Numerical integration and parallel microprocessors systems

NDHERICAL

RJTRGRATIoeJ AND PARALLZL HICROPROCRSSORS

‘)

Kwwskf

Siri

Abstraot:

Imsrioal

solution

the regicms, where toaow,

-

equations

digital systems represent

neither

optimum method for an extremely equatiaps

oaa be obtained by mesns

of parallel nioroprooessors.

Stiff systems, equations, problems

preoise

preoise auwrioal

of ordinary differential

differential

one of

repressnts

has not been prwided

yet. Up aor

spee-

- for suoh a kind of oomputations.

osmputers

A new

of differential

optimum user's oafort

oaPfemporary

dy enou&

SYSTMS

is presented. Eigh-spssd

partial differential

equatiam,

geaeratiop

systems

equations,

of algebra10

of funotions,

strmg

nonlinear

and transoeadeatal

linear and nonlinear

of oaxtrol have been suooessfully

solutim

operations

progrraimg,

solved and sLUrted

by the

given method. Key

words:

-A,

RQM%RICAL

TAYLOR'S

1. Nwerioal

PARALI2L WICROPROCRSSORS,

DIFFZRXXTIAL

of aumerioal

solution

EULER, RUNGE

EQUATIm

of ordinary differential

solution

Methods analyasd

INTEGRATION,

EXPARSIUN,

equatims

of differential

equations

were

in detail by Rrolge snd Kutta a& the end of the last osntury.

A lot of aey asettmds for ths aumerioal

solution

of differential

equations have been published sinoe. We have had sme-step -Kutta, Taylor)

and multi-step

.

(Euler, Rungs-

(sxplioit and implioit) methods at our

disposal. The aumerioal at our dspartnnt ooopsratiax. equations

possibility

solution a)

espsoially

of differential with respsot

equations

to parallel

Ve have found that the aumsrioal

represents

operations arwrioal

solution

a very attraotive

of starting and oompating independent

solution system

b) systr

almritim,

of differential

the 8-0,

wbioh gives us a shsoltamestm,

equatiam

Parallel

that the

must be divided

into the

systems:

of hsmo~ous

osmstant

qioroprooessor

soluficm

of eaoh other. It has been prwed

of differsatial

of two follow*

has bsea aaalyasd

lhear

differential

equatiam

with

owffioisnts

of asmljnoar

1) J.RmoIsIf,TBcHRIcAL 612 66 BBIO, &JR

differential

DBITERSITI,

lqaatiolls

DEP. OF CONPUTBRS,

BO&'i!CEOVA

2

453

2. System

of homogenou8

linear differential

equationm

with oon8tmt

ooeffioiant8 lolutian

Analytio oaastant

oo8ffioient8

numerioal

of homogenou8

linear differential

8olution ha8 not been malyzed

It wa8 just the numerioal tial 8quatian8 department

with oaa8tant

that prwided

ooeffioient8

th8 lsthudology

we oarvfully

analyzed at our for the realiza-

parallel operatiau

oaloulaticmo

8imulfanoou8ly

to determine

a new (next) value of 8olutiam

Rung8=Kutta,

3. Sy8tem

intogratiun Hilne)

to Taylor.8

of nonlinear

The 8olution

method

independent

M

initial oonditiap8) in aooordanoe

ntrerioal

oan be generalized expa~ion

of eaoh

of how to *tart and how

with a

(again 8imultaneou8ly integrati~

in leprrrte (Euler,

- the g8neralization

method.

differential

equation8

of a 8y8t8m of nonlinear

not been eatiafaotory method8

the give

All the algoritln8

Adam,

lead8 direotly

(fra

linear diff8r8n-

and prinoipl8

other. The nrplerioal lolufiar grvo ~8 the methodology

qioroprooe88oro).

with

in great detail yet.

lolution of homogenou8

tiaDl of the 8am0, 8imultaneou8,

suitable numerioal

equatioP8

ha8 b88n well known for a long time. That i8 why

differential

equatiaao hao

obtained yet. That i8 why a lot o@ numerioal

eriat.

It ha8 been przv8d again that, when 8olving a 8y8fr differential oooperation

equatiolu,

the ba8io problem of parallel mioroprooe88or

oau be 8olved by m8ak

tranofoxmation

of a lpeoial

(not oaplioated)

of a giv8n 8j8t8m. The tran8formatiar

generally noprutamomou8

of nonlinear

oonv8rt8 a

8y8tem Y'S f (t, Y)

to an autanamoua

ome with the 8p8oial form of z't F (z)

The tran8forration differential

i8 based on a 8yotem of 80 oalled goaerating

equaticm8.

4.sumary Colrorefe reoulf8 the leotur8.

Moreover,

(gfving aoouraoy aud th8 hardvare

ayetom will be di8ouo8ed.

lpood)

willb8

moonted

of a parallel nioroprooe88or

in