c05rbf is an easy-to-use routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.
c05rbf 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).
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: $\mathbf{fcn}$ – Subroutine, supplied by the user.External Procedure
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 $x$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.
3: $\mathbf{fvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{iflag}}=2$, fvec contains the function values ${f}_{i}\left(x\right)$ and must not be changed.
On exit: if ${\mathbf{iflag}}=1$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless iflag is set to a negative value by fcn).
4: $\mathbf{fjac}({\mathbf{n}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{iflag}}=1$, fjac contains the value of
$\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$, and must not be changed.
On exit: if ${\mathbf{iflag}}=2$ on entry,
${\mathbf{fjac}}(\mathit{i},\mathit{j})$ must contain the value of $\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$, (unless iflag is set to a negative value by fcn).
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.
fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05rbf is called. Arguments denoted as Input must not be changed by this procedure.
Note:fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05rbf. If your code inadvertently does return any NaNs or infinities, c05rbf is likely to produce unexpected results.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of equations.
Constraint:
${\mathbf{n}}>0$.
3: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
4: $\mathbf{fvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the function values at the final point returned in x.
5: $\mathbf{fjac}({\mathbf{n}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
6: $\mathbf{xtol}$ – Real (Kind=nag_wp)Input
On entry: the accuracy in x to which the solution is required.
Suggested value:
$\sqrt{\epsilon}$, where $\epsilon $ is the machine precision returned by x02ajf.
8: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by c05rbf, but are passed directly to fcn and may be used to pass information to this routine.
9: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=2$
There have been at least $100\times ({\mathbf{n}}+1)$ calls to fcn. Consider restarting the calculation from the point held in x.
${\mathbf{ifail}}=3$
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=4$
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 Section 7). Otherwise, rerunning c05rbf from a different starting point may avoid the region of difficulty.
${\mathbf{ifail}}=5$
iflag was set negative in fcn. ${\mathbf{iflag}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=12$
On entry, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
If $\hat{x}$ is the true solution, c05rbf tries to ensure that
If this condition is satisfied with ${\mathbf{xtol}}={10}^{-k}$, 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 c05rbf 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 routine exits with ${\mathbf{ifail}}={\mathbf{3}}$.
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 c05rbf may incorrectly indicate convergence. The coding of the Jacobian can be checked using c05zdf. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning c05rbf with a lower value for xtol.
8Parallelism and Performance
c05rbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rbf 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 routine. 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 c05rbf. The total size of these arrays amounts to $n\times (n+13)/2$ real elements.
The time required by c05rbf 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 c05rbf 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 c05rbf 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.
10Example
This example determines the values ${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations: