c05qcc 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).
4References
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
5Arguments
1: $\mathbf{fcn}$ – function, supplied by the userExternal Function
fcn must return the values of the functions ${f}_{i}$ at a point $x$, unless ${\mathbf{iflag}}=0$ on entry to fcn.
On entry: if ${\mathbf{iflag}}=0$, fvec contains the function values ${f}_{i}\left(x\right)$ and must not be changed.
On exit: if ${\mathbf{iflag}}>0$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless iflag is set to a negative value by fcn).
4: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to fcn.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling c05qcc you may allocate memory and initialize these pointers with various quantities for use by fcn when called from c05qcc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
5: $\mathbf{iflag}$ – Integer *Input/Output
On entry: ${\mathbf{iflag}}\ge 0$.
${\mathbf{iflag}}=0$
x and fvec are available for printing (see nprint).
On exit: in general, iflag should not be reset by fcn. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), iflag should be set to a negative integer.
Note:fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05qcc. If your code inadvertently does return any NaNs or infinities, c05qcc is likely to produce unexpected results.
On exit: the function values at the final point returned in x.
5: $\mathbf{xtol}$ – doubleInput
On entry: the accuracy in x to which the solution is required.
Suggested value:
$\sqrt{\epsilon}$, where $\epsilon $ is the machine precision returned by X02AJC.
Constraint:
${\mathbf{xtol}}\ge 0.0$.
6: $\mathbf{maxfev}$ – IntegerInput
On entry: the maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. c05qcc will exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_TOO_MANY_FEVALS, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
On entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ${\mathbf{ml}}={\mathbf{n}}-1$.)
Constraint:
${\mathbf{ml}}\ge 0$.
8: $\mathbf{mu}$ – IntegerInput
On entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ${\mathbf{mu}}={\mathbf{n}}-1$.)
Constraint:
${\mathbf{mu}}\ge 0$.
9: $\mathbf{epsfcn}$ – doubleInput
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 X02AJC) then machine precision is used. Consequently a value of $0.0$ will often be suitable.
Suggested value:
${\mathbf{epsfcn}}=0.0$.
10: $\mathbf{scale\_mode}$ – Nag_ScaleTypeInput
On entry: indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$, the scaling must have been specified in diag.
Otherwise, if ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$, the variables will be scaled internally.
Constraint:
${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$ or $\mathrm{Nag\_ScaleProvided}$.
On entry: if ${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$, diag must contain multiplicative scale factors for the variables.
If ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$, diag need not be set.
Constraint:
if ${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$, ${\mathbf{diag}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$).
12: $\mathbf{factor}$ – doubleInput
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 ${\mathbf{factor}}\times {\Vert {\mathbf{diag}}\times {\mathbf{x}}\Vert}_{2}$ if this is nonzero; otherwise the bound is factor.)
Suggested value:
${\mathbf{factor}}=100.0$.
Constraint:
${\mathbf{factor}}>0.0$.
13: $\mathbf{nprint}$ – IntegerInput
On entry: indicates whether (and how often) special calls to fcn, with iflag set to $0$, are to be made for printing purposes.
${\mathbf{nprint}}\le 0$
No calls are made.
${\mathbf{nprint}}>0$
fcn is called at the beginning of the first iteration, every nprint iterations thereafter and immediately before the return from c05qcc.
14: $\mathbf{nfev}$ – Integer *Output
On exit: the number of calls made to fcn with ${\mathbf{iflag}}>0$.
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
19: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_DIAG_ELEMENTS
On entry, ${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$ and diag contained a non-positive element.
NE_INT
On entry, ${\mathbf{maxfev}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{maxfev}}>0$.
On entry, ${\mathbf{ml}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ml}}\ge 0$.
On entry, ${\mathbf{mu}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{mu}}\ge 0$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
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 $\u27e8\mathit{\text{value}}\u27e9$ iterations.
The iteration is not making good progress, as measured by the improvement from the last $\u27e8\mathit{\text{value}}\u27e9$ 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, ${\mathbf{factor}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{factor}}>0.0$.
On entry, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least maxfev calls to fcn: ${\mathbf{maxfev}}=\u27e8\mathit{\text{value}}\u27e9$. Consider restarting the calculation from the final point held in x.
NE_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_USER_STOP
iflag was set negative in fcn. ${\mathbf{iflag}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
If $\hat{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05qcc tries to ensure that
If this condition is satisfied with ${\mathbf{xtol}}={10}^{-k}$, then the larger components of $Dx$ have $k$ significant decimal digits. There is a danger that the smaller components of $Dx$ may have large relative errors, but the fast rate of convergence of c05qcc 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$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 c05qcc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qcc with a lower value for xtol.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
c05qcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qcc 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.
9Further Comments
Local workspace arrays of fixed lengths are allocated internally by c05qcc. The total size of these arrays amounts to $4\times n$ double elements.
The time required by c05qcc 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 c05qcc to process each evaluation of the functions is approximately $11.5\times {n}^{2}$. The timing of c05qcc 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.
10Example
This example determines the values ${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations: