The Perturbation Bound for the Perron Vector of a Transition Probability Tensor

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1 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Liear Algebra Appl. ; : 6 Published olie i Wiley IterSciece DOI:./la The Perturbatio Boud for the Perro Vector of a Trasitio Probability Tesor We Li Lu-Bi Cui Michael K. Ng SUMMARY I this paper we study the perturbatio boud for the Perro vector of a m th -order -dimesioal trasitio probability tesor P = p i i...i m with p i i...i m ad i = p i i...i m =. The Perro vector x associated to the largest Z-eigevalue of P satisfies Px m = x where the etries x i of x are oegative ad i= x i =. The mai cotributio of this paper is to show that whe P is perturbed to a aother trasitio probability tesor P by P the -orm error betwee x ad x is bouded by m P ad the computable quatity related to the uiqueess coditio for the Perro vector x of P. Based o our aalysis we ca derive a ew perturbatio boud for the Perro vector of a trasitio probability matrix which refers to the case of m =. Numerical examples are preseted to illustrate the theoretical results of our perturbatio aalysis. Copyright c Joh Wiley & Sos Ltd. Received.... School of Mathematics South Chia Normal Uiversity Guagzhou 56 P. R. Chia. E- mail: liwe@scu.edu.c hzkc@6.com. Cetre for Mathematical Imagig ad Visio ad Departmet of Mathematics Hog Kog Baptist Uiversity Hog Kog. it : mg@math.hkbu.edu.hk.. INTRODUCTION Let R be the real field. We cosider a m th order -dimesioal tesor A cosistig of m etries ir: A =a i i...i m a i i i m R i i...i m. A is called o-egative or respectively positive if a i i i m or respectively a i i i m >. I the followig discussio we defie := { }. To a -dimesioal colum vector u=[u u u ] T R we defie a tesor-vector multiplicatio to be a -dimesioal colum vector as follows: A u := m a i i i m u i u im. i i m = i We deote a vector u is o-egative or respectively positive by u or respectively u>. I recet studies of umerical multi-liear algebra eigevalue problems for tesors have brought special attetio see e.g. [9 4 5]. Defiitio Let A be a m th -order -dimesioal tesor adcbe the set of all complex umbers. Assume that A x m is ot idetical to zero. We say λx C C \{} is a H-eigepair of A if A x m = λ x [m ] Copyright c Joh Wiley & Sos Ltd. Prepared usig laauth.cls [Versio: /5/ v.]

2 W. LI L.B. CUI AND M.K. NG where x [m ] :=[x m x m x m ] T. We sayλx C C \{} is a Z-eigepair of A if where x is the Euclidea orm of x. A x m = λ x with x = This defiitio was itroduced by Qi [5] whe m is eve ad A is symmetric. Idepedetly Lim [] gave such a defiitio but restricted x to be a real vector ad λ to be a real umber. I [] the Perro-Frobeius Theorem for o-egative matrices has bee geeralized to the class of o-egative tesors. Defiitio A m th -order -dimesioal tesor A is called reducible if there exists a oempty proper idex subset I such that a i i i m = i I i...i m / I. If A is ot reducible the we call A irreducible. Theorem [] If A is a m th -order -dimesioal irreducible o-egative tesor the there exist λ > ad x>such that A x m = λ x [m ]. 4 Moreover if λ is a eigevalue with a o-egative eigevector the λ = λ. If λ is a eigevalue of A the λ λ. I this paper we are iterested to study the perturbatio boud for the Perro vector of a m th - order -dimesioal trasitio probability tesor P =p i i...i m with p i i...i m ad p i i...i m =. 5 i = Trasitio probability tesors are a high-dimesioal aalog to trasitio probability matrices i Markov chai [7]. Such trasitio probability tesors arise i higher-order Markov chai [8 ] higher order coectivity i liked objects [6 4 9] ad higher-order graph matchig [5 7]. I [8] we have show the followig results for the largest Z-eigepair of P: Theorem If P is a m th -order -dimesioal trasitio probability tesor satisfyig 5 the there exists a ozero o-egative vector x such that Px m = x i= x i =. 6 I particular if P is irreducible the x must be positive. We ote that is the largest Z-eigevalue of P. Here x is called the Perro vector of P. We remark that the largest Z-eigevalue problem i 6 is differet from Defiitio where the Euclidea orm of x is equal to see [6]. The Z-eigevalue problem 6 is very importat to study the probability distributio of high-order Markov chais e.g. see [8]. Let S be a proper subset of S be its complemetary set i i.e. S = \S. For a give trasitio probability tesor P we defie δ m PS := mi p ii i m + mi p ii i m 7 i i m i S i i m i S Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

3 THE PERTURBATION BOUND FOR THE PERRON VECTOR ad δ m P := mi S δ mps. 8 With a suitable coditio o δ m P Li ad Ng showed that the Perro vector x i Theorem is uique. Theorem Suppose P is a m th -order -dimesioal trasitio probability tesor satisfyig 5. If δ m P> the x i Theorem is uique. m m Remark As i [8] a special kid of a m th -order -dimesioal trasitio probability tesor P =p i i...i m : p ii...i m p i j... j m < m ii...i m j... j m satisfy the required coditio i Theorem. The mai cotributio of this paper is to show that whe the trasitio probability tesor P is perturbed to a aother trasitio probability tesor P by P the -orm error betwee their Perro vectors x ad x is give by x x P m δ m P+ m. Here the -orms of a vector ad a m th order -dimesio tesor A are defied by ad x := A := max i= x i i...i m i = a i i...i m respectively. Accordig to our perturbatio aalysis we ca derive a ew perturbatio boud for the Perro vector of a trasitio probability matrix which refers to the case of m =. Numerical examples are preseted to illustrate the theoretical results of our perturbatio aalysis. The outlie of this paper is give as follows. I Sectio we preset our mai results. I Sectio we apply the results to study the perturbatio boud for the Perro vector of trasitio probability matrices. I Sectio 4 we show some umerical examples to validate our results. Fially cocludig remarks are give i Sectio 5.. THE PERTURBATION RESULTS I this sectio we will give perturbatio aalysis for the Perro vector of a m th -order -dimesioal trasitio probability tesor P =p i i...i m satisfyig 5. Let P = P+ P = p i i...i m be a perturbed trasitio probability tesor of P ad be the Perro vector of P i.e. x=[ x x x ] T P x m = x x x =. 9 I the followig discussio we give a boud for the differece betwee the origial eigevector ad the perturbed eigevector satisfyig 6 ad 9: x=[ x x x ] T = x x Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

4 4 W. LI L.B. CUI AND M.K. NG with Let x i = x i x i i. V ={i : x i > i } ad V be its complemetary set i i.e. V = \V. Here we assume that V / ad V / otherwise it ot ecessary to study the perturbatio boud. The reaso is that if V = / the x i for ay i. Sice x i = x i x i = i i i we must have x i = for ay i. The ext lemma is used to derive the perturbatio boud. Lemma Let V be oempty ad give by. The for ay k {... m} we have i V i...i m where δ m P is defied i 8. p i i...i m x i x ik x ik+ x im x ik δ m P x i Proof By the defiitio V ad V we obtai i V i...i m = { p i i...i m x i x ik x ik+...x im x ik i...i k i k+...i m i V { i...i k i k+...i m i V i...i k i k+...i m i...i k i k+...i m = i...i k i k+...i m i...i k i k+...i m max i V max p i i...i m x i x ik x ik+...x im } x ik p i i...i m x i x ik x ik+...x im } x ik p i i...i k...i m x i x ik x ik+...x im x ik mi i k p i V i...i k...i m x i x ik x ik+...x im x ik i V p i i...i k...i m i V x i x ik x ik+...x im x ik mi i k p i V i...i k...i m x i x ik x ik+...x im x ik i V Sice we obtai = i= x i = x i + x i x i = x i. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

5 THE PERTURBATION BOUND FOR THE PERRON VECTOR 5 Hece we have i V i...i m = i...i k i k+...i m max i...i k i k+...i m i...i k i k+...i m p i i...i m x i x ik x ik+...x im x ik max max i V i V p i i...i k...i m mi i k p i V i...i k...i m i V p i i...i k...i m mi i k p i V i...i k...i m i V x i x ik x ik+...x im x i which together with the followig equalities i= x i = i= x i = ad p i i...i m = i = x i x ik x ik+...x im x i gives = i V i...i m max i...i k i k+...i m max i...i k i k+...i m { p i i...i m x i x ik x ik+...x im x ik mi max i...i k i k+...i m δ m P x i i V p i i...i k...i m mi i V mi p i i...i k...i m mi i V i k V i V mi p i i...i k...i m + mi i V p i i...i k...i m x i i k V i V p i i...i k...i m x i } p i i...i k...i m x i which proves the lemma. Next we make use of a quatity to measure the magitude of P. Give a m th order - dimesio tesor A we defie where We ca establish the followig lemma. α m A:= max max α mas S i...i m α m AS := a i i...i m. i S Lemma Let P = p i i...i m be a perturbatio tesor of a trasitio probability tesor P. The we have α m P= P. Proof Assume that there are i...i m ad a subset U of such that α m P= i U p i i...i m. It is easy to see that U ={i : p ii...i m }. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

6 6 W. LI L.B. CUI AND M.K. NG Sice we have which implies that Therefore p i i...i m = i = = i = i = p i i...i m p i i...i m p i i...i m i = p i i...i m = = p i i...i m = p i i...i m p i i...i m = p i i...i m i = i U i U i U i U p ii...i m = α m P= max S i = p i i...i m P. max p ii...i m i...i m P. 4 O the other had there are i...i m such that P = i = p i i...i m. Let U = {i : p i i...i m }. By the first equality of we obtai P = p i i...i m max i U S which together with 4 gives. max p i i...i m =α m P i...i m i S It is oted that for ay trasitio probability tesor P we have α m P P. I particular if P is positive the α m P< P. Lemma Let x be a o-egative vector with x = ad S be a subset of. The we have Proof It is easy to see that Px m i i S Px m i i S P. 5 = The the result follows from 6 ad. i S i...i m Next we give the mai result i this paper. p ii...i m x i x im max p ii...i m i...i m i S i...i m x i x im α m P. 6 Theorem 4 Let P ad its perturbed tesor P = P+ P be m th -order -dimesioal trasitio probability tesors satisfyig 5. If δ m P> m m the the Perro vector x of P is uique ad for ay Perro vector x of P P x x m δ m P+ m 7 where δ m P is defied by 7. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

7 THE PERTURBATION BOUND FOR THE PERRON VECTOR 7 Proof By Theorem it is kow that x is uique. Let x= x x ad V be give by. If V is empty the x= this implies that the boud 7 holds. Hece we assume that V is oempty. By usig 9 it follows from Lemma that x+ x i i V = P x m i = i V i V = = i V i...i m i V i...i m i V i...i m + i V i...i m i V i...i m i V i...i m i...i m p i i...i m x i x im x im + p i i...i m x i x im x+ x im i V i...i m p i i...i m x i x im x im + δ m P x i p i i...i m x i x im x im x im p i i...i m x i x im x im p i i...i m x i x im x im x im + δ m P x i p i i...i m x i x im x im x im + δ m P x i i V p i i...i m x i x im +m δ m P x i. i V Sice p i i...i m x i x im i...i m = p i i...i m x i x im + p i i...i m x i x im i...i m i...i m = Px m i + x i we have It follows that x+ x i Px m i + x i +m δ m P x i. i V i V i V i V Accordig to Lemma we obtai which together with 8 gives m δm P x i Px m i. 8 i V i V Px m i i V P m+m δm P x i i V P. 9 Sice ad x = x i + x i x i = x i Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

8 8 W. LI L.B. CUI AND M.K. NG we have x i = x which together with 9 gives the desired boud. It is oted that P ca also be writte as the perturbed tesor of P i.e. P = P P. The by usig Theorem 4 we have the followig corollary. Corollary Let P ad its perturbed tesor P = P+ P be m th -order -dimesioal trasitio probability tesors satisfyig 5. If δ m P> m m the the Perro vector x of P is uique ad for ay Perro vector x of P P x x m δ m P+ m I the followig we provide several remarks about our perturbatio results. i The relative boud betwee x ad x is equal to the absolute oe because the -orms of x ad x are always equal to. ii Whe m= it has bee show i [8] that if { γp := mi S mi i mi i mi p i i i + mi i S i S i S i S mi p i i i S i + mi i S i S i S p i i i + p i i i }. is greater tha oe the P has the uique solutio x satisfyig P x = x. Also it has bee show that for ay trasitio probability tesor P we have γp δ P ad thus this gives a weaker uiqueess coditio: γp >. By usig similar argumet i Theorem 4 we ca establish the followig results: Theorem 5 Let P ad its perturbed tesor P = P+ P be th -order -dimesioal trasitio probability tesors satisfyig 5. If γ P > the the Perro vector x of P is uique ad for ay Perro vector x of P x x P γ P. Similarly P ca also be writte as the perturbed tesor of P by Theorem 5 we have the followig corollary. Corollary Let P ad its perturbed tesor P = P+ P be th -order -dimesioal trasitio probability tesors satisfyig 5. If γp > the the Perro vector x of P is uique ad for ay Perro vector x of P x x P γp. iii The bouds give i Theorems 4 ad 5 are sharp. For example let P ε be a th order - dimesio tesor give by + ε P ε = + ε ε ε ε + ε Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

9 THE PERTURBATION BOUND FOR THE PERRON VECTOR 9 where ε > ad P =. By a simple computatio γ P= 7 δ P= 7 6 ad P = P ε P = ε. By usig Theorem 4 or 5 we get x ε. A simple calculatio reveals that the equality holds whe ε =. Ideed the eigevector solutios are give by x= T ε ε ad x=/ / T ε ε respectively. Whe ε = / we have x x = ε =. Figure shows the relatioship betwee our boud ad the value of x x whe ε /..9.8 True Distace Our Boud ε Figure. The relatioship betwee our boud ad error x x.. THE RESULTS FOR TRANSITION PROBABILITY MATRICES I this sectio we apply the results i the previous sectio to a trasitio probability matrix P i.e. m=. We ote that a trasitio probability matrix correspod to a stadard Markov chai. Whe P is irreducible the Perro vector x with x = of the trasitio probability matrix P is uique ad it is called statioary probability distributio. I the literature there are some results for the perturbatio boud of the -orm of the Perro vector x of a trasitio probability matrix see for istace i [4]: I [8] it is give as follows: where Z :=I P+xe T ad e is a vector of all oes. I [] it is give as follows: wherei P # is the group iverse of I P. x Z P 4 x I P # P 5 Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

10 W. LI L.B. CUI AND M.K. NG I [9] it is give as follows: x where ηp := sup v =v T e= Pv. I [] it is give as follows: ηp P 6 x ηi P # P = ηz # P. 7 By usig the trick i Theorem 4 we obtai the followig result. Theorem 6 Let P ad its perturbed matrix P=P+ P be trasitio probability matrices. If δ P> the the Perro vector x with x = of P is uique ad for ay Perro vector x with x = of P we have α PS x x max P S δ PS δ P. 8 Proof Assume that there exists x ad y such that Px = x ad Py = y x = y =. Therefore we have z i = p i j z j j= where z =[ z z z ]=y x. Let Z = {i : z i > i }. It is clear that Z must be o-empty otherwise x=y. We ca compute It follows from Lemma that ad hece i Z j= z i = i Z i Z p i j z j. j= p i j z j δ P z j j Z δ P which cotradicts the assumptio. The Perro vector of P must be uique. Now let x be a o-egative vector satisfyig P x = x x = ad let x = [ x x x ]= x x. We ote that x i = j= p i j x j + Let V be a o-empty set give i. The we obtai x i = j= By usig the proof of Lemmas ad we have ad j= p i j x j + p i j x j. j= p i j x j α PV p i j x j δ PV x j j= j V p i j x j. 9 j= Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

11 THE PERTURBATION BOUND FOR THE PERRON VECTOR which together with 9 gives δ PV x i α PV. It is clear that δ PV δ P. By the assumptio of the theorem we have δ PV> ad thus which implies that x i α PV δ PV α P S x max S δ PS. The last iequality i 8 follows immediately from Lemma ad the fact that The theorem is proved. α P S max S δ PS max S α PS mi S δ PS. Here we remark that the coditio i Theorem 6 ad the irreducibility coditio do ot iclude each other. The followig example shows that P satisfies the coditio of Theorem 6 but P is reducible. Let P= The δ = 6 > but P is reducible. The followig example shows that P is irreducible but δ =. Let P=. The δ = but P is irreducible. I additio it may be very difficult to compute the bouds i 4 5 ad 6. For 4 we eed to compute the Perro vector x of P; for 5 we eed to compute the group iverse of I P; for 6 we eed to solve a optimizatio problem. However our boud ca be computed based o the etries of P oly. Although we caot show our boud is always better tha the other bouds i ad 7 the followig example shows that our boud is better: P= / / / / P= ad the bouds i ad our boud are ad.9 respectively. 4. NUMERICAL RESULTS I this sectio six trasitio probability tesors for differet applicatios cosidered i [8] are used to illustrate the results. By usig the Matlab multi-dimesioal array otatio these trasitio Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

12 W. LI L.B. CUI AND M.K. NG probability tesors are give by P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::= P::4= ; ; ; These trasitio probability tesors are positive ad hece are all irreducible. A simple computatio shows δ > / for Examples ad 4 ad δ 4 > / for Example. Hece the coditio i Theorem is satisfied for all examples ad the they have the uique ad positive Perro vector. Other two examples are give below: 5 P::= P::= P::= P::= P::= P::= For these two tesors their quatities δ are less tha /. But their quatities γ are greater tha. Therefore they have the uique ad positive Perro vector accordig to the remark of Theorem 5 i Sectio. I the test each etry of P is radomly geerated by a Gaussia distributio such that. i p i i i m = i i m. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la ;

13 THE PERTURBATION BOUND FOR THE PERRON VECTOR Example δ m P γp m δ m P+ m γp * *.79 Table I. Numerical results of Example -6. * meas the value does ot provide a valid upper boud. Therefore P is still a trasitio probability tesor uder perturbatio. The computed quatities i Examples -6 are show i Table I. For Example γp is ot valid as the order of P is 4. For Examples 5 ad 6 their δ values are less tha / ad therefore the boud i Corollary is ot valid. However the boud i Corollary ca be applied. I Figures -7 we show the chage of our bouds ad the true differece of the Perro vectors with respect to the chage of P. It is oted that the bouds are computed by Theorem 4 for 4 th order tesor ad by Theorem 5 for th order tesor respectively ad their Perro vectors are give by the algorithm i [8]. The perturbatio bouds i our results have liear correlatio with P. From Figures ad 6 it is see that our perturbatio bouds are quite close to the actual differeces for Examples ad 5..5 x The differece betwee our boud ad true boud 8 Our boud Ture boud P x 8 Figure. The differece betwee our boud ad true boud of Example. 5. CONCLUDING REMARKS I summary we have studied the perturbatio boud for the Perro vector of a m th -order - dimesioal trasitio probability tesor P =p i i...i m with p i i...i m ad i = p i i...i m =. We have show that the -orm error betwee x ad the perturbed eigevector x is bouded by a quatity that depeds o liearly o P. Our experimetal results have illustrated the theoretical results of this perturbatio aalysis. REFERENCES. A. Berma ad R. Plemmos Noegative Matrices i the Mathematical Scieces SIAM press K. C. Chag K. Pearso ad T. Zhag Perro-Frobeius theorem for oegative tesors Commu. Math. Sci Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

14 4 W. LI L.B. CUI AND M.K. NG.5 x The differece betwee our boud ad true boud 8 Our boud Ture boud P x 8 Figure. The differece betwee our boud ad true boud of Example. x The differece betwee our boud ad true boud 7 Our boud.9 Ture boud P..4.6 x 8 Figure 4. The differece betwee our boud ad true boud of Example. 5 x The differece betwee our boud ad true boud Our boud Ture boud P x 8 Figure 5. The differece betwee our boud ad true boud of Example 4.. W. Chig ad M. Ng Markov Chais: Models Algorithms ad Applicatios Iteratioal Series o Operatios Research ad Maagemet Sciece Spriger 6. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

15 THE PERTURBATION BOUND FOR THE PERRON VECTOR 5 9 x The differece betwee our boud ad true boud Our boud Ture boud P x 8 Figure 6. The differece betwee our boud ad true boud of Example 5. x The differece betwee our boud ad true boud 7 Our boud.8 Ture boud P x 8 Figure 7. The differece betwee our boud ad true boud of Example G. E. Cho ad C. D. Meyer Compariso of perturbatio bouds for the statioary distributio of a Markov chai Liear Algebra Appl O. Duchee F. Bach I. Kweo ad J. Poce A tesor-based algorithm for high-order graph matchig CVPR D. Galway ad I. Park TOPDIS: Tesor-based Rak- ig for Data Search ad Navigatio J. Lee M. Cho ad K. Lee Hypergraph matchig via reweighted radom walks CVPR. 8. W. Li ad M. Ng O the limitig probability distributio of a trasitio probability tesor preprit o November mg/tesor-research/highermarkov.pdf 9. X. Li M. Ng ad Y. Ye HAR: hub authority ad relevace scores i multi-relatioal data for query search The SIAM Iteratioal Coferece o Data Miig.. L. H. Lim Sigular values ad eigevalues of tesors: a variatioal approach Proceedigs of the IEEE Iteratioal Workshop o Computatioal Advaces i Multi-Sesor Adaptive Processig CAMSAP L. H. Lim Multiliear pagerak: measurig higher order coectivity i liked objects The Iteret: Today ad Tomorrow July 5.. C. D. Meyer The coditio of a fiite Markov chai ad perturbatio bouds for the limitig probabilities SIAM J. Alg. Disc. Meth M. Ng L. Qi ad G. Zhou Fidig the largest eigevalue of a o-egative tesor SIAM J. Matrix Aal. Appl M. Ng X. Li ad Y. Ye MultiRak: co-rakig for objects ad relatios i multi-relatioal data The 7th ACM SIGKDD Coferece o Kowledge Discovery ad Data Miig KDD- August -4 Sa Diego CA. 5. L. Qi Eigevalues of a real supersymmetric tesor Joural of Symbolic Computatio L. Qi Eigevalues ad ivariats of tesor Joural of Mathematical Aalysis ad Applicatios S. Ross Itroductio to Probability Models Academic Press. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la

16 6 W. LI L.B. CUI AND M.K. NG 8. P. Schweitzer Perturbatio theory ad fiite Markov chais J. Appl. Probability E. Seeta Perturbatio of the statioary distributio measured by ergodicity coefficiets Adv. Appl. Pro E. Seeta Sesitivity aalysis ergodicity coefficiets ad rak-oe updates for fiite Markov chais i W.J. Stewart ed. Numerical Solutio of Markov Chais Marcel Dekker NY J. Su Matrix perturbatio aalysis i Chiese Sciece press. Copyright c Joh Wiley & Sos Ltd. Numer. Liear Algebra Appl. Prepared usig laauth.cls DOI:./la