NAG Library Function Document
nag_zero_nonlin_eqns_aa_rcomm (c05mdc)
1
Purpose
nag_zero_nonlin_eqns_aa_rcomm (c05mdc) is a comprehensive reverse communication function that finds a solution of a system of nonlinear equations by fixedpoint iteration using Anderson acceleration.
2
Specification
#include <nag.h> 
#include <nagc05.h> 
void 
nag_zero_nonlin_eqns_aa_rcomm (Integer *irevcm,
Integer n,
double x[],
double fvec[],
double atol,
double rtol,
Integer m,
double cndtol,
Integer astart,
Integer iwsav[],
double rwsav[],
NagError *fail) 

3
Description
The system of equations is defined as:
This homogeneous system can readily be reformulated as
A standard fixedpoint iteration approach is to start with an approximate solution
${\hat{x}}_{0}$ and repeatedly apply the function
$g$ until possible convergence; i.e.,
${\hat{x}}_{i+1}=g\left({\hat{x}}_{i}\right)$, until
$\Vert {\hat{x}}_{i+1}{\hat{x}}_{i}\Vert <\text{tol}$. Anderson acceleration uses up to
$m$ previous values of
$\hat{x}$ to obtain an improved estimate
${\hat{x}}_{i+1}$. If a standard fixedpoint iteration converges, then Anderson acceleration usually results in convergence in far fewer iterations (and therefore using far fewer function evaluations).
Full details of Anderson acceleration are provided in
Anderson (1965). In summary, the previous
$m$ iterates are combined to form a succession of least squares problems. These are solved using a QR decomposition, which is updated at each iteration.
You are free to choose any value for $m$, provided $m\le n$. A typical choice is $m=4$.
4
References
Anderson D G (1965) Iterative Procedures for Nonlinear Integral Equations J. Assoc. Comput. Mach. 12 547–560
5
Arguments
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and reentries,
all arguments other than fvec must remain unchanged.
 1:
$\mathbf{irevcm}$ – Integer *Input/Output

On initial entry: must have the value $0$.
On intermediate exit:
specifies what action you must take before reentering
nag_zero_nonlin_eqns_aa_rcomm (c05mdc) 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, and a limit on the number of iterations can be applied.
 ${\mathbf{irevcm}}=2$
 Indicates that before reentry to nag_zero_nonlin_eqns_aa_rcomm (c05mdc), fvec must contain the function values $f\left({\hat{x}}_{i}\right)$.
On final exit: ${\mathbf{irevcm}}=0$ and the algorithm has terminated.
Constraint:
${\mathbf{irevcm}}=0$, $1$ or $2$.
Note: any values you return to nag_zero_nonlin_eqns_aa_rcomm (c05mdc) as part of the reverse communication procedure should not include floatingpoint NaN (Not a Number) or infinity values, since these are not handled by nag_zero_nonlin_eqns_aa_rcomm (c05mdc). If your code inadvertently does return any NaNs or infinities, nag_zero_nonlin_eqns_aa_rcomm (c05mdc) 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 initial entry: an initial guess at the solution vector, ${\hat{x}}_{0}$.
On intermediate exit:
contains the current point.
On final exit: the final estimate of the solution vector.
 4:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – doubleInput/Output

On initial entry: need not be set.
On intermediate reentry: 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,
$f\left({\hat{x}}_{i}\right)$.
On final exit: the function values at the final point,
x.
 5:
$\mathbf{atol}$ – doubleInput

On initial entry: the absolute convergence criterion; see below.
Suggested value:
$\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision returned by
nag_machine_precision (X02AJC).
Constraint:
${\mathbf{atol}}\ge 0.0$.
 6:
$\mathbf{rtol}$ – doubleInput

On initial entry: the relative convergence criterion. At each iteration $\Vert f\left({\hat{x}}_{i}\right)\Vert $ is computed. The iteration is deemed to have converged if $\Vert f\left({\hat{x}}_{i}\right)\Vert \le \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{atol}},{\mathbf{rtol}}\times \Vert f\left({\hat{x}}_{0}\right)\Vert \right)$.
Suggested value:
$\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision returned by
nag_machine_precision (X02AJC).
Constraint:
${\mathbf{rtol}}\ge 0.0$.
 7:
$\mathbf{m}$ – IntegerInput

On initial entry: $m$, the number of previous iterates to use in Anderson acceleration. If $m=0$, Anderson acceleration is not used.
Suggested value:
${\mathbf{m}}=4$.
Constraint:
$0\le {\mathbf{m}}\le {\mathbf{n}}$.
 8:
$\mathbf{cndtol}$ – doubleInput

On initial entry: the maximum allowable condition number for the triangular
QR factor generated during Anderson acceleration. At each iteration, if the condition number exceeds
cndtol, columns are deleted until it is sufficiently small.
If ${\mathbf{cndtol}}=0.0$, no condition number tests are performed.
Suggested value:
${\mathbf{cndtol}}=0.0$. If condition number tests are required, a suggested value is ${\mathbf{cndtol}}=1.0/\sqrt{\epsilon}$.
Constraint:
${\mathbf{cndtol}}\ge 0.0$.
 9:
$\mathbf{astart}$ – IntegerInput

On initial entry: the number of iterations by which to delay the start of Anderson acceleration.
Suggested value:
${\mathbf{astart}}=0$.
Constraint:
${\mathbf{astart}}\ge 0$.
 10:
$\mathbf{iwsav}\left[14+{\mathbf{m}}\right]$ – IntegerCommunication Array
 11:
$\mathbf{rwsav}\left[2\times {\mathbf{m}}\times {\mathbf{n}}+{{\mathbf{m}}}^{2}+{\mathbf{m}}+2\times {\mathbf{n}}+1+\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},1\right)\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},3\times {\mathbf{m}}\right)\right]$ – doubleCommunication Array

The arrays
iwsav and
rwsav MUST NOT be altered between calls to
nag_zero_nonlin_eqns_aa_rcomm (c05mdc).
The size of
rwsav is bounded above by
$3\times {\mathbf{n}}\times \left({\mathbf{m}}+2\right)+1$.
 12:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_CONVERGENCE

An error occurred in evaluating the QR decomposition during Anderson acceleration. This may be due to slow convergence of the iteration. Try setting the value of
cndtol. If condition number tests are already performed, try decreasing
cndtol.
 NE_INT

On entry, ${\mathbf{astart}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{astart}}\ge 0$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>0$.
On initial entry, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate entry, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=1$ or $2$.
 NE_INT_2

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{n}}$.
 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, as measured by the reduction in the norm of $f\left(x\right)$ in the last $\u2329\mathit{\text{value}}\u232a$ iterations.
 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, ${\mathbf{atol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{atol}}\ge 0.0$.
On entry, ${\mathbf{cndtol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{cndtol}}\ge 0.0$.
On entry, ${\mathbf{rtol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{rtol}}\ge 0.0$.
7
Accuracy
There are no theoretical guarantees of global or local convergence for Anderson acceleration. However, extensive numerical tests show that, in practice, Anderson acceleration leads to significant improvements over the underlying fixedpoint methods (which may only converge linearly), and in some cases can even alleviate divergence.
At each iteration,
nag_zero_nonlin_eqns_aa_rcomm (c05mdc) checks whether
$\Vert f\left({\hat{x}}_{i}\right)\Vert \le \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{atol}},{\mathbf{rtol}}\times \Vert f\left({\hat{x}}_{0}\right)\Vert \right)$. If the inequality is satisfied, then the iteration is deemed to have converged. The validity of the answer may be checked by inspecting the value of
fvec on final exit from
nag_zero_nonlin_eqns_aa_rcomm (c05mdc).
8
Parallelism and Performance
nag_zero_nonlin_eqns_aa_rcomm (c05mdc) 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 implementationspecific information.
During each iteration, Anderson acceleration updates the factors of a QR decomposition and uses the decomposition to solve a linear least squares problem. This involves an additional $\mathit{O}\left(mn\right)$ floatingpoint operations per iteration compared with the unaccelerated fixedpoint iteration.
nag_zero_nonlin_eqns_aa_rcomm (c05mdc) does not count the number of iterations. Thus, it is up to you to add a limit on the number of iterations and check if this limit has been exceeded when nag_zero_nonlin_eqns_aa_rcomm (c05mdc) is called. This is illustrated in the example program below.
10
Example
This example determines the values
${x}_{1},\dots ,{x}_{4}$ which satisfy the equations
10.1
Program Text
Program Text (c05mdce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (c05mdce.r)