# NAG Library Routine Document

## 1Purpose

c05rcf is a comprehensive routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

## 2Specification

Fortran Interface
 Subroutine c05rcf ( fcn, n, x, fvec, fjac, xtol, mode, diag, nfev, njev, r, qtf,
 Integer, Intent (In) :: n, maxfev, mode, nprint Integer, Intent (Inout) :: iuser(*), ifail Integer, Intent (Out) :: nfev, njev Real (Kind=nag_wp), Intent (In) :: xtol, 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 c05rcf_ (void (NAG_CALL *fcn)(const Integer *n, const double x[], double fvec[], double fjac[], Integer iuser[], double ruser[], Integer *iflag),const Integer *n, double x[], double fvec[], double fjac[], const double *xtol, const Integer *maxfev, const Integer *mode, double diag[], const double *factor, const Integer *nprint, Integer *nfev, Integer *njev, double r[], double qtf[], Integer iuser[], double ruser[], Integer *ifail)

## 3Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
c05rcf is based on the MINPACK routine HYBRJ (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 requested, but it is not asked for 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
Depending upon the value of iflag, fcn must either return the values of the functions ${f}_{i}$ at a point $x$ or return the Jacobian at $x$.
The specification of fcn is:
Fortran Interface
 Subroutine fcn ( n, x, fvec, fjac,
 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), fjac(n,n), ruser(*)
#include nagmk26.h
 void fcn (const Integer *n, const double x[], double fvec[], double fjac[], 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 or the Jacobian must be evaluated.
3:     $\mathbf{fvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{iflag}}=0$ or $2$, fvec contains the function values ${f}_{i}\left(x\right)$ and must not be changed.
On exit: if ${\mathbf{iflag}}=1$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless iflag is set to a negative value by fcn).
4:     $\mathbf{fjac}\left({\mathbf{n}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{iflag}}=0$, ${\mathbf{fjac}}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$. When ${\mathbf{iflag}}=0$ or $1$, fjac must not be changed.
On exit: if ${\mathbf{iflag}}=2$ on entry, ${\mathbf{fjac}}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$, (unless iflag is set to a negative value by fcn).
5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
fcn is called with the arguments iuser and ruser as supplied to c05rcf. You should use the arrays iuser and ruser to supply information to fcn.
7:     $\mathbf{iflag}$ – IntegerInput/Output
On entry: ${\mathbf{iflag}}=0$, $1$ or $2$.
${\mathbf{iflag}}=0$
x, fvec and fjac are available for printing (see nprint).
${\mathbf{iflag}}=1$
fvec is to be updated.
${\mathbf{iflag}}=2$
fjac is to 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 value.
fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05rcf 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 c05rcf. If your code inadvertently does return any NaNs or infinities, c05rcf 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:     $\mathbf{fjac}\left({\mathbf{n}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
6:     $\mathbf{xtol}$ – 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: ${\mathbf{xtol}}\ge 0.0$.
7:     $\mathbf{maxfev}$ – IntegerInput
On entry: the maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. c05rcf will exit with ${\mathbf{ifail}}={\mathbf{2}}$, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Suggested value: ${\mathbf{maxfev}}=100×\left({\mathbf{n}}+1\right)$.
Constraint: ${\mathbf{maxfev}}>0$.
8:     $\mathbf{mode}$ – 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 $2$.
9:     $\mathbf{diag}\left({\mathbf{n}}\right)$ – 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 ${\mathbf{mode}}=1$).
10:   $\mathbf{factor}$ – 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: ${\mathbf{factor}}>0.0$.
11:   $\mathbf{nprint}$ – 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 c05rcf.
12:   $\mathbf{nfev}$ – IntegerOutput
On exit: the number of calls made to fcn to evaluate the functions.
13:   $\mathbf{njev}$ – IntegerOutput
On exit: the number of calls made to fcn to evaluate the Jacobian.
14:   $\mathbf{r}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – 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.
15:   $\mathbf{qtf}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the vector ${Q}^{\mathrm{T}}f$.
16:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
17:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by c05rcf, but are passed directly to fcn and may be used to pass information to this routine.
18:   $\mathbf{ifail}$ – 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. This failure exit 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 c05rcf from a different starting point may avoid the region of difficulty.
${\mathbf{ifail}}=5$
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations. This failure exit 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 c05rcf from a different starting point may avoid the region of difficulty.
${\mathbf{ifail}}=6$
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.
${\mathbf{ifail}}=18$
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.

## 7Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05rcf 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 c05rcf 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 and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then c05rcf may incorrectly indicate convergence. The coding of the Jacobian can be checked using c05zdf. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning c05rcf with a lower value for xtol.

## 8Parallelism and Performance

c05rcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rcf 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 c05rcf. The total size of these arrays amounts to $4×n$ real elements.
The time required by c05rcf 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 c05rcf is approximately $11.5×{n}^{2}$ to process each evaluation of the functions and approximately $1.3×{n}^{3}$ to process each evaluation of the Jacobian. The timing of c05rcf 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.

## 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 (c05rcfe.f90)

None.

### 10.3Program Results

Program Results (c05rcfe.r)

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