Lecture 7. Norms and Condition Numbers


 Mervyn Andrews
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1 Lecture 7 Norms ad Codto Numbers To dscuss the errors umerca probems vovg vectors, t s usefu to empo orms. Vector Norm O a vector space V, a orm s a fucto from V to the set of oegatve reas that obes three postuates: > 0 0, C λ = λ λ R, V + +, V ( Traguar Iequat) we ca thk of as the egth or magtude of the vector. The most famar orm o R s the Eucdea orm defed b orm defed b = ma = / orm defed b = I geera porm, defed b p orm defed b p = p / p for p >0advector Eampe : Usg the orm, compare the egths of the foowg three vectors R. Repeat t for other orms =(,,,) T, v = (0, 5, 5, 5) T, w = (6, 0, 0, 0) T Souto: X 6 8 v w 6 6 6
2 To uderstad these orms better, t s structve to cosder R. For the three orms gve above, we sketches Fgure of the set : R, { } Ths set s caed the ut ce or the ut ba twodmesoa vector space. (0,) (,) (0,) (,0) (,0) (,) Fgure : Ut ces R for three orms I geera, for a vector R, Matr Norm Matr orm correspodg to gve vector orm defed b = ma 0 Norm of matr measures mamum stretchg matr does to a vector gve vector orm. Matr orm correspodg to vector orm s mamum absoute coum sum = ma a j j Matr orm correspodg to vector  orm s mamum absoute row sum, Propertes of Matr Norm matr orm satsfes:. > O O. γ = γ for a scaar vaue γ = ma j= a j
3 3. + B + B Matr orm aso satsfes. B B 5. for a vector Matr Codto Number Codto umber of square osguar matr defed b cod( ) = B coveto, cod () = sguar Eampe: = 0 = 6 = = =.5 = cod ()=6.5=7 cod ()=83.5=8 The umerca vaue of the codto umber of a matr depeds o the partcuar orm used (dcated b the correspodg subscrpt), but because of the equvaece of the uderg vector orms, these vaues ca dfer b at most a fed costat (whch depeds o ), ad hece the are equa usefu as quattatve measure of codtog. Sce = ma 0 m 0 The codto umber of the matr measures the rato of the mamum reatve stretchg to the mamum reatve shrkg that matr does to a o ero vectors. other wa to sa that the codto umber of a matr measures the amout of dstorto of the ut sphere ( the correspodg vector orm) uder the trasformato b the matr. The arger the codto umber, the more dstorted (reatve og ad th) the ut sphere becomes whe trasformed b the matr.
4 I two dmesos, for eampe, the ut crce the orm becomes ad creasg cgar shaped epse, ad wth the orm or  orm, the ut sphere s trasformed from a square to creasg skewed paraeogram as the codto umber creases. The codto umber a measure of how cose a matr s to beg sguar: a matr wth arge codto umber s ear sguar, whereas a matr wth codto umber cose to s far from beg sguar. It s obvous from the defto that a osguar matr ad ts verse have the same codto umber. Note: Large codto umber of mea s ear sguar. Propertes of the codto umber. For a matr, cod (). For dett matr, cod (I) = 3. For a matr ad scaar γ, cod (γ ) = cod (). For a dagoa matr D = Dag(d ), cod (D) = (ma d )/(m d ) Computg Codto umber Defto of codto umber voves matr verse, so otrva to compute Computg codto umber from the defto woud requre much more work tha computg souto whose accurac to be assessed. I practce, codto umber estmated epesve as bproduct of souto process Matr orm ca be eas computed as mamum absoute coum sum (or row sum, depedg o orm used) But, estmatg at ow cost more chaegg Computg Codto Number We w ow see the usefuess of the codto umber assessg the accurac of the souto to ear sstem. I fact, to compute the codto umber drect from defto woud requre substata more work tha sovg the ear sstem whose accurac s to be assessed. I practce, therefore the codto umber s mere estmated, to perhaps wth a order of magtude, as a reatve epesve bproduct of the souto procedure. From the propertes of orm, we kow that s the souto of = the
5 = So that ad ths boud s assocate for some optma chose vector. Thus, we ca choose a vector such that the rato w have reasoabe estmate for. s as arge as possbe, the we Eampe: 0.93 = If we choose = [ 0,.5] T, the = [7780, 0780] T So that ad hece cod.38 0 ( ) = = whch turs out to be eact to the umber of dgts show. The vector ths eampe was chose to produce the mamum possbe rato,ad hece the correct vaue for. Fdg such a optmum vaue woud be prohbtve epesve. I geera, but a usefu appromato ca be obtaed much more cheap. Oe heurstc s to choose as the souto to the sstem T =c. c s a vector whose compoets are ± wth sg chose successve to make as arge as possbe.
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