c05qb is an easy-to-use method that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

Syntax

C#
public static void c05qb( C05..::..C05QB_FCN fcn, int n, double[] x, double[] fvec, double xtol, out int ifail )

Visual Basic
Public Shared Sub c05qb ( _ fcn As C05..::..C05QB_FCN, _ n As Integer, _ x As Double(), _ fvec As Double(), _ xtol As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void c05qb( C05..::..C05QB_FCN^ fcn, int n, array<double>^ x, array<double>^ fvec, double xtol, [OutAttribute] int% ifail )

F#
static member c05qb : fcn : C05..::..C05QB_FCN * n : int * x : float[] * fvec : float[] * xtol : float * ifail : int byref -> unit

Parameters

fcn: Type: NagLibrary..::..C05..::..C05QB_FCN
fcn must return the values of the functions $f_{i}$ at a point $x$ .
A delegate of type C05QB_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: $xtol \geq 0.0$ .

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 .

c05qb is based on the MINPACK routine HYBRD1 (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:

$ifail = 2$: There have been at least $200 \times (n + 1)$ calls to fcn. Consider restarting the calculation from the 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. 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 c05qb from a different starting point may avoid the region of difficulty.

$ifail = 5$: iflag was set negative in fcn. $iflag = 〈value〉$ .

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

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

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

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

Accuracy

\hat{x}

is the true solution, c05qb tries to ensure that

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

If this condition is satisfied with

xtol = 10^{- k}

, then the larger components of

x

have

k

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

x

may have large relative errors, but the fast rate of convergence of c05qb 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 c05qb may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qb with a lower value for xtol.

Parallelism and Performance

None.

Further Comments

Local workspace arrays of fixed lengths are allocated internally by c05qb. The total size of these arrays amounts to

n \times (3 \times n + 13) / 2

real elements.

The time required by c05qb 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 c05qb to process each evaluation of the functions is approximately

11.5 \times n^{2}

. The timing of c05qb 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#): c05qbe.cs

Example program results: c05qbe.r