MACHINEEPSILON Eps

Eps 1224 6 to 8

EpsDp 2 S3 1.0b

fl IR f Mapsreal sontothefloatingpointsetcountably infiniteset

For ell X EIR Te x'EF s t

tx s eps1 1

i For all x EIR f E s t lets eps

flex X Its

Thedifferencebetween a floatingpointnumber and

a rednumber closestto it isalways smaller thanEps

FLOATINGPOINTARITHMETIC

The operations t X have

computer analogues 0 I

X yAtx y

for all x'y c IF F E lekeps st

x y's fx g He

where x y EIR

i even fl arithmeticoperation is exactuptoa relative error of size atmost Eps

Rgb Complex flockypointarithmetic isdone via software

LECTURE14 StabilityBigoh G Little 0 Notation

Spee Exn Can 2 sequences

if xn GlanBig Oh

Xu An

s T

no

F K constant and no constant SvtHnl Ek Hnl for n no

F Yaitsk pIf Xu du littleoh

taNo

if chm HI son ses Hnl

Evey 0 is 6 but not theotherway around.g

ex hfi lt4e2gcnl.eu

E f E 4

had.co

gcnDexfln engCn se2n

ftp.tnsEzeE.oifcnsolgCnl

We mightbe looking at whetheppesuhene 70

say gce offles

Eino so

e.g.EE ty

IsogACCURDCYFIDELITY

Let f be an algorithm that computes fDefinition

Abserr absolute error s HEA fG HRel err relative error s HFGI.sc

fw sore xtf H

Relative Accuracy11 1 5 11

1154111wantthistobeGceps A

STABILITY f is relativelySTABLE if

It f G faith6311

Oceps

for som I ltx 06ps

Wenotethat we compute ftp.fcxlfflxgenerally

BACKWARD STABILITY

The algorithm f is backward stable if

fG s f G for sure x with lk Okeps

Gives theright answer is y a nearby input

R Since all moms are equivalent on finitediners.wel problems an algorithm willbestable or otherwise regardless oftle unweighted mom used

LECTURE 15

Example of backwardstability

Subtraction

f X 72

fail fKx 0 fl ka

fl x X CHE KfcEpsfl1 1 72 Hea Iedreps

fG skate Kath CltE3letups

i fl FIGD XCited ItsXz HED Ite

X Itc Ez 1EiE3

Xz It Ez1EastE2E3

E Es Erez G Epswe ignore

i fl ftGD X Hey XzClxEs

i ftp GD of CxDisegualtotle differenceof I and I where

I t.sep.sk j seps

BeIewcrdstebhupeEhien.f

ex A Xy team yeah

Ps AAcy ft flagg

fl Ix Ext Sx

flip ytSy

flex FICg

Cxt8x CytsyCxtSx ytSy CIt 8

so let x sxtsxy yi.su

I g Its could find neighboring

x RigbIgeneallyHis 8 destroysthe

rate I nature ottle exactproduct

AXIOMSOF FLOATING POINTARITHMETIC

xeR I e IdeepsSt fl G skate

for all x y EIR TE ldeeps

x y y Ite

ACCURACY OF BACKWARD STABLEALGORITHMS

Than Spse BWstablealgorithm Apply itto

solve a problem f X Y w andihunumber k on acomputer satisfymy FLAXIOMS

Rel err llffxt.sc1fG7H

0lkcx7epsx

Thismeansthat algorithmic choices that

improvethe cmdihwy bymeting it

smaller

canhave a significant input on accuracyaswellas an stability