# NAG CL Interfacec05rcc (sys_​deriv_​expert)

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## 1Purpose

c05rcc is a comprehensive function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

## 2Specification

 #include
void  c05rcc (
 void (*fcn)(Integer n, const double x[], double fvec[], double fjac[], Nag_Comm *comm, Integer *iflag),
Integer n, double x[], double fvec[], double fjac[], double xtol, Integer maxfev, Nag_ScaleType scale_mode, double diag[], double factor, Integer nprint, Integer *nfev, Integer *njev, double r[], double qtf[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: c05rcc, nag_roots_sys_deriv_expert or nag_zero_nonlin_eqns_deriv_expert.

## 3Description

The system of equations is defined as:
 $fi (x1,x2,…,xn) = 0 , ​ i= 1, 2, …, n .$
c05rcc 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).
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: $\mathbf{fcn}$function, supplied by the user External Function
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:
 void fcn (Integer n, const double x[], double fvec[], double fjac[], Nag_Comm *comm, Integer *iflag)
1: $\mathbf{n}$Integer Input
On entry: $n$, the number of equations.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.
3: $\mathbf{fvec}\left[{\mathbf{n}}\right]$double Input/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 ${f}_{i}\left(x\right)$.
4: $\mathbf{fjac}\left[{\mathbf{n}}×{\mathbf{n}}\right]$double Input/Output
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j-1\right)×{\mathbf{n}}+i-1\right]$.
On entry: if ${\mathbf{iflag}}=0$, ${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)×{\mathbf{n}}+\mathit{i}-1\right]$ 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, ${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)×{\mathbf{n}}+\mathit{i}-1\right]$ 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: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to fcn.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling c05rcc you may allocate memory and initialize these pointers with various quantities for use by fcn when called from c05rcc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
6: $\mathbf{iflag}$Integer * Input/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.
Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05rcc. If your code inadvertently does return any NaNs or infinities, c05rcc is likely to produce unexpected results.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of equations.
Constraint: ${\mathbf{n}}>0$.
3: $\mathbf{x}\left[{\mathbf{n}}\right]$double Input/Output
On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
4: $\mathbf{fvec}\left[{\mathbf{n}}\right]$double Output
On exit: the function values at the final point returned in x.
5: $\mathbf{fjac}\left[{\mathbf{n}}×{\mathbf{n}}\right]$double Output
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j-1\right)×{\mathbf{n}}+i-1\right]$.
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.
6: $\mathbf{xtol}$double 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 X02AJC.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
7: $\mathbf{maxfev}$Integer Input
On entry: the maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. c05rcc will exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_MANY_FEVALS, 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: $\mathbf{scale_mode}$Nag_ScaleType Input
On entry: indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$, the scaling must have been specified in diag.
Otherwise, if ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$, the variables will be scaled internally.
Constraint: ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$ or $\mathrm{Nag_ScaleProvided}$.
9: $\mathbf{diag}\left[{\mathbf{n}}\right]$double Input/Output
On entry: if ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$, diag must contain multiplicative scale factors for the variables.
If ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$, diag need not be set.
Constraint: if ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$, ${\mathbf{diag}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$).
10: $\mathbf{factor}$double 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: $\mathbf{nprint}$Integer Input
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 c05rcc.
12: $\mathbf{nfev}$Integer * Output
On exit: the number of calls made to fcn to evaluate the functions.
13: $\mathbf{njev}$Integer * Output
On exit: the number of calls made to fcn to evaluate the Jacobian.
14: $\mathbf{r}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$double Output
On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.
15: $\mathbf{qtf}\left[{\mathbf{n}}\right]$double Output
On exit: the vector ${Q}^{\mathrm{T}}f$.
16: $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
17: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_DIAG_ELEMENTS
On entry, ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$ and diag contained a non-positive element.
NE_INT
On entry, ${\mathbf{maxfev}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{maxfev}}>0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_IMPROVEMENT
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 c05rcc from a different starting point may avoid the region of difficulty.
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 c05rcc from a different starting point may avoid the region of difficulty.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{factor}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{factor}}>0.0$.
On entry, ${\mathbf{xtol}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
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.
NE_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=⟨\mathit{\text{value}}⟩$.
NE_USER_STOP
iflag was set negative in fcn. ${\mathbf{iflag}}=⟨\mathit{\text{value}}⟩$.

## 7Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05rcc tries to ensure that
 $‖D(x-x^)‖2 ≤ xtol × ‖Dx^‖2 .$
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 c05rcc 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 function exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_SMALL.
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 c05rcc may incorrectly indicate convergence. The coding of the Jacobian can be checked using c05zdc. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning c05rcc with a lower value for xtol.

## 8Parallelism and Performance

c05rcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rcc 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 function. 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 c05rcc. The total size of these arrays amounts to $4×n$ double elements.
The time required by c05rcc 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 c05rcc 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 c05rcc 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:
 $(3-2x1)x1-2x2 = −1, -xi-1+(3-2xi)xi-2xi+1 = −1, i=2,3,…,8 -x8+(3-2x9)x9 = −1.$

### 10.1Program Text

Program Text (c05rcce.c)

None.

### 10.3Program Results

Program Results (c05rcce.r)