A C T A U N I V E R S I T A T I S L O IJ Z I E N S I S
F O LIA OECONOMICA 48, 10P5 ______
Dózef S i a ł a ś * , W ła d y s ła w H i l o * * . Z b ig n ie w ' J o a t l e w s k i * * * ■ ON THE NUMERICAL PRO PERT IES OP THE M-P GENERALIZED
IN VERSE ALGORITHMS
1
. I n t f o d u c tlo nA la r g o number o f a lg o r ith m s f o r the c a l c u l a t i o n o f a ge n e r a liz e d in v e r s e h e
3
been sug g ested in ro c e n t y e a r s . I n t h if i pa p er we s h a l l c o n fin o our a t t e n t i o n to the a lg o r it h m s o f th e H ooro-Ponroso g e n e r a liz e d in v e r s e (t h e f’V-P g e n e r a liz e d i n v e r s e ) . These a lg o r it h m s in tu rn can be d iv id e d in t o two c la s 3 c s t n o n - i t e r a t i v e and i t e r a t i v e a lg o r it h m s .In fo r m u la tin g n o n - i t e r a t i v e a lg o r ith m s f o r the c a l c u l a t i o n o f the M»P g e n e r a liz e d in v o r s e A* o f the r e a l m a trix A w it h m rows and n colum ns, i . e . A « Rr (w hore A i s such a m a trix t h a t A*AA* - A * ; AA*A - A ; ( A A * )' « AA+; CA+A ) ' » A+A ) the f a c t o r i z a t i o n p r i n c i p l e i s u sed . T h is p r i n c i p l o c o n s is t s in the f a c t o r i z a t i o n o f g iv e n m a tr ix A w it h r a n k ( A ) ■ in t o th e p ro d u ct o f two ( o r m o re ), i . e .
mxk k xn
( l . l ) A » OF, C « R ° , F « R ° , r a n k (D ) * r a n k ( F ) » kQ.
Oue to the p r o p e r t ie s of the M*P g e n e r a liz e d in v e r s e and tho d e f i n i t i o n s of D, F we have
* D r . , L e c t u r e r a t the I n s t i t u t e o f M a th e m a tic s , U n i v e r s i t y
o f Łó d ź . . _
# i f D r. , L e c t u r e r a t the I n s t i t u t e o f E c o n o m e tric s and S t a t i s t i c s . U n i v e r s i t y o f Ló d ź .
# * ' S e n i o r A s s is t a n t a t th e I n e t i t u t e o f E c o n o m e tric s and S t a t i s t i c s , U n i v e r s i t y o f L ó d ż .
J o T.bt b ia la c , Wtad^alaw K ilo , Zbip/iiow Wa£itl»wskl
( 1 . 2 ) A* - F *D +.
( 1 . 3 ) F * - F ' ( F F ' ) “ 1 , D+ - ( D 'O ) “ 1p ' ,
Throo ty p e s o f f a c t o r i z a t i o n » a ro v e r y w e ll known. Tha f i r s t typ o has th e form
( f l ) A - LU,
mxk
where L « R l a a lo w e r t r a p e z o id a l m a tr ix w it h u n i t s on the L XII
d ia g o n a l and z e ro s above the d ia g o n a l, U <a R o tp the upper t r a p e z o id a l. T h e r e fo r e , due to ( l , 2 ) , ( 1 . 3 ) i t i s
( 1 . 4 ) A* - U V - U ' ( U U ' T 1 ( L ' L r V .
The second typ e has th e form
< f2 ) A • QS,
mxk
Q * R ° , Q'Q ■ I ^ k y S i s the upper t r i a n g u l a r , and
( 1 . 5 ) A* « S ' ( S S ' ) " V - S“ 1^ '
( t h e d e c o m p o s itio n ( f 2 ) can be done by the use o f H o u sh o ld er tran s fo rm a tio n s o r th e n o d lf ie d G ra m -Sch ald t p r o c e d u r e ). The t h i r d ty p e has th e f o r e < fl> A - U Л V ', mxk , nxk к л хк U e R ° , V - R , Л. ■ d ia g ( X j , . . . . \ ) « R ° U ' U • О ■ V 'V * 1 ^ y , l a th e d ia g o n a l m a tr ix o f n onzero sq u are r o o t s of e i g e n v a l u e s o f A 'A ( o r A A * ), and ( 1 . 6 ) A* • V A mlU . '■ Л ' /V '■■■• - "" 4 •■V's
The f a c t o r i z e t i o n ( f l ) i e u se d , among o t h e r s . In c a l c u l a t i n g tbe М-Р g e n e r a liz e d in v e r s e , when th e Gaues e lim i n a t i o n method w ith com p lete p i v o t i n g , l a used (« е е th e d e s c r i p t i o n o f th e a lg o
The f a c t o r i z a t i o n ( f 2 ) l a u se d , among o t h e r s , In the con s t r u c t i o n o f th e a lg o r it h m o f I u a c k e [ 3 ] .
Th* f a c t o r i z a t i o n ( f 3 ) i s u sed , among o t h e r s , in th a f o r m u la tio n o f th s a lg o r it h m SVDZ d e s c r ib e d in t h i s p a p e r.
I t e r a t i v e a lg o r ith m s ( t o w hich th e s tu d ie d a lg o r it h m A B I be lo n g s ) o ra baaed on th a Id e a of l t o r a t l v o m onotonlc tw o - sid e d op- p ro x lm a tio n e o f S c h u lt z .
Tha e x is te n c e o f th a g r e a t number o f th e a lg o r ith m s f o r c a l c u l a t i o n o f the M oore-Penrose g e n e r a liz e d In v e r s e o f th e m a trix im p lie s th a need o f th e c o m p a ra tiv e s tu d y o f t h e i r p r o p e r t ie s . I n the p a p e r the a u th o rs g iv e a d e s c r ip t io n o f fo u r such a lg o r ith m s and t r y to compare them by use o f the M o n re-C arlo e x p e rim e n ts . Tha m easures o f a lg o r ith m p r e c i s i o n , used in the e x p e rim e n ts , w are based on f o u r c o n d it io n s w hich d e f in e the M oore-Penrose g e n e r a liz e d i n v e r s e . They a re o f th e form«
I A * > - A II II A?AA* - A t | ■ •> .... i * T »> c ) II A* H l ( A A * r - A A ? | H (aT A )* - A * A 8 ---- --- — A— a ) — 1 - ,---1— Ia a* I | aJa | whore A* d e n o te s , g iv e n by th e i - t h a lg o r it h m , e s t im a te o f th e M-P g e n e r a liz e d in v e r s e o f th e m a tr ix A ( i ■ i , 4 ) , and de note«} E u c lid e a n norm.
B e s id e s t h i s , we t r y to answ er the q u e e tlo n how much th e chongea in i l l c o n d it io n in g o f the in v e r t e d m a tr ix in f lu e n c e th e form o f th e g e n e r a liz e d In v e r s e o f t h i s m a t r ix , r e a liz e d by aeons o f the 1-th a lg o r it h m . T h is has boon done by means o f a com parison o f th e E u c lid e a n norms o f th e M-P g e n e r a liz e d In v e r « i s « aJ , 1 • 1 , 4 f o r v a r i o u s d e g re e s o f i l l - c o n d i t i o n i n g o f th e in v e r t e d m a tr ix A.
0 Jû a a f 8U1&», VtadysUw M ilo , Zbigniew V aallew ski
*--- ---- -— --- ;---;— 2 . The D e s c r ip t io n o f th e M g o rlth m e
Wo c o n fin e d o u r a n a l y s i s to ona i t e r a t i v e a lg o r it h m , n ark ed h e re a * A B I , and th r o e n o n - i t e r a t i v e AWMEL, G E IN
1
V, SVOZ bas ed on th e Id e e o f th e f a c t o r i z a t i o n o f th e g iv e n m a tr ix A o f ronk k Q. A lg o rith m s AIWIEL and GEINW a r e based on th e fa c - t o r i z e t i o n o f th e fo rm tA • LU, A « R * * n , r a n k ( A ) ■ kfl < « i n (■ , n ) ,
•xk
w here L e R i s l o n e r t r e p e z o i d e l m a tr ix w it h u n i t s in main d ia g o n a l and z e r o s above i t , U « r?ko * “
1
q u p p er t r a p e z o ld e l m a tr ix and A* a ■ U '( U U * ) ~ * ( L %L ) " ‘* L * .A lg o rith m SVOZ ie based on th e Id e a o f th e e p e e t r s l f a c t o r i z a t i o n o f th e form
mxk_ nxk k x k
O IJ '«/ _ i.'.É _ A - U A . V ', Ü « R ° , V « R A « R ° ° , V 'V • U 'U - I
o where A d e n o te s the d ia g o n a l m a tr ix o f n onzero s q u a re r o o t s o f
* - A \
th e s i n g u l a r v a lu e s o f th e m a tr ix A 'A , and A » V K~ U *.
B e lo w we g iv e a c o n e ls e d e s c r i p t i o n o f th e s u c c e s s iv e s te p s o f th e H o o re -Pe n ro se in v e r s e c o m p u ta tio n by means o f each from th e a n a ly s e d a lg o r it h m s .
» -V ’ • . • •
2 .1 , A lg o rith m A 8 I ( c f . B e n - I s r a e 1 [
2
] )S l : U s in g one o f th e a lg o r ith m e f o r d e a lin g w it h e ig e n v a lu e p rob lem ( e . g . th e method o f 3 e c o b l) f in d th e g r e a t e s t e ig e n v a lu e o f th e m a tr ix A*A and d e n o te I t by ^ B X . S 2 t f i n d such a v a lu e a t h e t
0
" » < S 3 t Porm - o A * . i o S 4 i F o r J • 1 , 2 . . . compute /u n t t l
l x - x A — i ± < r^,
whero ^ d e n o te a tho rank of th e a c c u r a c y o f tho i t e r a t i v e p ro c e s s . I f the c o n ve rg e n c e c r i t e r i o n le s a t i s f i e d f o r J • J z thon p u t A+ j ■ X j z .
2 .2 . A l g o r i t hm AWMEL ( c f . __XI i l l e r [9 . ] )
S ix U sin g the row e le m e n ta r y t r a n s f o r m a t io n s red u ce tho i n v e r t e d m a tr ix A e Rnxn to the norm al H orm ltoon form , i . e . f in d
the m a tr ix H - (•£•). H e Rm* n , G e Rko * n , kQ «* rank ( A ) - rank ( H ) , w h e re :
. f i r s t k rows of m a tr ix H a re nonzero rows and e le m e n ts o
of th e rows kQ ♦ 1 , . . . . n a ra a l l e q u a l z e r o ,
. . the f i r s t nonzero elem ent o f the i - t h row o f the m a tr ix H ( i ■ 1, . . . . k Q) i s e q u a l 1 and b e lo n g s to column n^, nx < n2 < < • • ><n. i
o
. . . th e o n ly nonzero elem en t in column n^ o f th e m a trix H i s
1 in th e i - t h row. (nx|<.
S 2 i U cln g th e m a tr ic e s A and H from m a tr ix P e R 0 a c c o rd in g to th e schome
P ^ î * *=» H ^ ) - 1 ^ J * 1» *••* 1 * I f •••# •
where P ^ , d e n o te r e s p e c t i v e l y the i - t h and J - t h colum ns o f the ir a t r i c e s P . A and H j d e n o te s th e u n it v e c t o r w it h 1-th component e q u a l 1.
2 .3 . Al g o r i t hm G E IN W ( c f . W a r m u s [
7
] )S i s U sin g row nnd column e le m e n ta r y t r a n s f o r m a t io n s red u ce the in v e r t e d r a t r l x A e Rl,xn o f rank k o s£ mln (m, n ) to th e mo» t r i x o f th e form
0 \
ranking r e s p e c t iv e row tro n o fo r m a tlo n e on tho m a tr ix G t ■ I and column t r i '•.uforumtlone on the m a tr ix F : « I .n
S 2 : U sin g t r a n s f ore.ed M t r l c e s G end F , o b ta in e d In S I , com-p u te
G~1 " (rm n :
v m, n • m « w- n /« n)!
V m (V'r )’lr'A .
* F " 1 ®( “ •"•■•I ••Y
U » 'V t f 't y f T 1 ,y^n-m.n
J
W - UV and forra th e - m a tr ix n- ko ' ko ko * m- ko W n- kO*l"“ kO; S 3 : Compute A* » FYG, 2 .4 . A lg o rith m SVÜZ ( c f . __ G^ o 1 u b ,___ R e i n e _ c _ h _ [ 6 ] M l 1_ W a e1
1 e w s k 1 _ 1.5 ] )3 1 ; U sin g one o f the a lg o r it h m s f o r d e a lin g w it h e ig e n v a lu e ¡>robie® ( e . g . the method o f O ^ c o b i) f in d th e e ig e n v a lu e s
md r j s p e c t i v e e ig e n v e c t o r s v x o f th e m a tr ix A #A o f rank ko . SS. •. O rd er e ig e n v a lu e s , o b ta in e d in S I , in a d e c r e a s in g
man-2 2
♦ - ° „ S 3 : Compute A « V O A
3. N u m e ric a l A n a l y s i s or tho A lg o r it h m s ' P r o p e r t ie s r--r--- — - -tct,;---—- . . . . {a n « № r & u 'Jia s a a ^ - <- - ■ .a .- rjg < i.n » r.i»n , •- - ^cxra
The com p ariso n s of tho p r o p e r t ie s o f tho fo u r a lg o r it h m s dov- c r ib e d a b o ve , w ere c o n fln o d to o n ly one s p e c i f i c n u m e ric a l s t r u c t u r e o f tho In v e r t e d m i i t r i c e i , Thu A n a ly s is coir-prisos sym m etric and r e c t a n g u la r n o t r ic o 's , and f o r each typ e of m a tr ic e s th e re •.'»ere d is t i n g u is h e d v a r i o u s d e g re e s o f i l l - c o n d i t io n in g and c o l l l n e a i i t y . Kach n u itrrx A was g e n e ra te d in two s ta g e s . A t f i r s t ttio re was g e n e ra te d n o n s in g u la r t r iiiit t g o n a ! b a s is m a tr ix w it h u n it s on d ia g o n a ls and z e ro s e ls e w h a r? (d e n o ted f u r t h e r as T I n th e next s ta g e the m a tr ix T war. e n la r g e d . I l l - c o n d i t i o n i n g was in tr o d u c e d throu g h sym m etric e n la rg e m e n t of the m a tr ix T by columna and rows d i f f e r i n g from r e s p e c t iv e co lumns and rows o f the m a trix T o n ly in m a in - d ia g o n a l ele m e n t by n e a r ze ro v a lu e t. S i n g u l a r i t y , i n tho caso o f sym m e tric m a t r ic e s , woo in tr o d u c e d throu g h e n la rg e m e n t of tho m a tr ix l by columns find rows b ein g the l i n e a r c o m b in a tio n s o f tho s u c c o s o lv o two co lumns and rows o f t h i s m a tr ix . R e c t a n g u la r m a tr ic e s were o b t a in ed by ad d in g to m a tr ix T i t s colum ns w it h In c r e a s e d by th e con s t a n t v a lu e
(3
e le m e n ts o f the l a s t row , and by ad d in g colum ns b e in g l i n e a r c o m b in a tio n o f th e s u c c e s s iv e two colum ns o f the m a trix T. Such a g e n e r a t io n o f th e m a tr ix A w i l l f i x in on a r b i t r a r y way th e number o f l l n e a r l l y dependent colum ns (t h e de g re e o f th e s i n g u l a r i t y o f tho m a tr ix A ) as w e l l as th e number o f colum ns b e in g th e c a r r i o r o f i l l - c o n d l t l o n l n g and i t s magnitu d e . A t th e same tim e the g e n e r a l s t r u c t u r e o f the In v e r t e d ma t r i c e s s t i l l rem ain s th e some. D e s c r ib in g th e r e s u l t s o f the e x p e rim e n ts we t i i l l use the f o llo w in g p a r a m e t r ic n o te c h a r a c t e
w h e re j ,
a , n ,k - r e s p e c t i v e l y th® d im e n s io n s and rank o f th « m a tr ix A,
1
,C - ro - .*p e rtiv o ly th « number o f columns w hich ore the c a r r i e r o f I l l - c o n d i t i o n i n g and p a ra m e te r d e f in in g the d e g re e of t h i s i11
- c o n d i t i o n i n g ,Cj .Cj, - c o e f f i c i e n t s of th e l i n e a r co m b in a tio n of the 'o a sis colum ns of A,
|3
_ c o n s ta n t v a lu e used in form in g r o c t a n g u la r m o tric e s in the ‘way d o s c r ib e d in th« t e x t ab o ve , ‘ .A t f i r s t we con r.id erod th-» m a tr ix o f tho form
A. - A ( l l , 11, 0 , 0 , O. 0 , 1, 0 )
i . e . o s in g u l a r but w u ll- c o n d it io n e d s y m m e tric a l m a tr ix , f o r such d e fin e d rn o trix A o i l a n a ly e e d a lg o r ith m s c h a r a c t e r iz e d the same p r o c is io r i b u t i t e r a t i v e a lg o r it h m A B I appeared the most tlm e- -consum ing and a lg o r it h m AWMEL the f a s t e s t of them. C o n s id e r in g m a tr ic e s A? - A ( U , 11, 0 . 0 . 0 , O .O O O l, 1) and A^ - A ( l l , 1 1 ,0 ,
0
,3
,0
. ?X , l ) we3
to tn d r a t h e r s m a ll changes in the v a lu e s or |j/*H and f; A* |j in com p ariso n w ith the v a lu e o f II A^ II w hich i n d i c a t e s the s t a b le p e rfo rm a n ce o f tho a lg o r it h m s w it h re g a rd to u n s i Q M f J c a n t changes in tho v a lu e s o f e le m e n t» in n o n b a s is colum ns o f the in v e r t e d m a tr ix A. The next group of a n a ly s e d m a tr ic e s c o n s is t e d of the s i n g u l a r s y m m e tric a l m a tr ic e s w hich c h a r a c t e r is e d the v a r i o u s d eg ree o f i l l - c o n d i t i o n i n g , I . e . ma t r i c e s o f the formA (1 1 , 11,
8
, I , C , 2 , 1, O ) -w h o re :
1
e1
*2 ,3
and £ *0
,0 1
,0
.0 0 1
,0
.0001
,F o r such a d e fln s d group o i in v e r t e d m a tr ic e s we o b se rv e d sub s t a n t i a l d i f f e r e n c e s betw een r e s p e c t iv e a lg o r it h m s in dependence
on th e d eg ree o f I l l - c o n d i t i o n i n g o f th e se m a t r ic e s . I n the c a re of s m a ll l e v e l of i l l - c o n d i t i o n i n g (£ <
0
.0 1
) , d e s p it e the number of colum ns b e in g I t s c a r r i e r ( i - 1 , 2 , 3 ) . the r e s u l t s of io m p u ta tlo n a f o r a l l a lg o r ith m s w e re , f o r each v a lu e o f1
, v e r y s i m i l a r C !j A + il computed by means of r e s p e c t iv e a lg o r it h m s d i f f e r s o n ly in th e f o u r t h p la c e a f t e r p o i n t ) . The p r e c i s i o n m easure«, wehad a c c e p te d , i n d i c a t e a b i t le 3 8 p r e c i s i o n o f the conput 'io n ;; «■-•hich w er« done by SVDZ, J u s t in the c<it»d o f £ * 0.0 0 1 ai-J £ ■ 0.0001 C onputptxons done by 'ronm o f GVIJZ 1 ud to quit-: d i f f e r e n t form of I! A*
S
in com p ariso n w it h th e se o b ta in e d by moony o f a lg o r it h m s AlVHEL, ADI and G£XN\V ( c f . T a b le t ) . Tho v o l i > of d e c lin e d from 1 6 4 .<ir.J5 to 2.92117 when tho v a lu e - of tho i l l - c o n d i t i o n i n g porom otwr £ d e c lin e d from0 .0 1
to0 .0 0 1
and rem ained s t a b l e d e s p it e f u r t h e r d e c lin e in £ . Y e t , the valur>i.and « ♦ II GEINW AWMEL propor t i o n a l l y to th e d e c lin e in C . in c r e a s e d a l l the tim e T a b l e
The in f lu e n c e o£ 1 1 1 - c o n d itio n in g on th e v a lu o o f S A*
3
o b ta in e d by meanso f a lg o r it h m s Al/t-iU , A L il,
6
VDZ and GEINWA lg o The v a lu e of c o o f f i c i e n r f rith m
0 .1
0 .0 1
0.0010.0001
1
B1
AWMEL ? 14.93664965 147. 4?73f!999 t 475.11192473 14 406.37441319 A6
I 14.93065005 147.427940701
475.96259026 14 761.67377477 ASV0Z2 14.93865165 147.42873272 3. .51496101 3.31490505 GEINW 14.93865005 147. 4;> 794049 I1
1
» 4 7 b .06259010 3 14 761.67157197 A.VMEL2 26.71292776 267.031450072
679.08125581 27 992.02097376 AOI 26.71292755 267.832015912
C0O.O35C5124 26 8 0 2 .172163CS A3VDZ2 26.71292739 26"/. 83226000 1.02713*58 1.82710446 GEINW 26.71292756 267.032016132
600.0 3567281 26 802.174054 31D e s p ite such s u b s t a n t i a l d i f f e r e n c e s betw een the v a lu e o f jA £ V D i;i and the v a lu e s o f I a* q][ | , | . II \ E I N i ; L a c c e p te d m easures o f the c o m p u ta tio n s ' p r o c ie io n In d i c o t e n s i m i l a r and q u it e good p r e c i s i o n o f a l l the a lg o r it h m s ( c f . T a b le 2 ) .
The com p ariso n o f v u lu o o o f the p r e c i s i o n m easures f o r v a r io u s v a lu e s o f £ i n d i c a t e a s n a i l decline- in p r e c i s i o n when the v a lu e o f £ d e c l i n e s . The most s i g n i f i c a n t d e c l i n e ap p eared in tho ca se o f the a lg o r it h m AvVMEL. On the o t h e r hand, th e b eat
r o b l o a Tho I n f lu e n c e o f i l l - c o n d i t i o n i n g o f the in v e r t e d a o t r i x A on th e c o m p u ta tio n a l p r e c is io n o f th e r e s p e c t iv e a lg o r ith m s P r e c i s i o n m easures A lg o
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A B I 0.00000000 0.00000001 0.00000000 0.00000000 ASVDZ2 0.00000001 0.00000011 0.00000000 0.00000011 GEINW 0.00000000 0.00000000 0.00000000 0.00000000 E - 0.0 1 AWMEL2 0.00000634 0.00000379 0.00000102 0.00001191 A B I 0.00000000 0.00000013 0.00000000
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0.00027984 0.00057636 0.00013251 0.00045671 A B I O.0OOOOOO2 0.00000118 0.00000004 0.00000712 ASVDZ2 0.00004995 O.OO'OOOOOO 0.00000000 0.00000000 GEINW 0.00000005 0.00000001 0.00000004 0.00000004 E - 0.0001 AWMEL2
0.01744044 0.02406894 0.00361627 0.04149807 A B I 0.00000012 0.00000987 0.00000027 0.00257619 ASVDZ2 0.00000499 0.00000000 0.00000000 0.00000000 GEXNY7 0.00000032 0.00000014 0.00000050 0.00000028p r e c i s i o n c h a r a c t e r iz e d a lg o r it h m GEINW. A lg o rith m A B I w as, ae compared vuith o t h e r s , n e a r l y 6-8 tim e s more tim o-consum in g , but i t s p r e c i s i o n was r a t h e r h ig h .
The l a s t group o f in v e r t e d m a tr ic e s c o n s is t s o f r e c t a n g u la r m a tr ic e s w it h d i f f e r e n t degro© o f i l l - c o n d i t i o n i n g . We c o n s id e re d n in e m a tr ic e s o f th e form
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, 1 1 ,4 + 1 ,1 ,£ ,2 ,1 ,1 ) » A l iCw here« I - 1 , 2 , 3 : E - O .O l, 0 .0 0 1 , 0 ,0 0 0 1 .
I n the caso when o n ly one column wos the c a r r i e r o f I l l - c o n d i t i o n i n g , «11 the a lg o r it h m s w ere c h n r o c t e r lz o d by h ig h p r e c i s io n and a l l o f then le d to M-P g e n e r a liz e d in v e r s e s o f s l n l - U r norms. In th e ca se when two o r th ro e columna wore* th e c a r r i e r s o f i l l - c o n d i t i o n i n g , th e s i t u a t i o n was s i m i l a r to th n t c o n c e rn in g s y m m e tric a l m a t r ic e s . The h ig h e s t p r e c i s i o n was cho- r a c t e r i e t i c of a lg o r it h m GEINIV and th e w or
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t one o f a lg o r it h m AVJMEL.4. F l n o l Remarks M g m ~ ri-a x.'.aa
Our i n v e s t i g a t i o n s a llo w u o, d e s p it e t h e i r v e r y l im it e d sc o p e , to fo rm u la te some rem arks c o n c e rn in g th e a n a ly s e d a lg o r it h m s f o r co m p u ta tio n o f M-P in v e r s e s .
1 ° I t I s supposed t h a t from the n u m e ric a l p o in t o f v ie w , mea s u re s o f a l g o r i t h m 's p r e c i s i o n boned on f o u r M oore-Penrose con d i t i o n s w hich were a c c e p te d In the p a p e r do not a llo w to i n d i c a te u n iq u e ly th e b e s t n u m e r ic a l a p p ro x im a tio n o f th o g e n e r a liz e d M-P in v e r s e o f the I l l - c o n d i t i o n e d m a tr ix .
2 ° A l l a lg o r it h m s w h ich w ere a n a ly s e d e x c e p t SVDZ were cha r a c t e r i z e d by s m a ll ro b u s tn e s s on h ig h i l l - c o n d i t i o n i n g o f th ' in v e r t e d m a tr ix .
3 ° From the c o m p u ta tio n s ' p r e c i s i o n p o in t o f v ie w a lg o r it h m GEIN'.*/ was th e b e s t and a lg o r it h m AIVMEL wao th e w o r s t , but a t the same tim e th e f a s t e s t among th e f o u r a lg o r it h m s b oing a n a ly s ed.
4 ° The i l l - c o n d i t i o n i n g o f h ig h d e g re e caused a d e c lin e in the a lg o r it h m s ' p r e c i s i o n , e x c e p t in the c a s e o f SVDZ.
5 ° From T a b lo 1 ,2 I t i s seen t h a t th e SVDZ i s s t a b l e both in th e sense o f II A* I as w o l l as In th e sense o f th e d e g re e o f f u l f ilm e n t o f th e fo u r c o n d it io n s d e f in in g th e M-P g e n e r a liz e d I n v e r s e A+ when the i l l - c o n d i t i o n i n g i s in c r e a s in g ( t h a t i s when E i s d e c l i n i n g from C * 0.0 0 1 to £ ■ 0 .0 0 0 1 ). I t i s co u sed , p r o b a b ly , by th e in h e r e n t p r o p e r t i e s o f D aco b l p ro c e d u re and the
assu m p tio n th a t U^ ■ AVj/v"* , where U ■ (U^ 5 Ug ) , U 'U * U U ' a l ^ y u ; u i ■ ■<!<„)■v ■ ( v i i v z >- v ' u ■ w ' • • (» )■ v i v i • j ( k o )- F " r d raw in g d e c i s i v e c o n c lu s io n s about i n s t a b i l i t y and I t s c a u s e * we a r e c a r r y in g ou t more s t r u c t u r a l e x p e rim e n ts . BIBLIO G RAPH Y • • [ 1 ] A l b e r t A. (1 9 7 2 ), R e g r e s s io n and th e M ooro-Ponrose P s e u d o in v e r s e , N .Y . , Academ ic P r e s s . [ 2 ] B e n - I a r a e 1 A . (1 9 6 6 ), A Note on an I t e r a t i v e Me thod f o r G e n e r a liz e d I n v e r s i o n o f M a t r i c e s , M ath, o f C om p ut., p . 439-440.
[ 3 ] L u e c k e G. (1 9 7 9 ), A N u m e ric a l P ro c e d u re f o r Comput in g th e M o ore-Penrose I n v e r s e , Numer. M ath. 32, p . 129-137. [ 4 ] M a r c u s M. , M i n c H. (1 9 6 4 ), A S u rv o y o f M a t r ix
T h e o ry and M a t r ix I n e q u a l i t i e s , B o s to n , A l l y n and Bacon I n c , [ 5 ] M i 1 o W ., W a s i l e w s k i 2. ( i 9 6 0 ) , G e n e r a liz e d In v e r s e A lg o rith m s and T h e ir A p p l i c a t i o n s . P a r t I , G ra n t R. I I I . 9 . 5 . 7 , u n p u b lis h e d m a n u s c rip t, [ 6 ] M l t r a S . , R a o C. (1 9 7 1 ), G e n e r a liz e d In v e r s e o f M a t r ic e s and I t a A p p l i c a t i o n s , N . Y . , W i le y . [ 7 } W a r m u s M. (1 9 7 2 ), U o g ó ln io n e o d w ro tn o ó c i m a c ie rz y . W arszaw a, PWN. . [O ] W i l k i n s o n J . , R a i n s c h 3. ( l 9 7 l ) . Handbook f o r A u to m a tic C o m p u ta tio n . V o l. I I I : L in e a r A lg e b r a , B e r l i n , S p r in g e r - V e r la g . [ 9 ] W i 1 1 n e r L . (1 9 6 7 ), An E l i m i n a t i o n Method f o r Com p u tin g the G e n e r a liz e d I n v e r s e , M ath. C om p ut., p . 227-229.
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O WŁASNOŚCIACH NUMERYCZNYCH ALGORYTMÓW MOO«E'A-Pł.NRPSH'A UOGÓLNIONYCH ODWROTNOŚCI MACIERZY
P r a c a z a w ie ra k r ó t k i o o la t r z e c h n l e i t o r a c y j n y c h i je d n e yo 1- t e r a c y jn e g o a lg o ry tm u o b l ic z a n ia u o g ó ln io n y c h o d w ró tn o fici "d a n e j m a c ie rz y o ra z r e z u lt a t ó w eksperym entów num erycznych z m ie rz a ją c y c h do u s t a l e n i a n ie k t ó r y c h num erycznych w ła s n o ś c i ty c h a lg o rytm ów. u tw ie rd z o n o , iż
a ) w e z y s tk io a n a liz o w a n o a lg o r y tm y , op rócz b a z u ją c e g o na zm°~ d y fik o w a rte j d e k o m p o zycji w a r t o ś c i w łn i;n a j, w yk a z yw a ły mołsj od p o rn o ść na z ł « uw arunkow anie o d w ra c a n e j m a c ie rz y ;
b ) n a js z y b s z y a lg o ry tm W l lln a r o b v ł n a jm n ie j p r e c y z y jn y ; c ) a lg o ry tm GEINW Warmuaa b y ł n a j b a r d z ie j p r e c y z y jn y pod względem s p o h n ie n ia c z te r e c h warunków o k r e ś la ją c y c h u o g ó ln io n y od w rotność m a c ie rz y A*-;
d ) w przyp ad ku w z r a s ta ją c e g o z łe g o uwarunkow ania n a j b a r d z ie j s t a b i l n y o k a z a ł Bi<j (w e s n i i e <£ 1 s p e ł n i a n i a d o f l n l c j i A ) a lg o rytm o p a r t y na zm o d yflko w anej d e k o m p o zycji w a r t o ś c i w ła s n e j.
