A standard fixed-point 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 fixed-point iteration converges, Anderson acceleration usually results in convergence in far fewer iterations (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 $m=4$.
Anderson acceleration is particularly useful if evaluating $g\left(x\right)$ is very expensive, in which case functions such as c05rbc or c05qbc, which require the Jacobian or its approximation, may perform poorly.
4References
Anderson D G (1965) Iterative Procedures for Nonlinear Integral Equations J. Assoc. Comput. Mach.12 547–560
5Arguments
1: $\mathbf{f}$ – function, supplied by the userExternal Function
f must return the values of the functions ${f}_{k}$ at a point $x$.
On exit: the function values ${f}_{k}\left(x\right)$ (unless iflag is set to a negative value by f).
4: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling c05mbc you may allocate memory and initialize these pointers with various quantities for use by f when called from c05mbc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
5: $\mathbf{iflag}$ – Integer *Input/Output
On entry: ${\mathbf{iflag}}\ge 0$.
On exit: in general, iflag should not be reset by f. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), iflag should be set to a negative integer. This value will be returned through fail.
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05mbc. If your code inadvertently does return any NaNs or infinities, c05mbc is likely to produce unexpected results.
On exit: the function values at the final point, x.
5: $\mathbf{atol}$ – doubleInput
On entry: the absolute convergence criterion; see rtol.
Suggested value:
$\sqrt{\epsilon}$, where $\epsilon $ is the machine precision returned by X02AJC.
Constraint:
${\mathbf{atol}}\ge 0.0$.
6: $\mathbf{rtol}$ – doubleInput
On 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 $\Vert f\left({\hat{x}}_{i}\right)\Vert \le \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{atol}},{\mathbf{rtol}}\times \Vert f\left({\hat{x}}_{0}\right)\Vert )$.
Suggested value:
$\sqrt{\epsilon}$, where $\epsilon $ is the machine precision returned by X02AJC.
Constraint:
${\mathbf{rtol}}\ge 0.0$.
7: $\mathbf{m}$ – IntegerInput
On 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 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 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: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
11: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ 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_DIVERGENCE
The iteration has diverged and subsequent iterates are too large to be computed in floating-point arithmetic.
NE_INT
On entry, ${\mathbf{astart}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{astart}}\ge 0$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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 7.5 in the Introduction to the NAG Library CL Interface 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. Rerunning c05mbc from a different starting point may avoid the region of difficulty.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{atol}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{atol}}\ge 0.0$.
On entry, ${\mathbf{cndtol}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{cndtol}}\ge 0.0$.
On entry, ${\mathbf{rtol}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{rtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least $200\times ({\mathbf{n}}+1)$ calls to f. Consider restarting the calculation from the point held in x.
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 fixed-point methods (which may only converge linearly), and in some cases can even alleviate divergence.
At each iteration, c05mbc checks whether $\Vert f\left({\hat{x}}_{i}\right)\Vert \le \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{atol}},{\mathbf{rtol}}\times \Vert f\left({\hat{x}}_{0}\right)\Vert )$. 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 exit from c05mbc.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
c05mbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05mbc 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
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 $\mathit{O}\left(mn\right)$ floating-point operations per iteration compared with the unaccelerated fixed-point iteration.
c05mdc also performs a fixed-point iteration with Anderson acceleration. It has a reverse communication interface, so may be preferred to c05mbc when function evaluations are difficult to encapsulate in a function argument.
10Example
This example determines the values ${x}_{1},\dots ,{x}_{4}$ which satisfy the equations