ETNA Kent State University

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1 Elecronic Tranacion on Numerical Analyi. Volume 37, pp. 7-86,. Copyrigh, Ken Sae Univeriy. ISSN ETNA Ken Sae Univeriy hp://ena.mah.ken.edu THE ANALYTIC SVD: ON THE NON-GENERIC POINTS ON THE PATH DÁŠA JANOVSKÁ AND VLADIMíR JANOVSKÝ Abrac. A new echnique for compuing he Analyic SVD i propoed. The idea i o follow a branch of ju one imple ingular value and he correponding lef/righ ingular vecor. Numerical compuaion may collape a non-generic poin; we will conider he cae when he coninuaion ge uck due o a nonzero muliple ingular value. We inerpre uch a poin a a ingulariy of he branch. We employ ingulariy heory in order o decribe and claify hi poin. Since i codimenion i one, we mee uch a poin rarely. Key word. SVD, ASVD, coninuaion, ingulariy heory. AMS ubjec claificaion. 65F5. Inroducion. A ingular value decompoiion SVD of a real marix A R m n, m n, i a facorizaion A = UΣV T, where U R m m and V R n n are orhogonal marice and Σ = diag,..., n R m n. The value i, i =,...,n, are called ingular value. They may be defined o be nonnegaive and o be arranged in nonincreaing order. Le A depend moohly on a parameer R, [a, b]. The aim i o conruc a pah of SVD. A = UΣV T, where U, Σ and V depend moohly on [a, b]. If A i a real analyic marix funcion on [a, b], hen here exi an Analyic Singular Value Decompoiion ASVD [], a facorizaion. ha inerpolae he claical SVD defined a = a, i.e. he facor U, V and Σ are real analyic on [a, b], for each [a, b], boh U R m m and V R n n are orhogonal marice, and Σ = diag,..., n R m n i a diagonal marix, a = a, he marice Ua, Σa and V a are he facor of he claical SVD of he marix Aa. The diagonal enrie i R of Σ are called ingular value. Due o he requiremen of moohne, ingular value may be negaive, and heir ordering may by arbirary. Under cerain aumpion, he ASVD may be uniquely deermined by he facor a = a. For heoreical background, ee [9]. A far a he compuaion i concerned, an incremenal echnique i propoed in []. Given a poin on he pah, one compue a claical SVD for a neighboring parameer value. Nex, one compue permuaion marice which link he claical SVD o he nex poin on he pah. The procedure i approximaive wih a local error of order Oh, where h i he ep ize. An alernaive echnique for compuing he ASVD i preened in [3]. A non-auonomou vecor field H : R R N R N of large dimenion N = n + n + m can be conruced in uch a way ha he oluion of he iniial value problem for he yem x = H, x i linked o he pah of he ASVD. Moreover, [3] conribue o he analyi of non-generic poin of he ASVD pah; ee []. Thee poin could be, in fac, inerpreed a ingulariie of he vecor field R N. In [], boh approache are compared. Received November 8, 6. Acceped November, 9. Publihed online April 7,. Recommended by Bruno Lang. The reearch of boh auhor wa parially uppored by he Gran Agency of he Czech Republic gran No. /6/356. The fir and he econd auhor acknowledge financial uppor by he reearch projec MSM and MSM 6839, repecively, of The Miniry of Educaion, Youh and Spor, Czech Republic. Iniue of Chemical Technology, Prague janovkd@vch.cz. Charle Univeriy, Faculy of Mahemaic and Phyic, Prague janovky@karlin.mff.cuni.cz. 7

2 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS 7 A coninuaion algorihm for compuing he ASVD i preened in [7]. I follow a pah of a few eleced ingular value and lef/righ ingular vecor. I i appropriae for large pare marice. The coninuaion algorihm i of predicor-correcor ype. The relevan predicor i baed on he Euler mehod, hence on an ODE olver. In hi repec, here i a link o [3]. Neverhele, he Newon-ype correcor guaranee he oluion wih precribed preciion. The coninuaion may ge uck a poin where a nonimple ingular value i appear for a paricular parameer and index i. In [, 3], uch poin are called non-generic poin of he pah. They are relaed o branching of he ingular value pah. The code in [7] incorporae exrapolaion raegie in order o jump over uch poin. In hi paper, we inveigae non-generic poin. In Secion, we give an example ha moivaed our reearch. Then we inroduce a pah-following mehod for coninuaion of a imple ingular value and he correponding lef/righ ingular vecor. In Secion 4, we define and analyze a ingulariy on he pah. Nex, we perurb hi ingulariy; ee Secion 5. We ummarize our concluion in Secion 6. Finally, in Appendix A, we provide deail of he expanion ued in our aympoic analyi.. Moivaion. Le A = ; + ee [, Example ]. The relevan ASVD, A = UΣV T,.5.5, can be compued explicily: U = V =, =, = +. Obviouly, = = i a nonimple muliple ingular value of A. We will ak he following queion: doe he ASVD-pah peri for an arbirary ufficienly mall perurbaion? Le. A = [ + ε + / /4 Conider he relevan ASVD. Thi ime, we compue i numerically uing he echnique decribed in [7]. We how he reul for he unperurbed and perurbed marice in Figure. and., repecively. Noice ha he branche in Figure. and in Figure. are qualiaively differen. We oberve a eniive dependence on he iniial condiion of he branche. 3. Coninuaion of a imple ingular value. 3.. Preliminarie. Le u recall he noion of a ingular value of a marix A R m n, m n. DEFINITION 3.. Le R. We ay ha i a ingular value of he marix A if here exi u R m and v R n uch ha 3. Av u =, A T u v =, u = v =. The vecor v and u are called he righ and he lef ingular vecor of he marix A. Noe ha i defined up o i ign: if he riple, u, v aifie 3. hen a lea hree more riple, u, v,, u, v,, u, v, ].

3 Ken Sae Univeriy hp://ena.mah.ken.edu 7 D. JANOVSKÁ AND V. JANOVSKÝ v v FIGURE.. Perurbaion ε =. Lef: Branche of ingular value and in red and blue a funcion of. Righ: The relevan righ ingular vecor in red and blue v v FIGURE.. Perurbaion ε =.. Lef: Branche of ingular value and in red and blue a funcion of. Righ: The relevan righ ingular vecor in red and blue. can be inerpreed a ingular value and lef and righ ingular vecor of A. DEFINITION 3.. Le R. We ay ha i a imple ingular value of a marix A if here exi u R m, u, and v R n, v, where, u, v,, u, v,, u, v,, u, v are he only oluion o 3.. A ingular value which i no imple i called a nonimple muliple ingular value. REMARK 3.3. Le. Then i a imple ingular value of A if and only if i a imple eigenvalue of A T A. In paricular, v R n and u R m uch ha A T Av = v, v =, u = Av are he relevan righ and lef ingular vecor of A. REMARK 3.4. = i a imple ingular value of A if and only if m = n and dimkera =. REMARK 3.5. Le i, j be imple ingular value of A wih i j. Then i j and i j. Le u recall he idea of [7]. The branche of eleced ingular value i and he correponding lef/righ ingular vecor U i R m, V i R n are conidered, i.e., AV i = i U i, A T U i = i V i, U i T U i = V i T V i =, for [a, b]. The naural orhogonaliy condiion U i T U j = V i T V j =, i j, [a, b], are added. For p n, he eleced ingular value S =,..., p R p and he correponding lef/righ ingular vecor U = [U,..., U p ] R m p and V = [V,..., V p ] R n p are followed for [a, b].

4 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS The pah of imple ingular value. In hi ecion, we conider he idea of pahfollowing for one ingular value and he correponding lef/righ ingular vecor. We expec he pah o be locally a branch of i, U i R m, V i R n, aifying condiion 3. and 3.3 for [a, b]. We conider he ih branch, i n, namely, he branch which i iniialized by i a, U i a R m, V i a R n, compued by he claical SVD [4]. Noe ha he SVD algorihm order all ingular value in decending order a i a n a. We aume ha i a i imple; ee Remark 3.3 and 3.4. DEFINITION 3.6. For a given [a, b] and R, le u e Im A 3.4 M, A T, I n where I m R m m and I n R n n are ideniie. REMARK 3.7. Le, [a, b].. i a ingular value of A if and only if dim Ker M,.. i a imple ingular value of A if and only if dim Ker M, =. REMARK 3.8. Le [, [a, b]. u. If M, = hen u v] T u = v T v. [ũ ]. If in addiion M, = hen u ṽ T ũ = v T ṽ. Noe ha if i hen due o Remark 3.8 one of he caling condiion 3.3 i redundan. Thi moivae he following definiion. DEFINITION 3.9. Conider he mapping f : R R +m+n R +m+n, R, x =, u, v R R m R n f, x R +m+n, where u + Av 3.5 f, x A T u v. v T v A an alernaive o 3.5 we will alo ue u + Av 3.6 f, x A T u v. u T u + v T v The equaion 3.7 f, x =, x =, u, v, may locally define a branch x =, u, v R +m+n of ingular value and lef/righ ingular vecor u and v. The branch i iniialized a, which play he role of = a. I i aumed ha here exi x R +m+n uch ha f, x =. The iniial condiion x =, u, v R +m+n play he role of already compued SVD facor i a R, U i a R m and V i a R n. We olve 3.7 on an open inerval J of parameer uch ha J. THEOREM 3.. Conider 3.5. Le, x J R +m+n, x =, u, v, be a roo of f, x =. Aume ha i a imple ingular value of A. Then

5 Ken Sae Univeriy hp://ena.mah.ken.edu 74 D. JANOVSKÁ AND V. JANOVSKÝ here exi an open ubinerval I J conaining and a unique funcion I x R +m+n uch ha f, x = for all I and ha x = x. Moreover, if A C k I, R m n, k, hen x C k I, R +m+n. If A C ω I, R m n hen x C ω I, R +m+n. Proof. We will how ha he aumpion imply ha he parial differenial f x a he poin, x i a regular + m + n + m + n marix. Le δx = δ, δu, δv R +m+n, 3.8 f x, x δx = u I m A v A T I n δ δu = R +m+n. T m v T δv Thi i equivalen o he yem 3.9 M, δu u = δ δv v, v T δv =. Projecing he fir equaion on u, v, and uing he ymmery of he marix M,, yield u T T 3. v M, δu δu = M δv δv, u v = δ u + v. u By definiion 3.5, M, v = R m+n. Therefore, δ =. δu u Due o Remark 3.7, here exi a conan c uch ha = c δv v. The econd condiion in 3.9 implie ha c =. Hence δx = δ, δu, δv =, which prove he claim. Auming ha A C k I, R m n, k, he aemen i a conequence of he Implici Funcion Theorem; ee, e.g., [6]. In cae ha A C ω I, R m n, i.e., when A i real analyic, he reul again follow from he Implici Funcion Theorem; ee []. REMARK 3.. The above aemen alo hold for he alernaive caling 3.6. The argumen i imilar. The pracical advanage of 3.7 i ha we can ue andard package for coninuaion of an implicily defined curve. In paricular, we ue he MATLAB oolbox MATCONT [3]. Pah-following of he oluion e of 3.7 via MATCONT i very robu. However, one ha o be careful when inerpreing he reul. In paricular, he lower bound for he ep ize, MinSepize, hould be choen ufficienly mall. We will commen on hi obervaion in Secion 6. In order o illurae he performance, we conider he ame problem a in [7], namely, he homoopy 3. A = A + A, [, ], where he marice A well33.mx, A illc33.mx are aken from he Marix Marke []. Noe ha A, A R 33 3 are pare and A, A are well- and illcondiioned, repecively. The aim i o coninue he en malle ingular value and correponding lef/righ ingular vecor of A. The coninuaion i iniialized a =. The iniial decompoiion of A i compued via vd; ee he MATLAB Funcion Reference. The reul of coninuaion are diplayed in Figure 3.. We run MATCONT en ime, once for each ingular value. The branche do no cro. The compuaion complie wih Theorem 3.. Each curve i compued a a equence of iolaed poin marked by circle; ee he zoom on he righ. The adapive epize conrol refine he epize individually for each branch.

6 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS 75. x FIGURE 3.. Branche of ingular value 3,..., 3 a funcion of. On he righ: The relevan zoom. 4. Singular poin on he pah. Le f, x =, x =, u, v,. Le u dicu he cae when he aumpion of Theorem 3. do no hold; namely, aume ha i a nonimple muliple ingular value of A. Then we conclude ha dimker M, ; ee Remark 3.7. In paricular, we will aume ha dimker M, =. Then, here exi δu R m, δu =, and δv R n, δv =, uch ha δu M, =, v δv T δv =. Noe ha hi implie u T δu =. Compuing Kerf x, x, ee , we conclude ha dimkerf x, = and 4. Kerf x, x = pan δu. δv 4.. Dimenional reducion. Our aim i o analyze he ingular roo, x of he parameer dependen equaion 3.7. A andard echnique i Lyapunov-Schmid reducion [6]. We apply a verion baed on bordered marice [5]. We aume ha dimkerf x, x =, i.e., he corank of he marix f x, x i one. Uing he proper erminology, we deal wih a corank = ingulariy. The algorihm of he reducion i a follow. Le u fix vecor B, C R +m+n. Find ξ R, τ R, x R +m+n and ϕ R uch ha f + τ, x + x + ϕb =, C T x = ξ. We define an operaor relaed o he above equaion: 4.4 Le u aume ha 4.5 de F : R R R +m+n R R +m+n R, f Fτ, ξ, x, ϕ + τ, x + x + ϕb C T. x ξ fx, x B C T.

7 Ken Sae Univeriy hp://ena.mah.ken.edu 76 D. JANOVSKÁ AND V. JANOVSKÝ I can be hown, ee [5], ha hi aumpion i aified for a generic choice of bordering vecor B and C. Neverhele, laer on we pecify B and C. Obviouly, ξ, τ, x, ϕ =,,, R R R +m+n R i a roo of F, i.e., F,,, =. The parial differenial of F wih repec o he variable x, ϕ, namely, he marix F x,ϕ,,, R +m+n +m+n, i regular a he origin,,, by he aumpion 4.5. Due o he Implici Funcion Theorem [6], he oluion manifold of Fξ, τ, x, ϕ = can be locally parameerized by τ and ξ; ha i, here exi funcion 4.6 ϕ = ϕτ, ξ R, x = xτ, ξ R +m+n, uch ha Fτ, ξ, xτ, ξ, ϕτ, ξ = for all τ and ξ being mall. From 4. and he fac ha B due o 4.5, we conclude ha 4.7 f + τ, x + xτ, ξ = if and only if 4.8 ϕτ, ξ =. The calar equaion 4.8 i called he bifurcaion equaion. There i a one-o-one link beween he oluion τ, ξ R of he bifurcaion equaion 4.8 and he oluion, x R R +m+n of he equaion 3.7: 4.9 = + τ, x = x + xτ, ξ. The aemen ha an obviou local meaning: i decribe all roo of 3.7 in a neighborhood of, x. A a rule, he oluion of he bifurcaion equaion can be approximaed only numerically. The uual echnique i. approximae he mapping τ, ξ ϕτ, ξ via i Taylor expanion a he origin,. olve a runcaed bifurcaion equaion, i.e., he equaion wih runcaed higher order erm. The Taylor expanion read 4. ϕτ, ξ = ϕ + ϕ τ τ + ϕ ξ ξ + ϕ ττ τ + ϕ ξτ ξτ + ϕ ξξ ξ + h.o.., where he parial derivaive of ϕ = ϕτ, ξ are underood o be evaluaed a he origin, e.g., ϕ ϕ, or ϕ ξτ ϕ ξτ,. Noe ha ϕ, =. The ymbol h.o.. denoe higher order erm. We alo need o expand 4. xτ, ξ = x+ x τ τ+ x ξ ξ+ x ττ τ + x ξτ ξτ+ x ξξ ξ +h.o... The parial derivaive of x = xτ, ξ are underood o be evaluaed a he origin. Thi expanion i needed o approximae he link 4.9 beween 4.8 and 3.7. Compuing he coefficien of boh expanion 4. and 4. i a rouine procedure; ee [5, Secion 6.]. For example, he coefficien x ξ, ϕ ξ aify a linear yem wih he marix from 4.5, [ fx, x B xξ C T =. ϕ ξ ]

8 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS 77 The coefficien x ξξ, ϕ ξξ are defined via a linear yem wih he ame marix, fx, x B xξξ C T = ϕ ξξ [ fxx, x x ξ x ξ ec. We conidered he paricular funcion f defined in 3.6. In our compuaion, we pecified B and C a 4. B = δu δv, C = δu. δv Noe ha condiion 4.5 i aified. Moreover, he compuaion are implified. In Appendix A, we li Taylor coefficien of 4. and 4. up o he econd order. The coefficien are compued by uing he pecific daa of he problem:, x, δu, δv, A and higher derivaive of A a. 4.. Bifurcaion analyi. Le u analyze he bifurcaion equaion 4.8 in a neighborhood of he origin. Le u ar wih a heuriic. Due o A., he bifurcaion equaion can be facored a 4.3 ϕτ, ξ = τ ητ, ξ =, ητ, ξ ϕ τ + ϕ ττ τ + ϕ ξτ ξ + h.o.., where h.o.. are of econd order. The oluion τ, ξ of 4.3 are linked o he oluion, x of 3.7 via he ranformaion 4.9, where he incremen xτ, ξ are expanded a ], τ, ξ = τ τ + ττ τ + ξτ ξτ + h.o.., uτ, ξ = u τ τ + δu ξ 8 u ξ + u ττ τ + u ξτ ξτ + h.o.., vτ, ξ = v τ τ + δv ξ 8 v ξ + v ττ τ + v ξτ ξτ + h.o... Noe ha we exploied A., A.6, and A.9. The h.o.. are of hird order. Obviouly, τ = i a rivial oluion of 4.3. In cae ϕ τ, hi i he only oluion locally available. In wha follow, le u conider he cae ϕ τ =. We olve 4.3 on an open inerval J of parameer auming ha J. THEOREM 4.. Le, x J R +m+n, x =, u, v be a roo of f, x =. Le dim Kerf x, x = ; ha i, le δu R m, δu =, δv R n, δv =, be uch ha 4. hold. Aume ha ϕ τ =, ϕ ττ. ϕ τ δut A v + u T A δv, For ϕ ττ, ee A.4. Le A = A be mooh, i.e., A C ω J, R +m+n. Then here exi an open ubinerval I J conaining and a unique funcion I x R +m+n, x C ω I, R +m+n, uch ha f, x = for all I wih x = x. Proof. By virue of he facorizaion 4.3 we have o olve ητ, ξ = for τ and ξ. The aumpion 4.8 make i poible o apply he Implici Funcion Theorem for real analyic funcion [].

9 Ken Sae Univeriy hp://ena.mah.ken.edu 78 D. JANOVSKÁ AND V. JANOVSKÝ In order o inroduce required erminology, le u briefly review Singulariy Theory [5, Chaper 6]. Le, x R R +m+n be a roo of f, i.e., f, x = ; ee 3.6 in hi paricular conex. If hi roo, x aifie he aumpion of Theorem 3., hen we ay ha i i a regular roo. If no, hen, x i aid o be a ingular roo. In oher word,, x i a ingular poin of he mapping f : R R +m+n R +m+n. In Secion 4. we have already menioned he codimenion of a ingular poin. Theorem 4. claifie codim = ingular poin. The codimenion i no he only apec of he claificaion. In Theorem 4. we require he equaliy 4.7 and he inequaliy 4.8. The former condiion i called he defining condiion and he laer one i he nondegeneracy condiion. The number of defining condiion i called he codimenion. Theorem 4. deal wih a ingular poin of codim =. The nex iem of he claificaion li, namely, a ingular poin of codim =, would be defined by he condiion ϕ τ = ϕ ττ = and he nondegeneracy condiion ϕ τττ, ec. Wih ome abue of language, we noe he following: REMARK 4.. Equaion 3.7 define a pah of parameer dependen ingular value and correponding lef/righ ingular vecor. There are ingular poin on hi pah. One of hem could be claified a corank =, codim = ingular poin. Thi paricular poin, x, x =, u, v, i relaed o a nonimple muliple ingular value. PROPOSITION 4.3. Le he aumpion of Theorem 4. be aified. Le, x be a ingular poin of f of corank = and codim =. Le δu R m, δu =, δv R n, δv =, pan he kernel 4.. Then he poin, y R R +m+n, y, δu, δv, i alo a ingular poin of f of corank = and codim =. Moreover, he Taylor expanion a, y can be obained from ha a he poin, x, i.e., from 4.3, More preciely, le he coefficien of he Taylor expanion a, y be marked by ilde, i.e., ϕ ξ, ϕ τ, ϕ ξτ, ϕ ξτ, j eϕ ξ,, and j f j ξ,. Then j 4.9 ϕ τ = ϕ τ, ϕ ξτ = ϕ ξτ, ϕ ττ = ϕ ττ and ϕ ξ = ϕ ξξ = = j ϕ, = =, ξj ξ = ξξ = = j, = = ξj for j =,,.... Proof. The aemen concerning he ingulariy a, y follow from he properie of he kernel of 3.4. The formulae can be readily verified. Hence, he claificaion of, y follow from he aumpion 4.7 and 4.8. A menioned in Secion, he paper [3] conain a deailed analyi of non-generic poin on he pah. I wa noed ha [3] aemp o rack all ingular value, i.e., o conruc he pah of Σ, U and V ; ee.. Neverhele, on he analyical level, one can peak abou pah of ingular value j and k. REMARK 4.4. In [3, Secion 3], he auhor inveigae imple cro over of he pah. He propoed a e of defining condiion for hi phenomenon. Neverhele, he doe no reolve hi yem explicily. I i inuiively clear ha hi imple cro over of he pah hould be he ingular poin reaed in Theorem 4.. In our analyi, we can poin ou compuable conan A. and A.4 o decide abou he cae.

10 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS Example. We conider [3, Example ]; ee alo [, Example ]. Le A USU, R, where S = diag.5 +,,,, c U = c c c c 3 3, 3 c 3 wih c = co, = in, c = co+, = in+, c 3 = co+, 3 = in +. There are five ingular poin of all pah a =.5,.5,.75,,.5. Thee are manifeed a imple cro over of he pah or imple cro over in erm of [3] relaed o paricular pair of pah; ee Figure 4. on he lef. Noe ha if =.5, hen = =.5. Due o Remark 3.5, eiher or i nonimple acually, boh of hem are nonimple. We are able o coninue eparae branche of ingular value and lef/righ ingular vecor. Le u e =., compue he claical SVD of A = UΣV T and iniialize he coninuaion a 3, U 3, V 3, which are he hird ingular value and he hird column of U and V a =.. We acually coninue a curve in R R In Figure 4. and Figure 4., and v 4 are depiced in green. On hi curve here are ju wo ingular poin, one a =.5 and he oher a =.75. Le u perform an aympoic analyi of he former poin. Uing he noaion of Theorem 4., =.5, =.75, u = [.9689;.474;.;.], v = [.9689;.78;.475;.87], δu = [.475;.5776;.98;.778], δv = [;.949;.98;.453]. The leading Taylor coefficien of ϕτ, ξ and τ, ξ are ϕ ττ =.687, ϕ ξτ =, τ =, ξτ =, ττ = Similarly, we can compue he leading coefficien of uτ, ξ R 4 and vτ, ξ R 4. Neglecing quadraic erm in 4.3, , we ge a local approximaion of he green branch being parameerized by τ. In paricular, we e τ =. :.5 :.5 regularly paced poin on he inerval [.,.5] wih incremen.5 and mark he reuling poin by black diamond; ee he zoom in Figure 4. and Figure 4.. Due o Propoiion 4.3, we can ge an aympoic expanion of he blue branch for free. 5. An unfolding. Le 5. f : R R +m+n R R +m+n, R, x =, u, v R R m R n, ε R f, x, ε R +m+n, u + A + εzv f, x, ε A + εz T u v. u T u + v T v The mapping 5. i an example of an unfolding of he mapping 3.6. I i required ha f, x, via 5. complie wih f, x via 3.6, which i obviou. For a fixed value of ε,

11 Ken Sae Univeriy hp://ena.mah.ken.edu 8 D. JANOVSKÁ AND V. JANOVSKÝ 4 3 "green".8.6 "yellow".4..8 "green".6.4. "yellow" FIGURE 4.. Branche of ingular value,..., 4 in red, blue, green, and yellow a funcion of. Zoomed: The ingular poin, marked a quare, on he green branch. The approximaion via aympoic analyi are marked by diamond..5 "yellow".8.6 "yellow".4 "green".5. v 4 v 4..5 "green" FIGURE 4.. Red, blue, green, and yellow branche of he ninh oluion componen, i.e., v 4 = v 4. Zoomed: The relevan aympoic analyi of he green branch. we conider he equaion 5. f, x, ε =, x =, u, v, for R and x R +m+n. The unfolding may alo model an imperfecion of he original mapping f, x. Le, x, be a ingular poin of he above f. Le hi poin aify he aumpion of Theorem 4.. Our aim i o udy oluion of 5. for a mall fixed ε. 5.. Dimenional reducion reviied. We adap he dimenional reducion from Secion 4. o he unfolding 5.. Le u fix vecor B R +m+n, C R +m+n. Find ξ R, τ R, ε R, x R +m+n and ϕ R uch ha f + τ, x + x, ε + ϕb =, C T x = ξ. Under he aumpion 4.5, he oluion ϕ and x of 5.3 and 5.4 can be locally parameerized by τ, ξ, and ε, i.e., 5.5 ϕ = ϕτ, ξ, ε R, x = xτ, ξ, ε R +m+n. Uing he ame argumen a in Secion 4., we conclude ha 5.6 f + τ, x + xτ, ξ, ε, ε = if and only if 5.7 ϕτ, ξ, ε =

12 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS 8 for mall τ, ξ and ε. The aim i o compue Taylor expanion of he funcion ϕτ, ξ, eε and xτ, ξ, ε a he origin in order o approximae oluion of 5.7 and 5.6. In Appendix A, we li he relevan Taylor coefficien up o econd order. The bordering i choen a in Imperfec bifurcaion. Conider he expanion ϕτ, ξ, ε =τ ϕ τ ϕ ττ τ + ϕ ξτ ξ + ε ϕ ε + ϕ ξε ξ + ϕ τε τ + ϕ εε ε + h.o.. and τ, ξ, ε =τ τ + ττ τ + x ξτ ξ + ε ε + ξε ξ + τε τ + εε ε + h.o.., uτ, ξ, ε = u τ τ + δu ξ 8 u ξ + u ττ τ + u ξτ ξτ + ε u ε + u ξε ξ + u τε τ + u εε ε + h.o.., vτ, ξ, ε = v τ τ + δv ξ 8 v ξ + v ττ τ + v ξτ ξτ + ε v ε + v ξε ξ + v τε τ + v εε ε + h.o... The h.o.. are of hird order. Inead of 5.7, we olve he runcaed bifurcaion equaion 5. τ ϕ τ + ϕ ττ τ + ϕ ξτ ξ + ε ϕ ε + ϕ ξε ξ + ϕ τε τ + ϕ εε ε = for ξ and τ and fixed ε. If ϕ ξε, he oluion o 5. can be parameerized by τ. Hence, given a mall value of τ we compue ξ = ξτ a a oluion of he runcaed bifurcaion equaion 5.. Then we ubiue hi pair τ, ξτ ino he runcaed verion of We ge an approximaion of he roo of 5.6. Le u conider he funcion. from Secion and e ε =.. In Figure., in fac, he oluion e of 5. are depiced, namely, he oluion componen x =, x 4 = v and x 5 = v a.5.5, for boh red and blue branche. In Figure 5., here are zoom of boh red and blue branche howing he oluion componen, on he lef, and he componen v, on he righ. Thee oluion were compued numerically via pah-following; ee Secion 3.. Approximaion via 5. and he runcaed verion of are marked by red and blue diamond. They are reaonably accurae for mall Example coninued. Le u conider A a in Example 4.3. We model an imperfecion 5., n = m = 4. In paricular, we e ε =., Z

13 Ken Sae Univeriy hp://ena.mah.ken.edu 8 D. JANOVSKÁ AND V. JANOVSKÝ v FIGURE 5.. Analyi of moivaing example from Secion, ε =.: zoom of boh red and blue branche, namely, he oluion componen, v, compared wih reul of he runcaed approximaion marked by diamond "green" "yellow" FIGURE 5.. Example from Secion 5.3, ε =.: Four perurbed branche of he analyic SVD,. On he lef: An illuraion of he adapive ep lengh. We compue oluion of 5. for.. In Figure 5., here are four oluion branche, namely, he fir four componen of parameerized by,., colored by red, blue, green, and yellow. The iniial condiion of hee branche are he perurbaion of he iniial condiion from Example 4.3. Oberve ha he blue branch i cloe o he red one, and he green branch o he yellow one for.5 and.5, repecively. We hould have in mind our moivaion, a illuraed in Figure. and Figure.. The imple cro over branching degenerae o a ouching of differen branche like. A far a he lef/righ ingular vecor pah are concerned, a perurbaion implie wiing. Coming back o he perurbed Example 5.3, namely, o he coninuaion of he blue and he green branche, we oberved wi of lef/righ ingular vecor for.5 and.5, repecively. In Figure 5.3, on he lef, he u, u -oluion componen of he blue and red branche are hown near.5. The paricular componen wi. Similarly, in Figure 5.3, on he righ, we depic he u, u 3 -oluion componen of he green and yellow branche a.5. Again, here i a wi. Noe ha a imilar obervaion can be made a he blue and green branche nearly ouch for.5. Comparing Figure 4. and Figure 5., we conclude ha he global branching cenario may change dramaically under a perurbaion. 6. Concluion. We inroduced a new pah-following echnique o follow imple ingular value and he correponding lef/righ ingular vecor. We inveigaed ingulariie on he pah; namely, we claified a ingulariy wih corank = and codim =. Thi ingulariy i relaed o imple cro over in erm of [3]. We alo udied hi ingulariy ubjec o a perurbaion an imperfecion.

14 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS 83. u.5 u u u FIGURE 5.3. The oluion componen. Lef: The blue and red branche for.5. Righ: The green and yellow branche for FIGURE 6.. Branche of ingular value,..., a funcion of.4. The branche 6 and 7 are in magena and cyan. They cro on he lef while hey do no cro on he righ. For he zoom, ee Figure 6.. Thee inveigaion ry o uppor he claim ha he aforemenioned pah-following echnique work generically. Conider he homoopy 3. again. Le u compue he pah of he en large ingular value uing MATCONT [3]. The reul i ummarized in Figure 6.. The pah 6 and 7 inerec on he lef. The Coninuer wa run under a defaul parameer eing IniSepize =., MinSepize = 5, MaxSepize =.. If we increae he preciion eing IniSepize =., MinSepize = 6, MaxSepize =. he relevan branche do no inerec, a hown in Figure 6. on he righ. Figure 6. how zoom of he menioned branche compued wih defaul on he lef and he increaed on he righ preciion. The figure on he lef ugge ha he croing i a numerical arifac. We refer o he magena curve, i.e., he numerically compued branch 6. The cyan branch 7 no hown here exhibi imilar zig-zag ocillaion. The doed cyan line i a rimmed curve 7 wih removed ocillaion. Generically, parameer-dependen ingular value do no change he ordering excep if hey change he ign. In a forhcoming paper we will inveigae zero ingular value ubjec o perurbaion. Appendix A. Deail of Taylor expanion. We review leading erm of he expanion 4.3, , and he imperfec verion 5.8, Noe ha he compuaion of hee erm follow a rouine chain rule procedure indicaed a he end of Secion 4.. We ake advanage of he rucure of f x and higher parial differenial of f.

15 Ken Sae Univeriy hp://ena.mah.ken.edu 84 D. JANOVSKÁ AND V. JANOVSKÝ "magena" "magena" "cyan" FIGURE 6.. Zoom: The croing of he branche 6 and 7, hown in magena and cyan, on he lef of Figure 6. i acually a numerical arifac. If he preciion i increaed, hen he numerically compued branche do no cro. The Taylor coefficien depend on he following daa: u, v, δu, δv, A, A, A and higher derivaive of A, and on Z, Z and higher derivaive of Z. Noe ha Z i relaed o he unfolding 5.. Concerning 4.3: A. ϕ ξ = ϕ ξξ = = j ϕ, = =, j =,,..., ξj A. A.3 A.4 ϕ τ = δut A v u T A δv, ϕ ξτ = u T A v δu T A δv, ϕ ττ = δu T A v + u T A δv δu T A v τ + u τ T A δv, where A.5 uτ A = M v +, v τ A T u, M, i defined in 3.4. Concerning : A.6 ξ =, u ξ = δu, v ξ = δv, A.7 τ = u T A v, A.8 ξτ = ϕ τ, A.9 ξξ =, uτ = M v +, A v τ A T u, uξτ = A v ξτ M+, δv A T, δu uξξ = u v ξξ 4 v,

16 Ken Sae Univeriy hp://ena.mah.ken.edu ASVD: NON-GENERIC POINTS 85 A. ττ = u T A v + u T A v τ + u τ T A v, uττ v ττ A = M +, v [ A T u ] A M +, δv τ A T δu τ u T A v M +, [ A v A T u ], A. ξξξ =, uξξξ =. v ξξξ Moreover, A. ξ = ξξ = = j, =... =, j =,,.... ξj REMARK A.. The formulae A. and A. can be verified by inducion. Concerning : A.3 A.4 A.5 A.6 ϕ ε = δu T Z v + u T Z δv, ϕ ξε = u T Z v δu T Z δv, ϕ εε = δu T Z v ε u ε T Z δv, ϕ τε = δu T A v ε + δv T A T u ε δu T Z v + δv T Z T u δu T Z v τ + δv T Z T u τ, where u τ, v τ are defined by A.5, uε Z A.7 = M v +, v ε Z T u. A.8 ε = u T Z v, A.9 ξε = 4 u T Z δv + δu T Z v, A. A. τε = uξε = Z v ξε M+, δv Z T, δu u T A v ε + v T A T u ε + δu T Z v τ + v T Z T u τ + u T Z v,

17 Ken Sae Univeriy hp://ena.mah.ken.edu 86 D. JANOVSKÁ AND V. JANOVSKÝ A. uτε = M v +, A v ε τε A T u ε + τ M +, uε v ε [ M +, Z v τ Z T u τ M +, Z v Z T u ] v τ [ + ε M +, uτ ], A.3 εε = u T Z v ε + u ε T Z v, A.4 uεε = v ε M +, uε M εε v +, Z v ε ε Z T. u ε REFERENCES [] A. BUNSE-GERSTNER, R. BYERS, V. MEHRMANN, AND N. K. NICHOLS, Numerical compuaion of an analyic ingular value decompoiion of a marix valued funcion, Numer. Mah., 6 99, pp. 39. [] P. DEUFLHART AND A. HOHMANN, Numerical Analyi in Modern Scienific Compuing. An Inroducion, Springer, New York, 3. [3] A. DHOOGE, W. GOVAERTS, AND YU. A. KUZETSOV, MATCONT: A Malab package for numerical bifurcaion analyi of ODE, ACM Tran. Mah. Sofware, 3 3, pp [4] G. H. GOLUB AND C. F. VAN LOAN, Marix Compuaion, hird ed., The John Hopkin Univeriy Pre, Balimore, 996. [5] W. GOVAERTS, Numerical Mehod for Bifurcaion of Dynamical Equilibria, SIAM, Philadelphia,. [6] S. N. CHOW AND J. K. HALE, Mehod of Bifurcaion Theory, Springer, New York, 98. [7] V. JANOVSKÝ, D. JANOVSKÁ, AND K. TANABE, Compuing he Analyic Singular Value Decompoiion via a pahfollowing, in Proceeding of ENUMATH 5, A. B. de Caro, D. Gómez, P. Quinela, and P. Salgado, ed., Springer, New York, 6, pp [8] D. JANOVSKÁ AND V. JANOVSKÝ, On non-generic poin of he Analyic SVD, in Inernaional Conference on Numerical Analyi and Applied Mahemaic 6, T. E. Simo, G. Pihoyio, and Ch. Tioura, ed., WILEY-VCH, Weinheim, 6, pp [9] T. KATO, Perurbaion Theory for Linear Operaor, econd ed., Springer, New York, 976. [] S. KRANTZ AND H. PARKS, A Primer of Real Analyic Funcion. Birkhäuer, New York,. [] Marix Marke. Available a hp://mah.ni.gov/marixmarke/. [] V. MEHRMANN AND W. RATH, Numerical mehod for he compuaion of analyic ingular value decompoiion, Elecron. Tran. Numer. Anal., 993, pp hp://ena.mah.ken.edu/vol..993/pp7-88.dir/. [3] K. WRIGHT, Differenial equaion for he analyic ingular value decompoion of a marix. Numer. Mah., 63 99, pp