c05rdc is a comprehensive reverse communication function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.
c05rdc is based on the MINPACK routine HYBRJ (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. 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
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 thanfvec and fjac must remain unchanged.
1: $\mathbf{irevcm}$ – Integer *Input/Output
On initial entry: must have the value $0$.
On intermediate exit:
specifies what action you must take before re-entering c05rdc with irevcmunchanged. The value of irevcm should be interpreted as follows:
${\mathbf{irevcm}}=1$
Indicates the start of a new iteration. No action is required by you, but x and fvec are available for printing.
${\mathbf{irevcm}}=2$
Indicates that before re-entry to c05rdc, fvec must contain the function values ${f}_{i}\left(x\right)$.
${\mathbf{irevcm}}=3$
Indicates that before re-entry to c05rdc,
${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)\times {\mathbf{n}}+\mathit{i}-1\right]$ 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 $\mathit{j}=1,2,\dots ,n$.
On final exit: ${\mathbf{irevcm}}=0$ and the algorithm has terminated.
Constraint:
${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
Note: any values you return to c05rdc 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 c05rdc. If your code inadvertently does return any NaNs or infinities, c05rdc 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 initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}\ne 3$, fjac must not be changed.
If ${\mathbf{irevcm}}=3$,
${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)\times {\mathbf{n}}+\mathit{i}-1\right]$ 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 $\mathit{j}=1,2,\dots ,n$.
On final exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.
6: $\mathbf{xtol}$ – doubleInput
On initial 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: $\mathbf{scale\_mode}$ – Nag_ScaleTypeInput
On initial entry: indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$, the scaling must have been supplied 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 final exit: the scale factors actually used (computed internally if ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$).
9: $\mathbf{factor}$ – doubleInput
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 ${\mathbf{factor}}\times {\Vert {\mathbf{diag}}\times {\mathbf{x}}\Vert}_{2}$ if this is nonzero; otherwise the bound is factor.)
The arrays iwsav and rwsav MUST NOT be altered between calls to c05rdc.
14: $\mathbf{fail}$ – 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, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
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. 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 c05rdc from a different starting point may avoid the region of difficulty.
The iteration is not making good progress, as measured by the improvement from the last $\u27e8\mathit{\text{value}}\u27e9$ Jacobian evaluations. 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 c05rdc 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, ${\mathbf{factor}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{factor}}>0.0$.
On entry, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
If $\hat{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05rdc 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 c05rdc 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 c05rdc 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 c05rdc with a lower value for xtol.
8Parallelism and Performance
c05rdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rdc 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
The time required by c05rdc 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 c05rdc 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 c05rdc 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: