c05 Chapter Contents
c05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zero_nonlin_eqns_deriv (c05pbc)

## 1  Purpose

nag_zero_nonlin_eqns_deriv (c05pbc) finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

## 2  Specification

 #include #include
void  nag_zero_nonlin_eqns_deriv (Integer n, double x[], double fvec[], double fjac[], Integer tdfjac,
 void (*f)(Integer n, const double x[], double fvec[], double fjac[], Integer tdfjac, Integer *userflag),
double xtol, NagError *fail)

## 3  Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , for ​ i= 1, 2, …, n.$
nag_zero_nonlin_eqns_deriv (c05pbc) is based upon 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. Under reasonable conditions this guarantees global convergence for starting points far from the solution and a fast rate of convergence. 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  Arguments

1:     nIntegerInput
On entry: $n$, the number of equations.
Constraint: ${\mathbf{n}}>0$.
2:     x[n]doubleInput/Output
On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
3:     fvec[n]doubleOutput
On exit: the function values at the final point, x.
4:     fjac[${\mathbf{n}}×{\mathbf{tdfjac}}$]doubleOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
5:     tdfjacIntegerInput
On entry: the stride separating matrix column elements in the array fjac.
Constraint: ${\mathbf{tdfjac}}\ge {\mathbf{n}}$.
6:     ffunction, supplied by the userExternal Function
Depending upon the value of userflag, f must either return the values of the functions ${f}_{i}$ at a point $x$ or return the Jacobian at $x$.
The specification of f is:
 void f (Integer n, const double x[], double fvec[], double fjac[], Integer tdfjac, Integer *userflag)
1:     nIntegerInput
On entry: $n$, the number of equations.
2:     x[n]const doubleInput
On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.
3:     fvec[n]doubleOutput
On exit: if ${\mathbf{userflag}}=1$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless userflag is set to a negative value by f).
If ${\mathbf{userflag}}=2$ on entry, fvec must not be changed.
4:     fjac[${\mathbf{n}}×{\mathbf{tdfjac}}$]doubleOutput
On exit: if ${\mathbf{userflag}}=2$ on entry, ${\mathbf{fjac}}\left[\left(\mathit{i}-1\right)×{\mathbf{tdfjac}}+\mathit{j}-1\right]$ 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 userflag is set to a negative value by f).
If ${\mathbf{userflag}}=1$ on entry, fjac must not be changed.
5:     tdfjacIntegerInput
On entry: the stride separating matrix column elements in the array fjac.
6:     userflagInteger *Input/Output
On entry: ${\mathbf{userflag}}=1$ or $2$.
${\mathbf{userflag}}=1$
fvec is to be updated.
${\mathbf{userflag}}=2$
fjac is to be updated.
On exit: in general, userflag should not be reset by f. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), then userflag should be set to a negative integer. This value will be returned through ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$.
7:     xtoldoubleInput
On entry: the accuracy in x to which the solution is required.
Suggested value: the square root of the machine precision.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{tdfjac}}=〈\mathit{\text{value}}〉$ while ${\mathbf{n}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdfjac}}\ge {\mathbf{n}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LE
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
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 8). Otherwise, rerunning nag_zero_nonlin_eqns_deriv (c05pbc) from a different starting point may avoid the region of difficulty.
NE_REAL_ARG_LT
On entry, xtol must not be less than 0.0: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
NE_TOO_MANY_FUNC_EVAL
There have been at least $100×\left({\mathbf{n}}+1\right)$ evaluations of f().
Consider restarting the calculation from the point held in x.
NE_USER_STOP
User requested termination, user flag value $\text{}=〈\mathit{\text{value}}〉$.
NE_XTOL_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

If $\stackrel{^}{x}$ is the true solution, nag_zero_nonlin_eqns_deriv (c05pbc) tries to ensure that
 $x-x^ ≤ xtol × x^ .$
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 nag_zero_nonlin_eqns_deriv (c05pbc) usually avoids 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 NE_XTOL_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 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 nag_zero_nonlin_eqns_deriv (c05pbc) may incorrectly indicate convergence. The coding of the Jacobian can be checked using nag_check_deriv (c05zbc). If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning nag_zero_nonlin_eqns_deriv (c05pbc) with a tighter tolerance.

The time required by nag_zero_nonlin_eqns_deriv (c05pbc) 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 nag_zero_nonlin_eqns_deriv (c05pbc) 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 f can be evaluated quickly, the timing of nag_zero_nonlin_eqns_deriv (c05pbc) will be strongly influenced by the time spent in f.
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 (c05pbce.c)

None.

### 9.3  Program Results

Program Results (c05pbce.r)