nag_zero_sparse_nonlin_eqns_easy (c05qsc) (PDF version)

c05 Chapter Contents

c05 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_zero_sparse_nonlin_eqns_easy (c05qsc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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.

2 Specification

#include <nag.h>

#include <nagc05.h>

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)

3 Description

The system of equations is defined as:

f_{i} (x_{1}, x_{2}, \dots, x_{n}) = 0,   ​ i = 1, 2, \dots, 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).

4 References

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

5 Arguments

1: 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: n – IntegerInput

On entry:

n

, the number of equations.

2: lindf – IntegerInput

On entry: lindf specifies the number of indices

i

for which values of

f_{i} (x)

must be computed.

3: indf[lindf] – const IntegerInput

On entry: indf specifies the indices

i

for which values of

f_{i} (x)

must be computed. The indices are specified in strictly ascending order.

4: x[n] – const doubleInput

On entry: the components of the point

x

at which the functions must be evaluated.

x [i - 1]

contains the coordinate

x_{i}

.

5: fvec[n] – doubleOutput

On exit:

fvec [i - 1]

must contain the function values

f_{i} (x)

, for all indices

i

in indf.

6: comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to fcn.

user – double *
iuser – Integer *
p – Pointer: 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 3.2.1.1 in the Essential Introduction).

7: iflag – Integer *Input/Output

On entry:

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), then iflag should be set to a negative integer.

2: n – IntegerInput

On entry:

n

, the number of equations.

Constraint:

n > 0

.

3: x[n] – doubleInput/Output

On entry: an initial guess at the solution vector.

x [i - 1]

must contain the coordinate

x_{i}

.

On exit: the final estimate of the solution vector.

4: fvec[n] – doubleOutput

On exit: the function values at the final point returned in x.

fvec [i - 1]

contains the function values

f_{i}

.

5: xtol – doubleInput

On entry: the accuracy in x to which the solution is required.

Suggested value:

\sqrt{ε}

, where

ε

is the machine precision returned by nag_machine_precision (X02AJC).

Constraint:

xtol \geq 0.0

.

6: 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

fail . code =

NE_NOERROR, NE_NO_IMPROVEMENT, NE_TOO_MANY_FEVALS or NE_TOO_SMALL,

icomm [0]

contains

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: rcomm[lrcomm] – doubleCommunication Array

rcomm MUST NOT be altered between successive calls to nag_zero_sparse_nonlin_eqns_easy (c05qsc).

8: lrcomm – IntegerInput

On entry: the dimension of the array rcomm.

Constraint:

lrcomm \geq 12 + nnz

where

nnz

is the number of nonzero entries in the Jacobian, as computed by nag_zero_sparse_nonlin_eqns_easy (c05qsc).

9: icomm[licomm] – IntegerCommunication Array

If

fail . code =

NE_NOERROR, NE_NO_IMPROVEMENT, NE_TOO_MANY_FEVALS or NE_TOO_SMALL on exit,

icomm [0]

contains

nnz

where

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: licomm – IntegerInput

On entry: the dimension of the array icomm.

Constraint:

licomm \geq 8 \times n + 19 + nnz

where

nnz

is the number of nonzero entries in the Jacobian, as computed by nag_zero_sparse_nonlin_eqns_easy (c05qsc).

11: comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $licomm = ⟨value⟩$ .
Constraint: $licomm \geq ⟨value⟩$ .
On entry, $lrcomm = ⟨value⟩$ .
Constraint: $lrcomm \geq ⟨value⟩$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
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.
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 $⟨value⟩$ .
NE_REAL: On entry, $xtol = ⟨value⟩$ .
Constraint: $xtol \geq 0.0$ .
NE_TOO_MANY_FEVALS: There have been at least $200 \times (n + 1)$ calls to fcn. Consider setting $init = Nag_FALSE$ 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: $xtol = ⟨value⟩$ .
NE_USER_STOP: iflag was set negative in fcn. $iflag = ⟨value⟩$ .

7 Accuracy

If

\hat{x}

is the true solution, nag_zero_sparse_nonlin_eqns_easy (c05qsc) tries to ensure that

{‖x - \hat{x}‖}_{2} \leq xtol \times {‖\hat{x}‖}_{2} .

If this condition is satisfied with

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

fail . code =

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.

8 Parallelism 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further 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 \times n + 2 \times q

double elements and

10 \times n + 2 \times q + 5

integer elements where the integer

q

is bounded by

8 \times 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.

10 Example

This example determines the values

x_{1}, \dots, x_{9}

which satisfy the tridiagonal equations:

\begin{array}{rcl} (3 - 2 x_{1}) x_{1} - 2 x_{2} & = & - 1, \\ - x_{i - 1} + (3 - 2 x_{i}) x_{i} - 2 x_{i + 1} & = & - 1, i = 2, 3, \dots, 8 \\ - x_{8} + (3 - 2 x_{9}) x_{9} & = & - 1 . \end{array}

It then perturbs the equations by a small amount and solves the new system.

10.1 Program Text

Program Text (c05qsce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (c05qsce.r)

nag_zero_sparse_nonlin_eqns_easy (c05qsc) (PDF version)

c05 Chapter Contents

c05 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014