# NAG C Library Function Document

## 1Purpose

nag_zero_sparse_nonlin_eqns_easy (c05qsc) is an easy-to-use function that finds a solution of a sparse system of nonlinear equations by a modification of the Powell hybrid method.

## 2Specification

 #include #include
void  nag_zero_sparse_nonlin_eqns_easy (
 void (*fcn)(Integer n, Integer lindf, const Integer indf[], const double x[], double fvec[], Nag_Comm *comm, Integer *iflag),
Integer n, double x[], double fvec[], double xtol, Nag_Boolean init, double rcomm[], Integer lrcomm, Integer icomm[], Integer licomm, Nag_Comm *comm, NagError *fail)

## 3Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
nag_zero_sparse_nonlin_eqns_easy (c05qsc) is based on the MINPACK routine HYBRD1 (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 sparse rank-1 method of Schubert (see Schubert (1970)). At the starting point, the sparsity pattern is determined and the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. Then, the sparsity structure is used to recompute an approximation to the Jacobian by forward differences with the least number of function evaluations. The function you supply must be able to compute only the requested subset of the function values. The sparse Jacobian linear system is solved at each iteration with nag_superlu_lu_factorize (f11mec) computing the Newton step. For more details see Powell (1970) and Broyden (1965).
Broyden C G (1965) A class of methods for solving nonlinear simultaneous equations Mathematics of Computation 19(92) 577–593
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
Schubert L K (1970) Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian Mathematics of Computation 24(109) 27–30

## 5Arguments

1:    $\mathbf{fcn}$function, supplied by the userExternal Function
fcn must return the values of the functions ${f}_{i}$ at a point $x$.
The specification of fcn is:
 void fcn (Integer n, Integer lindf, const Integer indf[], const double x[], double fvec[], Nag_Comm *comm, Integer *iflag)
1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of equations.
2:    $\mathbf{lindf}$IntegerInput
On entry: lindf specifies the number of indices $i$ for which values of ${f}_{i}\left(x\right)$ must be computed.
3:    $\mathbf{indf}\left[\mathit{dim}\right]$const IntegerInput
On entry: indf specifies the indices $i$ for which values of ${f}_{i}\left(x\right)$ must be computed. The indices are specified in strictly ascending order.
4:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the components of the point $x$ at which the functions must be evaluated. ${\mathbf{x}}\left[i-1\right]$ contains the coordinate ${x}_{i}$.
5:    $\mathbf{fvec}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${\mathbf{fvec}}\left[i-1\right]$ must contain the function values ${f}_{i}\left(x\right)$, for all indices $i$ in indf.
6:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to fcn.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_zero_sparse_nonlin_eqns_easy (c05qsc) you may allocate memory and initialize these pointers with various quantities for use by fcn when called from nag_zero_sparse_nonlin_eqns_easy (c05qsc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
7:    $\mathbf{iflag}$Integer *Input/Output
On entry: ${\mathbf{iflag}}>0$.
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), iflag should be set to a negative integer.
Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_zero_sparse_nonlin_eqns_easy (c05qsc). If your code inadvertently does return any NaNs or infinities, nag_zero_sparse_nonlin_eqns_easy (c05qsc) is likely to produce unexpected results.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of equations.
Constraint: ${\mathbf{n}}>0$.
3:    $\mathbf{x}\left[{\mathbf{n}}\right]$doubleInput/Output
On entry: an initial guess at the solution vector. ${\mathbf{x}}\left[i-1\right]$ must contain the coordinate ${x}_{i}$.
On exit: the final estimate of the solution vector.
4:    $\mathbf{fvec}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the function values at the final point returned in x. ${\mathbf{fvec}}\left[i-1\right]$ contains the function values ${f}_{i}$.
5:    $\mathbf{xtol}$doubleInput
On entry: the accuracy in x to which the solution is required.
Suggested value: $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision returned by nag_machine_precision (X02AJC).
Constraint: ${\mathbf{xtol}}\ge 0.0$.
6:    $\mathbf{init}$Nag_BooleanInput
On entry: init must be set to Nag_TRUE to indicate that this is the first time nag_zero_sparse_nonlin_eqns_easy (c05qsc) is called for this specific problem. nag_zero_sparse_nonlin_eqns_easy (c05qsc) then computes the dense Jacobian and detects and stores its sparsity pattern (in rcomm and icomm) before proceeding with the iterations. This is noticeably time consuming when n is large. If not enough storage has been provided for rcomm or icomm, nag_zero_sparse_nonlin_eqns_easy (c05qsc) will fail. On exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_NO_IMPROVEMENT, NE_TOO_MANY_FEVALS or NE_TOO_SMALL, ${\mathbf{icomm}}\left[0\right]$ contains $\mathit{nnz}$, the number of nonzero entries found in the Jacobian. On subsequent calls, init can be set to Nag_FALSE if the problem has a Jacobian of the same sparsity pattern. In that case, the computation time required for the detection of the sparsity pattern will be smaller.
7:    $\mathbf{rcomm}\left[{\mathbf{lrcomm}}\right]$doubleCommunication Array
rcomm MUST NOT be altered between successive calls to nag_zero_sparse_nonlin_eqns_easy (c05qsc).
8:    $\mathbf{lrcomm}$IntegerInput
On entry: the dimension of the array rcomm.
Constraint: ${\mathbf{lrcomm}}\ge 12+\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian, as computed by nag_zero_sparse_nonlin_eqns_easy (c05qsc).
9:    $\mathbf{icomm}\left[{\mathbf{licomm}}\right]$IntegerCommunication Array
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_NO_IMPROVEMENT, NE_TOO_MANY_FEVALS or NE_TOO_SMALL on exit, ${\mathbf{icomm}}\left[0\right]$ contains $\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian.
icomm MUST NOT be altered between successive calls to nag_zero_sparse_nonlin_eqns_easy (c05qsc).
10:  $\mathbf{licomm}$IntegerInput
On entry: the dimension of the array icomm.
Constraint: ${\mathbf{licomm}}\ge 8×{\mathbf{n}}+19+\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian, as computed by nag_zero_sparse_nonlin_eqns_easy (c05qsc).
11:  $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{licomm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{licomm}}\ge 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{lrcomm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lrcomm}}\ge 〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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 7). Otherwise, rerunning nag_zero_sparse_nonlin_eqns_easy (c05qsc) from a different starting point may avoid the region of difficulty. The condition number of the Jacobian is $〈\mathit{\text{value}}〉$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least $200×\left({\mathbf{n}}+1\right)$ calls to fcn. Consider setting ${\mathbf{init}}=\mathrm{Nag_FALSE}$ and restarting the calculation from the point held in x.
NE_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
NE_USER_STOP
iflag was set negative in fcn. ${\mathbf{iflag}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

If $\stackrel{^}{x}$ is the true solution, nag_zero_sparse_nonlin_eqns_easy (c05qsc) tries to ensure that
 $x-x^ 2 ≤ xtol × x^ 2 .$
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_sparse_nonlin_eqns_easy (c05qsc) 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 are reasonably well behaved. If this condition is not satisfied, then nag_zero_sparse_nonlin_eqns_easy (c05qsc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_sparse_nonlin_eqns_easy (c05qsc) with a lower value for xtol.

## 8Parallelism and Performance

nag_zero_sparse_nonlin_eqns_easy (c05qsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zero_sparse_nonlin_eqns_easy (c05qsc) 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

Local workspace arrays of fixed lengths are allocated internally by nag_zero_sparse_nonlin_eqns_easy (c05qsc). The total size of these arrays amounts to $8×n+2×q$ double elements and $10×n+2×q+5$ integer elements where the integer $q$ is bounded by $8×\mathit{nnz}$ and ${n}^{2}$ and depends on the sparsity pattern of the Jacobian.
The time required by nag_zero_sparse_nonlin_eqns_easy (c05qsc) 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_sparse_nonlin_eqns_easy (c05qsc) to process each evaluation of the functions depends on the number of nonzero entries in the Jacobian. The timing of nag_zero_sparse_nonlin_eqns_easy (c05qsc) is strongly influenced by the time spent evaluating the functions.
When init is Nag_TRUE, the dense Jacobian is first evaluated and that will take time proportional to ${n}^{2}$.
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:
 $3-2x1x1-2x2 = -1, -xi-1+3-2xixi-2xi+1 = -1, i=2,3,…,8 -x8+3-2x9x9 = -1.$
It then perturbs the equations by a small amount and solves the new system.

### 10.1Program Text

Program Text (c05qsce.c)

None.

### 10.3Program Results

Program Results (c05qsce.r)