Determination

of

all

coherent

pairs

of

functionals

Report

H.G.

Meijer

Technische

Universiteit

Delft

Delft

University

of

Technology

Faculteit

der

Technische

Wiskunde

en

Informatica

Faculty

of

Technical

Mathematics

and

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Determination

of

all

coherent

pairs

of

functionals

H.G.

Meijer

Delft

University

of

Technology

Faculty

of

Technical

Mathematics

and

Informatics

Mekelweg

CD

Delft

the

Netherlands

Abstract

A

pair

of

quasi-definite

linear

functionals

on

the

set

of

polynomials

is

called

a

coherent

pair

if

their

corresponding

sequences

of

monic

orthogonal

polynomials

fPng

and

fTng

satisfy

a

relation

Tn

P

n

n

oen

P

n

n

n

with

oen

non-zero

constants

We

prove

that

if

is

a

coherent

pair

then

at

least

one

of

the

functionals

has

to

be

classical

i.e.

Hermite

Laguerre

Jacobi

or

Bessel

A

similar

result

is

derived

for

symmetrically

coherent

pairs

AMS

Subject

Classification

-LRB-

-RRB-

Introduction

The

concept

of

coherent

pair

was

introduced

by

Iserles

et

al

-LSB-

-RSB-

in

the

study

of

polynomials

orthogonal

with

respect

to

an

inner

product

of

the

form

-LRB-

-RRB-

hf

giS

Z

b

a

fg

d

Z

b

a

f

where

and

are

distribution

functions

and

Let

fPng

denote

the

monic

orthogonal

polynomial

sequence

-LRB-

MOPS

-RRB-

with

respect

to

d

and

fTng

the

MOPS

with

respect

to

d

then

fd

d

is

called

a

coherent

pair

if

there

exist

non-zero

constants

oen

such

that

-LRB-

-RRB-

Tn

P

n

n

oen

P

n

n

for

all

n

Iserles

et

al.

showed

that

if

fd

d

is

a

coherent

pair

then

the

sequence

of

polynomials

fS

ng

orthogonal

with

respect

to

the

inner

product

-LRB-

-RRB-

has

an

attractive

structure

See

also

-LSB-

-RSB-

Therefore

it

is

interesting

to

investigate

under

what

conditions

fd

d

is

a

coherent

pair

Marcell?an

and

Petronilho

-LSB-

-RSB-

studied

this

problem

in

a

more

general

setting

where

and

are

quasi-definite

linear

functionals

on

the

space

of

polynomials

and

the

corresponding

MOPS

satisfy

a

relation

of

the

form

-LRB-

-RRB-

They

solve

the

problem

completely

for

the

case

that

one

of

the

functionals

is

a

classical

one

i.e.

Hermite

Laguerre

Jacobi

or

Bessel

In

a

recent

paper

-LSB-

-RSB-

Marcell?an

P?erez

Pi

nar

showed

that

if

is

a

coherent

pair

of

quasi-definite

linear

functionals

then

both

are

semiclassical

i.e.

there

exist

polynomials

i

i

-LRB-

i

-RRB-

such

that

D

-LRB-

iui

-RRB-

iui

-LRB-

i

-RRB-

where

D

denotes

the

distributional

differentiation

Moreover

they

showed

that

there

exist

polynomials

A

and

B

such

that

with

degree

A

degree

B

It

is

the

aim

of

the

present

paper

to

solve

the

problem

completely

and

to

determine

all

coherent

pairs

of

quasi-definite

linear

functionals

We

will

prove

that

at

least

one

of

the

functionals

has

to

be

classical

so

all

coherent

pairs

of

functionals

are

already

determined

in

-LSB-

-RSB-

We

show

that

there

are

only

two

cases

I

The

functional

is

classical

there

exist

polynomials

ae

degree

degree

degree

ae

such

that

D

-LRB-

-RRB-

and

II

There

exists

a

such

that

-LRB-

x

-RRB-

is

classical

in

that

case

there

exist

polynomials

degree

degree

and

a

constant

ae

such

that

D

-LRB-

-RRB-

and

ae

then

is

classical

with

D

-LRB-

-RRB-

-LRB-

-RRB-

The

two

cases

are

not

completely

disjoint

as

will

be

shown

by

some

examples

If

is

a

coherent

pair

that

is

in

the

situation

described

under

I

as

well

as

in

the

situation

under

II

then

obviously

and

both

are

classical

In

section

we

give

the

basic

definitions

notations

and

known

results

on

functionals

and

coherent

pairs

of

functionals

In

section

we

show

that

every

coherent

pair

belongs

to

case

I

or

to

case

II

In

section

we

give

all

coherent

pairs

for

linear

functionals

which

can

be

represented

by

distribution

functions

In

section

the

functionals

are

symmetric

A

pair

of

symmetric

functionals

is

called

a

symmetrically

coherent

pair

if

their

corresponding

MOPS

fPng

and

fTng

satisfy

a

relation

Tn

P

n

n

oen

P

n

n

for

n

with

oen

non-zero

constants

We

prove

that

if

is

a

symmetrically

coherent

pair

then

at

least

one

of

the

functionals

has

to

be

classical

Moreover

a

division

in

two

cases

as

for

the

coherent

pairs

is

given

Basic

definitions

and

results

Let

u

denote

a

linear

functional

defined

on

the

space

of

polynomials

P

A

sequence

of

monic

polynomials

fPng

is

called

a

monic

orthogonal

polynomial

sequence

-LRB-

MOPS

-RRB-

with

respect

to

u

if

-LRB-

i

-RRB-

degree

Pn

n

n

-LRB-

ii

-RRB-

hu

PnPmi

n

m

n

m

-LRB-

iii

-RRB-

hu

P

i

pn

n

There

exists

a

MOPS

with

respect

to

u

if

and

only

if

u

is

quasi-definite

see

-LSB-

-RSB-

Ch.I

In

that

case

the

MOPS

is

unique

In

the

sequel

we

always

suppose

the

functionals

to

be

quasi-definite

The

MOPS

fPng

satisfies

a

three-term

recurrence

relation

of

the

form

-LRB-

see

-LSB-

-RSB-

p.

-RRB-

Pn

-LRB-

x

-RRB-

-LRB-

x

n

-RRB-

Pn

-LRB-

x

-RRB-

nPn

-LRB-

x

-RRB-

n

with

n

for

n

If

A

is

a

polynomial

and

u

a

functional

then

Au

is

defined

by

hAu

pi

hu

Api

p

P

If

the

polynomial

A

is

not

the

zero-polynomial

then

we

can

write

A

nP

k

ckPk

with

cn

so

hAu

Pni

cnpn

This

implies

that

if

A

and

B

are

polynomials

with

Au

Bu

then

h

-LRB-

A

B

-RRB-

u

pi

for

all

p

P

and

A

B

has

to

be

the

zero-polynomial

i.e.

A

B

The

distributional

derivative

Du

of

the

functional

u

is

defined

by

hDu

pi

hu

p

P

It

is

easy

to

check

that

we

have

for

an

arbitrary

polynomial

D

-LRB-

u

-RRB-

Du

A

functional

u

is

called

classical

if

it

satisfies

a

relation

D

-LRB-

u

-RRB-

u

with

and

polynomials

degree

degree

The

classical

functionals

and

corresponding

orthogonal

polynomial

sequences

are

the

following

ones

see

-LSB-

-RSB-

up

to

a

linear

transformation

of

the

variable

-LRB-

i

-RRB-

degree

Hermite

polynomials

fHng

with

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

ii

-RRB-

degree

Laguerre

polynomials

fL

-LRB-

-RRB-

n

g

with

f

g

-LRB-

x

-RRB-

x

-LRB-

x

-RRB-

x

-LRB-

iii

-RRB-

degree

and

has

two

different

roots

Jacobi

polynomials

fP

ff

fi

n

g

with

ff

f

g

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

x

-LRB-

iv

-RRB-

degree

and

has

a

double

root

Bessel

polynomials

fB

-LRB-

-RRB-

n

g

with

f

g

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

x

In

the

sequel

we

will

use

the

following

definition

and

notations

and

denote

quasi-definite

linear

functionals

on

P

fPng

the

MOPS

with

respect

to

fTng

the

MOPS

with

respect

to

P

i

pn

n

T

i

tn

n

The

pair

is

called

a

coherent

pair

if

there

exist

non-zero

constants

oen

such

that

-LRB-

-RRB-

Tn

P

n

n

oen

P

n

n

for

n

For

a

coherent

pair

we

introduce

the

polynomials

-LRB-

-RRB-

Cn

oen

Tn

tn

Tn

tn

n

The

following

two

propositions

are

due

to

Marcell?an

P?erez

Pi

nar

-LSB-

-RSB-

Since

they

are

fundamental

for

the

next

sections

we

give

here

a

somewhat

different

proof

of

them

Proposition

Let

denote

a

coherent

pair

then

nPn

pn

D

-LRB-

-RRB-

for

n

Proof

Relation

-LRB-

-RRB-

can

be

written

as

P

n

n

Tn

oen

P

n

n

and

by

repeated

application

and

P

we

have

for

n

-LRB-

-RRB-

P

n

n

Tn

oenTn

oenoen

oenoen

This

implies

for

n

P

n

n

Tk

tk

i

if

k

n

if

k

n

oen

oek

if

k

n

Then

for

n

P

n

n

-LRB-

oek

Tk

tk

Tk

tk

-RRB-

i

if

k

n

if

k

n

if

k

n

Using

the

notation

-LRB-

-RRB-

this

can

be

rewritten

as

-LRB-

n

k

-RRB-

-LRB-

-RRB-

hD

-LRB-

Ck

-RRB-

Pn

i

-LRB-

n

if

k

n

otherwise

Obviously

for

n

k

-LRB-

-RRB-

h

-LRB-

k

-RRB-

Pk

pk

Pn

i

-LRB-

n

if

k

n

otherwise

As

trivially

for

k

-LRB-

-RRB-

hD

-LRB-

Ck

-RRB-

hCk

i

h

-LRB-

k

-RRB-

Pk

pk

Since

fPng

is

a

basis

for

P

we

have

by

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

D

-LRB-

Ck

-RRB-

-LRB-

k

-RRB-

Pk

pk

k

Proposition

Let

denote

a

coherent

pair

Then

D

-LRB-

-RRB-

and

with

-LRB-

-RRB-

-LRB-

-RRB-

P

-LRB-

-RRB-

B

C

where

degree

degree

degree

B

Proof

Proposition

with

n

and

n

reads

C

C

Elimination

of

gives

the

first

result

and

elimination

of

the

second

one

The

coefficient

of

xn

in

the

polynomial

Cn

defined

by

-LRB-

-RRB-

is

oentn

thus

the

coefficient

of

in

the

polynomial

B

defined

in

-LRB-

-RRB-

is

then

B

has

degree

We

will

extend

the

notation

introduced

in

proposition

by

-LRB-

-RRB-

ss

C

C

then

degree

ss

and

-LRB-

-RRB-

Determination

of

coherent

pairs

In

this

section

we

suppose

that

is

a

coherent

pair

and

we

use

the

notations

of

section

It

follows

from

proposition

and

and

-LRB-

-RRB-

that

for

n

nPn

pn

nPn

pn

'D

-LRB-

-RRB-

C

-LRB-

C

n

Cnss

-RRB-

This

gives

the

basic

relation

for

our

investigations

-LRB-

-RRB-

nPn

pn

B

C

n

Cnss

n

We

will

prove

that

B

and

have

at

least

one

zero

in

common

Theorem

Suppose

that

B

in

-LRB-

-RRB-

has

two

different

zeros

then

at

least

one

of

them

is

also

a

zero

of

Proof

Let

denote

a

zero

of

B

with

-LRB-

-RRB-

then

-LRB-

-RRB-

implies

-LRB-

-RRB-

C

n

-LRB-

-RRB-

-LRB-

-RRB-

Cn

-LRB-

-RRB-

ss

-LRB-

-RRB-

n

In

particular

for

n

with

C

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

ss

-LRB-

-RRB-

Observe

that

since

-LRB-

-RRB-

and

also

-LRB-

-RRB-

and

ss

-LRB-

-RRB-

From

-LRB-

-RRB-

and

-LRB-

-RRB-

we

obtain

-LRB-

-RRB-

C

n

-LRB-

-RRB-

Cn

-LRB-

-RRB-

n

Using

the

definition

of

Cn

in

-LRB-

-RRB-

we

get

oen

-LRB-

-LRB-

-RRB-

T

n

-LRB-

-RRB-

tn

Tn

-LRB-

-RRB-

tn

-RRB-

-LRB-

-RRB-

T

n

-LRB-

-RRB-

tn

Tn

-LRB-

-RRB-

tn

Put

-LRB-

-RRB-

hn

-LRB-

-RRB-

-LRB-

-RRB-

T

n

-LRB-

-RRB-

tn

Tn

-LRB-

-RRB-

tn

n

then

-LRB-

-RRB-

oenhn

-LRB-

-RRB-

hn

-LRB-

-RRB-

n

Note

that

-LRB-

-RRB-

and

-LRB-

-RRB-

implies

hn

-LRB-

-RRB-

for

all

n

Now

suppose

that

B

has

two

different

zeros

and

with

-LRB-

-RRB-

and

-LRB-

-RRB-

Then

relation

-LRB-

-RRB-

holds

with

as

well

as

with

We

will

show

that

this

leads

to

a

contradiction

so

if

B

has

two

different

zeros

then

at

least

one

of

them

is

also

a

zero

of

It

follows

from

-LRB-

-RRB-

with

and

that

hn

-LRB-

-RRB-

hn

-LRB-

-RRB-

hn

-LRB-

-RRB-

hn

-LRB-

-RRB-

n

and

by

repeated

application

hn

-LRB-

-RRB-

hn

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

or

hn

-LRB-

-RRB-

hn

-LRB-

-RRB-

for

all

n

But

now

-LRB-

-RRB-

gives

-LRB-

-RRB-

-LRB-

-RRB-

T

n

-LRB-

-RRB-

Tn

-LRB-

-RRB-

-LRB-

-RRB-

T

n

-LRB-

-RRB-

Tn

-LRB-

-RRB-

for

all

n

We

will

show

that

since

such

a

relation

can

not

hold

for

all

n

with

gives

the

desired

contradiction

Suppose

that

there

exist

constants

different

from

zero

ff

ffi

such

that

-LRB-

-RRB-

ffT

n

-LRB-

-RRB-

Tn

-LRB-

-RRB-

T

n

-LRB-

-RRB-

ffiTn

-LRB-

-RRB-

for

all

n

Let

fTng

satisfy

the

recurrence

relation

Tn

-LRB-

x

-RRB-

-LRB-

x

n

-RRB-

Tn

-LRB-

x

-RRB-

nTn

-LRB-

x

-RRB-

n

From

the

recurrence

relation

and

-LRB-

-RRB-

with

n

n

and

n

we

obtain

that

also

should

hold

-LRB-

-RRB-

ff

n

-LRB-

-RRB-

-LRB-

-RRB-

Tn

-LRB-

-RRB-

n

-LRB-

-RRB-

-LRB-

ffi

-RRB-

Tn

-LRB-

-RRB-

n

Starting

with

-LRB-

-RRB-

-LRB-

-RRB-

ffi

and

making

the

step

from

-LRB-

-RRB-

to

-LRB-

-RRB-

three

times

we

obtain

a

system

of

equations

which

should

be

satisfied

for

Tn

for

all

n

The

determinant

of

the

system

however

is

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Hence

-LRB-

-RRB-

can

not

hold

and

thus

B

and

must

have

a

common

zero

Theorem

Let

denote

a

double

zero

of

B

in

-LRB-

-RRB-

then

-LRB-

-RRB-

Proof

From

-LRB-

-RRB-

we

obtain

-LRB-

-RRB-

-LRB-

-RRB-

C

-LRB-

-RRB-

-LRB-

-RRB-

Hence

-LRB-

-RRB-

Then

applying

again

-LRB-

-RRB-

we

have

B

-LRB-

-RRB-

C

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

so

-LRB-

-RRB-

Then

-LRB-

-RRB-

implies

-LRB-

-RRB-

We

have

seen

in

theorem

and

theorem

that

and

B

in

the

relation

from

proposition

have

at

least

one

zero

in

common

So

if

is

a

coherent

pair

a

relation

holds

with

polynomials

and

B

with

degree

degree

B

The

next

theorem

gives

a

more

precise

description

of

the

situation

Theorem

Let

denote

a

coherent

pair

Let

denote

a

common

zero

of

and

B.

Put

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

Then

there

are

two

possibilities

-LRB-

i

-RRB-

is

classical

D

-LRB-

-RRB-

for

some

polynomial

of

degree

and

with

degree

ae

-LRB-

ii

-RRB-

-LRB-

x

-RRB-

is

classical

D

-LRB-

-RRB-

for

some

polynomial

with

degree

and

is

classical

with

D

-LRB-

-RRB-

-LRB-

-RRB-

Proof

Put

B

-LRB-

-RRB-

-LRB-

-RRB-

in

-LRB-

-RRB-

then

Cn

-LRB-

-RRB-

ss

-LRB-

-RRB-

for

n

Now

we

have

to

distinguish

two

possibilities

ss

-LRB-

-RRB-

and

ss

-LRB-

-RRB-

-LRB-

i

-RRB-

Suppose

ss

-LRB-

-RRB-

Write

ss

-LRB-

x

-RRB-

-LRB-

x

-RRB-

ss

-LRB-

x

-RRB-

Then

-LRB-

-RRB-

reduces

to

and

D

-LRB-

-RRB-

-LRB-

ss

-RRB-

with

ss

and

degree

Hence

is

classical

Write

B

-LRB-

x

-RRB-

ae

then

degree

ae

and

reduces

to

-LRB-

ii

-RRB-

Suppose

ss

-LRB-

-RRB-

Then

Cn

-LRB-

-RRB-

for

all

n

It

follows

from

-LRB-

-RRB-

that

-LRB-

-RRB-

Write

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

then

degree

and

D

-LRB-

-RRB-

is

equal

to

D

-LRB-

-RRB-

i.e.

is

classical

Since

-LRB-

-RRB-

-LRB-

-RRB-

it

follows

from

-LRB-

-RRB-

that

B

-LRB-

-RRB-

-LRB-

-RRB-

hence

B

-LRB-

x

-RRB-

-LRB-

x

-RRB-

Then

reduces

to

Finally

is

classical

since

D

-LRB-

-RRB-

D

-LRB-

-RRB-

-LRB-

-RRB-

Remark

The

situations

under

-LRB-

i

-RRB-

and

-LRB-

ii

-RRB-

in

theorem

are

not

completely

disjoint

It

is

possible

that

is

classical

and

that

-LRB-

x

-RRB-

for

some

is

classical

again

In

that

case

both

and

are

classical

For

examples

see

section

Coherent

pairs

of

distribution

functions

A

linear

functional

is

positive-definite

if

and

only

if

it

can

be

represented

by

a

distribution

function

as

-LRB-

see

-LSB-

-RSB-

Ch

II

-RRB-

hu

pi

Z

b

a

p

-LRB-

x

-RRB-

d

-LRB-

x

-RRB-

p

P

Then

a

coherent

pair

of

positive-definite

functionals

corresponds

to

a

coherent

pair

of

distribution

functions

fd

d

In

this

section

we

mention

all

coherent

pairs

of

distribution

functions

which

follow

from

theorem

The

classical

polynomials

in

this

section

are

given

in

their

usual

notation

-LRB-

see

e.g.

Szego

-LSB-

-RSB-

-RRB-

and

not

in

their

monic

version

a

linear

change

in

the

variable

gives

again

a

coherent

pair

A.

Laguerre

case

The

distribution

function

d

-LRB-

x

-RRB-

x

e?xdx

with

on

-LRB-

-RRB-

defines

a

positive-definite

classical

functional

with

-LRB-

x

-RRB-

x.

Theorem

-LRB-

i

-RRB-

gives

the

coherent

pair

d

-LRB-

x

-RRB-

-LRB-

x

-RRB-

x

xdx

d

-LRB-

x

-RRB-

x

e?xdx

In

order

to

ensure

that

and

are

distribution

functions

on

-LSB-

-RRB-

we

have

to

take

or

It

is

easy

to

check

that

this

is

indeed

a

coherent

pair

-LRB-

see

-LSB-

-RSB-

-RRB-

By

theorem

-LRB-

ii

-RRB-

we

have

the

coherent

pair

d

-LRB-

x

-RRB-

x

e?xdx

d

-LRB-

x

-RRB-

x

x

e?xdx

M

ffi

-LRB-

-RRB-

where

we

have

to

take

M

The

spectrum

of

is

-LSB-

-RRB-

d

has

to

be

interpretted

as

Z

f

-LRB-

x

-RRB-

d

-LRB-

x

-RRB-

Z

f

-LRB-

x

-RRB-

x

x

e?xdx

Mf

-LRB-

-RRB-

so

the

spectrum

of

is

-LSB-

-RRB-

-LSB-

f?g

As

before

it

is

easy

to

check

that

this

is

a

coherent

pair

-LRB-

see

-LSB-

-RSB-

-RRB-

Observe

that

if

M

in

the

examples

above

then

we

obtain

the

coherent

pair

d

-LRB-

x

-RRB-

d

-LRB-

x

-RRB-

x

e?xdx

which

is

an

example

of

theorem

-LRB-

i

-RRB-

as

well

as

of

theorem

-LRB-

ii

-RRB-

It

corresponds

to

the

well-known

relation

L

-LRB-

-RRB-

n

-LRB-

x

-RRB-

d

dxL

-LRB-

-RRB-

n

-LRB-

x

-RRB-

d

dxL

-LRB-

-RRB-

n

-LRB-

x

-RRB-

B.

Jacobi

case

The

distribution

function

d

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dx

with

on

-LRB-

-RRB-

represents

a

positive-definite

classical

functional

with

-LRB-

x

-RRB-

Theorem

-LRB-

i

-RRB-

gives

the

coherent

pair

of

distribution

functions

on

-LSB-

-RSB-

d

-LRB-

x

-RRB-

jx

j

-LRB-

x

-RRB-

-LRB-

x

-RRB-

d

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dx

with

j?j

or

or

On

the

other

hand

theorem

-LRB-

ii

-RRB-

gives

the

coherent

pair

d

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dx

d

-LRB-

x

-RRB-

jx

j

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dx

M

ffi

-LRB-

-RRB-

with

j?j

M

has

-LSB-

-RSB-

as

its

spectrum

the

spectrum

of

is

-LSB-

-RSB-

-LSB-

f?g

Again

it

is

easy

to

check

that

these

pairs

are

indeed

coherent

pairs

Observe

that

the

pairs

fd

d

f

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dx

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dxg

and

f

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dx

-LRB-

x

-RRB-

-LRB-

x

-RRB-

dxg

with

occur

as

special

cases

of

both

theorem

-LRB-

i

-RRB-

and

theorem

-LRB-

ii

-RRB-

They

correspond

to

the

relations

-LRB-

see

e.g.

-LSB-

-RSB-

p.

-RRB-

-LRB-

-RRB-

P

fi

n

-LRB-

x

-RRB-

d

dxP

ff

fi

n

-LRB-

x

-RRB-

-LRB-

n

-RRB-

n

d

dxP

ff

fi

n

-LRB-

x

-RRB-

and

-LRB-

-RRB-

P

ff

fi

n

-LRB-

x

-RRB-

d

dxP

ff

fi

n

-LRB-

x

-RRB-

-LRB-

n

-RRB-

n

d

dxP

ff

fi

n

-LRB-

x

-RRB-

C.

Hermite

case

In

the

Hermite

case

the

distribution

function

is

d

-LRB-

x

-RRB-

on

-LRB-

-RRB-

with

-LRB-

x

-RRB-

Theorem

implies

that

there

can

not

exist

coherent

pairs

Symmetrically

coherent

pairs

In

this

section

and

denote

symmetric

functionals

and

fPng

and

fTng

the

corresponding

MOPS

Since

in

this

situation

-LRB-

-RRB-

only

can

hold

with

oen

for

all

n

Iserles

et

al

-LSB-

-RSB-

introduced

the

concept

of

symmetrically

coherent

pair

The

pair

of

symmetric

functionals

is

called

a

symmetrically

coherent

pair

if

there

exist

non-zero

constants

oen

such

that

Tn

P

n

n

oen

P

n

n

for

n

In

this

section

we

assume

to

be

a

symmetrically

coherent

pair

and

we

will

prove

that

again

at

least

one

of

the

functionals

has

to

be

classical

Therefore

we

will

use

the

polynomials

Cn

oen

Tn

tn

Tn

tn

n

Proposition

is

replaced

by

proposition

which

can

be

proved

in

the

same

way

Proposition

Let

denote

a

symmetrically

coherent

pair

then

nPn

pn

D

-LRB-

Cn

-RRB-

for

n

Observe

that

in

proposition

the

index

of

Cn

is

one

greater

than

that

of

Cn

in

proposition

the

same

should

be

done

in

the

corollaries

of

proposition

Proposition

Let

denote

a

symmetrically

coherent

pair

then

D

-LRB-

-RRB-

and

with

-LRB-

-RRB-

-LRB-

-RRB-

P

-LRB-

-RRB-

B

C

where

degree

degree

degree

B

This

result

is

also

given

in

-LSB-

-RSB-

We

will

also

use

ss

C

C

Then

-LRB-

-RRB-

and

ss

is

an

odd

polynomial

of

degree

we

often

write

ss

-LRB-

x

-RRB-

-LRB-

x

-RRB-

so

that

is

an

even

polynomial

of

degree

The

basic

relation

for

our

investigations

compare

-LRB-

-RRB-

is

-LRB-

-RRB-

nPn

pn

B

C

n

Cn

ss

n

All

above

mentioned

polynomials

are

either

even

or

odd

Then

all

zeros

apart

from

x

in

the

odd

polynomials

appear

in

pairs

f

g.

First

we

prove

that

the

even

polynomials

B

and

have

at

least

one

pair

of

zeros

in

common

Theorem

Suppose

that

B

in

-LRB-

-RRB-

has

two

different

pairs

of

zeros

f

and

f

then

at

least

one

of

them

is

also

a

pair

of

zeros

of

Proof

The

proof

follows

the

lines

of

the

proof

of

theorem

It

follows

from

-LRB-

-RRB-

recall

ss

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

B

C

x

n

Let

denote

a

zero

of

B

with

-LRB-

-RRB-

Then

-LRB-

-RRB-

C

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

n

Here

C

is

an

odd

polynomial

and

if

then

one

has

to

read

C

-LRB-

-RRB-

lim

x

C

-LRB-

x

-RRB-

x

For

n

-LRB-

-RRB-

becomes

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

also

-LRB-

-RRB-

-LRB-

-RRB-

With

-LRB-

-RRB-

we

can

simplify

-LRB-

-RRB-

by

eliminating

-LRB-

-RRB-

Put

-LRB-

-RRB-

-LRB-

-RRB-

T

-LRB-

-RRB-

-LRB-

-RRB-

n

then

we

obtain

from

-LRB-

-RRB-

compare

the

proof

of

theorem

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

n

Since

-LRB-

-RRB-

we

have

-LRB-

-RRB-

for

all

n

Suppose

that

B

has

two

zeros

and

with

such

that

-LRB-

-RRB-

and

-LRB-

-RRB-

Then

-LRB-

-RRB-

should

hold

with

as

well

as

with

which

should

imply

-LRB-

-RRB-

-LRB-

-RRB-

for

all

n

As

in

the

proof

of

theorem

this

leads

to

a

contradiction

so

at

least

one

of

the

zeros

is

also

a

zero

of

Theorem

Suppose

that

B

in

-LRB-

-RRB-

is

of

the

form

B

-LRB-

-RRB-

then

-LRB-

-RRB-

Proof

-LRB-

i

-RRB-

Let

i.e.

B

Then

B

-LRB-

-RRB-

-LRB-

-RRB-

C

-LRB-

-RRB-

and

-LRB-

-RRB-

C

-LRB-

-RRB-

C

-LRB-

-RRB-

-LRB-

-RRB-

C

-LRB-

-RRB-

give

-LRB-

-RRB-

Now

-LRB-

-RRB-

gives

-LRB-

-RRB-

-LRB-

ii

-RRB-

Let

Consider

the

basic

relation

-LRB-

-RRB-

with

n

and

n

and

eliminate

the

polynomial

ss

then

one

obtains

-LRB-

-RRB-

B

-LRB-

-RRB-

Put

-LRB-

x

-RRB-

-LRB-

-RRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

where

and

are

polynomials

of

degree

and

respectively

then

-LRB-

-RRB-

-LRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

-RRB-

B

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

Suppose

-LRB-

-RRB-

We

will

show

that

this

leads

to

a

contradiction

From

-LRB-

-RRB-

we

conclude

that

f

g

is

not

a

pair

of

zeros

of

hence

B

-LRB-

x

-RRB-

const

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

or

with

t

-LRB-

t

-RRB-

const

-LRB-

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-RRB-

From

the

last

relation

it

is

easy

to

derive

compare

the

proof

of

theorem

that

-LRB-

-RRB-

-LRB-

-RRB-

Divide

-LRB-

-RRB-

by

B

-LRB-

x

-RRB-

and

substitute

x

then

it

follows

-LRB-

-RRB-

which

is

a

contradiction

Hence

-LRB-

-RRB-

From

theorem

and

theorem

it

follows

that

and

B

in

the

relation

in

proposition

have

at

least

one

pair

of

zeros

in

common

We

will

now

prove

that

at

least

one

of

the

functionals

is

a

classical

one

First

we

derive

a

lemma

to

discuss

a

special

situation

Lemma

Suppose

-LRB-

-RRB-

B

-LRB-

-RRB-

and

-LRB-

-RRB-

then

B

Proof

It

follows

from

-LRB-

-RRB-

that

for

all

n

Take

the

basic

relation

-LRB-

-RRB-

for

n

and

n

and

eliminate

the

polynomial

ss

then

one

obtains

-LRB-

-RRB-

-LRB-

-RRB-

B

-LRB-

-RRB-

With

-LRB-

-RRB-

-LRB-

-RRB-

it

follows

Since

-LRB-

-RRB-

B

-LRB-

-RRB-

we

can

write

B

where

and

are

even

polynomials

of

degree

respectively

at

most

Then

it

follows

from

-LRB-

-RRB-

that

divides

There

are

two

possibilities

i

-RRB-

or

ii

-RRB-

const

We

will

prove

that

the

second

case

can

not

occur

which

will

prove

the

theorem

Relation

-LRB-

-RRB-

with

n

reads

B

Since

-LRB-

-RRB-

this

reduces

to

Now

suppose

const

then

ff

for

some

constant

But

then

relation

-LRB-

-RRB-

reduces

to

When

we

apply

this

relation

on

we

conclude

that

all

moments

are

zero

with

exception

of

at

most

one

moment

and

is

not

quasi-definite

Now

we

can

prove

the

main

result

of

this

section

which

is

similar

to

theorem

for

coherent

pairs

Theorem

Let

denote

a

symmetrically

coherent

pair

Let

f

g

denote

a

common

pair

of

zeros

of

and

B.

Put

-LRB-

x

-RRB-

-LRB-

-RRB-

-LRB-

x

-RRB-

Then

there

are

two

possibilities

-LRB-

i

-RRB-

is

classical

D

-LRB-

-RRB-

for

some

odd

polynomial

of

degree

and

where

ae

is

an

even

polynomial

with

degree

ae

-LRB-

ii

-RRB-

-LRB-

-RRB-

is

classical

D

-LRB-

-RRB-

for

some

odd

polynomial

with

degree

and

is

classical

with

D

-LRB-

-RRB-

-LRB-

-RRB-

Proof

Put

B

-LRB-

-RRB-

-LRB-

-RRB-

in

-LRB-

-RRB-

then

Cn

-LRB-

-RRB-

-LRB-

-RRB-

for

n

Now

we

have

to

distinguish

three

possibilities

a

-RRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

a

Suppose

-LRB-

-RRB-

then

-LRB-

x

-RRB-

const

-LRB-

-RRB-

Write

ss

-LRB-

x

-RRB-

-LRB-

-RRB-

ss

-LRB-

x

-RRB-

then

-LRB-

-RRB-

reduces

to

which

implies

as

in

the

proof

of

theorem

-LRB-

i

-RRB-

that

is

classical

Moreover

write

B

-LRB-

-RRB-

ae

then

reduces

to

and

degree

ae

Suppose

-LRB-

-RRB-

then

Cn

-LRB-

-RRB-

for

all

n

It

follows

from

-LRB-

-RRB-

that

-LRB-

-RRB-

Write

-LRB-

x

-RRB-

-LRB-

-RRB-

-LRB-

x

-RRB-

then

-LRB-

x

-RRB-

constx

Relation

D

-LRB-

-RRB-

is

equal

to

D

-LRB-

-RRB-

and

is

classical

Moreover

since

-LRB-

-RRB-

-LRB-

-RRB-

also

-LRB-

-RRB-

-LRB-

-RRB-

C

-LRB-

-RRB-

C

-LRB-

-RRB-

-LRB-

-RRB-

thus

B

-LRB-

-RRB-

Then

reduces

to

It

follows

as

in

the

proof

of

theorem

-LRB-

ii

-RRB-

that

is

classical

Suppose

-LRB-

-RRB-

Now

we

can

apply

the

lemma

and

conclude

B

Then

B

-LRB-

-RRB-

-LRB-

-RRB-

implies

-LRB-

-RRB-

C

-LRB-

-RRB-

Then

-LRB-

-RRB-

gives

-LRB-

-RRB-

thus

-LRB-

x

-RRB-

Write

-LRB-

x

-RRB-

-LRB-

x

-RRB-

Now

we

can

proceed

as

under

-RRB-

with

Theorem

enables

us

to

give

all

symmetrically

coherent

pairs

which

can

be

represented

by

distribution

functions

In

theorem

the

may

be

complex

in

the

distribution

functions

below

we

always

assume

the

to

be

real

D.

Hermite

case

The

classical

distribution

function

is

d

-LRB-

x

-RRB-

on

-LRB-

-RRB-

with

-LRB-

x

-RRB-

Theorem

gives

the

following

symmetrically

coherent

pairs

of

distribution

functions

on

-LRB-

-RRB-

f

-LRB-

-RRB-

and

dxg

with

E.

Gegenbauer

case

The

classical

distribution

function

is

d

-LRB-

x

-RRB-

-LRB-

-RRB-

dx

on

-LRB-

-RRB-

with

and

-LRB-

x

-RRB-

Theorem

gives

the

following

symmetrically

coherent

pairs

of

distribution

functions

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

dxg

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

dxg

with

j?j

or

j?j

f

-LRB-

-RRB-

-LRB-

-RRB-

dxg

and

f

-LRB-

-RRB-

-LRB-

-RRB-

dx

M

ffi

-LRB-

-RRB-

M

ffi

-LRB-

-RRB-

g

with

j?j

M

If

M

the

last

mentioned

distribution

function

has

and

in

its

spectrum

all

other

distribution

functions

are

only

defined

on

-LSB-

-RSB-

Remark

In

-LSB-

-RSB-

the

concept

of

generalized

coherent

pair

has

been

introduced

It

reads

for

linear

functionals

let

and

denote

quasi-definite

linear

functionals

and

let

fPng

and

fTng

denote

their

MOPS

then

is

called

a

generalized

coherent

pair

if

there

exist

constants

oen

on

such

that

Tn

P

n

n

oen

P

n

n

on

P

n

n

for

n

Let

M

then

-LRB-

see

section

-RRB-

fx

e?xdx

x

x

e?xdx

M

ffi

-LRB-

-RRB-

g

and

f

-LRB-

x

-RRB-

x

e?xdx

x

e?xdxg

are

coherent

pairs

From

this

observation

it

easily

follows

that

f

-LRB-

x

-RRB-

x

e?xdx

x

x

e?xdx

M

ffi

-LRB-

-RRB-

g

is

a

generalized

coherent

pair

-LRB-

Obviously

the

dx-terms

are

distribution

functions

on

-LSB-

-RRB-

and

if

M

the

last

term

gives

a

contribution

from

outside

-LRB-

-RRB-

-RRB-

Here

of

the

distribution

functions

is

a

classical

one

so

the

results

of

this

paper

can

not

be

generalized

to

generalized

coherent

pairs

References

-LSB-

-RSB-

M.

Abramowitz

I.A.

Stegun

Handbook

of

Mathematical

Functions

Dover

Publications

New

York

-LSB-

-RSB-

M.G.

de

Bruin

H.G.

Meijer

Zeros

of

orthogonal

polynomials

in

a

non-discrete

Sobolev

space

Annals

of

Numerical

Mathematics

-LRB-

-RRB-

to

appear

-LSB-

-RSB-

T.S.

Chihara

An

Introduction

to

Orthogonal

Polynomials

Gordon

and

Breach

New

York

-LSB-

-RSB-

A.

Iserles

P.E.

Koch

S.P.

NOErsett

J.M.

Sanz-Serna

On

polynomials

orthogonal

with

respect

to

certain

Sobolev

inner

products

J.

Approx

Theory

-LRB-

-RRB-

-LSB-

-RSB-

F.

Marcell?an

T.E.

P?erez

M.A.

Pi

nar

Orthogonal

Polynomials

on

Weighted

Sobolev

Spaces

The

semiclassical

case

Annals

of

Numerical

Mathematics

to

appear

-LSB-

-RSB-

F.

Marcell?an

J.C.

Petronilho

Orthogonal

Polynomials

and

Coherent

Pairs

The

Classical

Case

Indag

Math

N.S.

to

appear

-LSB-

-RSB-

F.

Marcell?an

J.C.

Petronilho

T.E.

P?erez

M.A.

Pi

nar

What

is

Beyond

Coherent

Pairs

of

Orthogonal

Polynomials

J.

Comput

Appl

Math.

to

appear

-LSB-

-RSB-

P.

Maroni

Une

th?eorie

algebrique

des

polyn

omes

orthogonaux

Application

aux

polyn

omes

orthogonaux

semi-classiques

In

Orthogonal

Polynomials

and

their

applications

C.

Brezinski

L.

Gori

A.

Ronveaux

Eds

IMACS

Annals

on

Comp

and

Appl

Math

Vol

J.C.

Baltzer

AG

Publ

Basel

-LRB-

-RRB-

-LSB-

-RSB-

H.G.

Meijer

Coherent

pairs

and

zeros

of

Sobolev-type

orthogonal

polynomials

Indag

Math.

N.S.

-LRB-

-RRB-

-LSB-

-RSB-

G.

Szego

Orthogonal

polynomials

Amer

Math

Soc

Colloq

Publ.

vol

Providence

R.I.

-LRB-

-RRB-

O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O 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Chapter 10 The effectiveness of recombination Antoine H. C. van Kampen , Lutgarde M. C. Buydens Laboratory for Analytical Chemistry , Catholic University of Nijmegen NL-6525 ED the Netherlands E-mail : avkampen@sci.kun.nl Abstract . The determination of the spatial structure of biomacromolecules via optimisation of its internal coordinates , i.e. , torsion angles , comprises a complex optimisation problem . The search space contains many nearly optimal solutions , and the torsion angles are highly non-linearly correlated . The genetic algorithm and simulated annealing were applied to this problem , and there performances were compared . The contribution of the crossover operation to the performance of the genetic algorithm was examined by comparing a mutation-selection scheme to a crossover-mutation-selection scheme . Thereupon , the so-called random crossover operator was applied to reveal whether building blocks were recombined in a constructive way . It was shown that the current implementation of the genetic algorithm did not promote the recombination of building blocks , and therefore was easily outperformed by simulated annealing . The complete title of this article : \ The Effectiveness of Recombination in the Genetic Algorithm Methodology . A Comparison to Simulated Annealing '' . The article is available via anonymous ftp at site ftp.uwasa.fi directory cs/2NWGA as file Kampen.ps.Z . 115 116 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 10.1 Introduction Biological macromolecules comprise classes of large molecules like proteins , nucleic acids , polysaccharides and lipid-containing complexes , which have biological interest . Proteins and nucleic acids play a central role in all known life forms , and consequently , knowledge about their three dimensional structure and function is considered to be of importance as it contributes to a more thorough understanding of biological and chemical processes involving these molecules . Multidimensional NMR spectroscopy has become the state-of-the-art method for determining the spatial structure -LSB- 127 , 598 , 490 -RSB- . Distance constraints obtained from these experiments , complemented with knowledge about the covalent structure of the molecule may lead to a description of the structure . The determination of the spatial structure by optimisation comprises a difficult problem because the the search space may contain many nearly optimal solutions , and in addition , the set of experimental distance constraints may not be fully self-consistent due to experimental error , and flexibility of the molecule in solution during measurement . During the last few years the genetic algorithm -LRB- GA -RRB- -LSB- 212 , 275 , 272 -RSB- raised interest as an alternative method for structure elucidation -LSB- 120 , 518 , 407 , 316 , 142 , 149 -RSB- . Recent investigations revealed that a GA may not always be the best choice for solving complex optimisation problems , due to the ineffectiveness of the recombination operator -LSB- 312 , 456 -RSB- . Although it is relatively simple to test the effect of recombination , it is more difficult to change the design of the GA when crossover does not to contribute to the search performance . This paper presents a comparative study between a GA and simulated annealing applied to the elucidation of the spatial structure of a heptapeptide in torsion angle space . Different operators and values for the GA control parameters were tested , and the best found parametrisation was compared to the performance of a simulated annealing approach . Thereupon , the effectiveness of recombination was tested . This paper does not claim to make any generalisation of its conclusions to other GA applications , but emphasises an additional pitfall in the GA design . The functionality of the recombination operator should not be taken for granted . 10.2 Methods Building block processing . The crossover operation is the mechanism that distinguishes the GA from other -LRB- stochastic -RRB- global optimisation methods , like simulated annealing , that often comprise a sophisticated mutation and selection scheme -LSB- 556 , 94 -RSB- . The basic idea of crossover is to recombine trial solutions such that improved solutions emerge . More precisely , the crossover operator establishes the communication between strings via the exchange of the so-called building blocks -LRB- sub-solutions -RRB- , which are highly fit schemata of low order and short defining length -LSB- 212 , 193 -RSB- . A schema comprises a similarity template describing a A. H. C. van Kampen : The effectiveness of recombination . 117 subset of strings with similarities at certain string positions . A low order schema only specifies a few bits , and the defining length specifies the distance between the first and last specified string position . The defining length of a building block is of minor importance if uniform crossover is used because instead of choosing breaking points that easily disrupt building blocks of long defining length , bits are exchanged between mated strings . Consequently , building blocks of identical order have the same chance of being disrupted -LSB- 533 -RSB- . According to the building block hypothesis , low order -LRB- short defining length -RRB- and highly fit schemata are sampled , recombined , and resampled to form strings of potentially higher fitness . In a way by working with the particular schemata -LRB- building blocks -RRB- , the complexity of the optimisation problem is reduced ; instead of building high-performance strings by trying every conceivable combination , the strings are constructed from the best partial solutions of past samplings . It is generally believed that this process of communication between solutions in the population should make the GA an efficient optimiser in comparison to other strategies . Deceptive optimisation problems -LSB- 212 , 225 , 373 -RSB- represent a class of problems in which recombination of low order -LRB- short defining length -RRB- building blocks lead to incorrect -LRB- sub-optimal -RRB- higher order building blocks . This is caused by non-linear interactions of the string elements e.g. , bits or parameters . As stated by Whitley -LSB- 588 -RSB- , deceptive problems are the only interesting problems for application of GAs . If no individual string element is affected by the value of other elements , a bit-wise optimisation with a more greedy algorithm will most likely perform more effectively than a GA. . However , a problem occurs when the deceptiveness of the optimisation problem becomes too severe , and none or only few combinations of fit building blocks lead to improvements . Consequently , the search degrades to an inefficient process in which trial solutions are only improved by accidental changes . In between these two extremes problem types exist for which a GA might be a good choice . It is important that constructive building blocks are contained in the string , which can be processed by the GA. . Accordingly , the combination of the fitness function , encoding of the problem parameters -LRB- e.g. , real or binary -RRB- , and type of crossover -LRB- e.g. , uniform or n-point -RRB- must ensure the presence of these building blocks . However , for complex non-linear problems these building blocks will not necessarily have a -LRB- physical -RRB- interpretation , and may therefore be difficult -LRB- if possible at all -RRB- to identify , which can make the design of a GA a laborious process . It should be noted that the non-linear correlations between the string elements are actually diminished to some extent by a proper choice of fitness function , encoding and crossover operator , i.e. , the building blocks become additive . For complex optimisation problems it is unlikely that these correlations can be diminished sufficiently by a transformation imposed by the GA design . Even if such a transformation exist , it will probably be quite difficult to find a suitable combination . Therefore , one should at least check the contribution of the crossover operator . To this end two experiments are available . The first one 118 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 is a comparison of a CMS scheme , which comprises a GA with crossover , mutation and selection , to a MS scheme , which comprises a GA without crossover . If this experiment reveals no significant contribution of recombination , then the GA design should be changed . If , on the other hand , crossover does contribute , then a second experiment is needed to test whether solutions are really constructed from building blocks . For this experiment the so-called random crossover operator -LSB- 312 -RSB- can be used . If this experiment reveals that recombination is without the desired effect , the resulting algorithm is left with only mutation and selection , and consequently , will probably be easily outperformed by a more sophisticated mutation selection scheme like simulated annealing . The structure elucidation problem . The experiments presented in this paper describe the structure determination of the natural heptapeptide optoid ? - selective dermorphin -LRB- Tyr-dAla-Phe-Gly-Tyr-Pro-Ser-NH2 -RRB- , which consists of 7 amino acids , and is described by 31 torsion angles -LSB- 457 , 96 , 427 -RSB- -LRB- Figure 10.2 -RRB- . A set of 19 distance constraints was available from -LSB- 520 -RSB- , and used in the experiments presented in this paper . Figure 10.1 : The natural heptapeptide optoid ? - selective dermorphin -LRB- Tyr-dAlaPhe-Gly-Tyr-Pro-Ser-NH2 -RRB- A peptide consist of a sequence of amino acids , which may be represented by their internal coordinates , i.e. , torsion angles OE , , ! , and a set of torsion angles O that describe the side chain . Figure 10.2 depicts a sequence of 2 amino acids in which the structure of the side chain R determines its type . The torsion angles give the angle between four atoms , and define the spatial conformation of the molecule . NMR experiments result in a set of distance constraints , which are defined between pairs of H-atoms of the molecule , which are in close proximity -LRB- < = 5 ? A -RRB- . The torsion angles are the subject of optimisation , and an optimal set of values describe a structure that has a minimum of experimental distance constraint A. H. C. van Kampen : The effectiveness of recombination . 119 N C C N C C H H O H H R R i i +1 c y wj O Figure 10.2 : A sequence of 2 amino acids . The torsion angles are indicated by OE , , ! and O . The side chain R may contain multiple torsion angles O , and determines the amino acid type . violations , and simultaneously a minimum amount of atom overlap . Accordingly , the fitness function was defined as : F itness ? 1 = Error = A ? X -LRB- restraint violations -RRB- + B ? X -LRB- atom overlap -RRB- 2 where A and B were weight factors , which were set to 2.0 and 1.0 respectively , restraint violation is the magnitude of a single violation of an experimental distance constraint , and atom overlap is the magnitude of the overlap between two atoms . The advantage of using torsion angles is that the covalent structure , i.e. , bond lengths and bond angles between connected atoms , is preserved during optimisation . Simulated annealing . Simulated annealing -LRB- SA -RRB- -LSB- 339 , 138 , 361 -RSB- is an alternative stochastic optimisation technique that mimics the physical process by which a crystal is grown from a melt . Physical systems may be coaxed into a minimum energy conformation -LRB- e.g. , crystal -RRB- by a slow annealing process . If the reduction of the temperature is slow enough , the system is able to pass out of local energy minima , and arrives at the global minimum configuration . This principle may be applied to solve optimisation problems . In contrast to the GA , which maintains a population of trial solutions , SA generates a trajectory through the search space by making incremental changes to a single set of problem parameters , i.e. , the torsion angles . At the start of the run these parameters are initialised at random . In the experiments discussed in this paper , the problem parameters were adjusted according to the principles of fast simulated annealing -LSB- 543 -RSB- . Each parameter was modified by increasing or decreasing its current value with an amount ? , generated from a Cauchy distribution : pnew = pcurrent ? ? Subsequently , the modified set of parameters was evaluated by the objective function . The outcome of this evaluation determined whether the new set was accepted or rejected . If the error value -LRB- E -RRB- corresponding to the new set was lower 120 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 than the error value for the current set , an improvement was obtained , and accordingly , the new set of parameters was accepted . In case of a detrimental step , the parameters were not immediately rejected but were accepted with a probability Paccept = e ? -LRB- Enew?Ecurrent -RRB- = ct. . This ensured the possibility for uphill moves that allows the algorithm to escape from local minima . The control parameter ct is analog to the temperature of a real physical system , and a high value increases the probability of accepting detrimental moves . Consequently , a high initial value for ct in combination with a suitable parameter modification function ensures a sufficient exploration of the search space . At each ` temperature ' the algorithm is allowed to make N steps , after which the temperature is decreased according to cooling schedule ct +1 = ct ? . The parameter was taken 0.95 for the experiments described in this paper . Subsequently , a new trajectory was generated . Decreasing the temperature forces the algorithm to converge to a minimum that , hopefully , is the global minimum of the optimisation problem . The SA algorithm in this research was parameterised such that the c was decreased when N = 50 steps were accepted , or a maximum of 250 proposed moves was exceeded . The initial value of c was chosen such that the initial ratio between accepted moves and proposed moves was about 0.9 . Statistical evaluation of results . The resulting error values and number of function evaluations obtained from the experiments , were evaluated by the Kruskal-Wallis non-parametric analysis of variance -LRB- ANOVA -RRB- -LSB- 353 , 354 , 240 -RSB- . The p-values that result from such a calculation represents an increasing index of reliability that is involved in accepting the hypothesis that the tested groups of observations were sampled from an identical population . Typically , in many sciences , results that yield p < = :05 are considered borderline statistically significant . Results that are significant at p < = :01 are commonly considered statistically significant , and p < = :005 or p < = :001 are called highly significant . These categories were used for the interpretation of the data in this paper . Part of the results were visualized with box-whisker plots -LSB- 559 -RSB- . The box borders were defined by the 25 % and 75 % percentiles . The median value of the observations was indicated by a square inside the box , and the minimum and maximum values were indicated by the whiskers . Outliers were depicted by an open circle , and extreme values by a star -LRB- * -RRB- . A value was rated to be an outlier if the distance from the box border was larger than 1.5 times the length of the box . If the distance was larger than 3.0 times the length of the box , then that value was regarded as an extreme value . The values 1.5 and 3.0 were taken from -LSB- 559 -RSB- , and only served the visualisation of the results . The random crossover operator -LRB- RXO -RRB- . The RXO operator -LSB- 312 -RSB- was developed to examine whether building blocks are recombined productively by the crossover operator . A. H. C. van Kampen : The effectiveness of recombination . 121 Crossover Random string generator Parent Child Figure 10.3 : The random crossover operator -LRB- RXO -RRB- Figure 10.2 depicts the function of this operator . A parent string is selected from the population , and recombined with a string that was generated at random . Due to the application of this operator , communication between strings of the population is no longer possible , i.e. , no building blocks can be exchanged . Thereupon , no knowledge is available about potential building blocks on the random string because it was not evaluated by the fitness function . Consequently , this type of crossover is equivalent to a -LRB- macro -RRB- mutation operator . Suppose the following RXO operation with a 2-point crossover , where the j 's denote the breaking points : random string 100 j 001 j 1100 parent string 110 j 010 j 1001 ?! crossover ?! 110 001 1001 The same result could be obtained by application of a macro-mutation , which only mutates bits between 2 randomly chosen points : parent string 110 j 010 j 1001 ?! mutate ?! 110 001 1001 Therefore , if RXO performs comparable to a regular crossover operator , it can be concluded that also the latter acts like a mutation , and solutions do not emerge from building block recombination , which should lead to a superior performance . It must be remarked that the mutation imposed by RXO is not precisely identical to a possible mutation imposed by a regular crossover operator . This is due to the fact that the strings in the GA population become more similar as optimisation proceeds , which diminishes the effective mutation rate caused by crossover . The mutation rate imposed by RXO remains constant because before every application a new random string is generated . Accordingly , interpretation of results obtained with RXO should be done carefully . 10.3 Results and discussion Prior to comparing the GA performance to simulated annealing , and determining the effect of recombination , the parametrisation of the GA was optimised . Experimental designs were setup to determine the effects of several control parameters 122 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 and operators . The interactions between the GA parameters were neglected because their determination would require too many experiments . Consequently , the final parametrisation was at most near optimal if it is assumed that the interaction effects were small compared to the individual effects . The ` number of function evaluations ' for the GA experiments was calculated from the population size and the number of generations after which no further improvement of the best string in the population was observed . A maximum of 1000 generations was imposed on each GA run . The first experiment was set up to determine a suitable population size . Population sizes of 50 and 200 strings were tested with 10 experiments at each level , starting from different initial populations . During these experiments rank-based selection -LSB- 380 -RSB- was used with the threshold defining the best 25 % of the population from which strings were selected at random . Elitism selection was used to copy the best 5 % of the strings to the next population , which remained unaffected by crossover and mutation . The crossover and mutation probabilities were set to 0.70 and 0.05 respectively . Uniform crossover was used , and exchanged 30 % of the bits between the mated strings on application . A normal point mutation was used . Each problem parameter was encoded with a 11 bit Gray coding . The results are depicted in Figure 10.3 , which denotes the box-whisker plots for both levels . An ANOVA calculation revealed no significant difference -LRB- p = 1.000 -RRB- between the two population sizes for the error values . Consequently , the smaller population size was chosen for the remaining experiments , because it was about four times faster . Population size Error 2 2.4 2.8 3.2 3.6 4 4.4 4.8 50 200 Population size Function evaluations 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000 50 200 Figure 10.4 : Effect of the population size . For each level 10 experiments were performed The elitism percentage and threshold fraction were optimised by using a full factorial design -LSB- 123 -RSB- , which is depicted in Table 10.3 . The effects of the number of encoding bits , type of crossover , and crossover probability were determined by using a 3 ? 3 latin square design -LSB- 123 -RSB- , which is depicted in Table 10.3 . Each combination of levels was tested with 2 replicated experiments , giving a total of 18 experiments . From this each effect could be determined A. H. C. van Kampen : The effectiveness of recombination . 123 Threshold Elitism -LRB- % -RRB- #Exp fraction 0.25 5 3 0.25 10 3 0.10 5 3 0.10 10 3 0.40 5 3 0.40 10 3 Table 10.1 : Full factorial design for determining the effects of the threshold fraction and elitism percentage . For every combination three experiments were conducted . from 6 experiments . Uniform 1-point 2-point 7 bits Pc = :5 Pc = :7 Pc = :9 9 bits Pc = :7 Pc = :9 Pc = :5 11 bits Pc = :9 Pc = :5 Pc = :7 Table 10.2 : 3 ? 3 latin square to determine the effects of the crossover type , crossover probability , and number of encoding bits . At each cell 2 replicate experiments were conducted . Subsequently , the type of encoding and number of bit exchanges for uniform crossover , were tested in a fractional factorial design , which is depicted in table 10.3 . Encoding type Bit exchange -LRB- % -RRB- #Exp Gray binary 10 4 Gray binary 30 4 Gray binary 50 4 Regular binary 10 12 Table 10.3 : Fractional factorial design to determine the effect of the encoding type , and the percentage of bits exchanges on application of uniform crossover . An ANOVA revealed that none of these factors were highly significant , i.e. , both the error values and the number of function evaluations were comparable for all tested levels . Table 10.3 depicts the p-values for each factor for the error values and function evaluations . Only an elitism fraction of 10 % , and the use of a 7 bit regular binary encoding were slightly better , and consequently , were used in the remaining experiments . 124 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 Parameter p-value Error Evaluations Elitism percentage 0.05 0.60 Threshold fraction 0.19 0.55 #Bits 0.72 0.07 Bit exchange fraction 0.92 0.90 Type crossover 0.58 0.93 Pc 0.72 0.93 Encoding 0.02 0.69 Table 10.4 : p-values for each tested control parameter . Next the effect of the mutation probability -LRB- Pm -RRB- was determined . Five different levels were tested with 10 experiments at each level . The levels ranged from Pm = :0 -LRB- only crossover , i.e. , CS scheme -RRB- to Pm = :3 . The results are depicted in Figure 10.3 , which clearly demonstrates the impact of the mutation probability . The performance of the CS scheme is , as expected , poor from which it is concluded that mutation is necessary to introduce new -LRB- lost -RRB- information in the population . A mutation rate that is too low or too large also produced poor results . A comparison of the levels Pm = :01 , Pm = :05 , and Pm = :1 revealed that a mutation rate of 0.05 produced better results -LRB- p = 0.007 for the error value , p = 0.040 for the number of evaluations -RRB- . These results indicate that mutation may not considered to be a background operator , but plays an essential role during the search -LSB- 532 -RSB- . Mutation probability Error 4 8 12 16 20 24 0.0 0.001 0.01 0.05 0.1 0.3 Mutation probability Function evaluations -5000 5000 15000 25000 35000 45000 55000 0.0 0.001 0.01 0.05 0.1 0.3 Figure 10.5 : Effect of the mutation probability . A mutation probability of Pm = :0 denotes the CS scheme . At each level 10 experiments were conducted The final parametrisation of the optimised GA is depicted in Table 10.3 . To test the performance of this GA , 84 experiments were carried out in which only the initial random population was varied . Figure 10.3 depicts the box-whisker plots for the error values and number of function evaluations -LRB- the actual values A. H. C. van Kampen : The effectiveness of recombination . 125 Range of each parameter ? 180ffi ? +180 ffi Encoding type Regular binary Number of bits per parameter 7 Selection method Rank-based selection Threshold fraction 0.25 Elitism fraction 10 % Crossover type Uniform Crossover probability 0.7 Number of bit exchanges 30 % Mutation type Point mutation Mutation probability 0.05 Table 10.5 : Parametrisation of optimised GA Error 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Function evaluations 10000 20000 30000 40000 50000 60000 Figure 10.6 : Error values and function evaluations , resulting from 84 experiments with the optimised CMS scheme . are depicted in Table 10.3 -RRB- . It is clear that there is a large deviation for both quantities ranging from poor to well defined structures , which was determined by investigating the number of distance constraint violations and the amount of overlapping atoms . In order to reveal the effect of crossover on the search performance , 10 experiments were carried out without crossover -LRB- MS scheme -RRB- , and compared to the results of the CMS scheme . The results are shown in Figure 10.3 . An ANOVA revealed no significant difference between the CMS and MS scheme for the error values -LRB- p = 0.740 -RRB- , or number of function evaluations -LRB- p = 0.880 -RRB- . From this it seems fair to conclude that crossover did not contribute to the performance of the GA. . One could argue that the CMS scheme produced one solution -LRB- E = 2:30 -RRB- that was slightly better than the best solution obtained with the MS scheme -LRB- E = 2:54 -RRB- , and that this might be an effect of crossover . It is , however , more likely that this is an effect only of the initial population , and consequently , repeating the MS scheme more often is likely to result in an equally good solution . 126 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 Even if one wishes to conclude that the CMS scheme gave slightly better results , this would not be of much use if one is not prepared to repeat the CMS scheme for many times in order to increase the probability that such a solution is included . Crossover probability Error 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0.0 0.7 Crossover probability Function evaluations 10000 20000 30000 40000 50000 60000 0.0 0.7 Figure 10.7 : Effect of crossover operator . Comparison of the MS scheme -LRB- 10 experiments -RRB- to the CMS scheme -LRB- 84 experiments -RRB- Crossover type Error 2 3 4 5 6 7 RXO Uniform Crossover type Function evaluations 10000 20000 30000 40000 50000 60000 RXO Uniform Figure 10.8 : A comparison of a GA with RXO operator -LRB- 40 experiments -RRB- , and the optimised CMS scheme -LRB- 84 experiments -RRB- . An additional experiment with the RXO operator strengthens the conclusion that recombination was not effective . Figure 10.3 depicts the results of 40 experiments with the RXO operator in comparison to the CMS scheme . This figure indicates that RXO is capable of producing equally well-defined structures , but on average is inferior to uniform crossover -LRB- p = 0.005 -RRB- . Since the comparison of the CMS scheme to the MS scheme revealed no significant difference , it is likely that this is caused by the constant mutation rate imposed by RXO -LRB- see method section -RRB- . Furthermore , the presence of building block exchange should probably accelerate the convergence rate of the algorithm , which is not the case in the present experiment . Figure 10.3 depicts the error values of the best string for the first 150 generations the cms , ms , and RXO scheme . As expected no pronounced differences were observed . A. H. C. van Kampen : The effectiveness of recombination . 127 50 100 150 10 20 30 40 50 Generation Error RXO MS CMS Figure 10.9 : Evolution of the error values of the best string for the CMS , MS and RXO scheme for the first 150 generations . From these experiments it may be concluded that no building blocks are recombined productively . Consequently , one is left with a simple MS scheme that is likely to be outperformed by a more advanced MS scheme like SA . To test this hypothesis , thirty SA experiments were performed , and compared to the CMS scheme -LRB- Figure 10.3 -RRB- . There was no significant difference between the error values for the two search strategies -LRB- p = :200 -RRB- . Table 10.3 depicts the actual values for the Box plots . The number of function evaluations for SA was , however , about a factor of 3 times less than the number of evaluations required by the GA. . Again one could argue that the GA produced one better solution , but again this is probably an effect of the initial starting point in the search space , and repeating SA is likely to result in a comparable solution . Furthermore , a simple and straightforward instance of the SA algorithm was applied , and not optimised like the GA. . An additional SA experiment was conducted to investigate if an improved solution could be found if the cooling rate was decreased , and longer trajectories were allowed at each temperature . Accordingly , the cooling schedule was parameterised with = :98 , and the temperature was decreased when N = 1000 steps were accepted , or a maximum of 2000 proposed steps was exceeded . The final solution of this experiment -LRB- E = 2:83 -RRB- was no improvement of the previous SA experiments . Probably , the method for adjusting the problem parameters is more important , which was , however , not further investigated . To investigate whether the population size could be further reduced 40 GA experiments with a population size of 25 strings were conducted , and compared to the optimised CMS scheme . The resulting solutions for this small population size were inferior to those found with a population size of 50 strings -LRB- p = 0.001 -RRB- . 128 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 Search strategy Error 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 GA SA Search strategy Function evaluations 10000 20000 30000 40000 50000 60000 GA SA Figure 10.10 : A comparison of the genetic algorithm -LRB- CMS scheme with 84 experiments -RRB- and simulated annealing -LRB- 30 experiments -RRB- GA SA Error Evaluations Error Evaluations Mean 3.39 40734 3.45 14461 Median 3.56 44100 3.07 14093 P25 2.69 37875 2.71 12620 P75 4.03 47775 4.00 15869 Minimum 2.30 7100 2.69 11283 Maximum 5.95 49850 6.14 19073 Table 10.6 : Comparison of the error values and function evaluations for the GA -LRB- optimised CMS scheme -RRB- and SA Thereupon , the number of function evaluations needed to derive at the final solutions was still significantly larger than for SA . From this it was concluded that decreasing the size of the population did not lead to results that were more comparable to SA , and the present choice for the population size is adequate . Since the present combination of fitness function , parameter encoding , and crossover type did not lead to an effective recombination , the binary parameter encoding was changed to a real-value encoding . A uniform crossover , similar to its binary analog , was applied to exchange 5 randomly selected parameters between the mated strings . A new mutation operator was designed which modified 10 randomly selected parameters on application according to : pnew = pcurrent + N -LRB- 0 ; 1 -RRB- where N -LRB- 0 ; 1 -RRB- denotes that standard normal distribution . Since the torsion angles were encoded as radials rather than degrees , the magnitude of the modification was reasonable . In order to be able to determine the effect of recombination for this type of encoding , this real-valued GA was applied in 20 experiments with Pc = :70 , and 20 experiments with Pc = :0 . An ANOVA calculation A. H. C. van Kampen : The effectiveness of recombination . 129 revealed no significant difference between the error values -LRB- p = :465 -RRB- or number of function evaluations -LRB- p = :808 -RRB- . The complete set of 40 experiments was , therefore compared to the binary encoded CMS scheme . The result is shown in Figure 10.3 . Again there was no significant difference between the error values -LRB- p = :423 -RRB- , or number of function evaluations -LRB- p = :248 -RRB- . Although for realencoded parameters many more sophisticated operators are available , no further attempts were made to optimise the GA for this type of encoding . Parameter encoding Error 1.5 2.5 3.5 4.5 5.5 6.5 7.5 real binary Parameter encoding Function evaluations 10000 20000 30000 40000 50000 60000 real binary Figure 10.11 : A comparison of a real encoded GA -LRB- 40 experiments -RRB- with a binary encoded GA -LRB- CMS scheme , 84 experiments -RRB- The experiments in this paper demonstrated that the combination of fitness function , 7 bit regular binary encoding or real-valued encoding , and uniform crossover , did not lead to building blocks that could be recombined to improved solutions . Consequently , the GA was easily outperformed by SA . From this it seems fair to conclude that changes to the design of the GA are necessary . Although there are various possibilities to change the fitness function , encoding , or crossover operator , only few guidelines are available that may provide direction in this task , and thereupon these guidelines are not readily applicable to complex optimisation task . Consequently , experiments remain the main resource for setting up the GA. . Methods for finding an optimal mutation rate were suggested by Hesser and M?anner -LSB- 266 -RSB- , but were not tested for problems of large complexity . Davidor and Ben-Kiki -LSB- 146 -RSB- proposed the use of information theory to find optimal parameterizations , and to classify optimisation problems according to how efficiently/robustly a GA performs . This research also requires further investigations before it can be of assistance in designing a GA. . Manderick introduced the concept of forma analysis -LSB- 477 -RSB- , which might be helpful in designing operators for specific problems but is difficult to apply . Representational issues were , for instance , discussed by Liepins and Vose -LSB- 372 -RSB- and Tate and Smith -LSB- 546 -RSB- . Most promising seems the analysis of the search space , which might contribute to a better understanding of the GA search , and consequently , its design . However , this approach is still subject of many investigations -LSB- 326 , 324 , 325 , 128 , 282 , 313 , 193 , 395 -RSB- , and not yet applicable to real-world problems . 130 Proceedings of the 2NWGA , Vaasa , Finland , 19 - 23 August 1996 The results of the present study are hard to generalise to other problem domains , or even to the same type of problems . Penersen and Moult -LSB- 464 -RSB- gave a brief overview of GA applications for protein structure determination , and discuss a comparison to molecular dynamics and Monte Carlo methods . They suggest that the GA may be preferable in some situations , but no details were given on the effectiveness of recombination , and no remarks were made about simulated annealing . Sun -LSB- 540 -RSB- also performed a comparative study on GAs and SA for protein structure determination , and showed that the GA was about a factor 100-200 times faster than SA . Again no details were given on the effectiveness of recombination or the implementation of SA From this research it should be evident that every GA application should at least be tested for the effectiveness of crossover . This provides more knowledge about applications in which the crossover operator does contribute to the search performance , and thereby may provide useful information for the design of new applications . It should be noted that evolutionary algorithms -LSB- 109 -RSB- that incooperate mutation only -LRB- and no crossover -RRB- have been investigated for many years in the fields of evolutionary programming -LSB- 186 , 189 -RSB- , and evolutionary strategies -LSB- 481 -RSB- . These techniques have been applied various problems with equal success as genetic algorithms -LSB- 126 , 381 , 408 -RSB- . Recombination might enhance the search but a comparative studies of GAs in which building blocks are exchanged are necessary . 10.4 Conclusion This research demonstrated the necessity for testing the effectiveness of recombination . If the crossover operator does not contribute to the performance of the GA , it may be outperformed by SA . The ineffectiveness of crossover is caused by an inappropriate combination of fitness function , parameter encoding , and crossover type , which should therefore be given special attention during the design of a GA. . Although finding the optimal parametrisation of the GA is subject of lively research , experiments remain the main resource for setting up GA applications . Acknowledgments F.F.A. Hollander is acknowledged for his contribution in writing a large part of the software to manipulate structures in torsion angle space . 10.5 Bibliography See chapter bibliography at the end of this proceedings .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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