c05rbc is an easy-to-use function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.
c05rbc 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: $\mathbf{fcn}$ – function, supplied by the userExternal Function
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 $x$.
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).
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{fjac}}\left[(j-1)\times {\mathbf{n}}+i-1\right]$.
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,
${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)\times {\mathbf{n}}+\mathit{i}-1\right]$ 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).
5: $\mathbf{comm}$ – 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 c05rbc you may allocate memory and initialize these pointers with various quantities for use by fcn when called from c05rbc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
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 c05rbc. If your code inadvertently does return any NaNs or infinities, c05rbc is likely to produce unexpected results.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{fjac}}\left[(j-1)\times {\mathbf{n}}+i-1\right]$.
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.
6: $\mathbf{xtol}$ – 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:
${\mathbf{xtol}}\ge 0.0$.
7: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
8: $\mathbf{fail}$ – 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_INT
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. 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 c05rbc from a different starting point may avoid the region of difficulty.
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, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least $100\times ({\mathbf{n}}+1)$ calls to fcn. Consider restarting the calculation from the 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, c05rbc 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 c05rbc 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 and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then c05rbc may incorrectly indicate convergence. The coding of the Jacobian can be checked using c05zdc. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning c05rbc with a lower value for xtol.
8Parallelism and Performance
c05rbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rbc 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 c05rbc. The total size of these arrays amounts to $n\times (n+13)/2$ double elements.
The time required by c05rbc 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 c05rbc 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 c05rbc 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: