c05nc is a comprehensive method to find a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

Syntax

C#
public static void c05nc( C05..::.C05NC_FCN fcn, int n, double[] x, double[] fvec, double xtol, int maxfev, int ml, int mu, double epsfcn, int mode, double[] diag, double factor, int nprint, out int nfev, double[,] fjac, double[] r, double[] qtf, out int ifail )

public static void c05nc(
	C05..::.C05NC_FCN fcn,
	int n,
	double[] x,
	double[] fvec,
	double xtol,
	int maxfev,
	int ml,
	int mu,
	double epsfcn,
	int mode,
	double[] diag,
	double factor,
	int nprint,
	out int nfev,
	double[,] fjac,
	double[] r,
	double[] qtf,
	out int ifail
)

Visual Basic (Declaration)
Public Shared Sub c05nc ( _ fcn As C05..::.C05NC_FCN, _ n As Integer, _ x As Double(), _ fvec As Double(), _ xtol As Double, _ maxfev As Integer, _ ml As Integer, _ mu As Integer, _ epsfcn As Double, _ mode As Integer, _ diag As Double(), _ factor As Double, _ nprint As Integer, _ <OutAttribute> ByRef nfev As Integer, _ fjac As Double(,), _ r As Double(), _ qtf As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic (Declaration)

Public Shared Sub c05nc ( _
	fcn As C05..::.C05NC_FCN, _
	n As Integer, _
	x As Double(), _
	fvec As Double(), _
	xtol As Double, _
	maxfev As Integer, _
	ml As Integer, _
	mu As Integer, _
	epsfcn As Double, _
	mode As Integer, _
	diag As Double(), _
	factor As Double, _
	nprint As Integer, _
	<OutAttribute> ByRef nfev As Integer, _
	fjac As Double(,), _
	r As Double(), _
	qtf As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void c05nc( C05..::.C05NC_FCN^ fcn, int n, array<double>^ x, array<double>^ fvec, double xtol, int maxfev, int ml, int mu, double epsfcn, int mode, array<double>^ diag, double factor, int nprint, [OutAttribute] int% nfev, array<double,2>^ fjac, array<double>^ r, array<double>^ qtf, [OutAttribute] int% ifail )

Visual C++

public:
static void c05nc(
	C05..::.C05NC_FCN^ fcn, 
	int n, 
	array<double>^ x, 
	array<double>^ fvec, 
	double xtol, 
	int maxfev, 
	int ml, 
	int mu, 
	double epsfcn, 
	int mode, 
	array<double>^ diag, 
	double factor, 
	int nprint, 
	[OutAttribute] int% nfev, 
	array<double,2>^ fjac, 
	array<double>^ r, 
	array<double>^ qtf, 
	[OutAttribute] int% ifail
)

F#
static member c05nc : fcn:C05..::.C05NC_FCN * n:int * x:float[] * fvec:float[] * xtol:float * maxfev:int * ml:int * mu:int * epsfcn:float * mode:int * diag:float[] * factor:float * nprint:int * nfev:int byref * fjac:float[,] * r:float[] * qtf:float[] * ifail:int byref -> unit

static member c05nc : 
        fcn:C05..::.C05NC_FCN * 
        n:int * 
        x:float[] * 
        fvec:float[] * 
        xtol:float * 
        maxfev:int * 
        ml:int * 
        mu:int * 
        epsfcn:float * 
        mode:int * 
        diag:float[] * 
        factor:float * 
        nprint:int * 
        nfev:int byref * 
        fjac:float[,] * 
        r:float[] * 
        qtf:float[] * 
        ifail:int byref -> unit

Parameters

fcn: Type: NagLibrary..::.C05..::.C05NC_FCN

fcn must return the values of the functions fi at a point x.

A delegate of type C05NC_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, x.

xtol: Type: System..::.Double

On entry: the accuracy in x to which the solution is required.

Suggested value: the square root of the machine precision.

Constraint: xtol≥0.0 .

maxfev: Type: System..::.Int32

On entry: the maximum number of calls to fcn with iflag≠0 . c05nc will exit with ifail=2, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.

Suggested value: maxfev=200×n+1 .

Constraint: maxfev>0 .

ml: Type: System..::.Int32

On entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ml=n-1 .)

Constraint: ml≥0 .

mu: Type: System..::.Int32

On entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set mu=n-1 .)

Constraint: mu≥0 .

epsfcn: Type: System..::.Double

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 then machine precision is used. Consequently a value of 0.0 will often be suitable.

Suggested value: epsfcn=0.0 .

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, the variables will be scaled internally.

diag: Type: array< System..::.Double >[]()[]
An array of size [n]

On entry: if mode=2, diag must contain multiplicative scale factors for the variables.

Constraint: diag[i]>0.0 , for i=0,1,…,n-1.

On exit: the scale factors actually used (computed internally if mode≠2).

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×diag×x2 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 special calls to fcn, with iflag set to 0, are to be made for printing purposes.

nprint≤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 c05nc.

nfev: Type: System..::.Int32 %

On exit: the number of calls made to fcn.

fjac: Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldfjac, n]
Note: ldfjac must satisfy the constraint: ldfjac≥n

On exit: the orthogonal matrix Q produced by the QR factorization of the final approximate Jacobian.

r: Type: array< System..::.Double >[]()[]
An array of size [n×n+1/2]

On exit: the upper triangular matrix R produced by the QR factorization of the final approximate Jacobian, stored row-wise.

qtf: Type: array< System..::.Double >[]()[]
An array of size [n]

On exit: the vector QTf .

ifail: Type: System..::.Int32 %

On exit: ifail=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

The system of equations is defined as:

fi x1,x2,…,xn = 0 , for i= 1, 2, …, n .

c05nc 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. Under reasonable conditions this guarantees global convergence for starting points far from the solution and a fast rate of convergence. 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).

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, iuser, ruser) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

ifail<0: This indicates an exit from c05nc because you have set iflag negative in fcn. The value of ifail will be the same as your setting of iflag.

ifail=1: On entry, n≤0 ,
or xtol<0.0 ,
or maxfev≤0 ,
or ml<0 ,
or mu<0 ,
or factor≤0.0 ,
or mode=2 and diag[i]≤0.0 for some i, i= 1, 2, …, n .

ifail=2: There have been at least maxfev evaluations of fcn. Consider restarting the calculation from the final point held in x.

ifail=3: No further improvement in the approximate solution x is possible; xtol is too small.

ifail=4: The iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations.

ifail=5: The iteration is not making good progress, as measured by the improvement from the last ten iterations.

ifail=-999: Internal memory allocation failed.

ifail=-4000: Invalid dimension for array value
ifail=-8000: Negative dimension for array value
ifail=-6000: Invalid Parameters value

The values ifail=4 and 5 may indicate that the system does not have a zero, or that the solution is very close to the origin (see [Accuracy]). Otherwise, rerunning c05nc from a different starting point may avoid the region of difficulty.

Accuracy

If x^ is the true solution and D denotes the diagonal matrix whose entries are defined by the array diag, then c05nc tries to ensure that

D x-x^ 2 ≤ xtol × D x^ 2 .

If this condition is satisfied with 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 c05nc usually avoids 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 test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then c05nc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05nc with a tighter tolerance.

Further Comments

Local workspace arrays of fixed lengths are allocated internally by c05nc. The total size of these arrays amounts to 4×n real elements.

The time required by c05nc 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 c05nc to process each call of fcn is about 11.5×n2 . Unless fcn can be evaluated quickly, the timing of c05nc will be strongly influenced by the time spent in fcn.

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.

Example

This example determines the values x1 , … , x9 which satisfy the tridiagonal equations:

3-2x1x1-2x2 = -1, -xi-1+3-2xixi-2xi+1 = -1, i=2,3,…,8 -x8+3-2x9x9 = -1.

Example program (C#): c05nce.cs

Example program results: c05nce.r

On entry,	n≤0 ,
or	xtol<0.0 ,
or	maxfev≤0 ,
or	ml<0 ,
or	mu<0 ,
or	factor≤0.0 ,
or	mode=2 and diag[i]≤0.0 for some i, i= 1, 2, …, n .