﻿ e04ab Method
e04ab searches for a minimum, in a given finite interval, of a continuous function of a single variable, using function values only. The method (based on quadratic interpolation) is intended for functions which have a continuous first derivative (although it will usually work if the derivative has occasional discontinuities).

# Syntax

C#
```public static void e04ab(
E04..::..E04AB_FUNCT funct,
ref double e1,
ref double e2,
ref double a,
ref double b,
ref int maxcal,
out double x,
out double f,
out int ifail
)```
Visual Basic
```Public Shared Sub e04ab ( _
funct As E04..::..E04AB_FUNCT, _
ByRef e1 As Double, _
ByRef e2 As Double, _
ByRef a As Double, _
ByRef b As Double, _
ByRef maxcal As Integer, _
<OutAttribute> ByRef x As Double, _
<OutAttribute> ByRef f As Double, _
<OutAttribute> ByRef ifail As Integer _
)```
Visual C++
```public:
static void e04ab(
E04..::..E04AB_FUNCT^ funct,
double% e1,
double% e2,
double% a,
double% b,
int% maxcal,
[OutAttribute] double% x,
[OutAttribute] double% f,
[OutAttribute] int% ifail
)```
F#
```static member e04ab :
funct : E04..::..E04AB_FUNCT *
e1 : float byref *
e2 : float byref *
a : float byref *
b : float byref *
maxcal : int byref *
x : float byref *
f : float byref *
ifail : int byref -> unit
```

#### Parameters

funct
Type: NagLibrary..::..E04..::..E04AB_FUNCT
You must supply this method to calculate the value of the function $F\left(x\right)$ at any point $x$ in $\left[a,b\right]$. It should be tested separately before being used in conjunction with e04ab.

A delegate of type E04AB_FUNCT.

e1
Type: System..::..Double%
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.)
e1 should be no smaller than $2\epsilon$, and preferably not much less than $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision.
On exit: if you set e1 to $0.0$ (or to any value less than $\epsilon$), e1 will be reset to the default value $\sqrt{\epsilon }$ before starting the minimization process.
e2
Type: System..::..Double%
On entry: the absolute accuracy to which the position of a minimum is required. e2 should be no smaller than $2\epsilon$.
On exit: if you set e2 to $0.0$ (or to any value less than $\epsilon$), e2 will be reset to the default value $\sqrt{\epsilon }$.
a
Type: System..::..Double%
On entry: the lower bound $a$ of the interval containing a minimum.
On exit: an improved lower bound on the position of the minimum.
b
Type: System..::..Double%
On entry: the upper bound $b$ of the interval containing a minimum.
On exit: an improved upper bound on the position of the minimum.
maxcal
Type: System..::..Int32%
On entry: the maximum number of calls of $F\left(x\right)$ to be allowed.
Constraint: ${\mathbf{maxcal}}\ge 3$. (Few problems will require more than $30$.)
There will be an error exit (see [Error Indicators and Warnings]) after maxcal calls of funct
On exit: the total number of times that funct was actually called.
x
Type: System..::..Double%
On exit: the estimated position of the minimum.
f
Type: System..::..Double%
On exit: the function value at the final point given in x.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

# Description

e04ab is applicable to problems of the form:
 $Minimize Fx subject to a≤x≤b.$
It 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 $\left[a,b\right]$ 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 funct to evaluate $F\left(x\right)$. The parameters e1 and e2 together specify the accuracy
 $Tolx=e1×x+e2$
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 $\left[a,b\right]$ contains more than one minimum, e04ab will normally find one of the minima.

# References

Gill P E and Murray W (1973) Safeguarded steplength algorithms for optimization using descent methods NPL Report NAC 37 National Physical Laboratory

# Error Indicators and Warnings

Note: e04ab may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
 On entry, $\left({\mathbf{a}}+{\mathbf{e2}}\right)\ge {\mathbf{b}}$, or ${\mathbf{maxcal}}<3$,
${\mathbf{ifail}}=2$
The number of calls of funct has exceeded maxcal. This may have happened simply because maxcal was set too small for a particular problem, or may be due to a mistake in funct. If no mistake can be found in funct, restart e04ab (preferably with the values of a and b given on exit from the previous call of e04ab).
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# Accuracy

If $F\left(x\right)$ is $\delta$-unimodal for some $\delta <\mathit{Tol}\left(x\right)$, where $\mathit{Tol}\left(x\right)={\mathbf{e1}}×\left|x\right|+{\mathbf{e2}}$, then, on exit, $x$ approximates the minimum of $F\left(x\right)$ in the original interval $\left[a,b\right]$ with an error less than $3×\mathit{Tol}\left(x\right)$.

# Parallelism and Performance

None.

Timing depends on the behaviour of $F\left(x\right)$, the accuracy demanded and the length of the interval $\left[a,b\right]$. Unless $F\left(x\right)$ 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 $\left[a,b\right]$, e04ab will determine an approximation $x$ (and improved bounds $a$ and $b$) for one of the minima.
If e04ab finds an $x$ such that $F\left(x-{\delta }_{1}\right)>F\left(x\right) for some ${\delta }_{1},{\delta }_{2}\ge \mathit{Tol}\left(x\right)$, the interval $\left[x-{\delta }_{1},x+{\delta }_{2}\right]$ will be regarded as containing a minimum, even if $F\left(x\right)$ is less than $F\left(x-{\delta }_{1}\right)$ and $F\left(x+{\delta }_{2}\right)$ only due to rounding errors in the method. Therefore funct should be programmed to calculate $F\left(x\right)$ as accurately as possible, so that e04ab will not be liable to find a spurious minimum.

# Example

A sketch of the function
 $Fx=sin xx$
shows that it has a minimum somewhere in the range $\left[3.5,5.0\right]$. The following program shows how e04ab can be used to obtain a good approximation to the position of a minimum.

Example program (C#): e04abe.cs

Example program results: e04abe.r