c05rc is a comprehensive method that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

Syntax

C#
public static void c05rc( C05..::..C05RC_FCN fcn, int n, double[] x, double[] fvec, double[,] fjac, double xtol, int maxfev, int mode, double[] diag, double factor, int nprint, out int nfev, out int njev, double[] r, double[] qtf, out int ifail )

public static void c05rc(
	C05..::..C05RC_FCN fcn,
	int n,
	double[] x,
	double[] fvec,
	double[,] fjac,
	double xtol,
	int maxfev,
	int mode,
	double[] diag,
	double factor,
	int nprint,
	out int nfev,
	out int njev,
	double[] r,
	double[] qtf,
	out int ifail
)

Visual Basic
Public Shared Sub c05rc ( _ fcn As C05..::..C05RC_FCN, _ n As Integer, _ x As Double(), _ fvec As Double(), _ fjac As Double(,), _ xtol As Double, _ maxfev As Integer, _ mode As Integer, _ diag As Double(), _ factor As Double, _ nprint As Integer, _ <OutAttribute> ByRef nfev As Integer, _ <OutAttribute> ByRef njev As Integer, _ r As Double(), _ qtf As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub c05rc ( _
	fcn As C05..::..C05RC_FCN, _
	n As Integer, _
	x As Double(), _
	fvec As Double(), _
	fjac As Double(,), _
	xtol As Double, _
	maxfev As Integer, _
	mode As Integer, _
	diag As Double(), _
	factor As Double, _
	nprint As Integer, _
	<OutAttribute> ByRef nfev As Integer, _
	<OutAttribute> ByRef njev As Integer, _
	r As Double(), _
	qtf As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void c05rc( C05..::..C05RC_FCN^ fcn, int n, array<double>^ x, array<double>^ fvec, array<double,2>^ fjac, double xtol, int maxfev, int mode, array<double>^ diag, double factor, int nprint, [OutAttribute] int% nfev, [OutAttribute] int% njev, array<double>^ r, array<double>^ qtf, [OutAttribute] int% ifail )

Visual C++

public:
static void c05rc(
	C05..::..C05RC_FCN^ fcn, 
	int n, 
	array<double>^ x, 
	array<double>^ fvec, 
	array<double,2>^ fjac, 
	double xtol, 
	int maxfev, 
	int mode, 
	array<double>^ diag, 
	double factor, 
	int nprint, 
	[OutAttribute] int% nfev, 
	[OutAttribute] int% njev, 
	array<double>^ r, 
	array<double>^ qtf, 
	[OutAttribute] int% ifail
)

F#
static member c05rc : fcn : C05..::..C05RC_FCN * n : int * x : float[] * fvec : float[] * fjac : float[,] * xtol : float * maxfev : int * mode : int * diag : float[] * factor : float * nprint : int * nfev : int byref * njev : int byref * r : float[] * qtf : float[] * ifail : int byref -> unit

static member c05rc : 
        fcn : C05..::..C05RC_FCN * 
        n : int * 
        x : float[] * 
        fvec : float[] * 
        fjac : float[,] * 
        xtol : float * 
        maxfev : int * 
        mode : int * 
        diag : float[] * 
        factor : float * 
        nprint : int * 
        nfev : int byref * 
        njev : int byref * 
        r : float[] * 
        qtf : float[] * 
        ifail : int byref -> unit

Parameters

fcn: Type: NagLibrary..::..C05..::..C05RC_FCN
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 $x$ .
A delegate of type C05RC_FCN.

n: Type: System..::..Int32
On entry: $n$ , the number of equations.

Constraint: $n > 0$ .

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.

fvec: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the function values at the final point returned in x.

fjac: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, n]
Note: dim1 must satisfy the constraint:
On exit: the orthogonal matrix $Q$ produced by the $Q R$ factorization of the final approximate Jacobian.

xtol: Type: System..::..Double
On entry: the accuracy in x to which the solution is required.
Suggested value: $\sqrt{ε}$ , where $ε$ is the machine precision returned by x02aj.

Constraint: $xtol \geq 0.0$ .

maxfev: Type: System..::..Int32
On entry: the maximum number of calls to fcn with $iflag \neq 0$ . c05rc will exit with $ifail = 2$ , if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Suggested value: $maxfev = 100 \times (n + 1)$ .

Constraint: $maxfev > 0$ .

mode: Type: System..::..Int32
On entry: indicates whether or not you have provided scaling factors in diag.
If $mode = 2$ the scaling must have been specified in diag.

Otherwise, if $mode = 1$ , the variables will be scaled internally.

Constraint: $mode = 1$ or $2$ .

diag: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: if $mode = 2$ , diag must contain multiplicative scale factors for the variables.
If $mode = 1$ , diag need not be set.

Constraint: if $mode = 2$ , $diag [i - 1] > 0.0$ , for $i = 1, 2, \dots, n$ .
On exit: the scale factors actually used (computed internally if $mode = 1$ ).

factor: Type: System..::..Double
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 $factor \times {‖diag \times x‖}_{2}$ if this is nonzero; otherwise the bound is factor.)
Suggested value: $factor = 100.0$ .

Constraint: $factor > 0.0$ .

nprint

Type: System..::..Int32

On entry: indicates whether (and how often) special calls to fcn, with iflag set to

0

, are to be made for printing purposes.

$nprint \leq 0$: No calls are made.
$nprint > 0$: fcn is called at the beginning of the first iteration, every nprint iterations thereafter and immediately before the return from c05rc.

nfev: Type: System..::..Int32%
On exit: the number of calls made to fcn to evaluate the functions.

njev: Type: System..::..Int32%
On exit: the number of calls made to fcn to evaluate the Jacobian.

r: Type: array<System..::..Double>[]()[][]
An array of size [ $n \times (n + 1) / 2$ ]
On exit: the upper triangular matrix $R$ produced by the $Q R$ factorization of the final approximate Jacobian, stored row-wise.

qtf: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the vector $Q^{T} f$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The system of equations is defined as:

f_{i} (x_{1}, x_{2}, \dots, x_{n}) = 0,   ​ i = 1, 2, \dots, n .

c05rc 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).

References

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

Error Indicators and Warnings

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDFJAC) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 2$: There have been at least maxfev calls to fcn: $maxfev = 〈value〉$ . Consider restarting the calculation from the final point held in x.

$ifail = 3$: No further improvement in the solution is possible. xtol is too small: $xtol = 〈value〉$ .

$ifail = 4$: The iteration is not making good progress, as measured by the improvement from the last $〈value〉$ 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 [Accuracy]). Otherwise, rerunning c05rc from a different starting point may avoid the region of difficulty.

$ifail = 5$: The iteration is not making good progress, as measured by the improvement from the last $〈value〉$ 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 [Accuracy]). Otherwise, rerunning c05rc from a different starting point may avoid the region of difficulty.

$ifail = 6$: iflag was set negative in fcn. $iflag = 〈value〉$ .

$ifail = 11$: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .

$ifail = 12$: On entry, $xtol = 〈value〉$ .
Constraint: $xtol \geq 0.0$ .

$ifail = 13$: On entry, $mode = 〈value〉$ .
Constraint: $mode = 1$ or $2$ .

$ifail = 14$: On entry, $factor = 〈value〉$ .
Constraint: $factor > 0.0$ .

$ifail = 15$: On entry, $mode = 2$ and diag contained a non-positive element.

$ifail = 18$: On entry, $maxfev = 〈value〉$ .
Constraint: $maxfev > 0$ .

$ifail = - 999$: Dynamic memory allocation failed.

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

\hat{x}

is the true solution and

D

denotes the diagonal matrix whose entries are defined by the array diag, then c05rc tries to ensure that

{‖D (x - \hat{x})‖}_{2} \leq xtol \times {‖D \hat{x}‖}_{2} .

If this condition is satisfied with

xtol = 10^{- k}

, then the larger components of

D x

have

k

significant decimal digits. There is a danger that the smaller components of

D x

may have large relative errors, but the fast rate of convergence of c05rc 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 method exits with

ifail = 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 c05rc may incorrectly indicate convergence. The coding of the Jacobian can be checked using c05zd. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning c05rc with a lower value for xtol.

Parallelism and Performance

None.

Further Comments

Local workspace arrays of fixed lengths are allocated internally by c05rc. The total size of these arrays amounts to

4 \times n

real elements.

The time required by c05rc 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 c05rc is approximately

11.5 \times n^{2}

to process each evaluation of the functions and approximately

1.3 \times n^{3}

to process each evaluation of the Jacobian. The timing of c05rc 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.

Example

This example determines the values

x_{1}, \dots, x_{9}

which satisfy the tridiagonal equations:

\begin{array}{rcl} (3 - 2 x_{1}) x_{1} - 2 x_{2} & = & - 1, \\ - x_{i - 1} + (3 - 2 x_{i}) x_{i} - 2 x_{i + 1} & = & - 1, i = 2, 3, \dots, 8 \\ - x_{8} + (3 - 2 x_{9}) x_{9} & = & - 1 . \end{array}

Example program (C#): c05rce.cs

Example program results: c05rce.r