BALANCING REGULAR MATRIX PENCILS

Size: px
Start display at page:

Download "BALANCING REGULAR MATRIX PENCILS"

Transcription

1 BALANCING REGULAR MATRIX PENCILS DAMIEN LEMONNIER AND PAUL VAN DOOREN Abstract. In this paper we present a new diagona baancing technique for reguar matrix pencis λb A, which aims at reducing the sensitivity of the corresponding generaized eigenvaues. It is inspired from the baancing technique of a square matrix A and has a comparabe compexity. Upon convergence, the diagonay scaed penci has row and coumn norms that are baanced in a precise sense. We aso show that baancing a penci bois down to making it coser to some standardized norma penci. We give numerica exampes iustrating that the sensitivity of generaized eigenvaues of a penci may significanty improve after baancing. Key words. generaized eigenvaues, norma pencis, baancing AMS subject cassifications. 15A18, 15A22, 65F15, 65F35 1. Introduction. A matrix A with a norm that is severa orders of magnitude arger than the moduus of its eigenvaues typicay has eigenvaues that are sensitive to perturbations in the entries of A. It is shown in [4] that the Frobenius norm of a matrix can then often be reduced via a diagona scaing of the type D 1 AD. Such a scaing can be performed in exact arithmetic if the diagona eements are constrained to be integer powers of the base of the finite precision arithmetic (typicay 2 or 10). As a consequence the eigenvaues do not change, but their sensitivity can significanty be reduced. Such a diagona scaing is therefore typicay used before running any eigenvaue agorithm. In this paper we introduce a simiar scaing method for square pencis λb A with a determinant det(λb A) that is not identicay zero for a vaues of λ. For such pencis which are caed reguar one can define generaized eigenvaues via the zeros of the poynomia det(λb A). Our scaing method can be viewed as a natura extension of the baancing agorithm of [4] to reguar matrix pencis and is aimed at reducing the sensitivity of the generaized eigenvaues of the penci. This new method differs from that of Ward [7] whose aim it is to make the penci entries have magnitudes as cose to unity as possibe, whereas our aim is to make the penci as cose as possibe to some standardized norma penci. We first reca the cassica baancing method for matrices and some of its properties. We then introduce the new baancing method for pencis and derive its anaogous properties. We briefy discuss the compexity of the agorithm and finay give some numerica resuts iustrating the performance of the new scaing method. 2. Norma matrices and baancing. Norma matrices are known to have orthogona eigenvectors and hence we conditioned eigenvaues [4]. Therefore if one has to compute eigenvaues of an arbitrary n n matrix A, it is recommended to make it coser to a norma matrix by an error free transformation. Diagona scaing transformations with positive diagona eements that are integer powers of the base can be performed exacty since they ony amount to integer changes in the exponents of the matrix entries. And in order to preserve the eigenvaues one performs diagona simiarities D 1 AD. This paper presents research resuts of the Begian Programme on Interuniversity Attraction Poes, initiated by the Begian Federa Science Poicy Office. The scientific responsibiity rests with its authors. Univ. Cathoique de Louvain, B-1348 Louvain-a-Neuve, Begium (emonnier@csam.uc.ac.be). Univ. Cathoique de Louvain, B-1348 Louvain-a-Neuve, Begium (vdooren@csam.uc.ac.be) 1

2 2 D. LEMONNIER AND P. VAN DOOREN The basic question is thus how to characterize a diagona scaing D 1 AD that makes a matrix coser to a norma matrix. For this we consider two equivaent characterizations of norma matrices. A matrix A is norma iff (1) A has orthogona eigenvectors or, equivaenty, its Schur form A S : A S := U AU, U U = I n (2.1) is a diagona matrix Λ A (2) the so-caed defect from normaity γ(a) := σi 2 λ i 2 (2.2) is zero, where σ i and λ i are the singuar vaues and the eigenvaues of A, respectivey. The defect from normaity γ(a) is aways non-negative [2], which easiy foows from (2.1) and the fact γ(a) = γ(a S ), since unitary simiarities do not change the eigenvaues nor the singuar vaues of a matrix. Let Orb(A) denote the orbit of A, i.e. the set of matrices simiar to A. Then γ(a) is the minimum squared distance between A and any norma matrix simiar to A. Theorem 2.1. The optimization probem T T 1 AT F (2.3) has a norma matrix N a in the cosure of the orbit of A as soution. If A is diagonaizabe then there exists a bounded T such that N a = T 1 AT, otherwise T is unbounded. Proof. Use the (compex) Schur decomposition A S = U AU and choose a unitary matrix Q such that the matrix R := U T Q is trianguar. Since unitary transformations do not change the Frobenius norm, the above minimization is then equivaent to R R 1 A S R F, which has the diagona part Λ A of A S as soution. The transformation matrix R wi be bounded for a diagonaizabe matrix A, and it wi be unbounded otherwise (see aso [5] for more detais). It then foows that n σ2 i is the Frobenius norm squared of A S and therefore aso the sum of the entries squared of A S, whie n λ i 2 is just the sum of the diagona entries squared of A S. A diagona scaing D 1 AD, on the other hand, does not change the λ i s but does modify the σ i s. So one can reduce the gap γ by scaing A in order to diminish its Frobenius norm. This is exacty what the baancing agorithm [4] does : it soves D D 1 AD F. (2.4) Let e i denote the i-th unit vector, then it is shown in [4] that the optima scaing is achieved when D 1 AD satisfies (D 1 AD)e i 2 2 = e T i (D 1 AD) 2 2 i = 1... n (2.5)

3 BALANCING REGULAR MATRIX PENCILS 3 and an agorithm is provided for computing an approximate soution D with eements that are powers of the base of the finite precision arithmetic. Each step of that agorithm decreases the Frobenius norm of the scaed matrix and hence aso the distance to the norma matrices with the same spectrum as A. The aim of this paper is to generaize these baancing ideas to reguar matrix pencis (λb A). In other words we wi try to answer the foowing questions : (1) what is the property of reguar pencis that is equivaent to normaity in the standard eigenvaue probem, and (2) how to scae an arbitrary penci so that it gets as cose as possibe to achieving this property? 3. Norma pencis. We first reca a definition of norma pencis, given in [1]. Definition 3.1. An n n compex reguar penci λb A is said to be norma if it has orthogona right and eft eigenvectors, i.e. if it has a decomposition of the form U (λb A)U r = λλ B Λ A, where U, U r are unitary, and Λ A, Λ B are diagona. In order to reate this to a defect we reca the definition of generaized singuar vaues of two square matrices A and B. Definition 3.2. The right (resp. eft) singuar vaues σ ri (resp. σ i ) of λb A are defined to be the generaized eigenvaues of λb T B A T A (resp. λbb T AA T ). Since the invertibiity of B is not essentia in these definitions, we first make the simpifying assumption that B is invertibe. It then foows easiy that σ ri = σ i (AB 1 ), σ i = σ i (B 1 A). When B is invertibe, it is shown in [1] that the penci λb A is norma iff both AB 1 and B 1 A are norma. A good candidate for the defect from normaity of a reguar penci λb A appears then to be Γ(A, B) := σri 2 + σi 2 2 λ i 2 where λ i are the generaized eigenvaues of the penci. Ceary Γ(A, B) = γ(ab 1 ) + γ(b 1 A), which is aways positive and is zero iff both AB 1 and B 1 A are norma, and hence iff the penci λb A is norma. If B is not invertibe, we need another defect from normaity function since Γ(A, B) is then the difference between two inite quantities. We can then consider a transformed penci λ B Â := λ(cb sa) (sb + ca), c2 + s 2 = 1. (3.1) It is we-known (see e.g. [1]) that for a reguar penci λb A there aways exists a choice (c, s) for which B is invertibe. Since the above transformation does not affect the eft and right eigenvectors of a penci, it foows that λ B Â is norma λb A is norma. Rather than minimizing Γ(A, B) one can thus minimize Γ(Â, B) which wi reach a minimum when both λ B Â and λb A are norma pencis. Notice however that the vaue of this defect then changes even though normaity is preserved. Without oss of generaity, we assume from now on that B is invertibe.

4 4 D. LEMONNIER AND P. VAN DOOREN But orthogonaity of the eft and right eigenvectors is not sufficient to guarantee a ow sensitivity of the generaized eigenvaues of a reguar penci because eigenvaues can now be arbitrariy arge or sma, irrespective of the norm of A and B. Let x i and y i be respectivey the right and eft eigenvectors of a given eigenvaue λ i : Ax i = λ i Bx i, y i A = λ i y i B, and define the corresponding Rayeigh components : α i := y i Ax i /( y i 2 x i 2 ), β i := y i Bx i /( y i 2 x i 2 ), λ i = α i /β i. In [6] it is shown that a perturbation in A and B of reative size ɛ : δa 2 ɛ A 2, δb 2 ɛ B 2, (3.2) may yied a perturbed eigenvaue λ i, but that the chorda distance : χ(λ i, λ i ) := α i βi α i β i αi 2 + β i 2 α i 2 + β i 2 (3.3) between the origina and the perturbed eigenvaue is bounded by : χ(λ i, λ i ) ɛ ( A B 2 2) 1/2 ( α i 2 + β i 2 ) 1/2 + O(ɛ 2 ), and that there exist perturbations δa and δb, for which this bound is met. quantity The κ(λ i ) := ( A B 2 2) 1/2, (3.4) ( α i 2 + β i 2 ) 1/2 is thus a vaid reative condition number for λ i in the sense that it measures how a perturbation of reative size ɛ in A and B affects λ i in the (intrinsicay reative) chorda metric. The reason why such a reative metric is to be preferred for pencis is inked to the fact that eigenvaues are now given by ratios of computed quantities (see [6] for more detais). When using the QZ-agorithm to compute the generaized eigenvaues of the penci λb A one obtains the so-caed Schur form of this penci : A S := Q AZ, B S := Q BZ, Q Q = I n, Z Z = I n, (3.5) where A S and B S are both upper trianguar. This agorithm typicay induces errors δa and δb in A and B that are of the order of (3.2), where ɛ is the machine accuracy of the computer. Since the orthogona transformations Q and Z do not affect the quantities used in the definition (3.4) of κ(λ i ), we can as we anayze the effect of perturbations in the coordinate system of the Schur form. The right and eft eigenvectors x i, y i can then be normaized as foows : 0 x i := ξ 1.. ξ i , y i :=. 0 1 η i+1. η n.

5 BALANCING REGULAR MATRIX PENCILS 5 If we denote the diagona entries of the trianguar matrices A S, B S by a ii, b ii, respectivey, we then obtain the equaities : Since α i = a ii /n i, β i = b ii /n i, n i := ( y i 2 x i 2 ) 1. A 2 = A S 2 max i we finay obtain the inequaity a ii, B 2 = B S 2 max b ii i κ(λ i ) (max i a ii 2 + max i b ii 2 ) 1/2 ( a ii 2 + b ii 2 ) 1/2 (3.6) with equaity hoding ony for a norma penci since then x i and y i have norm 1 and A S and B S are diagona. But norma pencis can sti have a quantity κ(λ i ) that can be very arge if the pairs (a ii, b ii ) vary a ot in norm. This is not the case for the foowing subcass of norma pencis. Definition 3.3. A reguar penci λb A is standard norma if there exist unitary transformations U, U r and diagona matrices Λ A, Λ B, such that for some rea γ 0 U (λb A)U r = λλ B Λ A, Λ A 2 + Λ B 2 = γ 2 I. For this cass of pencis we obviousy have 1 κ(λ i ) = (max i a ii 2 + max i b ii 2 ) 1/2 ( a ii 2 + b ii 2 ) 1/2 2, (3.7) with the ower bound κ(λ i ) = 1 met for each i in the particuar case where Λ A = αi and Λ B = βi. Obviousy the cass of standard norma pencis is neary optima in terms of eigenvaue sensitivity since κ(λ i ) 2 for each eigenvaue λ i. The foowing theorem expains which pencis can be transformed to standard norma form using eft and right transformations. Theorem 3.4. Every reguar penci with a fu set of right and eft eigenvectors can be transformed into a standard norma form T 1 (λb A)T r = λλ B Λ A, Λ A 2 + Λ B 2 = γ 2 I with γ R 0. Proof. If λb A has a fu set of right and eft eigenvectors x i, y i, then putting x i as the coumns of T r and yi as the rows of T 1 wi diagonaize T 1 (λb A)T r. A simpe additiona diagona scaing which can be absorbed in either T r or T wi ensure that moreover Λ A 2 + Λ B 2 = γ 2 I, for some arbitrary rea positive γ. Remark 3.1. For non diagonaizabe (reguar) pencis, the theorem remains vaid in the imit, but then T, T r are unbounded. In this case we have that λλ B Λ A beongs to the cosure of the orbit of λb A under eft and right transformations T, T r [5]. 4. Baancing pencis. We now ook for scaing transformations that make a given penci get coser to a norma one. We coud use a scaing of the type D 1 (λb A)D r (4.1)

6 6 D. LEMONNIER AND P. VAN DOOREN where D r, D are rea positive diagona scaing matrices. This does not modify the generaized eigenvaues of the penci, but the defect from normaity Γ(A, B) becomes now Γ(D 1 AD r, D 1 BD r ) = σ 2 i (D 1 AB 1 D ) + σ 2 i (D 1 r B 1 AD r ) 2 It foows from Section 2 that the optima D r, D are soutions of λ i 2. D r D 1 r B 1 AD r F, D D 1 AB 1 D F. But such an approach woud require to invert the matrix B (at east impicity) and it is uncear how to proceed when B is singuar. We now define a new optimization probem inspired from Theorem 3.4 that avoids the inversion of B. It uses the so-caed Frobenius inner product for reguar pencis defined in a 2n 2 -dimensiona space of two n n compex matrices : λb 1 A 1, λb 2 A 2 F := tr(a 1 A 2 + B 1 B 2). It foows then that λb A 2 F := λb A, λb A F = A 2 F + B 2 F where. F denotes the usua Frobenius matrix norm. Theorem 4.1. The optimization probem det(t 1 T r )=1 T 1 (λb A)T r F (4.2) has a standard norma penci as soution. If λb A is diagonaizabe then T r, T have a bounded soution, otherwise they are unbounded. Proof. Using the Schur decomposition λb S A S = Q (λb A)Z we define trianguar matrices R r := Z T r Q r and R := Q T Q where Q r are Q are chosen to be unitary and det Q Q r = 1. Since unitary transformations do not change the Frobenius norm, the above minimization is then equivaent to det(r 1 R r)=1 R 1 (λb S A S )R r F where now a matrices are upper trianguar. Moreover, if we factor R r = D r U r and R = D U, where U r and U are unit upper trianguar and D r and D are diagona, then the probem spits in two subprobems. Ceary U r and U ony affect the eements above the diagona of R 1 (λb S A S )R r F and these can a be put equa to zero if the penci is diagonaizabe (e.g. when there are no repeated eigenvaues). In such a case the probem reduces further to det(d 1 D r )=1 D 1 (λλ B Λ A )D r F which is easiy soved using a Lagrange mutipier approach. The soution D 2 D 2 r(λ BΛ B + Λ AΛ A ) = γ 2 I, γ 2n = det(λ BΛ B + Λ AΛ A ) is equivaent to the condition that D 1 (λλ B Λ A )D r is a standard norma penci. If the penci is not diagonaizabe, it is sti possibe to find unbounded diagona scaings D r, D that wi make the eements that are above the diagona in the Schur form tend to zero.

7 BALANCING REGULAR MATRIX PENCILS 7 The above theorem suggests to use the same minimization probem but now restricted to rea positive diagona scaing matrices : det(d 1 D r)=1 D 1 (λb A)D r F (4.3) as a technique to baance reguar pencis. We wi show that this has a unique minimum that is attained when (D 1 AD r )e j (D 1 BD r )e j 2 2 = e T i (D 1 AD r ) e T i (D 1 BD r ) 2 2 = γ 2 for a i s and j s. This eads to the foowing generaization of (2.5). if Definition 4.2. An n n reguar compex penci λb A is said to be baanced Ae j Be j 2 2 = e T i A e T i B 2 2 = γ 2, i, j. (4.4) The foowing theorem proves that every baanced penci can be seen as the soution of an optimization probem very simiar to (4.3). Theorem 4.3. A reguar penci D 1 (λb A)D r with rea positive diagona scaings D, D r, is baanced if and ony if it is a soution of det(d 1 D r )=c D 1 (λb A)D r F. Proof. Denote the i-th diagona entry of D r and D by d ri and d i, respectivey, and et a ij, b ij be the entries of the matrices A, B. We want to minimize d i, d rj i,j=1 ( a ij 2 + b ij 2 )( d rj ) 2 dk, where ( d i drk ) 2 = c 2. With the change variabes d 2 ri = exp(u ri) and d 2 i = exp( u i) and when putting m ij := a ij 2 + b ij 2, this becomes u i, u rj i,j=1 m ij exp(u i + u rj ), where k (u k + u rk ) = 2 n c. This is a convex minimization probem with a inear constraint. Its soution can be found via the use of a Lagrange mutipier Γ : ( m ij exp(u i + u rj ) + Γ 2 n c ) (u k + u rk ). u i, u rj i,j=1 k This unconstrained minimization has therefore a minimum iff the first order conditions are satisfied. These are k (u k + u rk ) = 2 n c and m ij exp(u i + u rj ) = m ij exp(u i + u rj ) = Γ, i, j. j=1

8 8 D. LEMONNIER AND P. VAN DOOREN Putting Γ = γ 2 and rephrasing it in the origina variabes, this amounts to e T i (D 1 AD r ) e T i (D 1 BD r ) 2 2 = (D 1 AD r )e j (D 1 BD r )e j 2 2 = γ 2 for a i, j. The optima penci D 1 (λb A)D r is therefore baanced. The converse statement is easiy checked in a simiar manner. Remark 4.1. Notice that if the penci λb A can be permuted to a bock trianguar penci, then so can the matrix M with eements m ij. One then easiy checks that the scaings of the above theorem can be unbounded for this so-caed reducibe case. This case is typicay excuded in the scaing probem, since then the generaized eigenvaue probem can de defated to smaer dimensiona ones [7]. When such permutations do not exist, the scaing probem has a bounded soution. Remark 4.2. The above theorem does not prove that the diagona scaing procedure wi aways improve the sensitivity of the eigenvaue probem but the bound (3.4) for κ(λ i ) suggests that this wi be the case. We wi iustrate by numerica experiments that the scaing typicay improves the sensitivity of the eigenvaues. Remark 4.3. The above theorem aso aows to choose the parameter γ in (4.4) since modifying the constant c in the condition det(d 1 D r ) = c, automaticay scaes a the coumn and row norms. This is used in the numerica method described beow. 5. Numerica method. In order to baance a penci, we wi use a very simpe method rather than using convex optimization techniques. This method [ consists ] in A aternativey updating D r and D such that the compound matrices D B r and D 1 [ ] A B have coumn norms and row norms equa to 1, respectivey. By doing so we converge ineary to a baanced penci with γ = 1 in (4.4). The proposed method is essentiay a coordinate descent method where one aternates between computing the optimum in the coordinates of D r and D. The convergence is sow but when we restrict ourseves to powers of the base (2 or 10) for the diagona eements of D r and D, stagnation typicay occurs after two or three updates of both D r and D. Each joint update of D and Dr in fact requires ony 4n 2 foating point operations if one uses the matrix M with eements m ij := a ij 2 + b ij 2 : 2n 2 to compute the row and coumn norms and 2n 2 to perform the two scaings. (A MATLAB code is given in the Appendix for the base 2). The scaing procedure has therefore a margina cost in comparison to the eigenvaue computation. As in the standard eigenvaue probem one has to test aso if there exist permutations that reduce the penci to a bock trianguar form so that ower dimensiona eigenvaue probems can be isoated. Such a procedure is needed to guarantee that the diagona scaing wi remain bounded but the compexity is aso quadratic in n (see [7]). 6. Numerica exampes. In the tabes beow we have compared the precision of the computed eigenvaues without scaing, after appying our proposed scaing procedure and after appying Ward s method [7], which is currenty impemented in LAPACK. We considered in Tabe 6.1 randomy generated diagonaizabe pencis T 1 (λλ B Λ A )T r (where λλ B Λ A is in standard norma form), in Tabe 6.2 randomy generated non diagonaizabe pencis T 1 (λj B J A )T r (where J 1 B J A is in Jordan norma form), and in Tabe 6.3 pencis with eements of strongy varying order of magnitude. We used normay distributed random numbers for the free eements of Λ A, Λ B, J A and J B. We imposed the normaization in λλ B Λ A by choosing Λ B to satisfy Λ 2 B + Λ2 A = γ2 I, and the Jordan structure in λj B J A by choosing some repeated consecutive eements on the diagonas of J A and J B and assigning

9 BALANCING REGULAR MATRIX PENCILS 9 corresponding off-diagona 1 s in J A. The condition number of the random matrices T and T r was controed by taking the k-th power of normay distributed random numbers r i,j as their eements. A arger power k then typicay yieds a arger condition number for the transformation. For these experiments we used the QZ-agorithm [3] appied to different pencis of size n = 10. We computed the chorda distances c i := χ(λ i, λ i ) for a eigenvaues λ i and compared in each tabe the quantities c := [c 1,..., c n ] 2 for the origina penci (c orig ), for the baanced penci constructed by our agorithm (c ba ), and for the baanced penci using Ward s method (c ward ). In Tabes 6.1 and 6.2 we aso give the condition numbers κ(t r ) and κ(t ). In Tabe 6.1 we focus on diagonaizabe pencis. When κ(t r ) = κ(t ) = 1 we observe that baancing does not improve the precision of the cacuated eigenvaues, but otherwise it does in genera significanty improve the accuracy of the cacuated eigenvaues. Reca aso that we restrict the diagona eements of the baancing transformations D r, D to be powers of two. From the tabe it appears that the proposed baancing agorithm has a positive effect on the precision of the computed eigenvaues. The comparison factor c ward /c ba shows that in genera the new method outperforms Ward s agorithm. Tabe 6.1 Comparison for randomy generated diagonaizabe pencis n = 10 c orig c ba c ward c ward /c ba κ(t r ) = 3.27e + 07, κ(t ) = 2.58e e e e κ(t r ) = 8.24e + 12, κ(t ) = 4.21e e e e e+02 κ(t r ) = 6.81e + 08, κ(t ) = 1.75e e e e e+02 κ(t r ) = 1.06e + 07, κ(t ) = 7.82e e e e e-01 κ(t r ) = 1.46e + 05, κ(t ) = 4.08e e e e e+00 κ(t r ) = 1.92e + 03, κ(t ) = 7.72e e e e e+00 κ(t r ) = 3.95e + 01, κ(t ) = 1.75e e e e e+00 κ(t r ) = 1.00e + 00, κ(t ) = 1.00e e e e e+02 In Tabe 6.2 we ook at non-diagonaizabe pencis. We imposed the first exampe to have two Jordan bocks of size 2, and the second exampe to have one Jordan bock of size 3. The eigenvaue sensitivity is in principe inite and the cacuated eigenvaues have itte in common with the true eigenvaues. The tabe shows that both baancing agorithms do not ater the precision of the computed eigenvaues. In Tabe 6.2 Randomy generated non-diagonaizabe pencis n = 10 c orig c ba c ward c ward /c ba κ(t r ) = 1.15e + 09, κ(t ) = 3.27e e e e e+00 κ(t r ) = 4.68e + 02, κ(t ) = 4.79e e e e e+00 Tabe 6.3 we ook at pencis with entries of strongy varying size : the argest ones are of the order of 1, the smaest ones are much smaer. Ward s method tries to make the size of these eements equa and in doing so, it appies a scaing that often

10 10 D. LEMONNIER AND P. VAN DOOREN deteriorates the sensitivity instead of improving it. The new method, on the other hand, usuay significanty improves the sensitivity. Tabe 6.3 Pencis with eements of strongy varying order of magnitude c orig c ba c ward c ward /c ba 4.38e e e e e e e e e e e e Acknowedgment. We woud ike to acknowedge the hep of Yurii Nesterov in Theorem 4.3 who pointed out that this was a convex optimization probem. We aso thank Danie Kressner who sent us a 3 by 3 exampe from his thesis for which Ward s scaing significanty deteriorates the sensitivity of the computed eigenvaues. The exampes in Tabe 6.3 are inspired from this. 8. Concusion. In this paper we presented a new baancing method for matrix pencis. From the point of view of the sensitivity of the eigenvaues we showed that the standard norma pencis are near optima and that they can be viewed as a natura extension of norma matrices. A diagona baancing method was then proposed that makes a given penci as cose as possibe to a standard norma one. Moreover we showed that the compexity of the new method is comparabe to that of the cassica baancing of matrices. We aso gave numerica evidence that the accuracy of computed generaized eigenvaues may significanty improve after baancing a penci and that the method often outperforms the method of Ward impemented in LAPACK. REFERENCES [1] J.-P. Charier and P. Van Dooren. A Jacobi-ike agorithm for computing the generaized Schur form of a reguar penci. J. Comput. App. Math., 27(1-2):17 36, Reprinted in Parae agorithms for numerica inear agebra, 17 36, North-Hoand, Amsterdam, [2] R. A. Horn and C. R. Johnson. Topics in matrix anaysis. Cambridge University Press, Cambridge, [3] C. B. Moer and G. W. Stewart. An agorithm for generaized matrix eigenvaue probems. SIAM J. Numer. Ana., 10: , [4] B. N. Parett and C. Reinsch. Baancing a matrix for cacuation of eigenvaues and eigenvectors. Numer. Math., 13: , [5] A. Pokrzywa. On perturbations and the equivaence orbit of a matrix penci.. Linear Agebra App. 82:99 121, [6] G. W. Stewart. Perturbation theory for the generaized eigenvaue probem. In Recent Advances in Numerica Anaysis, Eds. de Boor C. and Goub G., Academic Press, New York, 1978, [7] R. C. Ward. Baancing the generaized eigenvaue probem. SIAM J. Sci. Stat. Comput., 2: , 1981.

11 BALANCING REGULAR MATRIX PENCILS 11 Appendix. function [D, Dr, iter] = baeig(a,b,max_iter) % Performs two-sided scaing D\A*Dr, D\B*Dr in order to improve % the sensitivity of generaized eigenvaues. The diagona matrices % D and Dr are constrained to powers of 2 and are computed iterativey % unti the number of iterations max_iter is met or unti the norms are % between 1/2 and 2. Convergence is often reached after 2 or 3 steps. % The diagonas of the scaing matrices are returned in D and Dr % and so is iter, the number of iterations steps used by the method. n=size(a,1); D=ones(1,n); Dr=ones(1,n); M=abs(A).^2+abs(B).^2; for iter=1:max_iter, emax=0;emin=0; for :n; % scae the rows of M to have approximate row sum 1 d=sum(m(i,:));e=-round(og2(abs(d))/2); M(i,:)=pow2(M(i,:),2*e); % appy the square root scaing aso to A, B and D D(i)=pow2(D(i),-e); if e > emax, emax=e; end; if e < emin, emin=e; end end for :n; % scae the coumns of M to have approximate coumn sum 1 d=sum(m(:,i));e=-round(og2(abs(d))/2); M(:,i)=pow2(M(:,i),2*e); % appy the square root scaing aso to A, B and Dr Dr(i)=pow2(Dr(i),e); if e > emax, emax=e; end; if e < emin, emin=e; end end % Stop if norms are a between 1/2 and 2 if (emax<=emin+2), break; end end

Lecture Note 3: Stationary Iterative Methods

Lecture Note 3: Stationary Iterative Methods MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or

More information

An explicit Jordan Decomposition of Companion matrices

An explicit Jordan Decomposition of Companion matrices An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC 88040-900 Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois 31057

More information

FRST Multivariate Statistics. Multivariate Discriminant Analysis (MDA)

FRST Multivariate Statistics. Multivariate Discriminant Analysis (MDA) 1 FRST 531 -- Mutivariate Statistics Mutivariate Discriminant Anaysis (MDA) Purpose: 1. To predict which group (Y) an observation beongs to based on the characteristics of p predictor (X) variabes, using

More information

CS229 Lecture notes. Andrew Ng

CS229 Lecture notes. Andrew Ng CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view

More information

First-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries

First-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische

More information

Problem set 6 The Perron Frobenius theorem.

Problem set 6 The Perron Frobenius theorem. Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator

More information

SUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS

SUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS ISEE 1 SUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS By Yingying Fan and Jinchi Lv University of Southern Caifornia This Suppementary Materia

More information

Partial permutation decoding for MacDonald codes

Partial permutation decoding for MacDonald codes Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics

More information

Algorithms to solve massively under-defined systems of multivariate quadratic equations

Algorithms to solve massively under-defined systems of multivariate quadratic equations Agorithms to sove massivey under-defined systems of mutivariate quadratic equations Yasufumi Hashimoto Abstract It is we known that the probem to sove a set of randomy chosen mutivariate quadratic equations

More information

THE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES

THE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES THE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES by Michae Neumann Department of Mathematics, University of Connecticut, Storrs, CT 06269 3009 and Ronad J. Stern Department of Mathematics, Concordia

More information

4 Separation of Variables

4 Separation of Variables 4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE

More information

A Brief Introduction to Markov Chains and Hidden Markov Models

A Brief Introduction to Markov Chains and Hidden Markov Models A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,

More information

MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES

MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is

More information

ASummaryofGaussianProcesses Coryn A.L. Bailer-Jones

ASummaryofGaussianProcesses Coryn A.L. Bailer-Jones ASummaryofGaussianProcesses Coryn A.L. Baier-Jones Cavendish Laboratory University of Cambridge caj@mrao.cam.ac.uk Introduction A genera prediction probem can be posed as foows. We consider that the variabe

More information

Separation of Variables and a Spherical Shell with Surface Charge

Separation of Variables and a Spherical Shell with Surface Charge Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation

More information

An implicit Jacobi-like method for computing generalized hyperbolic SVD

An implicit Jacobi-like method for computing generalized hyperbolic SVD Linear Agebra and its Appications 358 (2003) 293 307 wwweseviercom/ocate/aa An impicit Jacobi-ike method for computing generaized hyperboic SVD Adam W Bojanczyk Schoo of Eectrica and Computer Engineering

More information

4 1-D Boundary Value Problems Heat Equation

4 1-D Boundary Value Problems Heat Equation 4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x

More information

MARKOV CHAINS AND MARKOV DECISION THEORY. Contents

MARKOV CHAINS AND MARKOV DECISION THEORY. Contents MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After

More information

Fitting affine and orthogonal transformations between two sets of points

Fitting affine and orthogonal transformations between two sets of points Mathematica Communications 9(2004), 27-34 27 Fitting affine and orthogona transformations between two sets of points Hemuth Späth Abstract. Let two point sets P and Q be given in R n. We determine a transation

More information

Turbo Codes. Coding and Communication Laboratory. Dept. of Electrical Engineering, National Chung Hsing University

Turbo Codes. Coding and Communication Laboratory. Dept. of Electrical Engineering, National Chung Hsing University Turbo Codes Coding and Communication Laboratory Dept. of Eectrica Engineering, Nationa Chung Hsing University Turbo codes 1 Chapter 12: Turbo Codes 1. Introduction 2. Turbo code encoder 3. Design of intereaver

More information

CONGRUENCES. 1. History

CONGRUENCES. 1. History CONGRUENCES HAO BILLY LEE Abstract. These are notes I created for a seminar tak, foowing the papers of On the -adic Representations and Congruences for Coefficients of Moduar Forms by Swinnerton-Dyer and

More information

An approximate method for solving the inverse scattering problem with fixed-energy data

An approximate method for solving the inverse scattering problem with fixed-energy data J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999

More information

XSAT of linear CNF formulas

XSAT of linear CNF formulas XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D-50968 Kön, Germany; bernd.schuh@netcoogne.de eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open

More information

Bayesian Learning. You hear a which which could equally be Thanks or Tanks, which would you go with?

Bayesian Learning. You hear a which which could equally be Thanks or Tanks, which would you go with? Bayesian Learning A powerfu and growing approach in machine earning We use it in our own decision making a the time You hear a which which coud equay be Thanks or Tanks, which woud you go with? Combine

More information

Primal and dual active-set methods for convex quadratic programming

Primal and dual active-set methods for convex quadratic programming Math. Program., Ser. A 216) 159:469 58 DOI 1.17/s117-15-966-2 FULL LENGTH PAPER Prima and dua active-set methods for convex quadratic programming Anders Forsgren 1 Phiip E. Gi 2 Eizabeth Wong 2 Received:

More information

Power Control and Transmission Scheduling for Network Utility Maximization in Wireless Networks

Power Control and Transmission Scheduling for Network Utility Maximization in Wireless Networks ower Contro and Transmission Scheduing for Network Utiity Maximization in Wireess Networks Min Cao, Vivek Raghunathan, Stephen Hany, Vinod Sharma and. R. Kumar Abstract We consider a joint power contro

More information

Higher dimensional PDEs and multidimensional eigenvalue problems

Higher dimensional PDEs and multidimensional eigenvalue problems Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u

More information

ORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION

ORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,

More information

18-660: Numerical Methods for Engineering Design and Optimization

18-660: Numerical Methods for Engineering Design and Optimization 8-660: Numerica Methods for Engineering esign and Optimization in i epartment of ECE Carnegie Meon University Pittsburgh, PA 523 Side Overview Conjugate Gradient Method (Part 4) Pre-conditioning Noninear

More information

1D Heat Propagation Problems

1D Heat Propagation Problems Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2

More information

c 2000 Society for Industrial and Applied Mathematics

c 2000 Society for Industrial and Applied Mathematics SIAM J. SCI. COMPUT. Vo. 2, No. 5, pp. 909 926 c 2000 Society for Industria and Appied Mathematics A DEFLATED VERSION OF THE CONJUGATE GRADIENT ALGORITHM Y. SAAD, M. YEUNG, J. ERHEL, AND F. GUYOMARC H

More information

Statistical Learning Theory: A Primer

Statistical Learning Theory: A Primer Internationa Journa of Computer Vision 38(), 9 3, 2000 c 2000 uwer Academic Pubishers. Manufactured in The Netherands. Statistica Learning Theory: A Primer THEODOROS EVGENIOU, MASSIMILIANO PONTIL AND TOMASO

More information

Maximizing Sum Rate and Minimizing MSE on Multiuser Downlink: Optimality, Fast Algorithms and Equivalence via Max-min SIR

Maximizing Sum Rate and Minimizing MSE on Multiuser Downlink: Optimality, Fast Algorithms and Equivalence via Max-min SIR 1 Maximizing Sum Rate and Minimizing MSE on Mutiuser Downink: Optimaity, Fast Agorithms and Equivaence via Max-min SIR Chee Wei Tan 1,2, Mung Chiang 2 and R. Srikant 3 1 Caifornia Institute of Technoogy,

More information

SVM: Terminology 1(6) SVM: Terminology 2(6)

SVM: Terminology 1(6) SVM: Terminology 2(6) Andrew Kusiak Inteigent Systems Laboratory 39 Seamans Center he University of Iowa Iowa City, IA 54-57 SVM he maxima margin cassifier is simiar to the perceptron: It aso assumes that the data points are

More information

C. Fourier Sine Series Overview

C. Fourier Sine Series Overview 12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a

More information

Substructuring Preconditioners for the Bidomain Extracellular Potential Problem

Substructuring Preconditioners for the Bidomain Extracellular Potential Problem Substructuring Preconditioners for the Bidomain Extraceuar Potentia Probem Mico Pennacchio 1 and Vaeria Simoncini 2,1 1 IMATI - CNR, via Ferrata, 1, 27100 Pavia, Itay mico@imaticnrit 2 Dipartimento di

More information

Some Measures for Asymmetry of Distributions

Some Measures for Asymmetry of Distributions Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester

More information

Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph G l

Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph G l Investigation on spectrum of the adjacency matrix and Lapacian matrix of graph G SHUHUA YIN Computer Science and Information Technoogy Coege Zhejiang Wani University Ningbo 3500 PEOPLE S REPUBLIC OF CHINA

More information

$, (2.1) n="# #. (2.2)

$, (2.1) n=# #. (2.2) Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier

More information

CONJUGATE GRADIENT WITH SUBSPACE OPTIMIZATION

CONJUGATE GRADIENT WITH SUBSPACE OPTIMIZATION CONJUGATE GRADIENT WITH SUBSPACE OPTIMIZATION SAHAR KARIMI AND STEPHEN VAVASIS Abstract. In this paper we present a variant of the conjugate gradient (CG) agorithm in which we invoke a subspace minimization

More information

A proposed nonparametric mixture density estimation using B-spline functions

A proposed nonparametric mixture density estimation using B-spline functions A proposed nonparametric mixture density estimation using B-spine functions Atizez Hadrich a,b, Mourad Zribi a, Afif Masmoudi b a Laboratoire d Informatique Signa et Image de a Côte d Opae (LISIC-EA 4491),

More information

8 Digifl'.11 Cth:uits and devices

8 Digifl'.11 Cth:uits and devices 8 Digif'. Cth:uits and devices 8. Introduction In anaog eectronics, votage is a continuous variabe. This is usefu because most physica quantities we encounter are continuous: sound eves, ight intensity,

More information

(This is a sample cover image for this issue. The actual cover is not yet available at this time.)

(This is a sample cover image for this issue. The actual cover is not yet available at this time.) (This is a sampe cover image for this issue The actua cover is not yet avaiabe at this time) This artice appeared in a journa pubished by Esevier The attached copy is furnished to the author for interna

More information

Integrating Factor Methods as Exponential Integrators

Integrating Factor Methods as Exponential Integrators Integrating Factor Methods as Exponentia Integrators Borisav V. Minchev Department of Mathematica Science, NTNU, 7491 Trondheim, Norway Borko.Minchev@ii.uib.no Abstract. Recenty a ot of effort has been

More information

A unified framework for Regularization Networks and Support Vector Machines. Theodoros Evgeniou, Massimiliano Pontil, Tomaso Poggio

A unified framework for Regularization Networks and Support Vector Machines. Theodoros Evgeniou, Massimiliano Pontil, Tomaso Poggio MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARTIFICIAL INTELLIGENCE LABORATORY and CENTER FOR BIOLOGICAL AND COMPUTATIONAL LEARNING DEPARTMENT OF BRAIN AND COGNITIVE SCIENCES A.I. Memo No. 1654 March23, 1999

More information

MAT 167: Advanced Linear Algebra

MAT 167: Advanced Linear Algebra < Proem 1 (15 pts) MAT 167: Advanced Linear Agera Fina Exam Soutions (a) (5 pts) State the definition of a unitary matrix and expain the difference etween an orthogona matrix and an unitary matrix. Soution:

More information

6 Wave Equation on an Interval: Separation of Variables

6 Wave Equation on an Interval: Separation of Variables 6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.

More information

More Scattering: the Partial Wave Expansion

More Scattering: the Partial Wave Expansion More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction

More information

JENSEN S OPERATOR INEQUALITY FOR FUNCTIONS OF SEVERAL VARIABLES

JENSEN S OPERATOR INEQUALITY FOR FUNCTIONS OF SEVERAL VARIABLES PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Voume 128, Number 7, Pages 2075 2084 S 0002-99390005371-5 Artice eectronicay pubished on February 16, 2000 JENSEN S OPERATOR INEQUALITY FOR FUNCTIONS OF

More information

BASIC NOTIONS AND RESULTS IN TOPOLOGY. 1. Metric spaces. Sets with finite diameter are called bounded sets. For x X and r > 0 the set

BASIC NOTIONS AND RESULTS IN TOPOLOGY. 1. Metric spaces. Sets with finite diameter are called bounded sets. For x X and r > 0 the set BASIC NOTIONS AND RESULTS IN TOPOLOGY 1. Metric spaces A metric on a set X is a map d : X X R + with the properties: d(x, y) 0 and d(x, y) = 0 x = y, d(x, y) = d(y, x), d(x, y) d(x, z) + d(z, y), for a

More information

14 Separation of Variables Method

14 Separation of Variables Method 14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt

More information

Source and Relay Matrices Optimization for Multiuser Multi-Hop MIMO Relay Systems

Source and Relay Matrices Optimization for Multiuser Multi-Hop MIMO Relay Systems Source and Reay Matrices Optimization for Mutiuser Muti-Hop MIMO Reay Systems Yue Rong Department of Eectrica and Computer Engineering, Curtin University, Bentey, WA 6102, Austraia Abstract In this paper,

More information

The Symmetric and Antipersymmetric Solutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B A l X l B l = C and Its Optimal Approximation

The Symmetric and Antipersymmetric Solutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B A l X l B l = C and Its Optimal Approximation The Symmetric Antipersymmetric Soutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B 2 + + A X B C Its Optima Approximation Ying Zhang Member IAENG Abstract A matrix A (a ij) R n n is said to be symmetric

More information

Efficiently Generating Random Bits from Finite State Markov Chains

Efficiently Generating Random Bits from Finite State Markov Chains 1 Efficienty Generating Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown

More information

Convergence Property of the Iri-Imai Algorithm for Some Smooth Convex Programming Problems

Convergence Property of the Iri-Imai Algorithm for Some Smooth Convex Programming Problems Convergence Property of the Iri-Imai Agorithm for Some Smooth Convex Programming Probems S. Zhang Communicated by Z.Q. Luo Assistant Professor, Department of Econometrics, University of Groningen, Groningen,

More information

Smoothers for ecient multigrid methods in IGA

Smoothers for ecient multigrid methods in IGA Smoothers for ecient mutigrid methods in IGA Cemens Hofreither, Stefan Takacs, Water Zuehner DD23, Juy 2015 supported by The work was funded by the Austrian Science Fund (FWF): NFN S117 (rst and third

More information

Gauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law

Gauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s

More information

LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS

LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL

More information

A. Distribution of the test statistic

A. Distribution of the test statistic A. Distribution of the test statistic In the sequentia test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch

More information

LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS

LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS

More information

Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete

Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity

More information

Cryptanalysis of PKP: A New Approach

Cryptanalysis of PKP: A New Approach Cryptanaysis of PKP: A New Approach Éiane Jaumes and Antoine Joux DCSSI 18, rue du Dr. Zamenhoff F-92131 Issy-es-Mx Cedex France eiane.jaumes@wanadoo.fr Antoine.Joux@ens.fr Abstract. Quite recenty, in

More information

Theory and implementation behind: Universal surface creation - smallest unitcell

Theory and implementation behind: Universal surface creation - smallest unitcell Teory and impementation beind: Universa surface creation - smaest unitce Bjare Brin Buus, Jaob Howat & Tomas Bigaard September 15, 218 1 Construction of surface sabs Te aim for tis part of te project is

More information

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.mer.com Preconditioned Locay Harmonic Residua Method for Computing Interior Eigenpairs of Certain Casses of Hermitian Matrices Vecharynski, E.; Knyazev,

More information

The Group Structure on a Smooth Tropical Cubic

The Group Structure on a Smooth Tropical Cubic The Group Structure on a Smooth Tropica Cubic Ethan Lake Apri 20, 2015 Abstract Just as in in cassica agebraic geometry, it is possibe to define a group aw on a smooth tropica cubic curve. In this note,

More information

(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].

(f) is called a nearly holomorphic modular form of weight k + 2r as in [5]. PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform

More information

An Algorithm for Pruning Redundant Modules in Min-Max Modular Network

An Algorithm for Pruning Redundant Modules in Min-Max Modular Network An Agorithm for Pruning Redundant Modues in Min-Max Moduar Network Hui-Cheng Lian and Bao-Liang Lu Department of Computer Science and Engineering, Shanghai Jiao Tong University 1954 Hua Shan Rd., Shanghai

More information

On Some Basic Properties of Geometric Real Sequences

On Some Basic Properties of Geometric Real Sequences On Some Basic Properties of eometric Rea Sequences Khirod Boruah Research Schoar, Department of Mathematics, Rajiv andhi University Rono His, Doimukh-791112, Arunacha Pradesh, India Abstract Objective

More information

Lecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential

Lecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential Lecture 6 Povh Krane Enge Wiiams Properties of -nuceon potentia 16.1 4.4 3.6 9.9 Meson Theory of Nucear potentia 4.5 3.11 9.10 I recommend Eisberg and Resnik notes as distributed Probems, Lecture 6 1 Consider

More information

Asymptotic Properties of a Generalized Cross Entropy Optimization Algorithm

Asymptotic Properties of a Generalized Cross Entropy Optimization Algorithm 1 Asymptotic Properties of a Generaized Cross Entropy Optimization Agorithm Zijun Wu, Michae Koonko, Institute for Appied Stochastics and Operations Research, Caustha Technica University Abstract The discrete

More information

Approximated MLC shape matrix decomposition with interleaf collision constraint

Approximated MLC shape matrix decomposition with interleaf collision constraint Approximated MLC shape matrix decomposition with intereaf coision constraint Thomas Kainowski Antje Kiese Abstract Shape matrix decomposition is a subprobem in radiation therapy panning. A given fuence

More information

<C 2 2. λ 2 l. λ 1 l 1 < C 1

<C 2 2. λ 2 l. λ 1 l 1 < C 1 Teecommunication Network Contro and Management (EE E694) Prof. A. A. Lazar Notes for the ecture of 7/Feb/95 by Huayan Wang (this document was ast LaT E X-ed on May 9,995) Queueing Primer for Muticass Optima

More information

ETNA Kent State University

ETNA Kent State University Eectronic Transactions on Numerica Anaysis. Voume 7, 1998, pp. 90-103. Copyright 1998,. ISSN 1068-9613. ETNA A THEORETICAL COMPARISON BETWEEN INNER PRODUCTS IN THE SHIFT-INVERT ARNOLDI METHOD AND THE SPECTRAL

More information

Combining reaction kinetics to the multi-phase Gibbs energy calculation

Combining reaction kinetics to the multi-phase Gibbs energy calculation 7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the muti-phase Gibbs energy cacuation

More information

Physics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions

Physics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p

More information

Distributed average consensus: Beyond the realm of linearity

Distributed average consensus: Beyond the realm of linearity Distributed average consensus: Beyond the ream of inearity Usman A. Khan, Soummya Kar, and José M. F. Moura Department of Eectrica and Computer Engineering Carnegie Meon University 5 Forbes Ave, Pittsburgh,

More information

On the Goal Value of a Boolean Function

On the Goal Value of a Boolean Function On the Goa Vaue of a Booean Function Eric Bach Dept. of CS University of Wisconsin 1210 W. Dayton St. Madison, WI 53706 Lisa Heerstein Dept of CSE NYU Schoo of Engineering 2 Metrotech Center, 10th Foor

More information

T.C. Banwell, S. Galli. {bct, Telcordia Technologies, Inc., 445 South Street, Morristown, NJ 07960, USA

T.C. Banwell, S. Galli. {bct, Telcordia Technologies, Inc., 445 South Street, Morristown, NJ 07960, USA ON THE SYMMETRY OF THE POWER INE CHANNE T.C. Banwe, S. Gai {bct, sgai}@research.tecordia.com Tecordia Technoogies, Inc., 445 South Street, Morristown, NJ 07960, USA Abstract The indoor power ine network

More information

Multiscale Domain Decomposition Preconditioners for 2 Anisotropic High-Contrast Problems UNCORRECTED PROOF

Multiscale Domain Decomposition Preconditioners for 2 Anisotropic High-Contrast Problems UNCORRECTED PROOF AQ Mutiscae Domain Decomposition Preconditioners for 2 Anisotropic High-Contrast Probems Yachin Efendiev, Juan Gavis, Raytcho Lazarov, Svetozar Margenov 2, 4 and Jun Ren 5 Department of Mathematics, TAMU,

More information

SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l

SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed

More information

School of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY

School of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY The ogic of Booean matrices C. R. Edwards Schoo of Eectrica Engineering, Universit of Bath, Caverton Down, Bath BA2 7AY A Booean matrix agebra is described which enabes man ogica functions to be manipuated

More information

hole h vs. e configurations: l l for N > 2 l + 1 J = H as example of localization, delocalization, tunneling ikx k

hole h vs. e configurations: l l for N > 2 l + 1 J = H as example of localization, delocalization, tunneling ikx k Infinite 1-D Lattice CTDL, pages 1156-1168 37-1 LAST TIME: ( ) ( ) + N + 1 N hoe h vs. e configurations: for N > + 1 e rij unchanged ζ( NLS) ζ( NLS) [ ζn unchanged ] Hund s 3rd Rue (Lowest L - S term of

More information

School of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY

School of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY The ogic of Booean matrices C. R. Edwards Schoo of Eectrica Engineering, Universit of Bath, Caverton Down, Bath BA2 7AY A Booean matrix agebra is described which enabes man ogica functions to be manipuated

More information

Semidefinite relaxation and Branch-and-Bound Algorithm for LPECs

Semidefinite relaxation and Branch-and-Bound Algorithm for LPECs Semidefinite reaxation and Branch-and-Bound Agorithm for LPECs Marcia H. C. Fampa Universidade Federa do Rio de Janeiro Instituto de Matemática e COPPE. Caixa Posta 68530 Rio de Janeiro RJ 21941-590 Brasi

More information

VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES

VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES SARAH DAY, JEAN-PHILIPPE LESSARD, AND KONSTANTIN MISCHAIKOW Abstract. One of the most efficient methods for determining the equiibria of a continuous parameterized

More information

Melodic contour estimation with B-spline models using a MDL criterion

Melodic contour estimation with B-spline models using a MDL criterion Meodic contour estimation with B-spine modes using a MDL criterion Damien Loive, Ney Barbot, Oivier Boeffard IRISA / University of Rennes 1 - ENSSAT 6 rue de Kerampont, B.P. 80518, F-305 Lannion Cedex

More information

Lecture 6: Moderately Large Deflection Theory of Beams

Lecture 6: Moderately Large Deflection Theory of Beams Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey

More information

STA 216 Project: Spline Approach to Discrete Survival Analysis

STA 216 Project: Spline Approach to Discrete Survival Analysis : Spine Approach to Discrete Surviva Anaysis November 4, 005 1 Introduction Athough continuous surviva anaysis differs much from the discrete surviva anaysis, there is certain ink between the two modeing

More information

Akaike Information Criterion for ANOVA Model with a Simple Order Restriction

Akaike Information Criterion for ANOVA Model with a Simple Order Restriction Akaike Information Criterion for ANOVA Mode with a Simpe Order Restriction Yu Inatsu * Department of Mathematics, Graduate Schoo of Science, Hiroshima University ABSTRACT In this paper, we consider Akaike

More information

David Eigen. MA112 Final Paper. May 10, 2002

David Eigen. MA112 Final Paper. May 10, 2002 David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.

More information

Stochastic Variational Inference with Gradient Linearization

Stochastic Variational Inference with Gradient Linearization Stochastic Variationa Inference with Gradient Linearization Suppementa Materia Tobias Pötz * Anne S Wannenwetsch Stefan Roth Department of Computer Science, TU Darmstadt Preface In this suppementa materia,

More information

Using constraint preconditioners with regularized saddle-point problems

Using constraint preconditioners with regularized saddle-point problems Comput Optim App (27) 36: 249 27 DOI.7/s589-6-94-x Using constraint preconditioners with reguarized sadde-point probems H.S. Doar N.I.M. Goud W.H.A. Schiders A.J. Wathen Pubished onine: 22 February 27

More information

u(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0

u(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0 Bocconi University PhD in Economics - Microeconomics I Prof M Messner Probem Set 4 - Soution Probem : If an individua has an endowment instead of a monetary income his weath depends on price eves In particuar,

More information

Emmanuel Abbe Colin Sandon

Emmanuel Abbe Colin Sandon Detection in the stochastic bock mode with mutipe custers: proof of the achievabiity conjectures, acycic BP, and the information-computation gap Emmanue Abbe Coin Sandon Abstract In a paper that initiated

More information

Converting Z-number to Fuzzy Number using. Fuzzy Expected Value

Converting Z-number to Fuzzy Number using. Fuzzy Expected Value ISSN 1746-7659, Engand, UK Journa of Information and Computing Science Vo. 1, No. 4, 017, pp.91-303 Converting Z-number to Fuzzy Number using Fuzzy Expected Vaue Mahdieh Akhbari * Department of Industria

More information

221B Lecture Notes Notes on Spherical Bessel Functions

221B Lecture Notes Notes on Spherical Bessel Functions Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,

More information

Solution methods for linear discrete ill-posed problems for color image restoration

Solution methods for linear discrete ill-posed problems for color image restoration BIT manuscript No. (wi be inserted by the editor) Soution methods for inear discrete i-posed probems for coor image restoration A. H. Bentbib M. E Guide K. Jbiou E. Onunwor L. Reiche Received: date / Accepted:

More information

Recursive Constructions of Parallel FIFO and LIFO Queues with Switched Delay Lines

Recursive Constructions of Parallel FIFO and LIFO Queues with Switched Delay Lines Recursive Constructions of Parae FIFO and LIFO Queues with Switched Deay Lines Po-Kai Huang, Cheng-Shang Chang, Feow, IEEE, Jay Cheng, Member, IEEE, and Duan-Shin Lee, Senior Member, IEEE Abstract One

More information

The EM Algorithm applied to determining new limit points of Mahler measures

The EM Algorithm applied to determining new limit points of Mahler measures Contro and Cybernetics vo. 39 (2010) No. 4 The EM Agorithm appied to determining new imit points of Maher measures by Souad E Otmani, Georges Rhin and Jean-Marc Sac-Épée Université Pau Veraine-Metz, LMAM,

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This artice appeared in a journa pubished by Esevier. The attached copy is furnished to the author for interna non-commercia research and education use, incuding for instruction at the authors institution

More information