c05rb is an easy-to-use 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 c05rb( C05..::..C05RB_FCN fcn, int n, double[] x, double[] fvec, double[,] fjac, double xtol, out int ifail )

Visual Basic
Public Shared Sub c05rb ( _ fcn As C05..::..C05RB_FCN, _ n As Integer, _ x As Double(), _ fvec As Double(), _ fjac As Double(,), _ xtol As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void c05rb( C05..::..C05RB_FCN^ fcn, int n, array<double>^ x, array<double>^ fvec, array<double,2>^ fjac, double xtol, [OutAttribute] int% ifail )

F#
static member c05rb : fcn : C05..::..C05RB_FCN * n : int * x : float[] * fvec : float[] * fjac : float[,] * xtol : float * ifail : int byref -> unit

Parameters

fcn: Type: NagLibrary..::..C05..::..C05RB_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 C05RB_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: $xtol \geq 0.0$ .

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 .

c05rb is based on the MINPACK routine HYBRJ1 (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 $100 \times (n + 1)$ 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 c05rb 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.
$ifail = -6000$: 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, c05rb 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 c05rb 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 c05rb 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 c05rb with a lower value for xtol.

Parallelism and Performance

None.

Further Comments

Local workspace arrays of fixed lengths are allocated internally by c05rb. The total size of these arrays amounts to

n \times (n + 13) / 2

real elements.

The time required by c05rb 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 c05rb 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 c05rb 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#): c05rbe.cs

Example program results: c05rbe.r