Use of barycentric coordinates to solve certain computational problems

USE OF BARYCENTRIC COORDINATES TO SOLVE CERTAIN COMPUTATIONAL PROBLEMS* S. 6.

LAVROV

(Moscow) (Received

IT is

shown in this

formulae tional

problems

of

functions,

of

functions. 1.

position,

that,

for

functions

solution

of

November

using

algorithms

Let us recall 121). Let the

[ll , the

paper

and useful

12

barycentric

can be derived of

numerous

nonlinear

variables,

equation

be selected

arbitrarily

in the

1

viz.

systems

simple

several

computa-

interpolation

and finding

extrema

n-th

Euclidean

space,

so that

. ..I

=10*11

=

.

.

.

. * * Xl,, .

Zno=nl*

not

point

equal

is

t0

If

M,).

mass

equal

zero

(Xlj,

.

a mass ~j to

is

.

.

.

centre

Cartesian

of

.

.

(1.1)

.

. . r,,,

are

Xnj

placed

the

at each

Cartesian

M, point

the

of total

gravity,

-.*

M. of

this

+I&=

(1.2)

1,

system

of

masses

will

have

the

coordinates zi =

The quantities

POxi

$

I1lTil+

p,-,, uI,

** *

+

i =

l',~ri,,*

. . . , p, are termed

l’ych.

.Math.

Vol.

4.

No.

5,

905-910. 157

1,

2, . . ., n.

barycentric

Zh.

coordinates

so that

1 PO+!%+

.

solve

determinant

D

the

coordinates,

to

the determination of barycentric coordinates (see basic points, Mc, MI, . . . . M, which are in a common

1

iS

1963)

1964.

coordinates

(1.3) of

the point M. For an arbitrary point 4! with coordinates xl, . . . , x,, the barycentric coordinates ~~~ pl, . . . . )I,, are unambiguously determined from the system of equations (1.21 and (X.3) because the determinant of this system is equal to Ct and. by hypothesis, is not zero. Some of the ctj masses for an arbitrary point M will be negative. x,) be a polynomial of the n-th degree in the vari2. Let P(xl, .+., We obtain an algebraic polynomial of the same m-th degree. It ables xi* can be transformed into a homogeneous polynomial of m-th degree in the variables ~j by multiplying each term of degree s < tn by the polynomial co+!A+

a..

06% * . -1 p,)

exactly

+&J"-**

can be written

Q (/lo,

pl,

. . .,

equal

to I. The transformed

polynomial

as

P,) =

(2.1)

where the summation is carried out over all possible combinations of the number II as the sua of a + 1 integral non-negative items, among which with a varying order of some might be equal to zero. The combinations It is easy to prove that two homoitems are considered to be different. geneous polynomials Q~@o, . . ., p,) and Q&O, . . ., p,) of the same degree condition and of identical admissible values, i.e. values which satisfy coefficients. Thus, the prob(1.2) of the variables clj, have identical lem of constructing a polynomial P(zr, . . ., zn) of the m-th degree approximating (in some sense) to a certain function F(xl, . . ., 2,). is equivalent to the probiem of approximating this function by a homogeneous i_to, . . . , gn. polynomial Q&LO, . . ., p,) of the same degree in the variables We consider the problem of constructing for the function F with a special selection polatfon.

an interpolation Polynomial of the basic points of inter-

We will split the unit mass into m equal parts and distribute them between the points MO, hf~, . . . , hi,. The number of ways in which this can be done is also equal to the number of representations of A) in the form of the ordered sum of a + 1 of integral non-negative items and, consequently, is the same as the number of coefficients of polynomial (2.1). where The point with barycentric coordinates (kw’m, kdm, . . ., k,,lm), (j = 0, I,..., nl, are integers. ko -P kl + ... +kn=mrkj>O ko, kl, . . ., k, corresponds to this distribution of masses. this point are denoted in terms of YlX3...u 11.,.X -k. rime* 8, timer

...

Values of nn...n -’

k,,eimes

the function

F at

(2.“)

Use of

We determine

having

. . ., RI)

We restrict For

the

tions

the

tarvcentric

coefficients values

our

hi.,i,, .,,,i,,

epual

to

consideration

to

the

bio9i,,

of

the

case

it

type

of

is

notations

., i,, =

m = I,

the

polynomial

of

the

values

interpolation,

of

polynomial

m = 2.

to

such

small

values

introduce

(2.2),

points. of

second

m.

nota-

namely

.,. n,,...,, -c '

ilLtime*

takes

QlPl

Q (pO,pl,

F at all

actual

×

Q (po, . . ., p,,)

the

-i- ***

+

(2 and function

form (2.3)

~A*

F at the

basic

Point

we obtain aj =

If

of

Il...1 ---

-

Q (PO,~1, . . ., p,) = UOPO-IAdjusting

function

of

“00 . 0

159

the polynomial

convenient

i. time8

If

of

those

to

unknown coefficients

similar

coordinates

the

coefficients

0, 1, . . ., n.

i =

Yj7

of

the

Q Co, . . ., pn)

(2.4)

polynomial

= i

have

cljjfl”j2j

f

j-0

j

to

be determined

&‘ikp?k’

(2.5)

‘j
By the

same method

we obtain ajj = 2yjj,

j = 0,i , . . ., n,

(2.6)

and then

We note

that

formula

(2.6).

If

m = 3.

this

formula

polynomial

remains

Q

correct

@e,. . ., p,,)

also

contains

if

j = k.

terms

of

simplifying

three

to

different

types

Q Co, . . .

...)=& j=O

For

its

coefficients

=

-

5~;,~

we successively

+

18Y.. -gYjh Jfi

+

ajk#p#r.

(2.8)

j. k. r=o j
h$

ajjj = 6~;~~ a,,k

n

I,

x -&ujj&lr + x

Ujjjpj +

obtain

/ = 0, I, . . ., n, 2Ykkk,

j.

k =

0, i, . . . , n,

(2.9) i # k,

(2.10)

160

.s. Lavrov

A.

=jkr

2

=

Y,-).rP) -

Y&f f

(y,tjj +

$ (yjjk + Yjjr f

iv k, r = 0, 1, . . ., n,

Here we are formula

restricted

(2.10)

Formulae nomial

of

verified

to

and if

can be derived any degree.

that

all

for

to

such

formulae

k f

which,

if

formula

coefficients

For polynomials

r9

i*

(2.11)

changes

Y!&r $- Yjrr f

f

y&r) f

27Yjkrl (2.11)

i # k,

one formula

k = j,

yjkk

of

a particular

by all

possible

r = j.

becomes

(2.9). the

up to the

are

r.

interpolation

5th degree case

of

poly-

it the

has

been

general

formula

where

summation

I into

is

r positive

from

1 to

notes

the

carried

III), while

items

ql>

sum in the

out

subdivisions

of

the

. . . . 9r (r assumes all integral S Ql... Qr 0'1, . . .( qs >, --- >, q,.. gxpression

whole

91,

number values

i,)

form (2.13)

'Yip ,... jp,... jp ...jp.,

-L 91time* extended

to

all

de-

combinations

of

qr times

indices

pI,

. . . . pr,

satisfying

the

limitations 1
It but

can be assumed

it

is

hardly a = 1.

its

(2.12)

Let

if

q,=qp,

zz
application

correct

for

any value

of

m.

to large values of I can the particular cases where and will

be used

equation

system

in the

sections. a solution

be required

fr h where

formula

that

r,

be justified in practice. However, 2 are, no doubt, of practical interest

following 3.

that

so cumbersome

PaZPpp

Pa + Pp, a, p=l,2 ,...,

fi are sufficiently

*-

-9

4

for

= 09

regular

a linear i=

1, 2, . . .) n,

continuous

functions.

(3.1) Under favourable

conditions this can be achieved by a method which is a generalization of the secant method for functions of one variable. We assume that, by a given method, the (a + 11th point of Mj has been selected with the

Cse

oj

barycentric

161

coordlnaies

coordinates I ’ = 0, 1, . . ., n,

21jV”*1xnj9 where fi

these

at the

these

points basic

are

in a general

points

of

Mj are

we approximate

values,

position.

denoted

Values

by

of

yij = fi (zij,

fi by linear

functions

the

functions

. . ., z,.,~). Using

polynomials

i = 1, 2, . . ., n. We find

a point

purpose

the

M at

Y#o

together

with

coordinates quent ly, the

which

following +

Y&l

equation

of

the

values

yi

existence

of

all

system +

the

’

-

If

the fi

Ed are

one of

the

following

should

change

can

the

M,; point

given

of

be replaced

be effected that

of

M, at which

any other

with points the

norm of

should

determinant

also

M and the order,

$j +

should

(3.1).

one of

on replacing I...

1

the

1,

. . . , n) can also

‘**

+

.

. x k=o

= /

be replaced

that

the

new by

(3.4)

has the that

highest the

Since

point

Mj by point

. . .

1

value.

new system the

.

.

.

.

.

.

pk=lk .

.

M, is

1

* . . %I .

.

.

.

=

.

coordinate

~j

should

not

However, of

new value

P,D,

k=o

sufficient

The

M as the

YXj

position.

1

2-10

is

only

basic

repeated.

point

n

it

M and

If

the

calculations

taking

be taken

be in a general

D’

Subse-

at point

system

satisfied,

(yij, . . ., Y,j)

care

(1.1))

Cartesian

(1.3).

(3.3)

in the

not

= 0.

+

vector

any substitution

the

formulae

value

I/Y:j

or

is

by point

the

from

verified

errors

(3.3)

Mj (j

this

i = i, 2, . . ., n,

in a cyclic

point

For

(3.2)

has a solution, be calculated

inequality

permissible

inequalities

points point

the

system

should

zero.

solution

i = 1, 2, . . ., n,

O*

M can be determined

functions

become

requires

Yin!& =

+

-1Yi I < ei9 where

polynomials

equations

*

(1.2).

point of

these of

be zero.

equal

basic D’ of to

162

.s. s.

If

the

smallness hazard

of

determinant,

its

change

same time not

system of

points

the

of

These

a solution

approximation

to

(1.2)

and (3.2)

is

not

solved

owing

This

Mj,

where

points,

system

the

value

of

(3.4)

is

low,

should

apparently,

(3.1)

can be obtained

method

is

not

series

of

for

the

are close to the point which using them, a more satisfactory

and,

for

ge also systems

the

functions

to be solved, it from the solution

tion

of

the

next

cannot

fi

solution

of

computational

note

that

if

to

each

other

close

have left tions

numerical

new* but the

and flexible.

fi

in the

systems

system

not

one system (in

the

is advisable to retain of one system as the

system.

vicinity

of

for

the

nonlinear here

(3.1)

value

the

functions

values

at certain

near

Bittner

the

of

points ate.

of

obtained However,

by the method view, it

ally

in the

course

at each

step,

as recommended 4.

made a detailed study of the that method. He confirms

We will

of

(3.4)

the process

the

have

to

find

In the

the

function

direction

of

the

f allows There

to

Gauss.

fi at such of

these

and rate of basic

and Interrupt increasing

this

the

tendency

volume

of

than

to do

calculations,

is

the

local

extremum

(the

definite

maximum) of

which we will consider to be sufficiently reguin addition to the existence of a maximum* moving function

maximum point

increase,

sought

a satisfactory no need to

nomial. mate formulations. It is lem into two stages: the

As indicated

of

new system.

by Blttner.

function f(xl, . .., lar. Be assume that. of

are

solution

has a tendency to degenerhere described, is more satisfactory to Intervene occasion-

considerably

x,)

vicinity

func-

points,

conditions the system

a related

in our

this

the

solution.

141 recently

convergence

fi)

points the solu-

basic

a slight deviation in the values of the functions Therefore, points should not markedly affect the quality of approximation functions

simple

but a whole

of

the

new system

equa-

is

the set of basic initial basis for

in many cases

Moreover,

be rechecked

of

described

function e. g, at those where the values of the estimation these points are far from the sufficiently high. In fact. the previous system and probably far from the solution of

*

the

unknown point.

tions

of

to

an attempt should be made to make a hapcoordinates of one or several basic points. At the

in the

those

be affected.

gives

equations

Lolirov

define

for

it

is

approximation

with

more accurately

advisable to subdivide result in the vicinity

by A.M. Ostrovskii

[31,

possible

and that.

this

in this

to

a second these

reach

the

vicinity, degree

somewhat

polyapproxi-

the solution of the probof the maximum and final

method can be traced

back

lise

of

barycentrtc

conrdlnntes

determination of the point of maximum. In the first stage, search is conducted in a region where the function f allows a linear approximation to be made, in the second stage, such an approximation has already ceased to reflect the behaviour of the function (see [51). Let us again select the basic points Mo, Ml, . . . , M,, the coordinates of which satisfy the condition D f 0. We put

Using these polynomial

values

we approximate

of the function,

‘I -;I YOPO + YlPl i

function

f by a linear

(4.1)

* * * + ?f#,,*

In the space of variables pe, vi, , . . , &i,, the gradient of the function o has components ye, ~1, . . . , _vn. We exclude from the gradient the component which does not conform to the relationship (1.2) and term the vector with the components as affine gradient

gj = Yj -Lafr n-+-l

y,,

i=

0, 1,. . ., n.

ai0

The direction of this vector depends only on the selection of the “j and the values of the function f at these points and does not with any affine transformation of coordinates in the xl, . . . . X, Subsequently, we will be interested only in the direction of the gradient; we therefore normalize it as follows:

l,b 73

rj

=

gjlg,

g=

a gs t

j=

points vary space. affine

0, 1, . . ., n.

s=1

We assume that the points values of the function f:

i.e.

,Vj are numbered in the order

of

We proceed from the point M, in the direction of the affine we take the point M with barycentric coordinates pj = rljl Pn =

1 + tr*

j = 0, 1, . . ., n -

increasing

gradient,

1,

(4.21

for t = 1. If the value y of the function f at the point M appears to be higher than yn we exclude from the sequence of points Me, Ml, . . . , , M, the first, to which coordinate Llj of point M corresponds which is not

equal

to

points the If it

zero.

We will

adopt

and assume point

calculations

new numbers

M to

starting

from the

a certain

previously

the

set

second

of

given

to

points

case

it

by a linear

the

positive of

value

one of

can be considered

polynomial

(4.1)

that

does

not

Then,

of

the

t in formula t does

not

of

t,

will

(4.2)

until than

case

point

of

a satisfactory

in

In the

function result

f because

direction of insolving the prob-

be used.

We approximate

f by a second

function

in addition

purpose,

half

same method.

approximation

give

to

all

gradient.

smaller

first

the gradient of this polynomial does not determine the crease of function f. In this case the second stage of lem should

affine

be incorporated

them by the the

subsequent we repeat

become

77. In the

number,

determined

Mj instead

by 1 for M,.

determination

we will reduce Y d Yn* subsequently, either does not show that Y > Y,,, or

M, corresponding

reduced

be the new point,

to

the

values

of

degree

function

we will use the ~111 now denote by Yjj, middle of all segments which join pairs

=

Yjk

f

( =lj

The approximating

+

Ilk

2

)

.

.

z”j~ ‘Ilk)

.(

polynomial

polynomial.

f at points

values

of

of

basic

)

can be written

the

this

For this “j

which

function points

k = 0, 1, . . . , n.

j,

we

in the

(4.3)

as (4.4)

where

(see

formulae

(2.7)

“jk =

We will tionary are

find

point).

related

and equate

the

4Yjk -

Yjj

-

partial

purpose,

we derive

j, k = 0, 1, . . ., n.

Ykk*

extremum of

For this

by (1.2)

its

and (2. g))

the polynomial q (or at least its stabearing in mind that the variables pi

the

derivatives

(4.5)

function

l.~a,pi,. . .,p,,

to zero (4.61

These

equations

Using

the

should relations

be solved (4.5)

together

and (1.2)

this

with

equation

system

(1.2).

can be changed

to the

(‘se

of

barycentric

coordrnatcs

165

form

(4.7)

where

we have

If the

this

system

point

place

put

found

one

of

has is

the

Vj basic

points

If,

at

unique this

distance

of

In

practice

comparing mating

can

value

polynomial

quality (1.2)

this the

of

of

the the

or

its

presence

often

be

the

9 at

the

approximation.

overcome

stationary It

the

an

extremum

from

if re-

value set

of

a

of

the

are

If

other

function

process f.

at

points

a of

derivatives.

of

the

f

function

problem

sufficient

in

9.

thus

equations

of

the

evaluating (4.4).

(4.6)

at

itself.

reliability

value

and

have

random. by

course the

8 difficult the

not point,

at

there

partial

point

follows

does basic

simply

of

first

with

the new

elucidated

points,

with

f

function

or be in

9 is

that the

(4.7) a new

point

of

polynomial

of

find

should

basic

ascertnin we again

again.

difficulty the

to not,

For

system to

extremum

to

function of

point

it

If

ensuring

gradient

cause

any

f.

replaced. (4.7)

made

affine

the

examined

point

be

process, be

its

from

this

verification

a stationary

the

near

order

of

this

that

when.

one

discontinuity The

of

new

system

can

recurs, shows

arises

the

be

function

Mj to

solve of

should

the

by

attempt

direction

Experience

described

and

feature

for

point

stage an

the

special

means.

the

we derive a certain

in

points

to

solution,

moving

maximum

basic

= 0 corresponds

it

a solution,

the

by approxithe and

that

h

(Jo=-As the

a control, point

it with

is

recommended

barycentric

.

= -; that

the

quality

of

the

approximation

at

coordinates

p. =

PI

=

.

.

.

=

fin =

L n +I

should

also

In the ordinates

be

verified.

problem are

investigated

particularly

the clear.

advantages Uith

movement

of

using along

barycentric the

affine

cogradient

166

each

,‘. .>.

vaiue

of

definition

new

of

the

mation

coefficients

the

the

function

direction

calculated

of

of

Lavro1,

this

the

facilitates

approximating

polynomial

simple to calculate (and on changing fron equation (4.7) the need to calculate these coefficients is for only

each

succeeding the

(n f l),

calculated

which

gradient

and more accurate new value is

for

(i.e.

the

the

function what is

are

of

this

investigated,

approximation)

approxi-

extremely

(4.6) to equation generally avoided)

determination

minimum of

a linear

of

the more precise

With a mean square

gradient.

required

has to

to

and

polynomial,

calculate

by a conventional

be the

difference

method. Translated

by 2.

Semere

REFERENCES 1.

MGbius,

A. F. , Dcr

analytischen 2.

Byushgens. Part

3.

4.

S. S.,

I,

Ostrovskii,

A.M.,

nie

uxavnenii

Bittner,

L.,

Box.

Calcul,

dcr

Analyticat

Solution

i sistem

G. E. and Wilson.

conditions,

of

J,

Roy.

Sot.,

and

zur

132’7.

geometriyaf.

Equation

Moscow.

Izd-vo

zur Auflijsung

Mech.,

K. B. , On the Stat,

Hilfsmittel

Barth,

~Analiticheskaya

Equations

uravnenii),

Math.

neues

GONTI, 1939.

Mebrpunktverfahren Z. Angew.

ein

Leipzig,

Ceometrie.

Geometry

Moscow - Leningrad,

systemen. 5.

barycentrische

Eehandlung

43,

No.

No.

lit.,

1963.

von Gleichungs-

3,

experimental 3 13,

(Reshe-

Systems

in.

1,

111-126,

1963.

attainment l-45,

1951.

of

optimum