c05qdc is a comprehensive reverse communication function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

2 Specification

#include <nag.h>

void	c05qdc (Integer irevcm, Integer n, double x[], double fvec[], double xtol, Integer ml, Integer mu, double epsfcn, Nag_ScaleType scale_mode, double diag[], double factor, double fjac[], double r[], double qtf[], Integer iwsav[], double rwsav[], NagError fail)

The function may be called by the names: c05qdc, nag_roots_sys_func_rcomm or nag_zero_nonlin_eqns_rcomm.

3 Description

The system of equations is defined as:

f_{i} (x_{1}, x_{2}, \dots, x_{n}) = 0, i = 1, 2, \dots, n .

c05qdc 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).

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

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than fvec must remain unchanged.

1: $irevcm$ – Integer * Input/Output

On initial entry: must have the value

0

On intermediate exit: specifies what action you must take before re-entering c05qdc with irevcm unchanged. The value of irevcm should be interpreted as follows:

$irevcm = 1$: Indicates the start of a new iteration. No action is required by you, but x and fvec are available for printing.
$irevcm = 2$: Indicates that before re-entry to c05qdc, fvec must contain the function values $f_{i} (x)$ .

On final exit:

irevcm = 0

and the algorithm has terminated.

Constraint:

irevcm = 0

1

2

Note: any values you return to c05qdc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by c05qdc. If your code inadvertently does return any NaNs or infinities, c05qdc is likely to produce unexpected results.

2: $n$ – Integer Input

On entry:

n

, the number of equations.

Constraint:

n > 0

3: $x [n]$ – double Input/Output

On initial entry: an initial guess at the solution vector.

On intermediate exit: contains the current point.

On final exit: the final estimate of the solution vector.

4: $fvec [n]$ – double Input/Output

On initial entry: need not be set.

On intermediate re-entry: if

irevcm = 1

, fvec must not be changed.

irevcm = 2

, fvec must be set to the values of the functions computed at the current point x.

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

5: $xtol$ – double Input

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

Suggested value:

\sqrt{ε}

, where

ε

is the machine precision returned by X02AJC.

Constraint:

xtol \geq 0.0

6: $ml$ – Integer Input

On initial 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

7: $mu$ – Integer Input

On initial 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

8: $epsfcn$ – double Input

On initial entry: the order 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 X02AJC) then machine precision is used. Consequently a value of

0.0

will often be suitable.

Suggested value:

epsfcn = 0.0

9: $scale_mode$ – Nag_ScaleType Input

On initial entry: indicates whether or not you have provided scaling factors in diag.

scale_mode = Nag_ScaleProvided

, the scaling must have been supplied in diag.

Otherwise, if

scale_mode = Nag_NoScaleProvided

, the variables will be scaled internally.

Constraint:

scale_mode = Nag_NoScaleProvided

Nag_ScaleProvided

10: $diag [n]$ – double Input/Output

On entry: if

scale_mode = Nag_ScaleProvided

, diag must contain multiplicative scale factors for the variables.

scale_mode = Nag_NoScaleProvided

, diag need not be set.

Constraint: if

scale_mode = Nag_ScaleProvided

diag [i - 1] > 0.0

, for

i = 1, 2, \dots, n

On exit: the scale factors actually used (computed internally if

scale_mode = Nag_NoScaleProvided

11: $factor$ – double Input

On initial 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

12: $fjac [n \times n]$ – double Input/Output

Note: the

(i, j)

th element of the matrix is stored in

fjac [(j - 1) \times n + i - 1]

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the orthogonal matrix

Q

produced by the

Q R

factorization of the final approximate Jacobian.

13: $r [n \times (n + 1) / 2]$ – double Input/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the upper triangular matrix

R

produced by the

Q R

factorization of the final approximate Jacobian, stored row-wise.

14: $qtf [n]$ – double Input/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the vector

Q^{T} f

15: $iwsav [17]$ – Integer Communication Array

16: $rwsav [4 \times n + 10]$ – double Communication Array

The arrays iwsav and rwsav MUST NOT be altered between calls to c05qdc.

17: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_DIAG_ELEMENTS: On entry, $scale_mode = Nag_ScaleProvided$ and diag contained a non-positive element.
NE_INT: On entry, $irevcm = ⟨ value ⟩$ .
Constraint: $irevcm = 0$ , $1$ or $2$ .

On entry, $ml = ⟨ value ⟩$ .
Constraint: $ml \geq 0$ .

On entry, $mu = ⟨ value ⟩$ .
Constraint: $mu \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_IMPROVEMENT: The iteration is not making good progress, as measured by the improvement from the last $⟨ value ⟩$ iterations.

The iteration is not making good progress, as measured by the improvement from the last $⟨ value ⟩$ Jacobian evaluations.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $factor = ⟨ value ⟩$ .
Constraint: $factor > 0.0$ .

On entry, $xtol = ⟨ value ⟩$ .
Constraint: $xtol \geq 0.0$ .
NE_TOO_SMALL: No further improvement in the solution is possible. xtol is too small: $xtol = ⟨ value ⟩$ .

7 Accuracy

\hat{x}

is the true solution and

D

denotes the diagonal matrix whose entries are defined by the array diag, then c05qdc 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 c05qdc 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 function exits with

fail . code =

NE_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 convergence test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then c05qdc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qdc with a lower value for xtol.

8 Parallelism and Performance

c05qdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c05qdc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time required by c05qdc 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 c05qdc to process the evaluation of functions in the main program in each exit is approximately