c05qdf is a comprehensive reverse communication routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.
c05qdf 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
Note: this routine 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 must remain unchanged.
1: $\mathbf{irevcm}$ – IntegerInput/Output
On initial entry: must have the value $0$.
On intermediate exit:
specifies what action you must take before re-entering c05qdf 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 c05qdf, fvec must contain the function values ${f}_{i}\left(x\right)$.
On final exit: ${\mathbf{irevcm}}=0$ and the algorithm has terminated.
Constraint:
${\mathbf{irevcm}}=0$, $1$ or $2$.
Note: any values you return to c05qdf 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 c05qdf. If your code does inadvertently return any NaNs or infinities, c05qdf 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 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: $\mathbf{fvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=1$, fvec must not be changed.
If ${\mathbf{irevcm}}=2$, 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: $\mathbf{xtol}$ – Real (Kind=nag_wp)Input
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 x02ajf.
Constraint:
${\mathbf{xtol}}\ge 0.0$.
6: $\mathbf{ml}$ – IntegerInput
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 ${\mathbf{ml}}={\mathbf{n}}-1$.)
Constraint:
${\mathbf{ml}}\ge 0$.
7: $\mathbf{mu}$ – IntegerInput
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 ${\mathbf{mu}}={\mathbf{n}}-1$.)
Constraint:
${\mathbf{mu}}\ge 0$.
8: $\mathbf{epsfcn}$ – Real (Kind=nag_wp)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 x02ajf) then machine precision is used. Consequently a value of $0.0$ will often be suitable.
Suggested value:
${\mathbf{epsfcn}}=0.0$.
9: $\mathbf{mode}$ – IntegerInput
On initial entry: indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{mode}}=2$, the scaling must have been supplied in diag.
Otherwise, if ${\mathbf{mode}}=1$, the variables will be scaled internally.
Constraint:
${\mathbf{mode}}=1$ or $2$.
10: $\mathbf{diag}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{mode}}=2$, diag must contain multiplicative scale factors for the variables.
Constraint:
if ${\mathbf{mode}}=2$, ${\mathbf{diag}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{mode}}=1$).
11: $\mathbf{factor}$ – Real (Kind=nag_wp)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 ${\mathbf{factor}}\times {\Vert {\mathbf{diag}}\times {\mathbf{x}}\Vert}_{2}$ if this is nonzero; otherwise the bound is factor.)
Suggested value:
${\mathbf{factor}}=100.0$.
Constraint:
${\mathbf{factor}}>0.0$.
12: $\mathbf{fjac}({\mathbf{n}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
13: $\mathbf{r}\left({\mathbf{n}}\times ({\mathbf{n}}+1)/2\right)$ – Real (Kind=nag_wp) arrayInput/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 $QR$ factorization of the final approximate Jacobian, stored row-wise.
14: $\mathbf{qtf}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
16: $\mathbf{rwsav}\left(4\times {\mathbf{n}}+10\right)$ – Real (Kind=nag_wp) arrayCommunication Array
The arrays iwsav and rwsavmust not be altered between calls to c05qdf.
17: $\mathbf{ifail}$ – IntegerInput/Output
On initial entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On final 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 $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=2$
On entry, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{irevcm}}=0$, $1$ or $2$.
${\mathbf{ifail}}=3$
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, as measured by the improvement from the last $\u27e8\mathit{\text{value}}\u27e9$ Jacobian evaluations.
${\mathbf{ifail}}=5$
The iteration is not making good progress, as measured by the improvement from the last $\u27e8\mathit{\text{value}}\u27e9$ iterations.
A value of ${\mathbf{ifail}}={\mathbf{4}}$ or ${\mathbf{5}}$ 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 c05qdf from a different starting point may avoid the region of difficulty.
${\mathbf{ifail}}=11$
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: ${\mathbf{xtol}}\ge 0.0$.
${\mathbf{ifail}}=13$
On entry, ${\mathbf{mode}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{mode}}=1$ or $2$.
${\mathbf{ifail}}=14$
On entry, ${\mathbf{factor}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{factor}}>0.0$.
${\mathbf{ifail}}=15$
On entry, ${\mathbf{mode}}=2$ and diag contained a non-positive element.
${\mathbf{ifail}}=16$
On entry, ${\mathbf{ml}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ml}}\ge 0$.
${\mathbf{ifail}}=17$
On entry, ${\mathbf{mu}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{mu}}\ge 0$.
${\mathbf{ifail}}=-99$
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 and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05qdf 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 c05qdf 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 are reasonably well behaved. If this condition is not satisfied, then c05qdf may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qdf with a lower value for xtol.
8Parallelism and Performance
c05qdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qdf 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
The time required by c05qdf 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 c05qdf to process the evaluation of functions in the main program in each exit is approximately $11.5\times {n}^{2}$. The timing of c05qdf 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: