using burst recovery concept to obtain global stability and performance of discrete time adaptive controller for time varying systems in this note a new technique is proposed for the global stability analysis of adaptive control of time varying systems the proposed method is based on the evaluation of possible burstings in the adaptive loop its simplicity and effectiveness is demonstrated by analyzing global stability and performance of the discrete time direct adaptive controller the developed technique reflects physical phenomena inside the adaptive loop i e possible occurrence of instabilities and burstings precise upper bounds on the admissible speed of the parameter variations and on all the signals are established in terms of clearly defined physical quantities

discriminant subspace analysis a fukunaga koontz approach the fisher linear discriminant fld is commonly used in pattern recognition it finds a linear subspace that maximally separates class patterns according to the fisher criterion several methods of computing the fld have been proposed in the literature most of which require the calculation of the so called scatter matrices in this paper we bring a fresh perspective to fld via the fukunaga koontz transform fkt we do this by decomposing the whole data space into four subspaces with different discriminabilities as measured by eigenvalue ratios by connecting the eigenvalue ratio with the generalized eigenvalue we show where the fisher criterion is maximally satisfied we prove the relationship between fld and fkt analytically and propose a unified framework to understanding some existing work furthermore we extend our theory to the multiple discriminant analysis mda this is done by transforming the data into intraclass and extraclass spaces followed by maximizing the bhattacharyya distance based on our fkt analysis we identify the discriminant subspaces of mda fkt and propose an efficient algorithm which works even when the scatter matrices are singular or too large to be formed our method is general and may be applied to different pattern recognition problems we validate our method by experimenting on synthetic and real data

a new indirect adaptive control scheme for time varying plants structures are developed for the control and estimation parts of the indirect adaptive scheme which are more appropriate for time varying plants in contrast to the usual pointwise schemes the new controller structure can meet the control objective exactly for a wide class of plants with smooth but otherwise unrestricted parameter variations on the other hand the possibly available knowledge of the form or frequency of variation of the fast parameters is included in the new estimator structure so that for successful estimation the overall plant is not restricted to varying slowly with time the new estimator and control law are combined using the certainty equivalence principle to develop an indirect adaptive control scheme meeting the control objective for plants with time varying parameters whose fast varying parts are of known form and unknown parts are slow in the mean

relatively robust representations of symmetric tridiagonals let ldlt be the triangular factorization of an unreduced symmetric tridiagonal matrix t i small relative changes in the nontrivial entries of l and d may be represented by diagonal scaling matrices 1 and 2 ldlt 2l 1d 1lt 2 the effect of 2 on the eigenvalues i is benign in this paper we study the inner perturbations induced by 1 suitable condition numbers govern the relative changes in the eigenvalues i we show that when j is an eigenvalue then the relative condition number of m j m j is the same for all n twisted factorizations one of which is ldlt that could be used to represent t i see section 2 we prove that as j the smallest eigenvalue has relative condition number relcond 1 o j each relcond is a rational function of we identify the poles and then use orthogonal polynomial theory to develop upper bounds on the sum of the relconds of all the eigenvalues these bounds require o n operations for an n n matrix we show that the sum of all the relconds is bounded by trace l d lt and conjecture that n ldlt the quantity trace l d lt ldlt is a natural measure of element growth in the context of this paper an algorithm for computing numerically orthogonal eigenvectors without recourse to the gram schmidt process is sketched it requires that there exist values of close to each cluster of close eigenvalues such that all the relconds belonging to the cluster are modest say 10 the sensitivity of the other eigenvalues is not important for this reason we develop o n bounds on the sum of the relconds associated with a cluster none of our bounds makes reference to the nature of the distribution of the eigenvalues within a cluster which can be very complicated

a new stable bidiagonal reduction algorithm a new bidiagonal reduction method is proposed for x r m n f orm n it decomposes x into the product x ubv t where u r m n has orthonormal columns v r n n is orthogonal and b r n n is upper bidiagonal the matrix v is computed as a product of householder transformations the matrices u and b are constructed using a recurrence if u is desired from the computation the new procedure requires fewer operations than the golub kahan procedure siam j num anal ser b 2 1965 205 and similar procedures in floating point arithmetic the columns of u may be far from orthonormal but that departure from orthonormality is structured the application of any backward stable singular value decomposition procedure to b recovers the left singular vectors associated with the leading largest singular values of x to near orthogonality

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

relative perturbation theory for hyperbolic eigenvalue problem we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenspaces of a hyperbolic eigenvalue problem hx jx where h is a positive definite matrix and j is a diagonal matrix of signs we consider two types of perturbations when a graded matrix h d ad is perturbed in a graded sense to h h d a a d and the multiplicative perturbations of the form h h i e h i e our bounds are simple to compute compare well to the classical results and can be used when analyzing numerical algorithms

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

a relative perturbation bound for positive definite matrices abstract we give a sharp estimate for the eigenvectors of a positive definite hermitian matrix under a floating point perturbation the proof is elementary

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

stable adaptive regulation of arbitrary nth order plants this paper presents an algorithm for adaptively stabilizing and asymptotically regulating an arbitrary single input single output linear time invariant plant which is controllable and observable of known order n and has unknown parameters no further assumptions are made no external probing signal is required

rounding error and perturbation bounds for the indefinite qr factorization abstract indefinite qr factorization is a generalization of the well known qr factorization where q is a unitary matrix with respect to the given indefinite inner product matrix j this factorization can be used for accurate computation of eigenvalues of the hermitian matrix a g jg where g and j are initially given or naturally formed from initial data the classical example of such a matrix is a b b c c with given b and c in this paper we present the rounding error and perturbation bounds for the so called triangular case of the indefinite qr factorization these bounds fit well into the relative perturbation theory for hermitian matrices given in factorized form

highly accurate symmetric eigenvalue decomposition and hyperbolic svd let g be a m n real matrix with full column rank and let j be a n n diagonal matrix of signs jii 1 1 the hyperbolic singular value decomposition hsvd of the pair g j is defined as g u v 1 where u is orthogonal is positive definite diagonal and v is j orthogonal matrix vtjv j we analyze when it is possible to compute the hsvd with high relative accuracy this essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits even if they have widely varying magnitudes we show that one sided j orthogonal jacobi method method computes the hsvd with high relative accuracy more precisely let b gd 1 where d is diagonal such that the columns of b have unit norms essentially we show that the computed hyperbolic singular values of the pair g j will have log10 e min b correct decimal digits where e is machine precision we give the necessary relative perturbation bounds and error analysis of the algorithm our numerical tests confirmed all theoretical results for the symmetric non singular eigenvalue problem hx x we analyze the two step algorithm which consists of factorization h gjgt followed by the computation of the hsvd of the pair g j here g is square and non singular let b dg where d is diagonal such that the rows of b have unit norms and let b be defined as above essentially we show that the computed eigenvalues of h will have log10 e 2min b e min b correct decimal digits this accuracy can be much higher then the one obtained by the classical qr and jacobi methods applied to h where the accuracy depends on the spectral condition number of h particularly if the matrices b and b are well conditioned and we are interested in the accurate computation of tiny eigenvalues again we give the perturbation and error bounds and our theoretical predictions are confirmed by a series of numerical experiments we also give the corresponding results for eigenvectors and hyperbolic singular vectors

a robust indirect adaptive control approach this paper considers the robust design of an indirect adaptive control approach which is applicable when the unknown parameters of a linear time invariant plant lie in a known convex set throughout which no unstable pole zero cancellation occurs in order to achieve the robustness the use of a relative dead zone in the adaptive law is proposed it is shown that with a suitably designed relative dead zone the adaptive control system is globally stable even in the presence of small unmodeled plant uncertainties

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

higher order direct model reference adaptive control with generic uniform ultimate boundedness this paper proposes a new higher order model reference adaptive control ho mrac approach following direct adaptive control philosophy which estimates unknown time varying parameters this approach leads to a lyapunov based conventional mrac update law augmented by an observer type parameter predictor dynamics the predictor dynamics are composed of a stable known part a feedback of the parameter error and unknown higher order parameters which are updated using a lyapunov based adaptive design so this ho mrac can cope with rapidly changing parameters due to estimation of their time derivatives moreover for stability analysis a lyapunov based generic ultimate boundedness theorem is presented which allows for a computation of separate bounds for each state vector partition furthermore this theorem formulates the explicit specification of transient and ultimate bounds reaching time on the ultimate bounds and a set of admissible initial conditions two challenging illustrative examples demonstr

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

brief interval excitation through impulse sequences a technical lemma we present a technical lemma that establishes in a general framework an interval excitation property using a specific class of impulse sequences according to such an interval excitation property an impulse applied within a time interval generates excitation within the same interval this property does not hold for arbitrary exciting sequences the lemma is a unified statement of previous results corresponding to particular choices of the exciting sequence

accurate computation of the product induced singular value decomposition with applications we present a new algorithm for floating point computation of the singular value decomposition svd of the product b tau c where b and c are full row rank matrices the algorithm replaces the pair b c with an equivalent pair b c and then it uses the jacobi svd algorithm to compute the svd of the explicitly computed matrix b tau c in this way each nonzero singular value sigma is approximated with some sigma delta sigma where the relative error delta sigma sigma is up to a factor of the dimensions of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta b min delta in cal d kappa 2 delta c where cal d denotes the set of diagonal nonsingular matrices kappa 2 cdot denotes the spectral condition number and boldmath varepsilon unboldmath is the roundoff unit of floating point arithmetic the new algorithm is applied to the eigenvalue problem hm x lambda x with symmetric positive definite h and m it is shown that each eigenvalue lambda is computed with high relative accuracy and that the relative error delta lambda lambda of the computed approximation lambda delta lambda is up to a factor of the dimension of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta the new algorithm can also be used for accurate svd computation of a single matrix g that admits an accurate factorization g b tau c

an extension of analytic solutions of a class of constrained matrix minimization problems abstract in this paper we consider an extension of analytic solutions of a class of constrained matrix minimization problems and propose analytic solutions of the following matrix minimization problem min s k s k h i n l k l k h i m det c i m k 1 s a k s k b k l k min s k s k h i n l k l k h i m t r c i m k 1 s a k s k b k l k where a k c m n b k c n m c is a complex number i m denotes m order identity matrix and det t r denote matrix determinant and trace function the proposed results improve the corresponding existing ones in xu et al 2019 numerical examples are provided to illustrate the efficiency of the proposed theoretical results

perturbation analysis of the canonical subspaces abstract this paper presents a perturbation analysis of the canonical subspaces of a matrix pair a b it is shown that if the perturbations in the matrices a b do not change the ranks of each matrix then the perturbation bounds of the canonical subspaces depend upon the perturbations and the scaled condition numbers of the matrix pair as well as upon the gap of the corresponding canonical correlations

global stability of adaptive pole placement algorithms this paper presents direct and indirect adaptive control schemes for assigning the closed loop poles of a single input single output system in both the continuous and discrete time cases the resulting closed loop system is shown to be globally stable when driven by an external reference signal consisting of a sum of sinusoids in particular persistent excitation of the potentially unbounded closed loop input output data and hence convergence of a sequential least squares identification algorithm is proved the results are applicable to standard sequential least squares and least squares with covariance reset

perturbation analysis for block downdating of a cholesky decomposition let the cholesky decomposition of x rm t x be x rm t x r rm t r where r is upper triangular the cholesky block downdating problem is to determine tilde r such that tilde r rm t tilde r r rm t r z rm t z where z is a block of rows from the data matrix x we analyze the sensitivity of this block downdating problem of the cholesky factorization a measure of conditioning for the cholesky block downdating is presented and compared to the single row downdating case

relative residual bounds for indefinite hermitian matrices abstract we prove several residual bounds for relative perturbations of the eigenvalues of indefinite hermitian matrix the bounds fall into two categories the weyl type bounds and the hofmann wielandt type bounds the bounds are expressed in terms of sines of acute principal angles between certain subspaces associated with the indefinite decomposition of the given matrix the bounds are never worse than the classical residual bounds and can be much sharper in some cases the bounds generalize the existing relative residual bounds for positive definite matrices to indefinite case

model reference adaptive control of linear time varying plants the case of jump parameter variations in this paper we consider the model reference adaptive control problem of single input single output linear time varying plants we assume that the plant parameters are uniformly bounded and piecewise lipschitz continuous inside the intervals of continuity we assume that the plant satisfies certain controllability observability relative degree and zero dynamics assumptions but the speed of parameter variations is not restricted to be small under these general conditions we design a control law of the model reference type which guarantees the boundedness of the closed loop plant and small tracking error in the mean square sense provided that the number of discontinuities of the plant parameters is small on average furthermore in the case of partially unknown plant parameters we show that the adaptive controller obtained by combining this control law with a suitable estimation algorithm guarantees closed loop boundedness and small tracking error in the mean square sense provided that in additi

a simultaneous decomposition for seven matrices with applications abstract let h m n be the set of all m n matrices over the quaternion algebra h in this paper we construct a simultaneous decomposition for seven matrices with compatible sizes a h m n b h m p 1 c h m p 2 d h m p 3 e h q 1 n f h q 2 n and g h q 3 n as applications of the simultaneous matrix decomposition we give some solvability conditions general solutions as well as the range of ranks of the general solutions to the following two generalized sylvester matrix equations b x e c y f d z g a and b x w e c y f d z g a where a b c d e f and g are given quaternion matrices moreover we provide some numerical examples to illustrate the results of this paper

robust model reference adaptive control we propose a new model reference adaptive control algorithm and show that it provides the robust stability of the resulting closed loop adaptive control system with respect to unmodeled plant uncertainties the robustness is achieved by using a relative error signal in combination with a dead zone and a projection in the adaptive law the extra a priori information needed to design the adaptive law are bounds on the plant parameters and an exponential bound on the impulse response of the inverse plant transfer function

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

optimal perturbation bounds for the hermitian eigenvalue problem there is now a large literature on structured perturbation bounds for eigenvalue problems of the formvbox hx mx where h and m are hermitian these results give relative error bounds on the ith eigenvalue i of the form i i i and bound the error in the ith eigenvector in terms of the relative gap minj i i j i j 1 2 in general this theory usually restricts h to be nonsingular and m to be positive definite we relax this restriction by allowing h to be singular for our results on eigenvalues weallow m to be positive semi definite and for a few results we allow it to be more general for these problems for eigenvalues that are not zero or infinity under perturbation it is possible to obtain local relative error bounds thus a wider class of problems may be characterized by this theory although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

relative perturbation theory for hyperbolic singular value problem we give relative perturbation bounds for singular values and perturbation bounds for singular subspaces of a hyperbolic singular value problem for the pair g j where g is a full rank matrix and j is a diagonal matrix of signs we consider two types of relative perturbations g g b b d and g g d b b depending whether g has full column or full row rank respectively in both cases we also consider relative element wise perturbations of g which typically occur in numerical computations

a bound for the condition of a hyperbolic eigenvector matrix the hyperbolic eigenvector matrix is a matrix x which simultaneously diagonalizes the pair h j where h is hermitian positive definite and j diag 1 such that x hx and x jx j we prove that the spectral condition of x x is bounded byk x mink d hd where the minimum is taken over all non singular matrices d which commute with j this bound is attainable and it can be simply computed similar results hold for other signature matrices j like in the discretized klein gordon equation

weyl type relative perturbation bounds for eigensystems of hermitian matrices we present a weyl type relative bound for eigenvalues of hermitian perturbations a e of not necessarily definite hermitian matrices a this bound given in function of the quantity a 1 2ea 1 2 2 that was already known in the definite case is shown to be valid as well in the indefinite case we also extend to the indefinite case relative eigenvector bounds which depend on the same quantity as a consequence new relative perturbation bounds for singular values and vectors are also obtained using matrix differential calculus techniques we obtain for eigenvalues a sharper first order bound involving the logarithm matrix function which is smaller than not only for small e as expected but for any perturbation

the matrix equations ax b xc d with px sxp constraint abstract the matrix equations ax b xc d with the constraint px sxp are considered where p is a given hermitian matrix satisfying p 2 i and s 1 using an eigenvalue decomposition of p the constrained problem can be equivalently transformed to two independent unconstrained problems of matrix equations in similar forms and hence the constrained problem can be solved in terms of the eigenvectors of p in this paper a simple and eigenvector free formula of the general solutions to the constrained problem is presented using moore penrose generalized inverses of the coefficient matrices a and c a similar problem of the matrix equations with generalized constraint is discussed too we also consider the least norm solution of the constrained problems

floating point perturbations of hermitian matrices abstract we consider the perturbation properties of the eigensolution of hermitian matrices for the matrix entries and the eigenvalues we use the realistic floating point error measure a a recently demmel and veselic considered the same problem for a positive definite matrix h showing that the floating point perturbation theory holds with constants depending on the condition number of the matrix a dhd where aii 1 and d is a diagonal scaling we study the general hermitian case along the same lines thus obtaining new classes of well behaved matrices and matrix pairs our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors like those in symmetric indefinite decompositions we also obtain norm estimates for the perturbations of the eigenprojections and show that some of our techniques extend to non hermitian matrices however unlike in the positive definite case we are still unable to describe simply the set of all well behaved hermitian matrices

spectral sensitivity of products of bidiagonals abstract simple formulae are derived for certain relative condition numbers tridiagonal matrices may be represented as products of bidiagonals in various ways depending on properties such as symmetry and positive definiteness the condition numbers give the amplification factor for relative changes in a nonzero eigenvalue caused by relative changes in an entry of a bidiagonal factor the formulae show that in many but not all cases these condition numbers are of modest size several examples illustrate the results and raise new questions

direct multi grid methods for linear systems with harmonic aliasing patterns multi level numerical methods that obtain the exact solution of a linear system are presented the methods are devised by combining ideas from the full multi grid algorithm and perfect reconstruction filters the problem is stated as whether a direct solver is possible in a full multi grid scheme by avoiding smoothing iterations and using different coarse grids at each step the coarse grids must form a partition of the fine grid and thus establishes a strong connection with domain decomposition methods an important analogy is established between the conditions for direct solution in multi grid solvers and perfect reconstruction in filter banks furthermore simple solutions of these conditions for direct multi grid solvers are found by using mirror filters as a result different configurations of direct multi grid solvers are obtained and studied

an overview of relative sin t theorems for invariant subspaces of complex matrices abstract relative perturbation bounds for invariant subspaces of complex matrices are reviewed with emphasis on bounding the sines of the largest principal angle between two subspaces i e sin theorems the goal is to provide intuition as well as an idea for why the bounds hold and why they look the way they do relative bounds have the advantage of being better at exploiting structure in a perturbation than absolute bounds therefore the reaction of subspaces to relative perturbations can be different than to absolute perturbations in particular there are certain classes of relative perturbations to which subspaces of indefinite hermitian matrices can be more sensitive than subspaces of definite matrices

the general linear model and the generalized singular value decomposition abstract a new derivation is given for the generalized singular value decomposition of two matrices x and f having the same number of rows it is shown how this decomposition reveals the structure of the general gauss markov linear model y x 2 ff and exhibits the structure and solution of the generalized linear least squares problem used to provide the best linear unbiased estimator for the model the decomposition is used to prove optimality of the estimator and to reveal the structure of the covariance matrix of the error of the estimator

adaptive tracking with saturating input and controller integral action the authors consider the problem of controlling stable linear plants subject to input saturation constraints and parametric uncertainties a globally stable adaptive controller which performs well in the presence of such constraints is designed using a specific pole placement that accounts for the saturation constraint an integral action to deal with the tracking problem and an adaptive control law with an ad hoc excitation capability together with a parameter adaptation to cope with the parametric uncertainty the proposed controller allows one to track slowly varying reference sequences particularly in the case of a constant reference the output tracking error vanishes asymptotically and the control input does stop saturating after a finite time

quadratic convergence estimate of scaled iterates by j symmetric jacobi method abstract this paper estimates the quadratic convergence reduction of scaled iterates by j symmetric jacobi method numer math 64 1993 241 although the method is well defined for a general definite pair h j h ht j diag im in m the paper considers the most important case when h is positive definite in that case the method is an accurate floating point eigensolver for the pair h j as such it is used in a compound algorithm for accurate floating point computation of eigenvalues and eigenvectors of a non singular indefinite symmetric matrix the new result is proved for scaled diagonally dominant matrices in the general case of multiple eigenvalues it uses frobenius norm of the off diagonal part of symmetrically scaled iteration matrix and a relative gap in the spectrum of h j it can be effectively used in connection with stopping criterion of the method especially with its one sided version

a note on the perturbation bounds of eigenspaces for hermitian matrices let h be a hermitian matrix and h be its perturbed matrix in this paper both additive and multiplicative perturbations of eigenspace are studied and some absolute and relative bounds are presented which are better than the existing bounds in some sense

pole placement direct adaptive control for time varying ill modeled plants the problem is discussed of preserving the stability of pole placement direct adaptive control in spite of output bounded disturbances time varying plant model parameters and unmodeled dynamics assumed to be small in the mean the controller parameter estimates are shown to track in the mean their true time varying parameter values such a convergence property is achieved using an ad hoc internally generated excitation sequence that ensures persistent excitation in the ideal case the convergence of the parameter estimates is exponential avoiding in particular possible chaotic phenomena

two sided coupled generalized sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices abstract in this paper we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices a i c p i t i b i c s i q i c i c p i t i 1 d i c s i 1 q i and e i c p i q i i 1 2 3 we show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two sided coupled generalized sylvester matrix equations with four unknowns a i x i b i c i x i 1 d i e i i 1 2 3 apart from proving an expression for the general solutions to this system we derive the range of ranks of these solutions using the ranks of the given matrices a i b i c i d i and e i we provide some numerical examples to illustrate our results moreover we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two sided coupled generalized sylvester matrix equations with k 1 unknowns a i x i b i c i x i 1 d i e i i 1 k k 4 the main results are also valid over the real number field and the real quaternion algebra

an optimal linear solver for the jacobian system of the extreme type ii ginzburg landau problem this paper considers the extreme type ii ginzburg landau equations a nonlinear pde model for describing the states of a wide range of superconductors based on properties of the jacobian operator and an amg strategy a preconditioned newton krylov method is constructed after a finite volume type discretization numerical experiments are done for representative two and three dimensional domains strong numerical evidence is provided that the number of krylov iterations is independent of the dimension n of the solution space yielding an overall solver complexity of o n

relative perturbation theory for a class of diagonalizable hermitian matrix pairs abstract veselic and slapnicar gave a general perturbation result for the eigenvalues of the hermitian matrix pair h k where k is positive definite in this paper their result is generalized to a wider class of hermitian matrix pairs especially estimates for the relative perturbation of eigenvalues of definite pairs are also obtained

the matrix nearness problem for symmetric matrices associated with the matrix equation atxa btxb c d abstract a direct method based on the projection theorem in inner products spaces the generalized singular value decomposition and the canonical correlation decomposition is presented for finding the optimal approximate solution x in the set s e to a given matrix x where s e denotes the least squares symmetric solution set of the matrix equation atxa btxb c d the analytical expression of the optimal approximate solution x is obtained and an algorithm for finding this solution is also suggested

a tangent algorithm for computing the generalized singular value decomposition we present two new algorithms for floating point computation of the generalized singular values of a real pair a b of full column rank matrices and for floating point solution of the generalized eigenvalue problem hx lambda mx with symmetric positive definite matrices h and m the pair a b is replaced with an equivalent pair a b and the generalized singular values are computed as the singular values of the explicitly computed matrix f a b 1 the singular values of f are computed using the jacobi method the relative accuracy of the computed singular value approximations does not depend on column scalings of a and b that is the accuracy is nearly the same for all pairs ad 1 bd 2 with d 1 d 2 arbitrary diagonal nonsingular matrices similarly the pencil h lambda m is replaced with an equivalent pencil h lambda m and the eigenvalues of h lambda m are computed as the squares of the singular values of g l h l m 1 where l h l m are the cholesky factors of h m respectively and the matrix g is explicitly computed as the solution of a linear system of equations for the computed approximation lambda delta lambda of any exact eigenvalue lambda the relative error delta lambda lambda is of order p n mbox boldmath varepsilon max min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta where p n is a modestly growing polynomial of the dimension of the problem mbox boldmath varepsilon is the round off unit of floating point arithmetic cal d denotes the set of diagonal nonsingular matrices and kappa 2 cdot is the spectral condition number furthermore floating point computation corresponds to an exact computation with h delta h m delta m where for all i j delta h ij sqrt h ii h jj and delta m ij sqrt m ii m jj are of order of mbox boldmath varepsilon times a modest function of n

a new periodic adaptive control approach for time varying parameters with known periodicity a new adaptive control approach is proposed to deal with nonlinear systems with a class of time varying parametric uncertainties which are in an unknown compact set periodic nonvanishing rapid time varying and the only prior knowledge is the periodicity unlike most continuous time adaptation laws which are of differential types in this work a periodic type adaptation law is introduced for continuous time systems the new adaptive controller updates the parameters and the control signal periodically in a pointwise manner over one entire period in the sequel achieves the asymptotic tracking convergence

dimension reduction in text classification with support vector machines support vector machines svms have been recognized as one of the most successful classification methods for many applications including text classification even though the learning ability and computational complexity of training in support vector machines may be independent of the dimension of the feature space reducing computational complexity is an essential issue to efficiently handle a large number of terms in practical applications of text classification in this paper we adopt novel dimension reduction methods to reduce the dimension of the document vectors dramatically we also introduce decision functions for the centroid based classification algorithm and support vector classifiers to handle the classification problem where a document may belong to multiple classes our substantial experimental results show that with several dimension reduction methods that are designed particularly for clustered data higher efficiency for both training and testing can be achieved without sacrificing prediction accuracy of text classification even when the dimension of the input space is significantly reduced

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

an optimization criterion for generalized discriminant analysis on undersampled problems an optimization criterion is presented for discriminant analysis the criterion extends the optimization criteria of the classical linear discriminant analysis lda through the use of the pseudoinverse when the scatter matrices are singular it is applicable regardless of the relative sizes of the data dimension and sample size overcoming a limitation of classical lda the optimization problem can be solved analytically by applying the generalized singular value decomposition gsvd technique the pseudoinverse has been suggested and used for undersampled problems in the past where the data dimension exceeds the number of data points the criterion proposed in this paper provides a theoretical justification for this procedure an approximation algorithm for the gsvd based approach is also presented it reduces the computational complexity by finding subclusters of each cluster and uses their centroids to capture the structure of each cluster this reduced problem yields much smaller matrices to which the gsvd can be applied efficiently experiments on text data with up to 7 000 dimensions show that the approximation algorithm produces results that are close to those produced by the exact algorithm

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

the hyberbolic singular value decomposition and applications a new generalization of the singular value decomposition svd the hyperbolic svd is advanced and its existence is established under mild restrictions the hyperbolic svd accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products signal processing applications where this task arises include the covariance differencing algorithm for bearing estimation in sensor arrays sliding rectangular windowing and array calibration two algorithms for effecting this decomposition are detailed one is sequential and follows a similar pattern to the sequential bidiagonal based svd algorithm the other is for parallel implementation and mimics hestenes svd algorithm 1958 numerical examples demonstrate that like its conventional counterpart the hyperbolic svd exhibits superior numerical behavior relative to explicit formation and solution of the normal equations furthermore the hyperbolic svd applies in problems where the conventional svd cannot be employed

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

novel fisher discriminant classifiers at the present several applications need to classify high dimensional points belonging to highly unbalanced classes unfortunately when the training set cardinality is small compared to the data dimensionality small sample size problem the classification performance of several well known classifiers strongly decreases similarly the classification accuracy of several discriminative methods decreases when non linearly separable and unbalanced classes are treated in this paper we firstly survey state of the art methods that employ improved versions of linear discriminant analysis lda to deal with the above mentioned problems secondly we propose a family of classifiers based on the fisher subspace estimation which efficiently deal with the small sample size problem non linearly separable classes and unbalanced classes the promising results obtained by the proposed techniques on benchmark datasets and the comparison with state of the art predictors show the efficacy of the proposed techniques

absolute and relative perturbation bounds for invariant subspaces of matrices absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices in the two norm and in the frobenius norm the absolute bounds can be considered extensions of davis and kahan s sin theorem to general matrices and invariant subspaces of any dimension the relative bounds are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation when the perturbed subspace has dimension one the relative bound is implied by the absolute bound

approximate eigenvectors as preconditioner given approximate eigenvector matrix u of a hermitian nonsingular matrix h the spectral decomposition of h can be obtained by computing h u hu and then diagonalizing h this work addresses the issue of numerical stability of the transition from h to h in finite precision arithmetic our analysis shows that the eigenvalues will be computed with small relative error if i the approximate eigenvectors are sufficiently orthonormal and ii the matrix h h 2 is of the form dad with diagonal d and well conditioned a in that case h can be efficiently and accurately diagonalized by the jacobi method if u is computed by fast eigensolver based on tridiagonalization this procedure usually gives the eigensolution with high relative accuracy and it is more efficient than accurate jacobi type methods on their own

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

algebraic multigrid solvers for complex valued matrices in the mathematical modeling of real life applications systems of equations with complex coefficients often arise while many techniques of numerical linear algebra e g krylov subspace methods extend directly to the case of complex valued matrices some of the most effective preconditioning techniques and linear solvers are limited to the real valued case here we consider the extension of the popular algebraic multigrid method to such complex valued systems the choices for this generalization are motivated by classical multigrid considerations evaluated with the tools of local fourier analysis and verified on a selection of problems related to real life applications

perturbation theory for the eigenvalues of factorised symmetric matrices abstract we obtain eigenvalue perturbation results for a factorised hermitian matrix h gjg where j 2 i and g has full row rank and is perturbed into g g where g is small with respect to g this complements the earlier results on the easier case of g with full column rank applied to square factors g our results help to identify the so called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices

robust adaptive regulation with minimal prior knowledge a robust adaptive regulator is designed under minimal prior knowledge namely the order of a nominal plant model in a well defined sense the involved robustness means that the closed loop stability is guaranteed in the presence of some class of time varying parameters and unmodeled dynamics the main design feature of the proposed adaptive regulator consists in its self excitation capability together with an appropriate identification stabilization time splitting unlike in the available literature concerning robust adaptive control the involved self excitation is established independently of the parameter estimator properties the use of the robustness oriented modifications as deadzone and parameter projection or contraction which are aimed at achieving the parameter estimates boundedness is no longer required beforehand thereby reducing the underlying prior knowledge

persistency of excitation in the presence of possibly unbounded signals this note develops an explicit characterization of the concept of persistency of excitation for time invariant linear systems in the presence of possibly unbounded signals this result can be used to establish exponential convergence for a range of adaptive control algorithms a feature of our proof is a particular technique for dealing with the initial condition response

the m n symmetric procrustes problem abstract an p q matrix a is said to be m n symmetric if man man t for given m r n p n r q n in this paper the following m n symmetric procrustes problem is studied find the m n symmetric matrix a which minimizes the frobenius norm of ax b where x and b are given rectangular matrices we use project theorem the singular value decomposition and the generalized singular value decomposition of matrices to analysis the problem and to derive a stable method for its solution the related optimal approximation problem to a given matrix on the solution set is solved furthermore the algorithm to compute the optimal approximate solution and the numerical experiment are given

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

a jacobi eigenreduction algorithm for definite matrix pairs we propose a jacobi eigenreduction algorithm for symmetric definite matrix pairsa j of small to medium size real symmetric matrices withj 2 i j diagonal neitherj nora itself need be definite our jacobi reduction works only on one matrix and usesj orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process for the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization we prove the global convergence of the method the quadratic convergence was observed in all experiments we apply our method in two situations i eigenreducing a single real symmetric matrix and ii eigenreducing an overdamped quadratic matrix pencil in both cases our method is preceded by a symmetric indefinite decomposition and performed in its one sided variant on the thus obtained factors our method outdoes the standard methods like standard jacobi orqr ql in accuracy in spite of the use of hyperbolic transformations which are not orthogonal a theoretical justification of this behaviour is made elsewhere the accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order in addition in working with quadratic pencils our method is shown to either converge or to detect non overdampedness

on relative residual bounds for the eigenvalues of a hermitian matrix abstract let h be a hermitian matrix x an orthonormal matrix and m x hx then the eigenvalues of m approximate some eigenvalues of h with an absolute error bounded by hx xm 2 the main interest in this work is the relative distance between the eigenvalues of m and some part of the spectrum of h it is shown that distance depends on the angle between the ranges of x and hx

a general interpolation strategy for algebraic multigrid using energy minimization algebraic multigrid methods solve sparse linear systems ax b by automatic construction of a multilevel hierarchy this hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method we propose a methodology to improve grid transfers through energy minimization the proposed strategy is applicable to hermitian non hermitian definite and indefinite problems each column of the grid transfer operator p is minimized in an energy based norm while enforcing two types of constraints a defined sparsity pattern and preservation of specified modes in the range of p a krylov based strategy is used to minimize energy which is equivalent to solving a p j boldsymbol 0 for each column j of p with the constraints ensuring a nontrivial solution for the hermitian positive definite case a conjugate gradient cg based method is utilized to construct grid transfers while methods based on generalized minimum residual gmres and cg on the normal equations cgnr are explored for the general case the approach is flexible allowing for arbitrary coarsenings unrestricted sparsity patterns straightforward long distance interpolation and general use of constraints either user defined or auto generated we conclude with numerical evidence in support of the proposed framework

robust adaptive pole placement for linear time varying systems this paper studies the robustness properties of a basic adaptive control algorithm with respect to plant parameter variation as well as modeling errors and bounded disturbances the algorithm consists of a projected gradient estimator and a pole assignment controller a unified method of proof is presented for robust stability of both discrete and continuous time adaptively controlled time varying systems the robust performance of an adaptive pole assignment controller is also discussed

a continuous jacobi like approach to the simultaneous reduction of real matrices abstract the problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered based on the jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced the projected gradient method is used in this paper it is shown that the projected gradient of the objective function can be formulated explicitly this gives rise to a system of ordinary differential equations that can be readily solved by numerical software the advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary and that if a desired form is not attainable then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form the general procedure for deriving these differential equations is discussed the framework can be generalized to the complex valued case some applications are given

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

perturbations of the eigenprojections of a factorized hermitian matrix abstract we give the perturbation bounds for the eigenprojections of a hermitian matrix h gjg where g has full column rank and j is nonsingular under the perturbations of the factor g our bounds hold for example when g is given with elementwise relative error our bounds contain relative gaps between the eigenvalues and may thus be much less pessimistic than the standard norm estimates

a ghost fluid level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection in this paper we present the development of a sharp numerical scheme for multiphase electrohydrodynamic ehd flows for a high electric reynolds number regime the electric potential poisson equation contains ehd interface boundary conditions which are implemented using the ghost fluid method gfm the gfm is also used to solve the pressure poisson equation the methods detailed here are integrated with state of the art interface transport techniques and coupled to a robust high order fully conservative finite difference navier stokes solver test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the ehd numerical scheme the method is then applied to simulate a charged liquid kerosene jet

on the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem for standard eigenvalue problems closed form expressions for the condition numbers of a multiple eigenvalue are known in particular they are uniformly 1 in the hermitian case and generally take different values in the non hermitian case we consider the generalized eigenvalue problem and identify the condition numbers our main result is that a multiple eigenvalue generally has multiple condition numbers even in the hermitian definite case the condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors

perturbation theory for orthogonal projection methods with applications to least squares and total least squares the stabilized versions of the least squares ls and total least squares tls methods are two examples of orthogonal projection methods commonly used to solve the overdetermined system of linear equations ax b when a is nearly rank deficient in practice when this system represents the noisy version of an exact rank deficient zero residual problem tls usually yields a more accurate estimate of the exact solution however current perturbation theory does not justify the superiority of tls over ls in this paper we establish a model for orthogonal projection methods by reformulating the parameter estimation problem as an equivalent problem of nullspace determination when the method is based on the singular value decomposition of the matrix a b the model specializes to the well known tls method we derive new lower and upper perturbation bounds for orthogonal projection methods in terms of the subspace angle which shows how the perturbation of the approximate nullspace affects the accuracy of the solution in situations where tls is typically used such as in signal processing where the noise free compatible problem is exactly rank deficient our upper bounds suggest that the tls perturbation bound is usually smaller than the one for ls which means that tls is usually more robust than ls under perturbations of all the data also the bounds permit a comparison between the ls and tls solutions as well as for any two competing orthogonal projection methods we include numerical simulations to illustrate our conclusions