# NAG Library Routine Document

## 1Purpose

c05qcf is a comprehensive routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

## 2Specification

Fortran Interface
 Subroutine c05qcf ( fcn, n, x, fvec, xtol, ml, mu, mode, diag, nfev, fjac, r, qtf,
 Integer, Intent (In) :: n, maxfev, ml, mu, mode, nprint Integer, Intent (Inout) :: iuser(*), ifail Integer, Intent (Out) :: nfev Real (Kind=nag_wp), Intent (In) :: xtol, epsfcn, factor Real (Kind=nag_wp), Intent (Inout) :: x(n), diag(n), ruser(*) Real (Kind=nag_wp), Intent (Out) :: fvec(n), fjac(n,n), r(n*(n+1)/2), qtf(n) External :: fcn
#include nagmk26.h
 void c05qcf_ (void (NAG_CALL *fcn)(const Integer *n, const double x[], double fvec[], Integer iuser[], double ruser[], Integer *iflag),const Integer *n, double x[], double fvec[], const double *xtol, const Integer *maxfev, const Integer *ml, const Integer *mu, const double *epsfcn, const Integer *mode, double diag[], const double *factor, const Integer *nprint, Integer *nfev, double fjac[], double r[], double qtf[], Integer iuser[], double ruser[], Integer *ifail)

## 3Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
c05qcf is based on the MINPACK routine HYBRD (see Moré et al. (1980)). It chooses the correction at each step as a convex combination of the Newton and scaled gradient directions. The Jacobian is updated by the rank-1 method of Broyden. At the starting point, the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. For more details see Powell (1970).

## 4References

Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 Argonne National Laboratory
Powell M J D (1970) A hybrid method for nonlinear algebraic equations Numerical Methods for Nonlinear Algebraic Equations (ed P Rabinowitz) Gordon and Breach

## 5Arguments

1:     $\mathbf{fcn}$ – Subroutine, supplied by the user.External Procedure
fcn must return the values of the functions ${f}_{i}$ at a point $x$, unless ${\mathbf{iflag}}=0$ on entry to c05qcf.
The specification of fcn is:
Fortran Interface
 Subroutine fcn ( n, x, fvec,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*), iflag Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: fvec(n), ruser(*)
#include nagmk26.h
 void fcn (const Integer *n, const double x[], double fvec[], Integer iuser[], double ruser[], Integer *iflag)
1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of equations.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the components of the point $x$ at which the functions must be evaluated.
3:     $\mathbf{fvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{iflag}}=0$, fvec contains the function values ${f}_{i}\left(x\right)$ and must not be changed.
On exit: if ${\mathbf{iflag}}>0$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless iflag is set to a negative value by fcn).
4:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
5:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
fcn is called with the arguments iuser and ruser as supplied to c05qcf. You should use the arrays iuser and ruser to supply information to fcn.
6:     $\mathbf{iflag}$ – IntegerInput/Output
On entry: ${\mathbf{iflag}}\ge 0$.
${\mathbf{iflag}}=0$
x and fvec are available for printing (see nprint).
${\mathbf{iflag}}>0$
fvec must be updated.
On exit: in general, iflag should not be reset by fcn. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), iflag should be set to a negative integer.
fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05qcf is called. Arguments denoted as Input must not be changed by this procedure.
Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05qcf. If your code inadvertently does return any NaNs or infinities, c05qcf is likely to produce unexpected results.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of equations.
Constraint: ${\mathbf{n}}>0$.
3:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
4:     $\mathbf{fvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the function values at the final point returned in x.
5:     $\mathbf{xtol}$ – Real (Kind=nag_wp)Input
On entry: the accuracy in x to which the solution is required.
Suggested value: $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision returned by x02ajf.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
6:     $\mathbf{maxfev}$ – IntegerInput
On entry: the maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. c05qcf will exit with ${\mathbf{ifail}}={\mathbf{2}}$, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Suggested value: ${\mathbf{maxfev}}=200×\left({\mathbf{n}}+1\right)$.
Constraint: ${\mathbf{maxfev}}>0$.
7:     $\mathbf{ml}$ – IntegerInput
On entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ${\mathbf{ml}}={\mathbf{n}}-1$.)
Constraint: ${\mathbf{ml}}\ge 0$.
8:     $\mathbf{mu}$ – IntegerInput
On entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ${\mathbf{mu}}={\mathbf{n}}-1$.)
Constraint: ${\mathbf{mu}}\ge 0$.
9:     $\mathbf{epsfcn}$ – Real (Kind=nag_wp)Input
On entry: a rough estimate of the largest relative error in the functions. It is used in determining a suitable step for a forward difference approximation to the Jacobian. If epsfcn is less than machine precision (returned by x02ajf) then machine precision is used. Consequently a value of $0.0$ will often be suitable.
Suggested value: ${\mathbf{epsfcn}}=0.0$.
10:   $\mathbf{mode}$ – IntegerInput
On entry: indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{mode}}=2$, the scaling must have been specified in diag.
Otherwise, if ${\mathbf{mode}}=1$, the variables will be scaled internally.
Constraint: ${\mathbf{mode}}=1$ or $2$.
11:   $\mathbf{diag}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{mode}}=2$, diag must contain multiplicative scale factors for the variables.
If ${\mathbf{mode}}=1$, diag need not be set.
Constraint: if ${\mathbf{mode}}=2$, ${\mathbf{diag}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{mode}}=1$).
12:   $\mathbf{factor}$ – Real (Kind=nag_wp)Input
On entry: a quantity to be used in determining the initial step bound. In most cases, factor should lie between $0.1$ and $100.0$. (The step bound is ${\mathbf{factor}}×{‖{\mathbf{diag}}×{\mathbf{x}}‖}_{2}$ if this is nonzero; otherwise the bound is factor.)
Suggested value: ${\mathbf{factor}}=100.0$.
Constraint: ${\mathbf{factor}}>0.0$.
13:   $\mathbf{nprint}$ – IntegerInput
On entry: indicates whether (and how often) special calls to fcn, with iflag set to $0$, are to be made for printing purposes.
${\mathbf{nprint}}\le 0$
${\mathbf{nprint}}>0$
fcn is called at the beginning of the first iteration, every nprint iterations thereafter and immediately before the return from c05qcf.
14:   $\mathbf{nfev}$ – IntegerOutput
On exit: the number of calls made to fcn with ${\mathbf{iflag}}>0$.
15:   $\mathbf{fjac}\left({\mathbf{n}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
16:   $\mathbf{r}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.
17:   $\mathbf{qtf}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the vector ${Q}^{\mathrm{T}}f$.
18:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
19:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by c05qcf, but are passed directly to fcn and may be used to pass information to this routine.
20:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=2$
There have been at least maxfev calls to fcn: ${\mathbf{maxfev}}=〈\mathit{\text{value}}〉$. Consider restarting the calculation from the final point held in x.
${\mathbf{ifail}}=3$
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=4$
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ Jacobian evaluations.
${\mathbf{ifail}}=5$
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations.
${\mathbf{ifail}}=6$
iflag was set negative in fcn. ${\mathbf{iflag}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=12$
On entry, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
${\mathbf{ifail}}=13$
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=1$ or $2$.
${\mathbf{ifail}}=14$
On entry, ${\mathbf{factor}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{factor}}>0.0$.
${\mathbf{ifail}}=15$
On entry, ${\mathbf{mode}}=2$ and diag contained a non-positive element.
${\mathbf{ifail}}=16$
On entry, ${\mathbf{ml}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ml}}\ge 0$.
${\mathbf{ifail}}=17$
On entry, ${\mathbf{mu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mu}}\ge 0$.
${\mathbf{ifail}}=18$
On entry, ${\mathbf{maxfev}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxfev}}>0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
A value of ${\mathbf{ifail}}={\mathbf{4}}$ or ${\mathbf{5}}$ may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 7). Otherwise, rerunning c05qcf from a different starting point may avoid the region of difficulty.

## 7Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05qcf tries to ensure that
 $D x-x^ 2 ≤ xtol × D x^ 2 .$
If this condition is satisfied with ${\mathbf{xtol}}={10}^{-k}$, then the larger components of $Dx$ have $k$ significant decimal digits. There is a danger that the smaller components of $Dx$ may have large relative errors, but the fast rate of convergence of c05qcf usually obviates this possibility.
If xtol is less than machine precision and the above test is satisfied with the machine precision in place of xtol, then the routine exits with ${\mathbf{ifail}}={\mathbf{3}}$.
Note:  this convergence test is based purely on relative error, and may not indicate convergence if the solution is very close to the origin.
The convergence test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then c05qcf may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qcf with a lower value for xtol.

## 8Parallelism and Performance

c05qcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Local workspace arrays of fixed lengths are allocated internally by c05qcf. The total size of these arrays amounts to $4×n$ real elements.
The time required by c05qcf to solve a given problem depends on $n$, the behaviour of the functions, the accuracy requested and the starting point. The number of arithmetic operations executed by c05qcf to process each evaluation of the functions is approximately $11.5×{n}^{2}$. The timing of c05qcf is strongly influenced by the time spent evaluating the functions.
Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.
The number of function evaluations required to evaluate the Jacobian may be reduced if you can specify ml and mu accurately.

## 10Example

This example determines the values ${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
 $3-2x1x1-2x2 = -1, -xi-1+3-2xixi-2xi+1 = -1, i=2,3,…,8 -x8+3-2x9x9 = -1.$

### 10.1Program Text

Program Text (c05qcfe.f90)

None.

### 10.3Program Results

Program Results (c05qcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017