NAG Library Function Document

nag_zero_nonlin_eqns (c05nbc)

nag_zero_nonlin_eqns (c05nbc) is based upon 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. 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).

4 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

5 Arguments

1: n – IntegerInput

On entry:

n

, the number of equations.

Constraint:

n > 0

2: x[n] – doubleInput/Output

On entry: an initial guess at the solution vector.

On exit: the final estimate of the solution vector.

3: fvec[n] – doubleOutput

On exit: the function values at the final point, x.

4: f – function, supplied by the userExternal Function

f must return the values of the

f_{i}

at a point

x

The specification of f is:

void	f (Integer n, const double x[], double fvec[], Integer *userflag)

1: n – IntegerInput: On entry: $n$ , the number of equations.
2: x[n] – const doubleInput: On entry: the components of the point $x$ at which the functions must be evaluated.
3: fvec[n] – doubleOutput: On exit: the function values $f_{i} (x)$ (unless userflag is set to a negative value by f).
4: userflag – Integer *Input/Output: On entry: $userflag > 0$ .

On exit: in general, userflag should not be reset by f. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), then userflag should be set to a negative integer. This value will be returned through $fail . errnum$ .

5: xtol – doubleInput

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

Suggested value: the square root of the machine precision.

Constraint:

xtol \geq 0.0

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LE: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .
NE_NO_IMPROVEMENT: 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 Section 8). Otherwise, rerunning nag_zero_nonlin_eqns (c05nbc) from a different starting point may avoid the region of difficulty.
NE_REAL_ARG_LT: On entry, xtol must not be less than 0.0: $xtol = 〈value〉$ .
NE_TOO_MANY_FUNC_EVAL: There have been at least 200*(n+1) evaluations of f().
Consider restarting the calculation from the point held in x.
NE_USER_STOP: User requested termination, user flag value $= 〈value〉$ .
NE_XTOL_TOO_SMALL: No further improvement in the solution is possible. xtol is too small: $xtol = 〈value〉$ .

7 Accuracy

\hat{x}

is the true solution, nag_zero_nonlin_eqns (c05nbc) tries to ensure that

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

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 nag_zero_nonlin_eqns (c05nbc) 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 function exits with NE_XTOL_TOO_SMALL.

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 nag_zero_nonlin_eqns (c05nbc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns (c05nbc) with a tighter tolerance.

8 Further Comments

The time required by nag_zero_nonlin_eqns (c05nbc) 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 nag_zero_nonlin_eqns (c05nbc) to process each call of f is about