c05qd is a comprehensive reverse communication method that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

Syntax

C#
```public static void c05qd(
ref int irevcm,
int n,
double[] x,
double[] fvec,
double xtol,
int ml,
int mu,
double epsfcn,
int mode,
double[] diag,
double factor,
double[,] fjac,
double[] r,
double[] qtf,
C05..::..c05qdCommunications communications,
out int ifail
)```
Visual Basic
```Public Shared Sub c05qd ( _
ByRef irevcm As Integer, _
n As Integer, _
x As Double(), _
fvec As Double(), _
xtol As Double, _
ml As Integer, _
mu As Integer, _
epsfcn As Double, _
mode As Integer, _
diag As Double(), _
factor As Double, _
fjac As Double(,), _
r As Double(), _
qtf As Double(), _
communications As C05..::..c05qdCommunications, _
<OutAttribute> ByRef ifail As Integer _
)```
Visual C++
```public:
static void c05qd(
int% irevcm,
int n,
array<double>^ x,
array<double>^ fvec,
double xtol,
int ml,
int mu,
double epsfcn,
int mode,
array<double>^ diag,
double factor,
array<double,2>^ fjac,
array<double>^ r,
array<double>^ qtf,
C05..::..c05qdCommunications^ communications,
[OutAttribute] int% ifail
)```
F#
```static member c05qd :
irevcm : int byref *
n : int *
x : float[] *
fvec : float[] *
xtol : float *
ml : int *
mu : int *
epsfcn : float *
mode : int *
diag : float[] *
factor : float *
fjac : float[,] *
r : float[] *
qtf : float[] *
communications : C05..::..c05qdCommunications *
ifail : int byref -> unit
```

Parameters

irevcm
Type: System..::..Int32%
On initial entry: must have the value $0$.
On intermediate exit: specifies what action you must take before re-entering c05qd with irevcm unchanged. 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 c05qd, 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 $2$.
n
Type: System..::..Int32
On entry: $n$, the number of equations.
Constraint: ${\mathbf{n}}>0$.
x
Type: array<System..::..Double>[]()[][]
An array of size [n]
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.
fvec
Type: array<System..::..Double>[]()[][]
An array of size [n]
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.
xtol
Type: System..::..Double
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 x02aj.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
ml
Type: System..::..Int32
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: ${\mathbf{ml}}\ge 0$.
mu
Type: System..::..Int32
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: ${\mathbf{mu}}\ge 0$.
epsfcn
Type: System..::..Double
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 x02aj) then machine precision is used. Consequently a value of $0.0$ will often be suitable.
Suggested value: ${\mathbf{epsfcn}}=0.0$.
mode
Type: System..::..Int32
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 $2$.
diag
Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: if ${\mathbf{mode}}=2$, diag must contain multiplicative scale factors for the variables.
If ${\mathbf{mode}}=1$, diag need not be set.
Constraint: if ${\mathbf{mode}}=2$, ${\mathbf{diag}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{mode}}=1$).
factor
Type: System..::..Double
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 ${\mathbf{factor}}×{‖{\mathbf{diag}}×{\mathbf{x}}‖}_{2}$ if this is nonzero; otherwise the bound is factor.)
Suggested value: ${\mathbf{factor}}=100.0$.
Constraint: ${\mathbf{factor}}>0.0$.
fjac
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [n, n]
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.
r
Type: array<System..::..Double>[]()[][]
An array of size [${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$]
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.
qtf
Type: array<System..::..Double>[]()[][]
An array of size [n]
On initial entry: need not be set.
On intermediate exit: must not be changed.
On final exit: the vector ${Q}^{\mathrm{T}}f$.
communications
Type: NagLibrary..::..C05..::..c05qdCommunications
An Object of type C05.c05qdCommunications.
ifail
Type: System..::..Int32%
On initial entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Library Overview for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this method the values of the output parameters may be useful even if ${\mathbf{ifail}}\ne {0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
On final exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The system of equations is defined as:
 $fix1,x2,…,xn=0, i=1,2,…,n.$
c05qd 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).

References

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

Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=2$
On entry, ${\mathbf{irevcm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irevcm}}=0$, $1$ or $2$.
${\mathbf{ifail}}=3$
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=4$
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ Jacobian evaluations.
${\mathbf{ifail}}=5$
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=12$
On entry, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
${\mathbf{ifail}}=13$
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=1$ or $2$.
${\mathbf{ifail}}=14$
On entry, ${\mathbf{factor}}=〈\mathit{\text{value}}〉$.
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, ${\mathbf{ml}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ml}}\ge 0$.
${\mathbf{ifail}}=17$
On entry, ${\mathbf{mu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mu}}\ge 0$.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-3001$
parameter : n passed into the options/communications class constructor is invalid
${\mathbf{ifail}}=-4000$
Invalid dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then c05qd tries to ensure that
 $Dx-x^2≤xtol×Dx^2.$
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 c05qd 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 method exits with ${\mathbf{ifail}}={3}$.
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 c05qd may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qd with a lower value for xtol.

Parallelism and Performance

None.

The time required by c05qd 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 c05qd to process the evaluation of functions in the main program in each exit is approximately $11.5×{n}^{2}$. The timing of c05qd 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.

Example

This example determines the values ${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
 $3-2x1x1-2x2=-1,-xi-1+3-2xixi-2xi+1=-1, i=2,3,…,8-x8+3-2x9x9=-1.$

Example program (C#): c05qde.cs

Example program results: c05qde.r