Computer arithmetic and ill-conditioned ... - IEEE Computer Society

())MPlITER ARITHMETIC AND ILL-())NDITIONED ALGEBRAIC PROBLEMS

t

GUnter Schumacher Universit�t Karlsruhe Institut flir Angewandte Mathematik

12. D-7500 Karlsruhe

Kai ser s t r .

West-Germany

Abs t ract

Occasionally,

are afflicted

which

that the

with

tolerances.

with

algorithms

computers.

on

guarantees

It

w .......

exact result of an alg o r ithm lies within

In ill-conditioned be extremly wide and al­

the computed tolerance bounds. cases

may

bounds

these

though the sta t ement still remains valid.

s ibi l ity the

successively

of

Furthermore,

solution

( in

with

Kulisch

and

essential

and

tole r­

scalar

exact

for their implemeQtation.

sense)

truncation

a

of

real

machine representation is def i ned from the rea l

by

p roduc t

is

numbe r

to

VnT Mn T WT IT

set

. . . . . .

vectors T E {fR,S} square

ponents

. . . . . . . . . . . . . .

of

length

matrices

in

T E

wi th

{ffi,S}

power s et of T E

over

n n

rows

interval

see

tThis paper is an extented excerpt from

CH2419-0/87/0000/0270$01.00 © 1987 IEEE

S I �

x

[ a,b ] E

IS

inter­

3 a E A. b € B : 5: b V b � x � a }

�

A

B}

two

for

i.e.

deAl

€

:=

b-a

take any point

IS}

rounding from Pffi into IS with the pro­

perty

A

OA

E Pffi

vectors

=

{X E IS

n

ma t r i ces

and

A C X}

I

defini­

this

tion applies componentwise

interval operations in IS, IVnS and IMnS

defined for T E

this

{S,VnS,MnS}

A�B

/\

oE{+,-,-./}

paper

cons i de r

we

by :=

the

O(AoB)

following

problem: "Let

f

D

:

Vnffi be a continuously differentiable

-+

A

D C Vnffi. We seek an x E

A

IVnffi of x,

where t he

Vnffi

A

w i th f(x)=O,

we ask for an incl usi on X E

diameter d(X)

is sufficiently

a

a

set

and

com-

We restrict ourselves ponents of f tions

of

set

and

letter

com­

The al­

such

as

the

existence

and

f within X. These a zero x of the first is (E called E-methods the three German wo rd s "Existenz" for of

are

of

"Eindeutigkeit"

"Einschlie}3ung"

270

i n t e g er exponentiation.

c o ndi tions

of

existence,

[4J.

the

A

uniqueness

[lJ.[6J).

the case where

the solution x and pe r forms an automatic veri­

methods

of

to

arithmetic expression with opera­

to be in t roduc ed determines an inclusion X

fication

set

are

+,-.",1

gorithm

{ffi,S.Vnffi.VnS.MnrR.MnS}

analysis

=

or more practically,

of

set of intervals over an ordered T E {fR.S.VnfR,VnS.MnfR.MnS}, i.e. the of all [a.bJ := {x € T I a � x � b} properties special some (For

a € A, b E

(in our con t ext m e A ) may

small."

numbers

a

out of an i n terva l

Throughout

its

to be a mapping

floating-point

I

mid point of an i n terval

function,

of

b

0

{x E

:=

A,BEIT

numbers into the screen of floating­

given digital computer

{a

:=

mCo)

For

We will summar i z e some notations and concepts:

. . . .. . . .

W B

A

0.0, 0,

. . . . . . . . set of real numbers

A

B

/\

point numbers.

rR S

0

diameter of an interval.

fixed

on

In [8] a mathematical foundation has been laid of what we call "Computer Arithmetic". For examp l e , the a l geb ra i c structure on a digital computer may and the be described in t e r ms of a " s cr een ", of

A

convex union operation vals A.B E IS:

dee)

O:Pffi-+IS

Introduction

process

.

interval

def ined

based

are

They

an

of

concepts ari thmetic

computer

Miranker.

t heorems

point

O.

the

a mathematical

the

combine

methods

analysis

the

dim i n ishing

the exis tence and uniqueness of

is proved

by the computer. These

been

introduced as E-methods provide the pos­

r ecently

ance.

it is in

have

which

Methods

worthless.

practice

�.

For two in terva l s A.B € T an arithme t ic operation 0 E {+,-,-,/} is de f ined by

al­

reliable statements when applied in

ways pr ovides

numerical

computation

the

i.e.

arithmetic,

Interval numbers

a member

den o te

we will of an interval A by

for

for

inclusion.

uniqueness,

and

this purpose, the pr ob l em of f i nding a zero x into a fixed point equation g(x) ::: X (see [7], [11], [12]). An i nc lusi on of the solution of this fixed-point problem is com­ puted iteratively in the space IVnffi starting with

For of

n

f is transformed

E Vnffi.

ho l d s .

use of resi­ the dual correction techniques the diameter of resulting interval is di mi n ished more and more.

an

approximat e

1.

solution x

By

Lemma 3:

1:

Lemma

X

:

Proof : From

[5], e.g.

this lemma

Theorem 1:

f

with

A

If F(X) � X,

/\ / \ x k�O

€

�

where

then the equat ion

f(x):::x

,...

l e a s t one solution x

at

has

(1.1)

'*

f(X)

:=

F (X) ,

tence

of

a

solution

The rest is shown

we

can

uniqueness tions.

(X».

•

y

'

d

:=

( X)

by

x

€ X of f(x)

induction.

d(C · X) �


(Y 0 �

WZ

-R00f(x)

HCZ) then

S and G(Y)

00(I-RoJ(x � Y)

The definition of

:=

:

for Y € WZ. where R € MnS is an trary but fixed matrix. If

'"

€

J(Y)

(1.5)

be an '"

implies G(Y)

Y

0 R 0 Of{x)

f(x)

Let H

holds.

�

x}

G* (Y)

to assert

Of denotes any function

1\

-

only one

if rn is

:= x

�

€

hence

arithmetic evaluation of G,

(Y)

IVnS. € and Z VnS, € x 1 C «x0Z) 11 x). For every arbitrary

Let

which satisfies

G{Y)

that

in the following

x,

with A

the matrix R

Y C xtJX it is

summarized

solu­

an inclusion

to better results than

see that R " "

been

�

approximate

fixed Y C (x Z) � x, let J(Y) € IMnS and assume that for every set of n vectors Yl" " Yn € y,

the inverse of the Jacobian J{x). deve loped for application on computers. In order to approach the goal of c ompu te r application, a first observation is that has

an

but

approximation of

3

- x of

< 1

y - x =O�y =x .

Theorem

lead s

x

considering

M(y)o(y -

is regular and

M{y)

For a better

X-x

�

x-Rof(x)

J(x � X}

A

fCy)

theorem can be formulated

x itself. This is

f.

of

critical

Let y € X be solution. The generalized theorem im p l ies the exisA

it follows

tence of

x

imp lie s p(I-RoM)

eigenvalues of I-R o M we are

x

we have

-

Theorem

all M €

a

the

"

(I-RoJ(x � X)}-(X-x)

for

only

f €

a fixed with (1.4)

of

+

-Rf(x)

t ion

suc h that

g(Y)

Furthermore, '"

J{x 11 Y)} may be evaluated with

the absolute e rro r

have

x - Rf(x) + (I-RoM(y»- (y-x)

existence

Hence,

theorem

1 and (1. 4) then delivers

Theorem

•

term

the

directly appli­ understanding of the following corollary it should be r e marked that, in practical application, the inclusion of

R(f(x)+ M(Y)o(y-x»

y -

=

€

But

Y

every

=

the

value

a M(y) €

exists

R

in

as stated

resi dua l

tool

cable on computers.

Consider the (continuous) function g : X � x � vnrn , g(x) = - R - f(x).

Proof:

-

exact summation algorithm

such a '"

one rounding in each componen t .

solution x in X and

� O.

k

f (x)

equati on

one

only

(X) ,

€ G

[8]. Wi th

(1.4)

then the

and

one "

requ i res an

X

�

�

Ox) with

Of ( x )

1)

an

2)

an

approximation

step

to

determine

sufficiently good estimate x.

step

inclusion

enclosure Z of

w i th

respect

to

to

determine

a

t he

the absolute error x - x x.

In this chapter we will discuss some implementa­ tion aspects of these two points. Let us start with the inclusion step.

the scalar product a,b € VnS,

272

2.1

Implementation of the inclusion s tep

Example:

Accor d ing to [11] we define an c-inflation i nte r va l A € IS by

{

:=

Aoc

Here 11 number

the

the

c

0

•

for d(A) pos itive

smallest

in use .

computer

=

be com pu ted by

will

X Zl := l z2 := x3

0

floating-po int

For

z3

interval vec­

thi s definition applies componentwise.

tors

an

for d(A) '# 0

d(A}

[-T/, +T)]

+

A

is of

+ [-1, 1 ]

A

f or

The expression

a

to be

the

Using

obtained

good value

by

for c.

a

corollary,

the

performing

the space IVnS:

Y

0

:=

inclusion may

verif ie d

fo l lowing

iteration

in

Z

Y

:=

'-

Y

c ount +

0

e.;

The

c omputa tion

If the

Z

)

Qt

of

corollary

( count

sol uti on x lies

g

In the

used.

an

(In [2]

term

has

me thods .

residual

If

is

it

ter m

f(x

not

(k»

for

We

[9]).

e.g.

accuracy

is

possible

common

to

accur a t e l y

compute

we

.

stated above .

the

c o mponen ts f

i

of

zl

step.

1

i. e.

using

.

.

•

zn.

zn

This

Hence .

fi(x)

=

system

for

the

wi th a

have

f are

an

algori thm which performs f o rmu la after the othe r .

operation in the r esults of the operations are intermediate variables zl' .... zn.

s t or ed

say.

this nonlinear sy s tem was trans­

system of

a ss

tri8Ilgular

in

ume

g

k

•

.

·

z

.

n

linear equations by using

The

techniques.

to

[7].)

be .

·

form and

.

resulting

can

system be solved wi th

differentiable

in

terms

of

zn be the approximations and

the exact s o l ut ions .

We define liz .: = J

Z.

J

and seek an

the

-

j

z.

J

inclus ion

•

=

1.

.

.

n

.

correction

of the

€

no

j

=

1. .

.

.

n

Applying t he mean value theorem to gk with respect

con­

sidered as arithmetic expressions. The computation is usually done on a compu t er step by

of f. (x)

[3]

a

z .··.z · Let z • k l l

to al l

m easu r e how near we are to the exact solution. As

.

a

0

=

z2)

'

the principles f ro m

methods

since the pro b l em of eva lua t ing the function f at wi th high

l

rewriting

also

( see

A

for x

and

formed into

(2.6)

x(k)

•

form.

as

nonl i near

solution

following any of those pr in cip l es may be Here we describe on ly the "classical"

point

l

of

variable

N ewto n ' s method

a

system

the exact

.

)

of different

approximation

a

= 0

l ( Zl )

2

Performing the approximation step

ob ta i ni ng

con s i de red

be

may

tr i angular

has

g (z

count is used to prevent infinite lo opi ng in c ase s

There are a large number

. z5

.

s pe cial type of equation system:

Z is sa t isf i ed .

�

where the iteration is not convergent

2.2

equations

count.J1l8X ) ;

=

in x�Z and that no other l ies between these bounds. ( The integer "-

.

guaranteed ac curacy we must consider the following

i t is verified that

1

of

computation of an i nc l usi on of

t e r minat i on criterion Y

then by

process

evaluation

equations for the unknowns z

1

O(I-RoJ(x!.! (x0Z»)0Z

�

.

will state the principal way to arrive at a highly accurate i nclus i on of the function values:

-R0(>f(x) 0

unt i l ( Y

0

We

s ucces s i v e

:=

:=

using intermediate results zl'

be

count := 0

count

+

zl

z2 z4 = : Xs - x6 z5 := z3 / z4

c-in­

flation is indispensable to assure the convergence of the inter va l iteration. In practi ce , 0. 1 tur ned

out

x 2 x4

+

to

the

yie ld s

one

The

some

273

last vari abl e

�

in the

i n t e rval

� U

zk

A

�)

Co nt inuing

A

A

p rocess

this

with

resul ts in A

A

gk(zl····�)

accuracy

sharp

A� .

•

by

Ck €

zk-l'

zk-2'

applying

from

the

sufficiently possible

ari thmetic

� •

� J
10

then

an

accuracy. Column 5 displays the number of s i on s t ep s in (N2). The ini ttal value for

� (I-R-J(X!1 (X0Z»)0Z := (Y � Z) ; (j

In

mati on in step

n

m:.

Xl'

As

display only

Newton s t ep s in s t ep (N!) of the algorithm. Column 4 answers the question whether the Newton approxi­

6;

s uc cess

k

( )

we

values,

approximation R of

i=o := 0

were 10-12

th e 61

�(k);

ximation

(N2)

the solution vector x becomes small. The charac­ teristic results in table 3.1 where computed on a 68000 microcomputer in PASCAL SC using a 13 digit decimal arithmetic. In all examples the value s of

an

x

0

a

Xi

where H € Mn� is the Hilbert matrix of degree n. i.e. H == «h . . ». h .. ::: 1/(i+j-l). i.j:: l. ... n. IJ IJ � : Vnm � Vnffi is defined by

=

=

a

4·�·xi_l

•

The values Xi may be

(1 - xi-l)

275

=

1.

... n.

theoret ieally determined by

explicit evaluation of the best the even Nevertheless.

an

i

right-hand arithmetic

side. will

approximately lose 0.2 digit places Table 3. 2a compares the results for

per

x n

step.

References:

wi thout

[ 1J Alefeld. G . . Herzberger. J.: Einflihrung in die Intervallrechnung. Bibl. Inst . . Mannheim­ Wien-ZUrich ( 1974 )

exact scalar product with those in table 3.2b when we apply that feature. In the case of evalu- ation by simple real arithmetic, the correct digits are underlined. The computation was done on an IBM 4361. the verified results where obtained by using the ACRITH package. The ini tal values for the results of t able 3.2b were the values of column 2 of table 3.2a. We have used a 0.3. A 0.95. =

[ 2J Bahm. H.; Berechnung von Polynomnullstellen und Auswertung ari thmetischer Ausdrlicke mit garantier t er maximaler Genauigkeit. Doktor­ Dissertation. Universitat Karlsruhe ( 1983 )

=

[ 3 J Bahm,

H.. Evaluation

S.M.: Least Significant Bit Ari thmetic Expressions in Computing 30. 189-199 Single-Precision, ( 1983 )

explicit evaluation by n

real

10

0.9336088861037432

20

0.9484616645868601

30

0.9497466644900358

40

0.6148475096423984

80

0.8340590480693408 0.6913591756135743 0.5952532668408813 0.5520411381488361 0.7244782989827280

50 60 70 90

100

Table

n

10 20 30

40 50 60

----

.

3.2. a

; 0.948461664586�

� 0.948461664586818� 0.949746664490856� 0.6148475094604607 0.772273942734024� 0.8340589602250766 0.691351258545544 � O.597340812574404� O.210981469055217� 0.586638531833946 6

0.93360888610374

0.9497466644

93

89

i 0.7722739�

0.614847509

0.83405

91 88

80

0'

7

3

601

594

"inclusion failed" ----

ed

.

•

[ 7 J Kaucher. E Rump. S.M.: E-Methods for Fixed Point Equations f(x}=x. Computing 28. 31-42 ( 1982 )

----

with exact scalar product

2nd

[ 6J Kaucher, E.: Interval Analysis in the Exten­ ded Interval Space 11R. Computing Suppl. 2. 33-49 ( 1980 )

----

without exact scalar product

0.6913

100

[ 5 J Heuser, H.: Funktionalanalysis. Teubner Verlag. Stuttgart ( 1986 )

•

[ 8 J Kulisch. U.. Mirap,ker. W.: Computer Ari thmetic in Theory and Practice. Academic Press. New York ( 1981 )

solved as nonlinear sys tem using func tion evaluation

70

90

12 0.933608886 08 8 0.9484 4 43.3 -31.2 "overflow" ------ ----

0.7722739530781480

of

[ 4 J Bahm, H., Rump, S.M.. Schumacher, G.: E­ Methods for Nonlinear Problems. to appear in: "Computer Arithmetic: Scientific Computation and Programming Languages. Teubner Verlag. Stuttgart { 1987 }

interval arithmetic

ari thmetic

Rump.

[ 9 J Ortega. J.M . . Rheinboldt. W.C.: Iterative S o l ution of Nonlinear Equations in Several Variables. Academic Press. New York, 1970

[ 10J RaIl. L.B.: Automatic Differentiation. Tech­ niques and Applications. Lecture Notes in Computer Science. No 12. Springer. Berlin ( 1981 )

0.933608886103744

[ 11 J Rump, S. M.: Solving nonl i nea r sys tems with least significant bit accuracy. Computing 29. 183-200 ( 1982 ) [12J Rump. S.M.: Solving algebraic problems with high accuracy in: "A new Approach to Scien­ tific Computation". p. 51-120. Academic Press. New York ( 1983 ) [13] Stark.

J.: Maximal genaue Auswertung arith­ metischer AusdrUcke durch Losen nichtlinearer Gleichungssysteme, Diplomarbeit a t the Insti­ tut fur Angewandte Mathematik. Universi tat Karlsruhe

[14J Varga. Table 3.2b

Prentice

276

( 1985 )

R.S.:

Hall.

Matrix Englewood

Iterative Analysis. Cliffs ( 1962 )