e04bbc searches for a minimum, in a given finite interval, of a continuous function of a single variable, using function and first derivative values. The method (based on cubic interpolation) is intended for functions which have a continuous first derivative (although it will usually work if the derivative has occasional discontinuities).
The function may be called by the names: e04bbc or nag_opt_one_var_deriv.
3Description
e04bbc is applicable to problems of the form:
$$\text{Minimize \hspace{1em}}F\left(x\right)\text{\hspace{1em} subject to \hspace{1em}}a\le x\le b$$
when the first derivative $dF/dx$ can be calculated. e04bbc normally computes a sequence of $x$ values which tend in the limit to a minimum of $F\left(x\right)$ subject to the given bounds. It also progressively reduces the interval $[a,b]$ in which the minimum is known to lie. It uses the safeguarded quadratic-interpolation method described in Gill and Murray (1973).
You must supply a function funct to evaluate $F\left(x\right)$ and its first derivative. The arguments e1 and e2 together specify the accuracy:
to which the position of the minimum is required. Note that funct is never called at any point which is closer than $\mathit{Tol}\left(x\right)$ to a previous point.
If the original interval $[a,b]$ contains more than one minimum,e04bbc will normally find one of the minima.
4References
Gill P E and Murray W (1973) Safeguarded steplength algorithms for optimization using descent methods NPL Report NAC 37 National Physical Laboratory
5Arguments
1: $\mathbf{funct}$ – function, supplied by the userExternal Function
funct must calculate the values of $F\left(x\right)$ and $dF/dx$ at any point $x$ in $[a,b]$.
On entry: $x$, the point at which the values of $F$ and $dF/dx$ are required.
2: $\mathbf{fc}$ – double *Output
On exit: the value of the function $F$ at the current point $x$.
3: $\mathbf{gc}$ – double *Output
On exit: the value of the first derivative $dF/dx$ at the current point $x$.
4: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to funct.
first – Nag_BooleanInput
On entry: will be set to Nag_TRUE on the first call to funct and Nag_FALSE for all subsequent calls.
nf – IntegerInput
On entry: the number of calls made to funct so far.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void * with a C compiler that defines void * and char * otherwise. Before calling e04bbc these pointers may be allocated memory and initialized with various quantities for use by funct when called from e04bbc.
Note:funct should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04bbc. If your code inadvertently does return any NaNs or infinities, e04bbc is likely to produce unexpected results.
Note:funct should be tested separately before being used in conjunction with e04bbc.
2: $\mathbf{e1}$ – doubleInput
On entry: the relative accuracy to which the position of a minimum is required. (Note that since e1 is a relative tolerance, the scaling of $x$ is automatically taken into account.)
It is recommended that e1 should be no smaller than $2\epsilon $, and preferably not much less than $\sqrt{\epsilon}$, where $\epsilon $ is the machine precision.
If e1 is set to a value less than $\epsilon $, its value is ignored and the default value of $\sqrt{\epsilon}$ is used instead. In particular, you may set ${\mathbf{e1}}=0.0$ to ensure that the default value is used.
3: $\mathbf{e2}$ – doubleInput
On entry: the absolute accuracy to which the position of a minimum is required. It is recommended that e2 should be no smaller than $2\epsilon $.
If e2 is set to a value less than $\epsilon $, its value is ignored and the default value of $\sqrt{\epsilon}$ is used instead. In particular, you may set ${\mathbf{e2}}=0.0$ to ensure that the default value is used.
4: $\mathbf{a}$ – double *Input/Output
On entry: the lower bound $a$ of the interval containing a minimum.
On exit: an improved lower bound on the position of the minimum.
5: $\mathbf{b}$ – double *Input/Output
On entry: the upper bound $b$ of the interval containing a minimum.
On exit: an improved upper bound on the position of the minimum.
Note that the value ${\mathbf{e2}}=\sqrt{\epsilon}$ applies here if ${\mathbf{e2}}<\epsilon $ on entry to e04bbc.
6: $\mathbf{max\_fun}$ – IntegerInput
On entry: the maximum number of calls to funct which you are prepared to allow.
The number of calls to funct actually made by e04bbc may be determined by supplying a non-NULL argument comm (see below) and examining the structure member $\mathbf{comm}\mathbf{\to}\mathbf{nf}$ on exit.
Constraint:
${\mathbf{max\_fun}}\ge 2$
(Few problems will require more than 20 function calls.)
On exit: the value of the first derivative $dF/dx$ at the final point x.
10: $\mathbf{comm}$ – Nag_Comm *Input/Output
Note:comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
On entry/exit: structure containing pointers for communication to user-supplied functions; see the above description of funct for details. The number of times the function funct was called is returned in the member $\mathbf{comm}\mathbf{\to}\mathbf{nf}$.
If you do not need to make use of this communication feature, the null pointer NAGCOMM_NULL may be used in the call to e04bbc; comm will then be declared internally for use in calls to user-supplied functions.
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_2_REAL_ARG_GE
On entry, ${\mathbf{a}}+{\mathbf{e2}}=\u27e8\mathit{\text{value}}\u27e9$ while ${\mathbf{b}}=\u27e8\mathit{\text{value}}\u27e9$. These arguments must satisfy ${\mathbf{a}}+{\mathbf{e2}}<{\mathbf{b}}$.
NE_INT_ARG_LT
On entry, max_fun must not be less than 2: ${\mathbf{max\_fun}}=\u27e8\mathit{\text{value}}\u27e9$.
NW_MAX_FUN
The maximum number of function calls, $\u27e8\mathit{\text{value}}\u27e9$, have been performed.
This may have happened simply because max_fun was set too small for a particular problem, or may be due to a mistake in the user-supplied function, funct. If no mistake can be found in funct, restart e04bbc (preferably with the values of a and b given on exit from the previous call to e04bbc).
7Accuracy
If $F\left(x\right)$ is $\delta $-unimodal for some $\delta <\mathit{Tol}\left(x\right)$, where $\mathit{Tol}\left(x\right)={\mathbf{e1}}\times \left|x\right|+{\mathbf{e2}}$, then, on exit, $x$ approximates the minimum of $F\left(x\right)$ in the original interval $[a,b]$ with an error less than $3\times \mathit{Tol}\left(x\right)$.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
e04bbc is not threaded in any implementation.
9Further Comments
Timing depends on the behaviour of $F\left(x\right)$, the accuracy demanded, and the length of the interval $[a,b]$. Unless $F\left(x\right)$ and $dF/dx$ can be evaluated very quickly, the run time will usually be dominated by the time spent in funct.
If $F\left(x\right)$ has more than one minimum in the original interval $[a,b]$, e04bbc will determine an approximation $x$ (and improved bounds $a$ and $b$) for one of the minima.
If e04bbc finds an $x$ such that $F(x-{\delta}_{1})>F\left(x\right)<F(x+{\delta}_{2})$ for some ${\delta}_{1},{\delta}_{2}\ge \mathit{Tol}\left(x\right)$, the interval $[x-{\delta}_{1},x+{\delta}_{2}]$ will be regarded as containing a minimum, even if $F\left(x\right)$ is less than $F(x-{\delta}_{1})$ and $F(x+{\delta}_{2})$ only due to rounding errors in the user-supplied function. Therefore, funct should be programmed to calculate $F\left(x\right)$ as accurately as possible, so that e04bbc will not be liable to find a spurious minimum. (For similar reasons, $dF/dx$ should be evaluated as accurately as possible.)
10Example
A sketch of the function
$$F\left(x\right)=\frac{\mathrm{sin}x}{x}$$
shows that it has a minimum somewhere in the range $[3.5,5.0]$. The example program below shows how e04bbc can be used to obtain a good approximation to the position of a minimum.