p (k) ) is approximately a minimum with respect to α (k) .
The vector p (k) used depends upon the reduction in the sum of squares obtained during the last iteration. If the sum of squares was sufficiently reduced, then p (k) is the Gauss–Newton direction; otherwise the second derivatives of the f i (x) are taken into account using a quasi-Newton updating scheme.
The method is designed to ensure that steady progress is made whatever the starting point, and to have the rapid ultimate convergence of Newton's method.

4 References

Gill P E and Murray W (1978) Algorithms for the solution of the nonlinear least squares problem SIAM J. Numer. Anal. 15 977–992

5 Arguments

1: m Integer Input
On entry: m , the number of residuals, f i (x) .
2: n Integer Input
On entry: n , the number of variables, x j .
Constraint: 1 n m.
3: lsqfun function, supplied by the user External Function
lsqfun must calculate the vector of values f i (x) and their first derivatives fi xj at any point x . (However, if you do not wish to calculate the residuals at a particular x , there is the option of setting an argument to cause e04gbc to terminate immediately.)
The specification of lsqfun is:
void  lsqfun (Integer m, Integer n, const double x[], double fvec[], double fjac[], Integer tdfjac, Nag_Comm *comm)
1: m Integer Input
2: n Integer Input
On entry: the numbers m and n of residuals and variables, respectively.
3: x[n] const double Input
On entry: the point x at which the values of the f i and the fi xj are required.
4: fvec[m] double Output
On exit: unless commflag=1 on entry, or commflag is reset to a negative number, then fvec[i-1] must contain the value of f i at the point x , for i=1,2,,m.
5: fjac[m×tdfjac] double Output
On exit: unless commflag=0 on entry, or commflag is reset to a negative number, then fjac[ (i-1) × tdfjac + j - 1 ] must contain the value of the first derivative fi xj at the point x , for i=1,2,,m and j=1,2,,n.
6: tdfjac Integer Input
On entry: the stride separating matrix column elements in the array fjac.
7: comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to lsqfun.
On entry: commflag contains 0, 1 or 2. The value 0 indicates that only the residuals need to be evaluated, the value 1 indicates that only the Jacobian matrix needs to be evaluated, and the value 2 indicates that both the residuals and the Jacobian matrix must be calculated. (If the default value of the optional parameter options.minlin is used (i.e., options.minlin=Nag_Lin_Deriv), then lsqfun will always be called with commflag set to 2.)
On exit: if lsqfun resets commflag to some negative number then e04gbc will terminate immediately with the error indicator NE_USER_STOP. If fail is supplied to e04gbc, fail.errnum will be set to the user's setting of commflag.
On entry: will be set to Nag_TRUE on the first call to lsqfun and Nag_FALSE for all subsequent calls.
On entry: the number of calls made to lsqfun including the current one.
userdouble *
iuserInteger *
The type Pointer will be void * with a C compiler that defines void * and char * otherwise. Before calling e04gbc these pointers may be allocated memory and initialized with various quantities for use by lsqfun when called from e04gbc.
Note: lsqfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04gbc. If your code inadvertently does return any NaNs or infinities, e04gbc is likely to produce unexpected results.
Note: lsqfun should be tested separately before being used in conjunction with e04gbc. Function e04yac may be used to check the derivatives.
4: x[n] double Input/Output
On entry: x[j-1] must be set to a guess at the j th component of the position of the minimum, for j=1,2,,n.
On exit: the final point x * . On successful exit, x[j-1] is the j th component of the estimated position of the minimum.
5: fsumsq double * Output
On exit: the value of F (x) , the sum of squares of the residuals f i (x) , at the final point given in x.
6: fvec[m] double Output
On exit: fvec[i-1] is the value of the residual f i (x) at the final point given in x, for i=1,2,,m.
7: fjac[m×tdfjac] double Output
On exit: fjac[(i-1)×tdfjac+j-1] contains the value of the first derivative fi xj at the final point given in x, for i=1,2,,m and j=1,2,,n.
8: tdfjac Integer Input
On entry: the stride separating matrix column elements in the array fjac.
Constraint: tdfjacn .
9: 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 e04gbc. 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 in Section 11.2.
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 e04gbc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.
10: 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 to the user-supplied function; see the above description of lsqfun 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 e04gbc; comm will then be declared internally for use in calls to the user-supplied function.
11: 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

If one of NE_USER_STOP, NE_2_INT_ARG_LT, NE_DERIV_ERRORS, NE_OPT_NOT_INIT, NE_BAD_PARAM, NE_2_REAL_ARG_LT, NE_INVALID_INT_RANGE_1, NE_INVALID_REAL_RANGE_EF, NE_INVALID_REAL_RANGE_FF and NE_ALLOC_FAIL occurs, no values will have been assigned to fsumsq, or to the elements of fvec, fjac, options.s or options.v.
The exits NW_TOO_MANY_ITER, NW_COND_MIN, and NE_SVD_FAIL may also be caused by mistakes in lsqfun, by the formulation of the problem or by an awkward function. If there are no such mistakes it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure.
On entry, m=value while n=value . These arguments must satisfy mn .
On entry, options.tdv=value while n=value . These arguments must satisfy options.tdvn .
On entry, tdfjac=value while n=value . These arguments must satisfy tdfjacn .
On entry, options.step_max=value while options.optim_tol=value . These arguments must satisfy options.step_maxoptions.optim_tol .
Dynamic memory allocation failed.
On entry, argument options.minlin had an illegal value.
On entry, argument options.print_level had an illegal value.
Large errors were found in the derivatives of the objective function.
You should check carefully the derivation and programming of expressions for the fi xj , because it is very unlikely that lsqfun is calculating them correctly.
On entry, n=value.
Constraint: n1.
Value value given to options.max_iter not valid. Correct range is options.max_iter0 .
Value value given to options.optim_tol not valid. Correct range is value options.optim_tol < 1.0 .
Value value given to options.linesearch_tol not valid. Correct range is 0.0 options.linesearch_tol < 1.0 .
Cannot open file string for appending.
Cannot close file string .
Options structure not initialized.
The computation of the singular value decomposition of the Jacobian matrix has failed to converge in a reasonable number of sub-iterations.
It may be worth applying e04gbc again starting with an initial approximation which is not too close to the point at which the failure occurred.
User requested termination, user flag value =value .
This exit occurs if you set commflag to a negative value in lsqfun. If fail is supplied the value of fail.errnum will be the same as your setting of commflag .
Error occurred when writing to file string .
The conditions for a minimum have not all been satisfied, but a lower point could not be found.
This could be because options.optim_tol has been set so small that rounding errors in the evaluation of the residuals make attainment of the convergence conditions impossible. See Section 7 for further information.
The maximum number of iterations, value, have been performed.
If steady reductions in the sum of squares, F (x) , were monitored up to the point where this exit occurred, then the exit probably occurred simply because options.max_iter was set too small, so the calculations should be restarted from the final point held in x. This exit may also indicate that F (x) has no minimum.

7 Accuracy

If the problem is reasonably well scaled and a successful exit is made, then, for a computer with a mantissa of t decimals, one would expect to get about t / 2 - 1 decimals accuracy in the components of x and between t-1 (if F (x) is of order 1 at the minimum) and 2 t - 2 (if F (x) is close to zero at the minimum) decimals accuracy in F (x) .
A successful exit ( fail.code=NE_NOERROR ) is made from e04gbc when (B1, B2 and B3) or B4 or B5 hold, where
B1 α (k) × p (k) < (options.optim_tol+ε) × (1.0+ x (k) ) B2 | F (k) - F (k-1) | < (options.optim_tol+ε) 2 × (1.0+ F (k) ) B3 g (k) < ε 1/3 × (1.0+ F (k) ) B4 F (k) < ε 2 B5 g (k) < (ε× F (k) ) 1/2  
and where . , ε and the optional parameter options.optim_tol are as defined in Section 11.2, while F (k) and g (k) are the values of F (x) and its vector of first derivatives at x (k) .
If fail.code=NE_NOERROR then the vector in x on exit, x sol , is almost certainly an estimate of x true , the position of the minimum to the accuracy specified by options.optim_tol.
If fail.code=NW_COND_MIN , then x sol may still be a good estimate of x true , but to verify this you should make the following checks. If
  1. (a)the sequence {F( x (k) )} converges to F ( x sol ) at a superlinear or a fast linear rate, and
  2. (b) g ( x sol ) T g ( x sol ) < 10 ε ,
where T denotes transpose, then it is almost certain that x sol is a close approximation to the minimum. When (b) is true, then usually F ( x sol ) is a close approximation to F ( x true ) .
Further suggestions about confirmation of a computed solution are given in the E04 Chapter Introduction.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
e04gbc is not threaded in any implementation.

9 Further Comments

The number of iterations required depends on the number of variables, the number of residuals, the behaviour of F (x) , the accuracy demanded and the distance of the starting point from the solution. The number of multiplications performed per iteration of e04gbc varies, but for m > > n is approximately n × m 2 + O ( n 3 ) . In addition, each iteration makes at least one call of lsqfun. So, unless the residuals can be evaluated very quickly, the run time will be dominated by the time spent in lsqfun.
Ideally, the problem should be scaled so that, at the solution, F (x) and the corresponding values of the x j are each in the range (−1,+1) , and so that at points one unit away from the solution, F (x) differs from its value at the solution by approximately one unit. This will usually imply that the Hessian matrix of F (x) at the solution is well-conditioned. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that e04gbc will take less computer time.
When the sum of squares represents the goodness-of-fit of a nonlinear model to observed data, elements of the variance-covariance matrix of the estimated regression coefficients can be computed by a subsequent call to e04ycc, using information returned in the arrays options.s and options.v. See e04ycc for further details.

10 Example

This example finds the least squares estimates of x 1 , x 2 and x 3 in the model
y = x 1 + t 1 x 2 t 2 + x 3 t 3  
using the 15 sets of data given in the following table.
y t1 t2 t3
0.14 1.0 15.0 1.0
0.18 2.0 14.0 2.0
0.22 3.0 13.0 3.0
0.25 4.0 12.0 4.0
0.29 5.0 11.0 5.0
0.32 6.0 10.0 6.0
0.35 7.0 9.0 7.0
0.39 8.0 8.0 8.0
0.37 9.0 7.0 7.0
0.58 10.0 6.0 6.0
0.73 11.0 5.0 5.0
0.96 12.0 4.0 4.0
1.34 13.0 3.0 3.0
2.10 14.0 2.0 2.0
4.39 15.0 1.0 1.0
The program uses (0.5, 1.0, 1.5) as the initial guess at the position of the minimum.
The program shows the use of certain optional parameters, with some option values being assigned directly within the program text and by reading values from a data file. The options structure is declared and initialized by e04xxc. A value is then assigned directly to options options.outfile and three further options are read from the data file by use of e04xyc. The memory freeing function e04xzc is used to free the memory assigned to the pointers in the option structure. You must not use the standard C function free() for this purpose.

10.1 Program Text

Program Text (e04gbce.c)

10.2 Program Data

Program Data (e04gbce.d)
Program Options (e04gbce.opt)

10.3 Program Results

Program Results (e04gbce.r)

11 Optional Parameters

A number of optional input and output arguments to e04gbc 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 e04gbc; 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 e04gbc, the options structure may only be re-used for future calls of e04gbc 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.
Optional parameter 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, 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 e04gbc 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
Boolean deriv_check Nag_TRUE
Integer max_iter max(50, 5 n)
double optim_tol ε
Nag_LinFun minlin Nag_Lin_Deriv
double linesearch_tol 0.9 (0.0 if n=1 )
double step_max 100000.0
double *s size n
double *v size n×n
Integer tdv n
Integer grade
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 e04gbc will be printed.
print_level – Nag_PrintType Default =Nag_Soln_Iter
On entry: the level of results printout produced by e04gbc. 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.
Nag_Soln_Iter_Full The final solution and detailed printout at each iteration.
Details of each level of results printout are described in Section 11.3.
Constraint: options.print_level=Nag_NoPrint, Nag_Soln, Nag_Iter, Nag_Soln_Iter or Nag_Soln_Iter_Full.
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 is
void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);
See Section 11.3.1 for further details.
deriv_check – Nag_Boolean Default =Nag_TRUE
On entry: if options.deriv_check=Nag_TRUE a check of the derivatives defined by lsqfun will be made at the starting point x. The derivative check is carried out by a call to e04yac. A starting point of x=0 or x=1 should be avoided if this test is to be meaningful, but if either of these starting points is necessary then e04yac should be used to check lsqfun at a different point prior to calling e04gbc.
max_iter – Integer Default = max(50, 5 n)
On entry: the limit on the number of iterations allowed before termination.
Constraint: options.max_iter0 .
optim_tol – double Default = ε
On entry: the accuracy in x to which the solution is required. If x true is the true value of x at the minimum, then x sol , the estimated position prior to a normal exit, is such that
x sol - x true < options.optim_tol × (1.0+ x true ) ,  
where y = j=1 n y j 2 . For example, if the elements of x sol are not much larger than 1.0 in modulus and if options.optim_tol = 1.0 × 10 −5 , then x sol is usually accurate to about five decimal places. (For further details see Section 7.) If F (x) and the variables are scaled roughly as described in Section 9 and ε is the machine precision, then a setting of order options.optim_tol = ε will usually be appropriate.
Constraint: 10 ε options.optim_tol < 1.0 .
minlin – Nag_LinFun Default =Nag_Lin_Deriv
On entry: options.minlin specifies whether the linear minimizations (i.e., minimizations of F ( x (k) + α (k) p (k) ) with respect to α (k) ) are to be performed by a function which just requires the evaluation of the f i (x) , Nag_Lin_NoDeriv, or by a function which also requires the first derivatives of the f i (x) , Nag_Lin_Deriv.
It will often be possible to evaluate the first derivatives of the residuals in about the same amount of computer time that is required for the evaluation of the residuals themselves – if this is so then e04gbc should be called with options.minlin set to Nag_Lin_Deriv. However, if the evaluation of the derivatives takes more than about four times as long as the evaluation of the residuals, then a setting of Nag_Lin_NoDeriv will usually be preferable. If in doubt, use the default setting Nag_Lin_Deriv as it is slightly more robust.
Constraint: options.minlin=Nag_Lin_Deriv or Nag_Lin_NoDeriv.
linesearch_tol – double Default =0.9 . (If n=1 , default =0.0 )
If options.minlin=Nag_Lin_NoDeriv then the default value of options.linesearch_tol will be changed from 0.9 to 0.5 if n>1 .
On entry: options.linesearch_tol specifies how accurately the linear minimizations are to be performed.
Every iteration of e04gbc involves a linear minimization, i.e., minimization of F ( x (k) + α (k) p (k) ) with respect to α (k) . The minimum with respect to α (k) will be located more accurately for small values of options.linesearch_tol (say 0.01) than for large values (say 0.9). Although accurate linear minimizations will generally reduce the number of iterations performed by e04gbc, they will increase the number of calls of lsqfun made each iteration. On balance it is usually more efficient to perform a low accuracy minimization.
Constraint: 0.0 options.linesearch_tol < 1.0 .
step_max – double Default =100000.0
On entry: an estimate of the Euclidean distance between the solution and the starting point supplied. (For maximum efficiency, a slight overestimate is preferable.) e04gbc will ensure that, for each iteration,
j=1 n ( x j (k) - x j (k-1) ) 2 (options.step_max) 2  
where k is the iteration number. Thus, if the problem has more than one solution, e04gbc is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence x (k) entering a region where the problem is ill-behaved and can help avoid overflow in the evaluation of F (x) . However, an underestimate of options.step_max can lead to inefficiency.
Constraint: options.step_maxoptions.optim_tol .
s – double * Default memory =n
On entry: n values of memory will be automatically allocated by e04gbc and this is the recommended method of use of options.s. However, you may supply memory from the calling program.
On exit: the singular values of the Jacobian matrix at the final point. Thus options.s may be useful as information about the structure of your problem.
v – double * Default memory = n × n
On entry: n×n values of memory will be automatically allocated by e04gbc and this is the recommended method of use of options.v. However, you may supply memory from the calling program.
On exit: the matrix V associated with the singular value decomposition
J = USV T  
of the Jacobian matrix at the final point, stored by rows. This matrix may be useful for statistical purposes, since it is the matrix of orthonormalized eigenvectors of J T J .
tdv – Integer  Default =n
On entry: if memory is supplied then options.tdv must contain the last dimension of the array assigned to options.tdv as declared in the function from which e04gbc is called.
On exit: the trailing dimension used by options.v. If the NAG default memory allocation has been used this value will be n.
Constraint: options.tdvn .
grade – Integer 
On exit: the grade of the Jacobian at the final point. e04gbc estimates the dimension of the subspace for which the Jacobian matrix can be used as a valid approximation to the curvature (see Gill and Murray (1978)); this estimate is called the grade.
iter – Integer 
On exit: the number of iterations which have been performed in e04gbc.
nf – Integer 
On exit: the number of times the residuals have been evaluated (i.e., the number of calls of lsqfun).

11.3 Description of Printed Output

The level of printed output can be controlled with the structure members options.list and options.print_level (see Section 11.2). If options.list=Nag_TRUE then the argument values to e04gbc are listed, whereas the printout of 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 e04gbc.
When options.print_level=Nag_Iter or 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 lsqfun.
Objective the value of the objective function, F ( x (k) ) .
Norm g the Euclidean norm of the gradient of F ( 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 (k) taken along the computed search direction p (k) .
When options.print_level=Nag_Soln_Iter_Full more detailed results are given at each iteration. Additional values output are:
Grade the grade of the Jacobian matrix. (See description of options.grade, Section 9.)
x the current point x (k) .
g the current gradient of F ( x (k) ) .
Singular values the singular values of the current approximation to the Jacobian matrix.
If options.print_level=Nag_Soln, Nag_Soln_Iter or Nag_Soln_Iter_Full the final result consists of:
x the final point x * .
g the gradient of F at the final point.
Residuals the values of the residuals f i at the final point.
Sum of squares the value of F ( x * ) , the sum of squares of the residuals at the final point.
If options.print_level=Nag_NoPrint then printout will be suppressed; you can print the final solution when e04gbc 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 iteration results and the final solution by use of 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 e04gbc. Calls to the user-defined function are again controlled by means of the options.print_level member. Information is provided through st and comm, the two structure arguments to options.print_fun. The structure member commit_prt is relevant in this context. If commit_prt = Nag_TRUE then the results from the last iteration of e04gbc are in the following members of st:
The number of residuals.
The number of variables.
xdouble *
Points to the stn memory locations holding the current point x (k) .
fvecdouble *
Points to the stm memory locations holding the values of the residuals f i at the current point x (k) .
fjacdouble *
Points to stm × sttdfjac memory locations. stfjac[ (i-1) × sttdfjac + (j-1) ] contains the value of fi xj , for i=1,2,,m and j=1,2,,n at the current point x (k) .
The trailing dimension for stfjac[ ] .
The step α (k) taken along the search direction p (k) .
The Euclidean norm of x (k-1) - x (k) .
gdouble *
Points to the stn memory locations holding the gradient of F at the current point x (k) .
The grade of the Jacobian matrix.
sdouble *
Points to the stn memory locations holding the singular values of the current Jacobian.
The number of iterations, k , performed by e04gbc.
The cumulative number of calls made to lsqfun.
The relevant members of the structure comm are:
Will be Nag_TRUE when the print function is called with the result of the current iteration.
Will be Nag_TRUE when the print function is called with the final result.
userdouble *
iuserInteger *
Pointers for communication of user information. If used they must be allocated memory either before entry to e04gbc or during a call to lsqfun or options.print_fun. The type Pointer will be void * with a C compiler that defines void * and char * otherwise.