Pivoting and Backward Stability of Fast Algorithms for Solving Cauchy Linear Equations

Transcription

1 Pivoting nd Bckwrd Stility of Fst Algorithms for Solving uchy iner Equtions Tior Boros Arryomm Inc 141 nker Rod Sn ose A 14 USA Thoms Kilth Informtion Systems ortory Deprtment of Electricl Engineering Stnford University Stnford A 4-4 USA nd Vdim Olshevsky Deprtment of Mthemtics nd Sttistics Georgi Stte University Atlnt GA USA Astrct Three fst lgorithms for solving uchy liner systems of equtions re proposed A rounding error nlysis indictes tht the ckwrd stility of these new uchy solvers is similr to tht of Gussin elimintion thus suggesting to employ vrious pivoting techniques to chieve fvorle ckwrd stility It is shown tht uchy structure llows one to chieve in opertions prtil pivoting ordering of the rows nd severl other judicious orderings in dvnce without ctully performing the elimintion The nlysis lso shows tht for the importnt clss of totlly positive uchy mtrices it is dvntgeous to void pivoting which yields remrkle ckwrd stility of the suggested lgorithms It is shown tht Vndermonde nd heyshev-vndermonde mtrices cn e efficiently trnsformed into uchy mtrices using Discrete Fourier osine or Sine trnsforms This llows us to use the proposed lgorithms for uchy mtrices for rpid nd ccurte solution of Vndermonde nd heyshev-vndermonde liner systems The nlyticl results re illustrted y computed exmples AMS suject clssifiction: 6F 62 1A 1A2 Key words: Displcement structure uchy mtrix Vndermonde mtrix fst lgorithms pivoting rounding error nlysis ckwrd stility totl positivity This work ws supported in prt y NSF contrcts R nd 8222 we: wwwcsgsuedu/ mtvro emil: volshevsky@gsuedu 1

2 #####" X #" 4 #####" ###" X : 1 Introduction 11 Relted fcts Vndermonde nd relted mtrices iner systems with uchy nd Vndermonde coefficient mtrices! %'&)(+*& - % / (+* & - %'&)(+*)/ % / (+* / 21 4!6 78 : - : : : - +: re clssicl They re encountered in mny pplied prolems relted to polynomil nd rtionl function computtions Vndermonde nd uchy mtrices hve mny similr properties mong them one could mention the existence of explicit formuls for their determinnts nd inverses see eg [BKO] nd references therein Along with mny interesting lgeric properties these mtrices hve severl remrkle numericl properties often llowing us much more ccurte computtions thn those sed on the use of generl (structure-ignoring) lgorithms sy Gussin elimintion with pivoting At the sme time such fvorle numericl properties re much etter understood for Vndermonde nd relted mtrices (see for exmple [BP7] [TG81] [F88] [Hig87] [Hig88] [Hig] [RO1] [R2] [R] [V] [Ty4]) s compred to the nlysis of numericl issues relted to uchy mtrices (see [GK] [GK]) The Björck-Pereyr lgorithm for Vndermonde systems In prticulr most of the ove mentioned ppers were devoted to the nlysis of numericl properties nd extensions of the now well-known Björck-Pereyr lgorithm for solving Vndermonde liner systems [BP7] [GV8] This lgorithm is sed on the decomposition of the inverse of Vndermonde mtrix into product of idigonl fctors where ( D8E F 4 ( 6 ; = ( G E F G E F ( D8E F ( ( ( A-B ( ( HIE F E F : E F HKE F (11) (12) (1) This description llows one to solve the ssocited liner systems in only opertions which is y n order of mgnitude less thn the complexity MNP' of generl (structure-ignoring) methods Moreover the lgorithm requires only MQ loctions of memory Remrkle ccurcy for monotoniclly ordered nodes It turns out tht long with drmtic speed-up nd svings in storge the BP lgorithm often produces surprisingly high reltive ccurcy in the computed solution NHighm nlyzed in [Hig87] the numericl performnce of the BP lgorithm nd identified clss of Vndermonde mtrices viz those for which the nodes cn e strictly orderded RTS S : S UV S (14) with fvorle forwrd error ound XA[ X X]_^ M`4 ( ` (1) for the solution computed y the BP lgorithm Here denotes the unit roundoff in the stndrd model of floting point rithmetic nd the opertions of comprison nd of tking the solute vlue of mtrix re understood in componentwise sense It ws further shown c in ] [BKO] tht under the condition (14) the BP lgorithm is not only forwrd ut lso ckwrd stle ` : Od M : `4e 6 ` : f4g hc (16) 4i is the exct solution of nery system Here the computed solution The Björck-Pereyr-type lgorithm for uchy mtrices The ove nlytic error ounds indicte tht the ccurcy of the BP lgorithm cn e much higher thn could e expected from the condition numer of the MN:O 2

3 coefficient mtrix Such high ccurcy motivted severl uthors to ext the BP lgorithm to severl other clsses of mtrices see eg [TG81] [Hig88] [RO1] All these generliztions were developed for Vndermonde relted structures In recent pper [BKO] similr decomposition i( Nj 7;k ( MN:O -- ( ( ( ( -B ( ws written down for uchy mtrices thus leding to Björck-Pereyr-type lgorithm which will e referred to s the BP-type lgorithm This lgorithm requires opertions nd loctions of memory It ws further shown in [BKO] tht the following configurtion of the nodes l S VV S ] S S VU S +m is n pproprite nlog of (14) for uchy mtrices llowing to prove tht the error ounds ssocited with the BP-type lgorithm re entirely similr to (1) nd (16) viz Xi[ Xn^ d M o` 6 j 6 ( c 7 dlr M; d M p` j 7 MQ (17) (18) ` : (1) ` : (1) 4g6 B It is n interesting fct tht the conditions (14) nd (18) imply the totl positivity 1 of nd resp Totlly positive mtrices re usully extremely ill-conditioned so tht the Gussin elimintion procedure often fils to compute even one correct digit in the computed solution The ound (1) indictes tht lso in such difficult cses the BP-type lgorithm cn produce for specil right hnd sides ll possile reltive precision see eg [BKO] for discussion nd numericl illustrtions imittions for non-totlly-positive 6 mtrices 6 Of course reordering of the nodes q r nd q sr is equivlent to row nd column permuttion of respectively nu Therefore if the two sets of nodes re seprted from ech other S tl fv M (111) the remrkle error ounds (1) (1) suggest to reorder the nodes monotoniclly s in (18) nd to pply the BP-type lgorithm of [BKO] However numericl experiments show tht in the generic cse ie when (111) do not hold the monotonic ordering does not gurntee stisfctory ccurcy nd the corresponding ckwrd error of the fst BP-type lgorithm of [BKO] (nd of the use of the explicit inversion formul s well) my e worse thn tht of the slow Gussin elimintion with 6 pivoting 6 Employing other orderings of the nodes (for exmple the prtil pivoting ordering of the rows of ) does not seem to essentilly improve the ckwrd stility A heuristic explntion for this fct cn e drwn from the oservtion tht the usul im of pivoting is to reduce the size of the fctors in the U fctoriztion of mtrix Therefore it improves the stility properties of the Gussin elimintion procedure for which the ckwrd error ound involves the product (computed fctors) In contrst n exmintion of the error nlysis of the BP-type lgorithm in [BKO] indictes tht the corresponding ckwrd ound involves the quntity - xy ( z{] } ; ( x}zh - w (112) which ecuse of non-cncelltion property (ie quntity ) cn e much higher thn the more ttrctive In the totlly positive cse (18) the entries of Nj nd 7 in (112) re ll nonnegtive thus llowing one to remove the moduli nd to replce (112) y just cf with the fvorle ound (1) Unfortuntely in the generl cse the idigonl structure of the nd in (1) does not llow one to remove the moduli nd to esily deduce from it stisfctory ckwrd error ound These limittions suggest tht in one s ttempts to design ckwrd stle uchy solver it cn e etter to develop new lgorithms rther thn to look for stilizing techniques for the Björck-Pereyr-type lgorithm 12 Min results New lgorithms In this pper we develop severl lterntives to the BP-type lgorithm ll sed on fctoriztions Nj 7; - 1 Totlly positive mtrices re those for which the determinnt of every sumtrix is positive see the monogrphs [GK] nd [K72] ( ( - w'

4 c X #######" X #######" where the fctors re of the form (digonls with one nonzero row or column) ~ E F ` 6 E F ~ E F - ~ E F ~ l E F ` E F - ` E F ` E F l (11) Bckwrd error ounds e produce n error nlysis for these new methods otining ckwrd nd residul ounds lso involving the quntity (112) ut now with fctors of the form (11) In contrst to the idigonl fctors (1) of the BP-type lgorithmu of [BKO] the sprsity U pttern of the fctors (11) immeditely implies the equlity resulting in plesing ckwrd ounds of the form c 7n UV ]` ( for the new lgorithms Here the computed solution These ounds re similr to the well-known ounds for Gussin elimintion ct d H V V - ( ` : UV w (114) is the exct solution of nery system f N66 ƒ7 - H M` M` (11) see eg p 17 in [Hig6] Here nd denote the computed tringulr fctors Different pivoting techniques This resemlnce etween the ckwrd error ounds (114) nd (11) suggests to employ different row nd column pivoting techniques to reduce the size of nd nd to stilize the new lgorithms for uchy systems There is vriety MN:O of possile pivoting techniques tht cn e incorported into the new lgorithms without incresing their complexity including prtil row nd prtil column pivoting Gu s pivoting ( vrition of complete pivoting for uchy-like mtrices [Gu]) nd others Totl positivity There re clsses of mtrices for which it is dvntgeous not to pivot For exmple for totlly positive mtrices the exct tringulr fctors hve only positive entries De Boor nd APinkus pointed out in [DBP77] tht if the entries of the computed fctors nd remin nonnegtive thn the ckwrd error of Gussin elimintion without pivoting is plesntly smll cte]n H (116) see lso p176 in [Hig6] It turns out tht the sme recommtion to void pivoting cn e mde for the fst uchy solvers proposed here nd moreover ecuse the corresponding k 6 6 error 6 ounds (114) involve the exct tringulr fctors in the cse of totl positivity we hve implying tht remrkle ckwrd stility of the sme form (116) is gurnteed for the new fst lgorithms without ny dditionl ssumptions on the computed tringulr fctors 1 Outline of the pper It is well-known tht for structured mtrices the Gussin elimintion procedure cn e speeded-up In the next section we exploit the displcement structure of uchy mtrices to specify two such lgorithms Then in the section we exploit the qusi-uchy structure of the Schur complements of j 6 to derive one more lgorithm for solving uchy liner equtions Then in section 4 we perform rounding error nlysis for these lgorithms otining ckwrd nd residul ounds similr to those for Gussin elimintion This nlogy llows us in Section to crry over the stilizing techniques known for Gussin elimintion to the new uchy solvers The numericl properties of new lgorithms re illustrted in section 6 y vriety of exmples Then in section 7 we show how Vndermonde nd heyshev-vndermonde mtrices cn e efficiently trnsformed into uchy mtrices y using Discrete Fourier osine or Sine trnsforms thus llowing us to use the proposed uchy solvers for the rpid solution of Vndermonde nd heyshev-vndermonde liner systems 4

5 Š q : E : E µ Ÿ q Š R X R ½ q : E E Œ 2 Schur-type lgorithms for uchy mtrices 21 Displcement structure By now the displcement structure pproch is well-known to e useful in the development of vrious types of fst lgorithms for structured mtrices including Toeplitz Hnkel Toeplitz-plus- Hnkel Vndermonde heyshev-vndermonde nd severl others This pproch is sed on convenient wy to cpture ech of the ove prticulr structures y specifying suitle displcement opertors ˆ i QŠ Œ 7 In this pper we shll use opertors of the form for vrious choices of (sprse) mtrices q êžj Q 8 r et D [ šl 7œ [ ˆŽ6 Q êž6 Q ; ž Ÿi8 then one cn fctor ž6ÿ Š 7 žÿ r is clled q where oth mtrices on the right-hnd side of (22) hve only D columns ech : The numer êžj Q is clled q r -displcement rnk of nd the pir q r -genertor of The d MN: displcement rnk mesures the complexity of ecuse ll its entries re descried y smller numer D M entries of its genertor q ž6ÿ r e refer to surveys [KS] [HR84] [O7] [D1] for more complete informtion on displcement different pproches nd further references Here we restrict ourselves only with uchy mtrices The next lemm reclls their displcement structure D )š 7œ emm 21 et êžj Q Š + eœ % [ ª«š Then the uchy mtrix hs displcement êž6 Q 22 Structure of Schur complements et F :6: y tringulr resp sy then the q ¹ & ~ ` 7 VU j+ r e defined y (21) with [ * ª«š ]O VU js r f j]6 7jw - ± ² - ³± in (22) e prtitioned : its Schur complement Then if the mtrices nd µ ~ ` F :: (21) (22) (2) (24) nd denote in (22) re lower nd upper º»¼µ ½ ¾µ : : : : r -displcement rnk of the Schur complement is less thn or equl to D so tht we cn write : : : [ž : : ªU 6ƒ À6Ál ž : Ÿ : Š E ( F 7 (2) (26) ža 6Ÿ r ž : 6Ÿ : r The next lemm (cf with [GO4] [GKO] [KO]) provides one prticulr form for the genertor õ ` emm 22 et ~ F stisfy the displcement eqution (22) with tringulr :6:Ä º prtitioned F of is nonzero then the Schur complement :6: ¹ & ~ ` stisfies the The ltter fct ws first oserved y ASkhnovich in [S76] (see lso [S86]) nd y MMorf in [M8] See eg [KS] nd the references therein for vrious formuls showing how to run the genertor recursion q recursion s in (2) If the (11) entry displcement eqution (26) with where Å nd Æ µ R ž : [ža ¹ & ~ Å denote the top rows of ža nd Ÿ resp ' R Ÿ : ± [Ÿ ¹ & ` ± Æ ' (27)

6 Ì : E E Ï µ R Ð The stndrd Gussin elimintion procedure computes the U fctoriztion using flop The ove lemm llows us to exploit the structure of to compute this fctoriztion in only MN:' flops s descried next 2 Speed-up of the Gussin elimintion procedure The first step of Gussin elimintion procedure pplied to mtrix is descried y where µ column M ¹ & ~ fter F :: õ ¹ & ~«É Ǹ of nd the first row ~ ` F :: õ R ¹ & ~ ÈÇ ` : MNPO (28) is Schur complement of (11) entry in the mtrix This step provides the first ` ± in the U fctoriztion of Proceeding with : similrly steps one otins the whole U fctoriztion Algorithms tht exploit the displcement structure of mtrix to speed-up the Gussin elimintion procedure re clled Schur-type lgorithms ecuse the clssicl M Êú Ëũ Schur lgorithm [S17] ws shown (see eg [AK86] p: [KS]) to elong to the clss Insted of computing the entries of the Schur complement one hs to compute only D M entries of its q r -genertor q ž 6Ÿ r which requires much less computtions To run the genertor recursion (27) s well s to write down the corresponding entries of the nd fctors one needs only to specify how to recover the first row nd column of from its genertor q ž Ÿ r For Schur complement ÍÌ Î E F Ï tñð OÒ Ï t Ò of uchy mtrix this is esy to do : Î E F Ï t Å E F 'Æ E F Ï t (2) where q Å E F -ÓÅ E F Ð +Ì Æ E F -ÔÆ E F Ð r designtes the corresponding genertor 24 Tringulr 6 fctoriztion 6 for uchy mtrices In the rest of this section we formulte two Schur-type lgorithms for The first version is n immedite implementtion of (27) nd (2) nd its MATAB code is given next Algorithm 2 (Schur-type-uchy) function [U] = uchy GS (xy) % tringulr fctoriztion of the uchy mtrix Û ÙUÜ]ÝoÞ ßàâá'ÝoÞ ß Ú n=mx(size(x)); =eye(nn); U=zeros(nn); % Genertors % g=ones(n1); =ones(n1); for k=1:n-1 % omputing the ã -th row of nd U % U(kk) = ( g(k) * (k) ) / ( x(k) - y(k) ); for j=k+1:n (jk) = (( g(j) * (k) ) / ( x(j) - y(k) )) / U(kk); U(kj) = (( g(k) * (j) ) / ( x(k) - y(j) )); % omputing the new genertors % for j=k+1:n g(j) = g(j) - g(k) * (jk); (j) = (j) - (k) * U(kj) / U(kk); U(nn) = ( g(n) * (n) ) / ( x(n)-y(n) ); return omplexity: Õ)Ö Q eø@ù ÖÚ flops 2 Direct genertor recursion In fct the lgorithm 2 is vlid for the more generl clss of uchylike mtrices see eg [GKO] [KO] for detils nd pplictions However for the specil cse of ordinry uchy mtrices we cn exploit the fct tht the corresponding displcement rnk is equl to one to formulte more specific Schur-type lgorithm sed on the next emm emm 24 et ª«š UV j in ª«š + UV j ; ž Ÿ ÍÌ 6 Å E F --ÍÅ E F Ð Ì Æ E F --äæ E F

7 ##" #" ##" ##" Ì ì ì j e the successive Schur complements of the uchy mtrix Then the genertor recursion (27) cn e specilized to Å E å F å Å E å F %Oæ6ç & (+%'æ %'æjç & (+*æ7å E F å % / (+%'æ %/](*æ Å E F Æ E å F å 6 6 ; è The nonzero entries of the fctors in ~ 21 %'æo(*)æ Å E F 21 1 é` ~ %O/(+*æÅ E F -- -ÔÆ E å F re given y ÐÓÌ *æjç & (*)æ *)æ6ç & (+%'æ7æ E F nd ` ±ÍÌ % æ (+* æ Æ E F - The following MATAB code implements the lgorithm sed on emm 24 Algorithm 2 (Schur-type-direct-uchy) function [DU]= uchy GS d( x y) % tringulr fctoriztion of the uchy mtrix Û ÙUÜ ÝoÞ ß àâá ÝoÞ ß Ú n=mx(size(x)); =eye(nn); D=zeros(nn); U=eye(nn); % Genertors % g=ones(n1); =ones(n1); for k=1:n-1 % omputing the ã -th row of nd U % D(kk) = (x(k)-y(k)); for j=k:n (jk) = g(j)/(x(j) - y(k)); U(kj) = (j)/(x(k) - y(j)); % omputing the new genertors % for j=k+1:n g(j) = g(j) *(x(j)-x(k))/(x(j)-y(k)); (j) = (j) *(y(j)-y(k))/(y(j)-x(k)); D(nn) = 1/(x(n)-y(n)); (nn) = g(n); U(nn) = (n); return; The fst lgorithms 2 nd 2 require with MQ storge is descried next MN:O Exploiting qusi-uchy structure - */(*æ *)/(+%'æ7æ E F Ð (2) % æ (*)/ Æ E F Ð (211) omplexity: Õ)Ö eø@ù ÖÚ flops loctions of memory to store the tringulr fctors An lgorithm The concept of displcement structure ws initited y the pper [KKM7] where it ws first pplied to study Toeplitz mtrices using displcement opertor of the form ˆ Ž6ê êë+ where is the lower shift mtrix In this section we mke connection with [AK86] where the fct tht Toeplitz mtrices elong to the more generl clss of mtrices with ˆ Ž6ê ê ë -displcement rnk 2 ws used to introduce the nme qusi-toeplitz for such mtrices It is shown tht ny qusi-toeplitz mtrix mtrices ;í ; ì cn e represented s product of three Toeplitz where is lower nd is upper tringulr Toeplitz mtrices Ptterning ourselves upon the ove definition nd % 6 * r -displcement rnk 1 s qusi-uchy tking emm 21 s strting point we shll refer to mtrices with q mtrices The next simple lemm is n nlog of (1) (1) 7

8 ###" ##" Ý Ý õ á ï : Ý Å ï ###" Å Ý ####" Å õ Ý µ Å R õ µ R ###" Æ Æ R Ý ##" õ Æ µ Ý Æ R Ü Ç ï ï û emm 1 et % nd * e defined y (24) Then the unique solution of the eqution % * - ± - ± is given y [ ª«š + Å : UV 7 j 7 ª«š + Æ : VU emm 1 llows us to otin elow n explicit fctoriztion formul for uchy mtrices Indeed y emm 22 its Schur complement :6: in ; µ ¹ & ~ ~ ` ` :: lso hs displcement rnk 1 so y emm 1 its uchy structure cn e recovered y dropping digonl fctors s shown next ~ Ç emm 2 The uchy mtrix nd its inverse cn e fctored s 6 ]6 7; -B ( + ( - 8' 7ªVšl +6 UV O 6 where nd ñ ï-ò 1 ñ ï-ò Ý îïwð ó æ ò]ô 1 õ 1 ñ ïò æ õöõ 11 ó / ò Ý ó æ ùï8ð ñjï-ò ø á ï) á otined from the ove leds to the following lgo- ø áß ( The representtion for N66 the X inverse mtrix rithm for solving Algorithm (qusi-uchy) 2 function = uchy qusi (xyf) % Solving uchy liner system Û ÙUÜ ÝoÞ ß àá ÝoÞ ß Úý ðëþ n=mx(size(x)); =f; fork=1:n-1 for j=k:n (j) = (j) * ( x(j) - y(k) ); for j=k+1:n (j) = (j) - (k); for j=k+1:n (j) = (j) / ( x(j) - x(k)); for k=1:n-1 (k) = (k) / ( x(k) - y(k) ); (n) = (n) * ( x(n) - y(n) ); for k = n-1:-1:1 for j=k+1:n 1 11 ñjï-ò õúõúûpûjûüõ õ õ ¹ & : ñ ï-ò 7 ï) ` Ýø Ü Ý ó æ ò]ô æjç & Ü ß ø Ü Ý ó æ ò]ô / 1 11 à 21 1 (2) () (4) 2 Algorithm hs lower complexity nd etter error ounds thn its erlier vrint clled uchy-2 in [BKO4] omplexity: Õ)Ö eø@ù ÖÚ flops 8

9 X #" ` ÿ ~ q ##" r ##" X return (j) = (j) /(y(k)-y(j)); tmp = ; for j=n:-1:k+1 tmp = tmp + (j); (k) = (k) -tmp; for j=k:n (j) = (j) * ( x(k) - y(j) ); y vector is performed y ccumultion of the inner product from the lst to the first entry; this order is influenced y the error nlysis in the next section The reder should note tht the multipliction of the centrl fctor of ( 4 Rounding error nlyses 41 Stility of the lgorithm 2 The lgorithm 2 is specil cse of the more generl GKO lgorithm [GKO] which is pplicle to the wider clss of uchy-like mtrices A normwise rounding error nlysis for the GKO lgorithm ppered in [SB] Along with the usul fctor ÿ ÿsÿ ÿ ( cf with (11)) the ckwrd N error ound of [SB] involves lso so-clled genertor growth fctor of the form ª«š Å E F Æ E F ÿ (41) Å E F Æ E F In the context of [GKO] [SB] the quntities Å E F nd Æ E F were vectors of size equl to the displcement rnk of ; so if the quntity in (41) is lrge then the ckwrd stility of of the GKO lgorithm could e less fvorle thn tht of Gussin elimintion However the ordinry uchy mtrices considered in the present pper ll hve displcement rnk 1 so tht the constnt in (41) is unity suggesting tht the ckwrd stility of the lgorithm 2 is relted to tht of Gussin elimintion without pivoting 42 Stility of the lgorithm 2 {c hc hc c c c ] H ( : H ( : H M` M` (42) Nj 7 X Furthermore if this computed tringulr fctoriztion f is used 6 6 to 6 solve {c the ssocited liner system solves c nery ] system V with 6 OR M d o` ` : 6 m (4) X ] -«X 6 6 j OR M d o` ` : 6 m Theorem 41 Assume tht the lgorithm ` 2 (Schur-type-direct-uchy lgorithm) is crried out in floting point rithmetic with unit roundoff nd tht no overflows were encountered during the computtion Then the Ê j ; m } = computed fctors of stisfy where nd then the computed solution Proof Error in tringulr fctoriztion A strightforwrd error nlysis for the direct genertor recursion (2) nd for (211) implies tht the computed nd the exct columns of re relted y ~ - ~ 21 %'æo(+*æ Å E F % / (*æ Å E F 21 1 ~ - E F ~ E F 21 1 (44)

10 X X X X E X X c X c ` c X X X c c c so tht the componentwise error is nicely ounded : `jp ( : ` E F t `pp ( : ` This nd similr rguments for led to the fvorle ounds (42) Error in the computed solution Stndrd error nlysis see eg p 14 in [Hig6] for solving liner system y forwrd nd cksustitution yields tht the computed solution stisfies where c ] H c g- The plesnt ound in (4) is now deduced from (4) nd (42) 4 Stility of the lgorithm Theorem 42 Assume ` tht the lgorithm (qusi-uchy lgorithm) is crried out in floting point rithmetic with unit roundoff nd tht no overflows were encountered during the computtion Then the computed solution solves nery system with nd c H ( F {c â j 7 X ] 6 6 c ] 6M : c ] H : jm : l M c {c s M R o` ] â hc c g-f ] {c R o` ` V : 6 ` : 6 m H å ( H m Proof et us recll tht lgorithm solves uchy liner system y computing k j 6 ( ( VU ( ( ( ( ( where the q Ï Ï r re given y (4) The proof for (47) will e otined in the following steps UV ( -«X H M` M` (4) (46) (47) (48) (i) First we pply the stndrd error nlysis for ech elementry mtrix-vector multipliction in (4) to show tht the computed solution stisfies üf ( ½ 8- VU f ( ( ½ w ( ½ g ( ( ½ ( where the sterisk ½ denotes the Hdmrd ( or componentwise ) product UV ( ½ - (4) (4) (ii) Next the otined ounds for will e used to deduce further ounds for defined y {c hc {c ( ½ ( ( ½ g ( Ê f ( ½ ( = (iii) Finlly inverting (4) we shll otin ü hc VU ( hc â hc g f ( ( UV {c (411) which will led to the desired ounds in (48) nd in (47)

11 u u u t c E u u e strt with (i) nd (ii) in (4) nd (411) The sprse ower tringulr fctors e strt with otining ounds for nd nture of these mtrices see eg emm 2 implies the following plesing ound for the (ij) entry of : ` ` ` Ï ` (412) Moreover even smller ounds hold for the (kk) entry : ` : Since the ll hve exctly the sme sprsity c pttern ] s their inverses (412) nd (41) imply tht ` ` P c ] ` ` u Upper tringulr fctors The nlysis shows tht for hve ` P ` Ï t Digonl fctor The simple structure of If the inner product corresponding to the in the (kj) entry is ounded y ` : immeditely implies tht (41) (414) (ie excluding the entries of the ` ` -th row of is evluted from the lst to the ` t ( å P In prticulr the error in the (kk) entry is ounded y ` P t ` t ( å P ` P (41) -th row) we (416) -th entry then the error Agin since hve exctly the sme sprsity c ] pttern s their inverses (416) (417) nd (418) imply tht ` å : ` P c ] p8) : let d VU for then nc ] j c (417) (418) (41) e re now redy to turn to (iii) To prove (48) we shll use the following esily verified fct (see eg [Hig6] This nd (414) imply tht hc VU The sprsity pttern of ( ( hc d UV 6M ( UV ( ] ` ( F ` P E ( F (42) VU ( (421) ) llows us to remove the moduli in the product on the right-hnd side of (421) implying the first ound in (48) The third ound in (48) is deduced from (41) nd (42) nlogously The second ound in (48) follows from (41) esily Finlly the ound in (47) is deduced from (48) (cf (4)) Pivoting In the previous section we estlished tht the ckwrd stility of ll the fst uchy solvers suggested in the present pper is relted to tht of Gussin elimintion This nlogy will llow us to crry over the stilizing techniques of Gussin elimintion to the new uchy solvers First however we identify the cse when no pivoting is necessry 11

12 u X #" X u 1 Totlly positive mtrices If we ssume tht S VV S S S VU S ie the mtrix j is totlly positive so tht ll the entries of the exct fctors nd U re positive [GK] In this cse theorems 41 nd 42 imply tht the lgorithms 2 nd produce fvorle smll ckwrd error orollry 1 Assume tht condition (1) holds ie tht N66 is totlly positive nd ssume tht the lgorithms 2 (Schur-type-direct-uchy ` lgorithm) nd (qusi-uchy lgorithm) re performed in the floting point rithmetic with unit roundoff nd tht no overflows were encountered during the computtion If the tringulr fctoriztion of the Schur-type-direct-uchy lgorithm is used to solve the ssocited liner system then the computed solution solves nery system with c c ] hc â j j R M p` ` : j6 N66 7 The nlogous ckwrd ound for the qusi-uchy lgorithm is ] jm : l M R p` (1) (2) ` : j () The ove results show tht the ckwrd stility of the fst lgorithms 2 for totlly positive uchy mtrices is even more fvorle thn tht of the slow Gussin elimintion procedure see (116) Indeed the difference is tht the ound (116) is vlid only for the cse when the entries of the computed fctors nd remin positive (which is usully not the cse with ill-conditioned mtrices) wheres the fvorle ounds in the two ove corollries hold while there re no overflows For exmple for the Hilert mtrix Ï å t ( ± the condition numer : M grows exponentilly with the size so lredy for smll we hve : MQ MQ M Here is polynomil of smll degree in Then in ccordnce with [68] the mtrix will likely lose during the elimintion not only its totl positivity ut lso the weker property of eing positive definite orrespondingly the single precision APAK M{ routine SPOSV for holesky fctoriztion when pplied to the Hilert mtrix exits with n error flg lredy for wrning tht the entries of ecme negtive so the plesing ckwrd ound (116) is no longer vlid for Gussin elimintion In contrst the fvorle ounds (2) () re vlid for higher sizes s long s there re no overflows 2 Generl cse Predictive pivoting techniques Here we ssume tht the two sets of nodes q r nd q sr re not seprted from ech other The similrity of the ckwrdv ounds (11) for Gussin elimintion nd of (42) (47) for the new uchy solvers suggests to use the sme pivoting techniques for preventing instility More precisely ny row or column reordering tht reduces the size of ppering in the ounds (42) (47) will stilize the numericl performnce of the lgorithms 2 Moreover the normwise error nlysis of [SB] for the lgorithm 2 reviewed t the eginning of section 4 lso indictes tht the pivoting will enhnce the ccurcy of the lgorithm 2 Here we should note tht the prtil pivoting technique cn e directly incorported into the Schur-type lgorithms 2 nd 2 see eg [GKO] However the corresponding ordering of q r cn lso e computed in dvnce in MN:O flops Indeed the prtil pivoting technique determines permuttion mtrix such tht t ech elimintion step the pivot elements in re s lrge s possile lerly the determinnt of the leding sumtrix of is equl to so the ojective of prtil pivoting is the successive mximiztion of the determinnts of leding sumtrices This u u 1 VV 12

13 Ï oservtion nd the well-known formul [41] for the determinnt of uchy mtrix imply tht prtil pivoting on is equivlent to the successive mximiztion of the quntities Ï ( t Ï t Ï ( t Ï t Ï Ï Ï ( t Ï t Ï ( t t Ï -- jm e shll cll this procedure predictive prtil pivoting ecuse it cn e rpidly computed in dvnce y the following lgorithm (4) Algorithm 2 Predictive Prtil Pivoting omplexity:!)ö ø#" flops function x = prtil(xy) n = mx(size(x)); dist = ; m = 1; ux = zeros(1n); for i = 1:n ux(i) = s(1 / (x(i) - y(1))); if distux(i) m = i; dist = ux(i); x = swp(x1m); ux(m) = ux(1); if n=2 return; for i = 2:(n-1) dist = ; m = i; for j = i:n ux(j) = ux(j) * s((x(j) - x(i-1)) / (x(j) - y(i))); if distux(j) m = j; dist = ux(j); x = swp(xim); ux(m) = ux(i); return A similr row reordering technique for Vndermonde mtrices (nd fst lgorithm for chieving it) ws proposed in [Hig] nd in [R] it ws clled ej ordering Therefore PPP my lso e clled rtionl ej ordering y nlogy with (polynomil) ej ordering of [Hig] [R] In recent pper [Gu] vrition of complete pivoting ws suggested for the more generl uchy-like mtrices In the context of [Gu] the corresponding displcement rnk is 2 or higher For the ordinry uchy mtrices (displcement rnk = 1) Gu s pivoting cn e descried s follows At ech elimintion step one chooses the column of with the mximl mgnitude entry Æ E F Ÿ in its genertor (here we use the nottions of emm 24) Then one interchnges this column with the 1-st one Ÿ nd performs the prtil pivoting step The explicit expression (2) for the entries of the successive genertors redily suggests modifiction of the lgorithm 2 to perform the Gu s vrint of pivoting in dvnce leding to wht cn e clled predictive Gu pivoting MN:O 6 Numericl illustrtions e performed numerous numericl tests for the three lgorithms suggested nd nlyzed in this pper The results confirm theoreticl results (s perhps should e expected) In this section we illustrte with just few exmples the influence of different orderings on the numericl performnce of the following lgorithms : () Schur-type-uchy (Fst MQ lgorithm 2 requiring MN:O storge) () Schur-type-direct-uchy (Fst MQ lgorithm 2 requiring MN:' storge) (c) qusi-uchy (Fst MQ lgorithm requiring MQ storge) (d) BKO (Fst MN:O lgorithm of ([BKO]) requiring MQ storge) The suroutine swp(xim) in Algorithm 2 swps the -th nd % -th elements of the vector Ü 1

14 Æ d X X / X X X Ï X ÿ X ÿ ÿ Ï X ÿ X ÿ X / X (e) INV The use of the explicit inversion formul (Fst MQ 6 ( '& lgorithm requiring MQ storge) æ( & E % æ (*) F /æ+( æ- ( &) E *æo(+* ) F (f) GEPP Gussin elimintion with prtil pivoting (Slow %-)(*/) MNP' æ( & E %-)(* æ F /æ+( æ- ( & E % (+%'æ F lgorithm requiring Ò Ï t Ò MN:' storge) (61) r nd q r cnnot e e refer to [BKO] for the discussion nd computed exmples relted to the importnt cse of totlly positive uchy mtrices nd restrict ourselves here to the generic cse in which the two sets q seprted from ech other so tht they cnnot e reordered to chieve (18) e solved vrious uchy liner systems (62) equidistnt clustered or rndomly distriuted nodes nd with mny others configurtions) with different right-hnd sides (RHS) e lso solved the so-clled uchy-toeplitz liner systems with coefficient mtrices of the form (including interlced S S : S : S UV S S ÓÌ 2 å E Ï ( t F4 Ð (6) OR( with different choices for the prmeters nd Æ All the experiments were performed on DE /1 RIS worksttion in single precision (unit roundoff ) For GEPP we used the APAK routine SGESV nd ll the other lgorithms were implemented in In order to check the ccurcy we implemented ll the ove lgorithms in doule precision (unit roundoff d dld R7( 6 ) nd in ech exmple we determined two prticulr lgorithms providing solutions tht were the closest to ech other In ll cses these two solutions greed in more thn the 8 significnt digits needed to check the ccurcy for solution otined in single precision so we regrded one of these doule precision solutions ¹ s eing exct nd used it to compute the 2-norm reltive error for the solutions X 7 Ï ÿ [ ¹ ÿ : ¹ ÿ : Ï computed y ech of the ove lgorithms In ddition we computed the residul errors Î Ï {{ct nd the ckwrd errors Æ Ï [ªU 8;:= : : :! Æ Ï! X ÿ : ÿ ÿ : Ï ÿ : Ï ÿ : Ï ÿ : using the formul result proly first shown y ilkinson see eg [Hig6] The tles elow disply lso the condition numer of the coefficient mtrix norms for the solution ÿ ¹ ÿ : nd the right-hnd side ÿ ÿ : s well s some other useful : â 61 Exmple 1 ell-conditioned uchy-toeplitz mtrices In this exmple we solved the liner system (62) with uchy-toeplitz coefficient mtrix in (6) with ² nd with the right-hnd side - ± e used two orderings A The nodes were ordered using the predictive prtil pivoting (PPP) technique (18) A The nodes q r sorted in n incresing order nd the nodes q r sorted in decresing order; the difference with (18) is in tht now two sets of nodes q r q r re not seprted from ech other Forwrd error 14

15 ÿ X X ÿ ÿ ÿ c ÿ ÿ ÿ Bckwrd error Residul error Tle 1 Forwrd error Prtil pivoting ordering n B /ED4F G4Ḧ GI G K GI GNMOGI P & P4 P4Q PIR P4S P4T 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy GEPP 2e+ 16e+ 61e+ 4e+ 7e-8 1e-7 1e-7 6e-8 6e-8 8e-8 e+ 16e+ 16e+1 1e+1 1e-7 2e-1 e-8 2e-7 4e-7 1e-7 e+ 16e+ 24e+1 14e+1 4e-7 1e+ 1e-7 1e-7 e-7 1e-7 Tle 2 Forwrd error Monotonic ordering n B ED4F G4Ḧ G G K G GNMOG P & P P Q P R P S 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 1e+ 16e+ 4e+ 2e+ 8e-8 2e-7 e-6 1e-6 2e-6 2e+ 16e+ 61e+ 4e+ 1e-7 2e- 1e- 7e- 6e- 2 e+ 16e+ e+ 6e+ 1e-7 6e-2 e+4 e+4 e+4 e+ 16e+ 12e+1 77e+ 1e-7 7e+ e+16 6e+17 2e+16 e+ 16e+ 16e+1 1e+1 2e-7 1e+2 NN NN NN 6 e+ 16e+ 18e+1 11e+1 NN 4e+26 NN NN NN n UVX )E [ N ]^_IF UVX )` EH F & Tle Bckwrd error Prtil pivoting ordering Q R INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy GEPP 2e+ 1e+ 6e-8 1e-7 6e-8 6e-8 4e-8 e-8 2e+ 1e+ 1e-7 2e-1 8e-8 1e-7 2e-7 1e-7 2e+ 1e+ e-7 1e+ 1e-7 1e-7 2e-7 1e-7 Tle 4 Bckwrd error Monotonic ordering n UVX )E [ N ]^_4F UVX )` EH F & Q INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 6e+2 1e+ 6e-8 2e-7 e-6 1e-6 2e-6 e+6 1e+ 7e-8 2e- 1e- 7e- 6e- 2 1e+14 1e+ 1e-7 6e-2 1e+ 1e+ 1e+ 4e+21 1e+ 1e-7 1e+ 1e+ 1e+ 1e+ 6e+6 1e+ 2e-7 1e+ NN NN NN 6 2e+44 1e+ NN 1e+ NN NN NN Tle Residul error Prtil pivoting ordering UV4X ) [ = ]^=c K F UVX )` EH F d & d dnq d4r dns dnt INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy GEPP n e+ 1e+ 1e-7 e-7 1e-7 1e-7 e-8 1e-7 2e+1 1e+ 4e-7 e-1 2e-7 e-7 4e-7 e-7 e+1 1e+ 8e-7 e+ e-7 e-7 e-7 e-7 Tle 6 Residul error Monotonic ordering UV4X ) [ = ]^ec K F UVX )` EH F d & d dnq dr dns INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy n e+2 1e+ 1e-7 e-7 6e-6 e-6 4e-6 1e+7 1e+ 2e-7 4e- e- 2e-2 1e-2 2 7e+14 1e+ e-7 1e-1 2e+ 8e+4 1e+ e+22 1e+ e-7 2e+4 7e+16 1e+18 4e+16 7e+7 1e+ 4e-7 e+2 NN NN NN 6 e+4 1e+ NN e+26 NN NN NN ompring the dt in Tles 1-6 indictes tht the ordering of the nodes hs profound influence on the ccurcy of ll lgorithms designed in the present pper Specificlly let us recll tht the quntity pper in the ckwrd error ounds (4) nd (47) for the lgorithms Schur-type-direct-uchy nd qusi-uchy respectively The second columns of Tles nd 4 show tht the ltter quntity is huge with monotonic ordering nd moderte with PPP ordering orrespondingly the ckwrd errors shown in the tles re lrge with monotonic ordering nd plesntly smll with PPP ordering Anlogously comprison U«X è of the dt in the second columns of Tles nd 6 shows tht PPP technique reduces the quntity ppering in the residul ounds for the lgorithms Schur-type-direct-uchy nd qusi-uchy resulting in fvorle smll X [ residul error for these lgorithms Further it is well-known : â ÿ Since the coefficient mtrix in this exmple is quite well-conditioned (see eg the dt in the second column of Tle 1) the PPP technique yields plesnt forwrd ccurcy for ll lgorithms Schur-type-uchy Schur-typedirect-uchy nd qusi-uchy The PPP technique lso improves the numericl performnce of the BKOBP-type lgorithm however for this lgorithm the results re not s fvorle s for other lgorithms (see eg introduction for the explntion of this phenomen nd [BKO] for the discussion on extremely high ccurcy of this lgorithm for totlly positive uchy mtrices) The use of explicit inversion formul lso yields high ccurcy predicted y the nlysis of [GK] nd pprently this is the only lgorithm whose ccurcy does not dep upon the ordering of the nodes At the sme time R S S T - m 1

16 X X comprison of the dt in Tles 1 nd 2 s well s in other exmples indictes tht the use of the PPP technique prevents the INV lgorithm from overflows llowing to solve lrger liner systems Since in this exmple the coefficient mtrix is well-conditioned the MNPO GEPP lgorithm while slow lso provides good forwrd nd ckwrd ccurcy The results of mny other computed exmples re quite similr to those in Tles 1-6 nd the lgorithms Schurtype-uchy Schur-type-direct-uchy nd qusi-uchy lwys yield fvorle smll ckwrd nd residul errors which re often etter thn those of GEPP nd of the use of inversion formul As for the forwrd stility there seem to e no cler winner however the use of of inversion formul often provides provides smller forwrd error 6 6 especilly 7 when using the unit vectors for the right-hnd side (which mens tht one hs to find column of ( ) At the sme time new lgorithms cn provide smller forwrd error in other cses s illustrted y the next exmples 62 Exmple 2 Ill-conditioned uchy-toeplitz mtrices R The condition numer of uchy-toeplitz mtrices deps upon the choice of prmeters nd Æ in (6) In this exmple we chose Æ to otin uchy-toeplitz ` mtrix whose condition numer is severl order if R( mgnitude igger thn the reciprocl to the mchine precision The next tles 7-12 present the dt ² on the forwrd nd ckwrd errors for the corresponding liner system with the RHS - ± Along with the PPP ordering we lso consider the originl ordering (no ordering) nd the Gu s pivoting ordering of q r q sr Forwrd error Bckwrd error Tle 7 Forwrd error Prtil pivoting ordering n B /`D4F G4Ḧ GI G K+ GI GM-GI P & P P4Q P4R P4S P4T 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy GEPP 6 e+ 12e+1 27e+ 77e+ e- 4e+2 e- 2e-6 2e-4 1e+ 8 e+11 12e+1 71e+ 8e+ 2e-2 NN e- 1e- e-4 1e+ e+11 12e+1 1e+6 1e+1 2e-2 NN 8e- e-6 6e-4 1e+ P & P P Q P R P S Tle 8 Forwrd error Gu s pivoting INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy n 6 2e- 7e+1 8e-4 1e-4 e-4 8 6e- 4e+ 4e- e-4 2e- 1e-2 2e+ e- e-4 4e- P & P P Q P R P S Tle Forwrd error No pivoting INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy n 6 e- 1e+2 e-4 2e-4 7e- 8 e-2 2e+ 1e- e-4 1e-4 2e-2 1e+ 7e-4 6e-4 2e-4 Tle Bckwrd error Prtil pivoting ordering n UVX )E [ N ]^_IF UVX )` EH F & Q R INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy GEPP 6 1e+ 1e+1 4e-4 1e+ 2e-7 e-7 e-7 4e-7 8 1e+ 1e+1 1e- NN 4e-7 4e-7 4e-7 7e-7 2e+ 1e+1 6e- NN 6e-7 6e-7 7e-7 1e-6 n UV4X )E f e ]^IF & Tle 11 Bckwrd error Gu s pivoting Q INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 6 1e+4 2e-4 1e+ e- 1e-4 e- 8 e+4 2e- 1e+ 2e-4 e-4 1e-4 4e+4 2e- 1e+ e-4 e-4 2e-4 n UV4X )E f e ]^IF & Tle 12 Bckwrd error No pivoting INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 6 1e+4 1e- 1e+ 1e-4 2e-4 7e- 8 e+4 4e- 1e+ 4e-4 e-4 1e-4 e+4 2e- 1e+ 4e-4 6e-4 2e-4 Agin comprison of the dt in Tles -12 confirms the nlyticl results of sections 4 indicting tht n pproprite pivoting technique cn reduce the size of ckwrd errors for the new lgorithms mking them s fvorle s those of GEPP The coefficient mtrix in exmples nd 4 is quite ill-conditioned so the forwrd ccurcy of GEPP is less fvorle However the lgorithms Schur-type-uchy Schur-type-direct-uchy nd qusi-uchy comined with prtil pivoting provide smller forwrd errors thn GEPP (nd the use of inversion Q R R S S S T 16

17 ^ ^ ^ formul) showing tht the use of the structure often llows us not only to speed-up the computtion ut to lso chieve more ccurcy s compred to generl structure-ignoring methods It my seem to e quite unexpected tht for uchy-toeplitz mtrices the Gu s pivoting technique (comining row nd column permuttions) cn led to less ccurte solutions s compred to the PPP technique (sed on row permuttions only) Ì To understnd this occurrence it is useful to oserve tht the entries of the digonls of uchy- Toeplitz mtrices 2 åg E Ï ( t F Ð v] Ý X h dep hyperoliclly on the difference thus giving pick for the digonl v] with Æ e next disply the MATAB grphs for the severl permuted versions of the mtrix in Exmple 2 for M OR No pivoting Prtil pivoting 2 2 Gu s pivoting 2 Figure 1: Permuted versions of uchy-toeplitz mtrix corresponding to different pivoting techniques eij ei' One sees tht in the originl mtrix the mximl mgnitude entries (ll = ) occupy the 4-th uu sudigonl (ie in the lower tringulr prt of the mtrix) Applying prtil pivoting technique mens moving ech of the rows 4- three postions up so tht the mximl mgnitude entries re now ll locted on the min digonl In the next tle we list the condition numers for the leding sumtrices corresponding to the three pivoting techniques Tle 1 onditioning of leding sumtrices k No pivoting Gu s pivoting Prtil pivoting j k R 8 12 l d R : d R OR 1 R R d OR 4 1 m R d k OR 1 R Rn d OR 6 1 d Rn d OR 7 14 Rn R OR : OR : 8 R R 6 kb m OR P d OR R 6 OR 6 OR 6 e note however tht the motivtion for introducing the Gu s pivoting technique ws given in [Gu] where n ppliction of [GKO] with displcement rnk 2 or higher ws discussed 6 Exmple A trnsposed system However n immedite question is wht will hppen for trnsposed to the mtrix in Exmple 2 (clerly the trnsposed uchy mtrix is uchy mtrix itself) Therefore we consider here uchy-toeplitz mtrix with 17

18 X ^ ^ ^ ^ R the prmeters nd Æ For such mtrix the mximl mgnitude entries will now e locted ove the min digonl Therefore it is resonle to pply prtil column pivoting technique As in the ove exmple we next disply the permuted versions of mtrix corresponding to different pivoting techniques No pivoting Prtil pivoting Gu s pivoting Prtil column pivoting Figure 2: Permuted versions of uchy-toeplitz mtrix corresponding to different pivoting techniques In Tle 14 we list the corresponding condition numers for ll successive leding sumtrices Tle 14 onditioning of leding sumtrices k No pivoting Gu s pivoting Prtil column pivoting j k R d R OR 12 l d R : k OR 1 R R d OR 4 1 m R kb OR 1 R Rn 6 j OR 1 d Rn d OR 7 14 Rn OR : OR : 8 R R 6 d OR d OR R 6 OR 6 OR 6 e now turn to the numericl results compring the performnce of the lgorithms designed in the present ² pper e gin used the vector -- ± for the right-hnd side Forwrd error Tle 1 Forwrd error Prtil pivoting ordering n B /`D4F G4Ḧ GI G K+ GI GM-GI P & P P4Q P4R P4S P4T 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy GEPP 4 e+ 12e+1 71e+4 6e+ 2e- e- e- e-6 e- 2e+ 6 e+ 12e+1 27e+ 77e+ 4e- 1e-4 4e-4 7e-6 6e-4 1e+ 8 e+11 12e+1 71e+ 8e+ e- 6e+2 e- 6e-6 e-4 1e+ e+11 12e+1 1e+6 1e+1 e-2 e+4 2e- 6e-6 2e- 1e+ Tle 16 Forwrd error Prtil column pivoting n B /`D4F G4Ḧ GI G K+ GI GM-GI P & P P4Q P4R P4S 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 4 e+ 12e+1 71e+4 6e+ 2e- e+14 2e- 2e-6 2e-4 6 e+ 12e+1 27e+ 77e+ e- 1e+28 e- 2e-6 2e-4 8 e+11 12e+1 71e+ 8e+ 6e- NN e- 8e-6 2e- e+11 12e+1 1e+6 1e+1 1e-2 NN e- 8e-6 6e-4 Tle 17 Forwrd error Gu s pivoting n B /`D4F G4Ḧ GI G K+ GI GM-GI P & P P4Q P4R P4S 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 4 e+8 2e+1 e+ 6e+ 1e- e-7 e-4 e-6 e-4 6 e+8 12e+1 11e+ 77e+ 8e- e-4 1e- 1e-4 2e-4 8 2e+ 12e+1 e+2 8e+ 4e-2 7e+2 8e- 2e- 8e- 4e+11 12e+1 66e+ 1e+1 e-2 8e+ 2e- 6e- e- 18

19 q ###" Tle 18 Forwrd error No pivoting n B /`D4F G4Ḧ GI G K+ GI GM-GI P & P P4Q P4R P4S 2-norms INV BKO qusi-uchy Schur-type-uchy Schur-type-direct-uchy 4 e+ 12e+1 71e+4 6e+ 4e-4 1e-7 7e-4 2e- e-4 6 e+ 12e+1 27e+ 77e+ e- 1e-4 6e-4 8e- 1e- 8 e+11 12e+1 71e+ 8e+ 1e-2 2e+1 4e- 1e-4 6e- e+11 12e+1 1e+6 1e+1 2e-2 e+ 1e-2 4e-4 2e- In this exmple the forwrd ccurcy of the Schur-type-uchy lgorithm is etter thn tht of the Schurtype-direct-uchy nd qusi-uchy lgorithms Note tht there re mny other exmples however where these lgorithms hve roughly the sme ccurcy Bckwrd error It turns out tht for mny different orderings ll the lgorithms designed in this pper exhiit fvorle ckwrd stility Moreover for n vrying from to for prtil row pivoting prtil column pivoting the Gu s pivoting nd for no-pivoting the lgorithms Schur-type-uchy Schur-type-uchy-direct nd qusi-uchy produced ckwrd errors of the order of OR (po which is comprle to tht of GEPP e however found tht monotonic ordering defined in Sec 61 nd rndomized ordering produce poor results This indictes tht nlyticl error ounds otined for the fst lgorithms of this pper in fct my led to wide vriety of different pivoting techniques ech imed t the reduction of the quntity 7 Trnsformtion of Polynomil-Vndermonde Mtrix into uchy Mtrix In this section we shll show tht ll the fst uchy solvers suggested in the present pper cn e used to solve liner systems with polynomil Vndermonde mtrices is - ( 1 is 4q» 6 srut=v : : - ( : 11 (71) i - ( where w qx i + - M ( + r denotes sis in the liner spce ( zy { of ll complex polynomils whose degree does not exceed hen w 4q» 6 is the power sis then is the ordinry Vndermonde mtrix If stnds for the sis of heyshev polynomils MN:O 4qé (of the first or of the second kind) then is clled heyshev-vndermonde mtrix Fst lgorithms for solving heyshev-vndermonde systems were suggested in [Hig88] [HHR8] [R] [GO4d] Here we suggest n lterntive sed on the next result Proposition 71 et q VV 6p '' UV jl r d M òl e pirwise distinct complex numers nd let Ï Ï Then the following formul is vlid 4 q 78 digq ò- -- jò+7 r 6 ]6 7 e shll prove the ove proposition t the of this section digq }-~IÝô & - }-~I Ýô / r 4 q 7 4q» 4qé 6 Oserve tht formul (72) reltes nd or in other words it llows us to chnge the nodes from q sr to q sr while keeping the polynomil sis w qx i - ( r 6 Suitle choices of the new points r 4_q cn ensure tht hs low complexity In such cses Proposition 71 llows us to reduce the prolem of solving 4_q liner system with to the nlogous prolem of solving liner system with the uchy mtrix r for which ordinry Vndermonde mtrices nd In the next proposition we specify severl sets of points q heyshev-vndermonde mtrices hve low complexities Proposition 72 1 ) et 'téƒ -ÁsÀ t+ ˆ ve q is the (scled) Discrete osine Trnsform I mtrix (72) R - jm r í + e the extrem of Then 4 i ; ÁsÀ t ± t i (7) 1

20 q Ž q 4 Ž q Ž q 4 Ÿ Ž q 4 4 t Ž q Ð ###" 'téƒ -ÁsÀ : t ( 2 ) et : v! q -- 6M r í + e the zeros of the Then 4 8ÓÌ ÁsÀ E : t ( Fâ : Ð t (74) is the Discrete osine Trnsform II mtrix ) et tƒ -ÁsÀ t ve q - jm is the (scled) Discrete Sine Trnsform I mtrix 4 ) et 'téƒ -ÁsÀ : t ( : v! q r ; e the zeros of ( + Then 4 ( ; À6ªV t ± ( (7) -- 6M r í e the zeros of 4p 6 ; Ì ÀjªV E : t ( F is the (scled nd trnsposed) Discrete Sine Trnsform II mtrix 'té -Š Œ Ž : Ot ) et is the Discrete Fourier Trnsform mtrix Then : Ð t (76) ve q R VU jm r denote the roots of unity Then 4!]6 7;äÌ OŠ : Ot E ( F Ð ( t i (77) The ltter proposition is esily deduced from the definitions of heyshev polynomils Before proving Propo- M Ì+ sition 71 let us introduce the necessry nottions et denote the mximl degree of two polynomils nd + The ivrite function Q678 ò 7 Ì7 + is Ì+ clled the Bezoutin of nd + Now let qš il+ - ( ' r e nother sis in the liner spce ( y _{ The mtrix ` K - Æ Ï t ± ( Ï t i whose entries re determined y Kj7 ( œ Ï t i Æ Ï t Ï i is clled the Bezout mtrix of ò nd t - ( + ± Ÿ ` K - is ( 7 with respect to the two sets of polynomils w nd Proof of Proposition 71 The proof is sed on the following useful property of the Bezout mtrix 4qé 6 Ÿ ` K - j Ï j't ± Ï t which follows immeditely from (78) It is esy to see tht the mtrix on the right-hnd side of Eq (7) is qusi-uchy mtrix : 4qé 6 ºŸ ` K - jäì E % ) F E * F Ì- - jò7 %x) (*+ Ï t r digq - -- l7 r digq On the other hnd y using (7) nd the ovious reltion it is esy to check tht 4qé 6 ºŸ Ÿ ` K - Now sustituting the Ž ` K - 6; [`p7 7 6 p digq ò7 f7 ` - ` r otined from the lst eqution ck into (7) yields (72) (78) (7) (7)

21 8 onclusion MN:' In [BKO] we developed fst Björck-Pereyr-type uchy solver nd proved tht for the importnt clss of totlly positive coefficient mtrices it yields plesntly smll forwrd ckwrd nd residul errors However experience shows tht in the generic cse the numericl performnce of the BP-type lgorithm cn e less fvorle Since the use of explicit inversion formul for uchy mtrices lso cn produce lrge ckwrd error no fst nd ccurte methods methods were ville for solving uchy liner equtions In this pper we designed severl lterntive fst MN:' uchy solvers nd the rounding error nlysis suggests tht their ckwrd stility is similr to tht of Gussin elimintion (GE) so tht vrious pivoting techniques (so successful for GE) will stilize the numericl ehvior lso for these new lgorithms It is further shown tht the row ordering of prtil pivoting nd of the Gu s pivoting [Gu] cn e chieved in dvnce without ctully performing elimintion nd fst MN:' lgorithms for these purposes re suggested e lso identified clss of totlly positive uchy mtrices for which it is dvntgeous not to pivot when using the new lgorithms which yields remrkle ckwrd stility This mtches the conclusion of de Boor nd Pinkus who suggested to void pivoting when performing stndrd Gussin elimintion on totlly positive mtrices Anlyticl error ounds nd results of numericl experiments indicte tht the methods suggested in the present pper enjoy fvorle ckwrd stility References [BKO4] [BKO] TBoros TKilth nd VOlshevsky Fst lgorithms for solving uchy liner systems Stnford Informtion Systems ortory report 14 TBoros TKilth nd VOlshevsky Fst Björck-Pereyr-type lgorithm for prllel solution of uchy liner equtions iner Alger nd Its Applictions 2- (1) p26-2 [BP7] ABjörck nd VPereyr Solution of Vndermonde Systems of Equtions Mth omp 24 (17) 8- [41] A uchy Mémoires sur les fonctions lternées et sur les sommes lternées Exercices d nlyse et de phys mth ii (1841) 11-1 [F88] Thn nd DFoulser Effectively well-conditioned liner systems SIAM Sci Stt omputtion (188) 6 6 [R2] Dlvetti nd Reichel A heyshev-vndermonde solver in Alg Appl 172(12) [R] [D1] [DBP77] [GK] [GK] [GK] lvetti D nd Reichel : Fst inversion of Vndermonde-like mtrices involving orthogonl polynomils BIT 1 HDym Structured Mtrices Reproducing Kernels nd Interpoltion to pper in Structured Mtrices in Mthemtics omputer Science nd Engineering (VOlshevsky Editor) ontemporry Mthemtics Series vol 28 AMS Pulictons 21 de Boor nd A Pinkus Bckwrd error nlysis for totlly positive liner systems Numer Mth 27 (177) 48-4 FRGntmcher nd MGKrein Oscilltory mtrices nd kernels nd smll virtions of mechnicl systems second edition (in Russin) GITT Moscow 1 Germn trnsltion : Oszilltionsmtrizen Oszilltionskerne und kleine Schwingungen mechnischer Systeme Berlin Akdemie Verlg 16 IGoherg nd IKoltrcht On the inversion of uchy mtrices In Signl processing Scttering nd Opertor Theory nd Numericl methods Proc of the MTNS-8 (MAKshoek Hvn Schuppen nd AMRn eds) 81-2 Birkhäuser Boston MA IGoherg nd IKoltrcht Mixed componentwise nd structured condition numers SIAM Mtrix Anl 14(1)