C05 Chapter Contents
C05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC05PCF/C05PCA

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

C05PCF/C05PCA is a comprehensive routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

## 2  Specification

### 2.1  Specification for C05PCF

 SUBROUTINE C05PCF ( FCN, N, X, FVEC, FJAC, LDFJAC, XTOL, MAXFEV, DIAG, MODE, FACTOR, NPRINT, NFEV, NJEV, R, LR, QTF, W, IFAIL)
 INTEGER N, LDFJAC, MAXFEV, MODE, NPRINT, NFEV, NJEV, LR, IFAIL REAL (KIND=nag_wp) X(N), FVEC(N), FJAC(LDFJAC,N), XTOL, DIAG(N), FACTOR, R(N*(N+1)/2), QTF(N), W(1,1) EXTERNAL FCN

### 2.2  Specification for C05PCA

 SUBROUTINE C05PCA ( FCN, N, X, FVEC, FJAC, LDFJAC, XTOL, MAXFEV, DIAG, MODE, FACTOR, NPRINT, NFEV, NJEV, R, LR, QTF, W, IUSER, RUSER, IFAIL)
 INTEGER N, LDFJAC, MAXFEV, MODE, NPRINT, NFEV, NJEV, LR, IUSER(*), IFAIL REAL (KIND=nag_wp) X(N), FVEC(N), FJAC(LDFJAC,N), XTOL, DIAG(N), FACTOR, R(N*(N+1)/2), QTF(N), W(1,1), RUSER(*) EXTERNAL FCN

## 3  Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
C05PCF/C05PCA 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. At the starting point, the Jacobian is calculated, but it is not recalculated until the rank-1 method fails to produce satisfactory progress. For more details see Powell (1970).

## 4  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

## 5  Parameters

1:     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 $x$.
The specification of FCN for C05PCF is:
 SUBROUTINE FCN ( N, X, FVEC, FJAC, LDFJAC, IFLAG)
 INTEGER N, LDFJAC, IFLAG REAL (KIND=nag_wp) X(N), FVEC(N), FJAC(LDFJAC,N)
The specification of FCN for C05PCA is:
 SUBROUTINE FCN ( N, X, FVEC, FJAC, LDFJAC, IFLAG, IUSER, RUSER)
 INTEGER N, LDFJAC, IFLAG, IUSER(*) REAL (KIND=nag_wp) X(N), FVEC(N), FJAC(LDFJAC,N), RUSER(*)
1:     N – INTEGERInput
On entry: $n$, the number of equations.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.
3:     FVEC(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if ${\mathbf{IFLAG}}=0$ or $2$, 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:     FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if ${\mathbf{IFLAG}}=0$, ${\mathbf{FJAC}}\left(\mathit{i},\mathit{j}\right)$ 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 $\mathit{j}=1,2,\dots ,n$. When ${\mathbf{IFLAG}}=0$ or $1$, FJAC must not be changed.
On exit: if ${\mathbf{IFLAG}}=2$ on entry, ${\mathbf{FJAC}}\left(\mathit{i},\mathit{j}\right)$ 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:     LDFJAC – INTEGERInput
On entry: the first dimension of the array FJAC as declared in the (sub)program from which C05PCF/C05PCA is called.
6:     IFLAG – INTEGERInput/Output
On entry: ${\mathbf{IFLAG}}=0$, $1$ or $2$.
${\mathbf{IFLAG}}=0$
X and FVEC are available for printing (see NPRINT).
${\mathbf{IFLAG}}=1$
FVEC is to be updated.
${\mathbf{IFLAG}}=2$
FJAC is to be updated.
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), then IFLAG should be set to a negative integer. This value will be returned through IFAIL.
Note: the following are additional parameters for specific use with C05PCA. Users of C05PCF therefore need not read the remainder of this description.
7:     IUSER($*$) – INTEGER arrayUser Workspace
8:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
FCN is called with the parameters IUSER and RUSER as supplied to C05PCF/C05PCA. You are free to use the arrays IUSER and RUSER to supply information to FCN as an alternative to using COMMON global variables.
FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05PCF/C05PCA is called. Parameters denoted as Input must not be changed by this procedure.
2:     N – INTEGERInput
On initial entry: $n$, the number of equations.
Constraint: ${\mathbf{N}}>0$.
3:     X(N) – 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:     FVEC(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the function values at the final point returned in X.
5:     FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorisation of the final approximate Jacobian.
6:     LDFJAC – INTEGERInput
On entry: the first dimension of the array FJAC as declared in the (sub)program from which C05PCF/C05PCA is called.
Constraint: ${\mathbf{LDFJAC}}\ge {\mathbf{N}}$.
7:     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.
Constraint: ${\mathbf{XTOL}}\ge 0.0$.
8:     MAXFEV – INTEGERInput
On entry: the maximum number of calls to FCN with ${\mathbf{IFLAG}}\ne 0$. C05PCF/C05PCA will exit with ${\mathbf{IFAIL}}={\mathbf{2}}$, if, at the end of an iteration, the number of calls to FCN exceeds MAXFEV.
Suggested value: ${\mathbf{MAXFEV}}=100×\left({\mathbf{N}}+1\right)$.
Constraint: ${\mathbf{MAXFEV}}>0$.
9:     DIAG(N) – REAL (KIND=nag_wp) arrayInput/Output
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 ${\mathbf{MODE}}=2$, ${\mathbf{DIAG}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{MODE}}=1$).
10:   MODE – INTEGERInput
On entry: indicates whether or not you have provided scaling factors in DIAG.
If ${\mathbf{MODE}}=2$ the scaling must have been specified in DIAG.
Otherwise, the variables will be scaled internally.
11:   FACTOR – REAL (KIND=nag_wp)Input
On 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: ${\mathbf{FACTOR}}>0.0$.
12:   NPRINT – INTEGERInput
On entry: indicates whether (and how often) special calls to FCN, with IFLAG set to $0$, are to be made for printing purposes.
${\mathbf{NPRINT}}\le 0$
${\mathbf{NPRINT}}>0$
FCN is called at the beginning of the first iteration, every NPRINT iterations thereafter and immediately before the return from C05PCF/C05PCA.
13:   NFEV – INTEGEROutput
On exit: the number of calls made to FCN to evaluate the functions.
14:   NJEV – INTEGEROutput
On exit: the number of calls made to FCN to evaluate the Jacobian.
15:   R(${\mathbf{N}}×\left({\mathbf{N}}+1\right)/2$) – REAL (KIND=nag_wp) arrayOutput
On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.
16:   LR – INTEGERDummy
This parameter is no longer accessed by C05PCF/C05PCA.
17:   QTF(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the vector ${Q}^{\mathrm{T}}f$.
18:   W($1$,$1$) – REAL (KIND=nag_wp) arrayInput
This parameter is no longer accessed by C05PCF/C05PCA. Workspace is provided internally by dynamic allocation instead.
19:   IFAIL – INTEGERInput/Output
Note: for C05PCA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).
Note: the following are additional parameters for specific use with C05PCA. Users of C05PCF therefore need not read the remainder of this description.
19:   IUSER($*$) – INTEGER arrayUser Workspace
20:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
IUSER and RUSER are not used by C05PCF/C05PCA, but are passed directly to FCN and may be used to pass information to this routine as an alternative to using COMMON global variables.
21:   IFAIL – INTEGERInput/Output
Note: see the parameter description for IFAIL above.

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}<0$
This indicates an exit from C05PCF/C05PCA because you have set IFLAG negative in FCN. The value of IFAIL will be the same as your setting of IFLAG.
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{N}}\le 0$, or ${\mathbf{XTOL}}<0.0$, or ${\mathbf{MAXFEV}}\le 0$, or ${\mathbf{FACTOR}}\le 0.0$, or ${\mathbf{LDFJAC}}<{\mathbf{N}}$, or ${\mathbf{MODE}}=2$ and ${\mathbf{DIAG}}\left(i\right)\le 0.0$ for some $i$, $i=1,2,\dots ,{\mathbf{N}}$.
${\mathbf{IFAIL}}=2$
There have been MAXFEV evaluations of FCN to evaluate the functions. Consider restarting the calculation from the final point held in X.
${\mathbf{IFAIL}}=3$
No further improvement in the approximate solution X is possible; XTOL is too small.
${\mathbf{IFAIL}}=4$
The iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations.
${\mathbf{IFAIL}}=5$
The iteration is not making good progress, as measured by the improvement from the last ten iterations.
${\mathbf{IFAIL}}=-999$
Internal memory allocation failed.
The values ${\mathbf{IFAIL}}={\mathbf{4}}$ and ${\mathbf{5}}$ 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 C05PCF/C05PCA from a different starting point may avoid the region of difficulty.

## 7  Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array DIAG, then C05PCF/C05PCA tries to ensure that
 $D x-x^ 2 ≤ XTOL × D x^ 2 .$
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 C05PCF/C05PCA 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 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 C05PCF/C05PCA may incorrectly indicate convergence. The coding of the Jacobian can be checked using C05ZAF. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning C05PCF/C05PCA with a lower value for XTOL.

Local workspace arrays of fixed lengths are allocated internally by C05PCF/C05PCA. The total size of these arrays amounts to $4×{\mathbf{N}}$ real elements.
The time required by C05PCF/C05PCA 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 C05PCF/C05PCA is about $11.5×{n}^{2}$ to process each evaluation of the functions and about $1.3×{n}^{3}$ to process each evaluation of the Jacobian. Unless FCN can be evaluated quickly, the timing of C05PCF/C05PCA will be strongly influenced by the time spent in FCN.
Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.

## 9  Example

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

### 9.1  Program Text

Program Text (c05pcfe.f90)

Program Text (c05pcae.f90)

None.

### 9.3  Program Results

Program Results (c05pcfe.r)

Program Results (c05pcae.r)