NAG Library Routine Document

E04ABF/E04ABA

+− Contents

1 Purpose

+− 2 Specification

2.1 Specification for E04ABF

2.2 Specification for E04ABA

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E04ABF/E04ABA 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).

E04ABA is a version of E04ABF that has additional parameters in order to make it safe for use in multithreaded applications (see Section 5).

2 Specification

2.1 Specification for E04ABF

SUBROUTINE E04ABF (

FUNCT, E1, E2, A, B, MAXCAL, X, F, IFAIL)

INTEGER	MAXCAL, IFAIL
REAL (KIND=nag_wp)	E1, E2, A, B, X, F
EXTERNAL	FUNCT

2.2 Specification for E04ABA

SUBROUTINE E04ABA (

FUNCT, E1, E2, A, B, MAXCAL, X, F, IUSER, RUSER, IFAIL)

INTEGER	MAXCAL, IUSER(*), IFAIL
REAL (KIND=nag_wp)	E1, E2, A, B, X, F, RUSER(*)
EXTERNAL	FUNCT

3 Description

E04ABF/E04ABA is applicable to problems of the form:

Minimize F (x) subject to a \leq x \leq b .

It normally computes a sequence of

x

values which tend in the limit to a minimum of

F (x)

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 FUNCT to evaluate

F (x)

. The parameters E1 and E2 together specify the accuracy

Tol (x) = E1 \times |x| + E2

to which the position of the minimum is required. Note that FUNCT is never called at any point which is closer than

Tol (x)

to a previous point.

If the original interval

[a, b]

contains more than one minimum, E04ABF/E04ABA will normally find one of the minima.

4 References

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

5 Parameters

1: FUNCT – SUBROUTINE, supplied by the user.External Procedure

You must supply this routine to calculate the value of the function

F (x)

at any point

x

[a, b]

. It should be tested separately before being used in conjunction with E04ABF/E04ABA.

The specification of FUNCT for E04ABF is:

SUBROUTINE FUNCT (

XC, FC)

REAL (KIND=nag_wp)

XC, FC

The specification of FUNCT for E04ABA is:

SUBROUTINE FUNCT (

XC, FC, IUSER, RUSER)

INTEGER	IUSER(*)
REAL (KIND=nag_wp)	XC, FC, RUSER(*)

1: XC – REAL (KIND=nag_wp)Input: On entry: the point $x$ at which the value of $F$ is required.
2: FC – REAL (KIND=nag_wp)Output: On exit: must be set to the value of the function $F$ at the current point $x$ .; Note: the following are additional parameters for specific use with E04ABA. Users of E04ABF therefore need not read the remainder of this description.
3: IUSER( $*$ ) – INTEGER arrayUser Workspace
4: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: FUNCT is called with the parameters IUSER and RUSER as supplied to E04ABF/E04ABA. You are free to use the arrays IUSER and RUSER to supply information to FUNCT as an alternative to using COMMON global variables.

FUNCT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04ABF/E04ABA is called. Parameters denoted as Input must not be changed by this procedure.

2: E1 – REAL (KIND=nag_wp)Input/Output

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 ε

, and preferably not much less than

\sqrt{ε}

, where

ε

is the machine precision.

On exit: if you set E1 to

0.0

(or to any value less than

ε

), E1 will be reset to the default value

\sqrt{ε}

before starting the minimization process.

3: E2 – REAL (KIND=nag_wp)Input/Output

On entry: the absolute accuracy to which the position of a minimum is required. E2 should be no smaller than

2 ε

On exit: if you set E2 to

0.0

(or to any value less than

ε

), E2 will be reset to the default value

\sqrt{ε}

4: A – REAL (KIND=nag_wp)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: B – REAL (KIND=nag_wp)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.

6: MAXCAL – INTEGERInput/Output

On entry: the maximum number of calls of

F (x)

to be allowed.

Constraint:

MAXCAL \geq 3

. (Few problems will require more than

30

There will be an error exit (see Section 6) after MAXCAL calls of FUNCT

On exit: the total number of times that FUNCT was actually called.

7: X – REAL (KIND=nag_wp)Output

On exit: the estimated position of the minimum.

8: F – REAL (KIND=nag_wp)Output

On exit: the function value at the final point given in X.

9: IFAIL – INTEGERInput/Output

Note: for E04ABA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

Note: the following are additional parameters for specific use with E04ABA. Users of E04ABF therefore need not read the remainder of this description.

9: IUSER( $*$ ) – INTEGER arrayUser Workspace

10: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by E04ABF/E04ABA, but are passed directly to FUNCT and may be used to pass information to this routine as an alternative to using COMMON global variables.

11: IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: E04ABF/E04ABA may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$(A + E2) \geq B$ ,
or	$MAXCAL < 3$ ,

$IFAIL = 2$: 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 E04ABF/E04ABA (preferably with the values of A and B given on exit from the previous call of E04ABF/E04ABA).

7 Accuracy

F (x)

δ

-unimodal for some

δ < Tol (x)

, where

Tol (x) = E1 \times |x| + E2

, then, on exit,

x

approximates the minimum of

F (x)

in the original interval

[a, b]

with an error less than

3 \times Tol (x)

8 Further Comments

Timing depends on the behaviour of

F (x)

, the accuracy demanded and the length of the interval

[a, b]

. Unless

F (x)

can be evaluated very quickly, the run time will usually be dominated by the time spent in FUNCT.

F (x)

has more than one minimum in the original interval

[a, b]

, E04ABF/E04ABA will determine an approximation

x

(and improved bounds

a

and

b

) for one of the minima.

If E04ABF/E04ABA finds an

x

such that

F (x - δ_{1}) > F (x) < F (x + δ_{2})

for some

δ_{1}, δ_{2} \geq Tol (x)

, the interval

[x - δ_{1}, x + δ_{2}]

will be regarded as containing a minimum, even if

F (x)

is less than

F (x - δ_{1})

and

F (x + δ_{2})

only due to rounding errors in the subroutine. Therefore FUNCT should be programmed to calculate

F (x)

as accurately as possible, so that E04ABF/E04ABA will not be liable to find a spurious minimum.

9 Example

A sketch of the function

F (x) = \frac{\sin x}{x}

shows that it has a minimum somewhere in the range

[3.5, 5.0]

. The following program shows how E04ABF/E04ABA can be used to obtain a good approximation to the position of a minimum.

9.1 Program Text

Note: the following programs illustrate the use of E04ABF and E04ABA.

Program Text (e04abfe.f90)

Program Text (e04abae.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (e04abfe.r)

Program Results (e04abae.r)

E04ABF/E04ABA (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentE04ABF/E04ABA