c05qc is a comprehensive method that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

Syntax

C#
public static void c05qc( C05..::..C05QC_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 c05qc(
	C05..::..C05QC_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
Public Shared Sub c05qc ( _ fcn As C05..::..C05QC_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

Public Shared Sub c05qc ( _
	fcn As C05..::..C05QC_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 c05qc( C05..::..C05QC_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 c05qc(
	C05..::..C05QC_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 c05qc : fcn : C05..::..C05QC_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 c05qc : 
        fcn : C05..::..C05QC_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..::..C05QC_FCN
fcn must return the values of the functions $f_{i}$ at a point $x$ , unless $iflag = 0$ on entry to c05qc.
A delegate of type C05QC_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.

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 $iflag \neq 0$ . c05qc will exit with $ifail = 2$ , if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Suggested value: $maxfev = 200 \times (n + 1)$ .

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: $ml \geq 0$ .

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: $mu \geq 0$ .

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 (returned by x02aj) 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, 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 c05qc.

nfev: Type: System..::..Int32%
On exit: the number of calls made to fcn with $iflag > 0$ .

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 $Q R$ factorisation of the final approximate 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 .

c05qc 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).

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.

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

$ifail = 6$: 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.

$ifail = 16$: On entry, $ml = 〈value〉$ .
Constraint: $ml \geq 0$ .

$ifail = 17$: On entry, $mu = 〈value〉$ .
Constraint: $mu \geq 0$ .

$ifail = 18$: 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 c05qc 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 c05qc 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 are reasonably well behaved. If this condition is not satisfied, then c05qc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qc with a lower value for xtol.

Parallelism and Performance

None.

Further Comments

Local workspace arrays of fixed lengths are allocated internally by c05qc. The total size of these arrays amounts to

4 \times n

real elements.

The time required by c05qc 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 c05qc to process each evaluation of the functions is approximately

11.5 \times n^{2}

. The timing of c05qc 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.

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#): c05qce.cs

Example program results: c05qce.r