numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

the fast hartley transform algorithm the fast hartley transform fht is similar to the cooley tukey fast fourier transform fft but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the fft through use of the fht discrete cosine transforms dct and discrete fourier transforms dft can be obtained the recursive nature of the fht algorithm derived in this paper enables us to generate the next higher order fht from two identical lower order fht s in practice this recursive relationship offers flexibility in programming different sizes of transforms while the orderly structure of its signal flow graphs indicates an ease of implementation in vlsi

upper and lower bounds on norms of functions of matrices abstract given an n by n matrix a we look for a set s in the complex plane and positive scalars m and m such that for all functions p bounded and analytic on s and throughout a neighborhood of each eigenvalue of a the inequalities m inf f l s f a p a p a m inf f l s f a p a hold we show that for 2 by 2 matrices if s is the field of values then one can take m 1 and m 2 we show that for a perturbed jordan block a matrix a that is an n by n jordan block with eigenvalue 0 except that its n 1 entry is with 0 1 if s is the unit disk then m m 1 we argue however that in general due to the behavior of minimal norm interpolating functions it may be very difficult or impossible to find such a set s for which the ratio m m is of moderate size

the numerical range and the spectrum of a product of two orthogonal projections the aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum several improvements removing the closure of the numerical range of the operator using a parametrization after its eigenvalues are possible under additional assumptions an estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the friedrichs angle applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

recursive fast algorithm and the role of the tensor product the use of the tensor product for modeling and designing fft algorithms is addressed the benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures the scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms this greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the fft

decay properties for functions of matrices over c algebras abstract we extend previous results on the exponential off diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a c algebra

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

inverse scattering for transmission problem abstract this paper discusses the inverse obstacle scattering problem with transmission boundary condition in particular we shall show that transmission boundary is uniquely determined by the far field pattern

on the uniqueness of functions that maximize the crouzeix ratio abstract let a be an n by n matrix with numerical range w a q a q q c n q 2 1 we are interested in functions f that maximize f a 2 the matrix norm induced by the vector 2 norm over all functions f that are analytic in the interior of w a and continuous on the boundary and satisfy max z w a f z 1 it is known that there are functions f that achieve this maximum and that such functions are of the form b where is any conformal mapping from the interior of w a to the unit disk d extended to be continuous on the boundary of w a and b is a blaschke product of degree at most n 1 it is not known if a function f that achieves this maximum is unique up to multiplication by a scalar of modulus one we show that this is the case when a is a 2 2 nonnormal matrix or a jordan block but we give examples of some 3 3 matrices with elliptic numerical range for which two different functions f involving the same conformal mapping but blaschke products of different degrees achieve the same maximal value of f a 2

split manageable efficient algorithm for fourier and hadamard transforms in this paper a general efficient manageable split algorithm to compute one dimensional 1 d unitary transforms by using the special partitioning in the frequency domain is introduced the partitions determine fast transformations that split the n point unitary transform into a set of n sub i point transforms i 1 n n sub 1 n sub n n here we introduce a class of splitting transformations the so called paired transforms based on these transforms the decompositions of the fourier transforms of arbitrary orders are given and the corresponding algorithms are considered comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given in particular a proposed method of calculating the 2 sup r point fourier transform requires 2 sup r 1 r 3 2 multiplications and 2 sup r 1 r 9 r sup 2 3r 6 additions in terms of the paired transforms the splitting of the 2 sup r point hadamard transform is described as a result the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample

real linear kronecker product operations the kronecker product in the real linear matrix analytic setting is studied more versatile operations are proposed such generalizations are of interest for the same reasons the standard kronecker product is to give an example new preconditioning ideas are suggested in connection with this several formulae for the inverse are devised orthogonal decompositions of real entried matrices are derived through introducing new kronecker product svds matrix equations are given to illustrate how the kronecker product structures introduced can arise

crouzeix s conjecture and perturbed jordan blocks abstract crouzeix conjectured in m crouzeix bounds for analytical functions of matrices integr equ oper theory 48 2004 461 477 that for any square matrix a and any polynomial p p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for matrices of the form 1 1 where and are complex numbers the technique for 1 and for 1 is to show that if g is a bijective conformal mapping from the field of values of such a matrix a to the unit disk mapping to 0 then g a is similar to a contraction via a similarity transformation with condition number at most 2

generalized sliding fft and its application to implementation of block lms adaptive filters this paper has two contributions first the concept of the generalized sliding fast fourier transform gsfft as an efficient implementation of the hopping fft is introduced application of the gsfft is broad and not limited to what has been considered in this paper the frequency domain block lms fblms adaptive filters are then revised and their implementations for block lengths less than the length of the adaptive filter are studied the gsfft and the available pruned ffts are used to give an efficient implementation of these filters in the particular case of the block length equal to one where the fblms algorithm reduces to the frequency domain lms flms algorithm it is shown that the latter can be implemented with the order of m complexity where m is the length of the adaptive filter

lemniscates and k spectral sets abstract we show how multicentric representation of functions provides a simple way to generalize the von neumann result that the unit disc is a spectral set for contractions in hilbert spaces in particular the sets need not be connected and the results can be applied to bounding riesz spectral projections

preconditioners based on the alternating direction implicit algorithm for the 2d steady state diffusion equation with orthotropic heterogeneous coefficients in this paper we combine the alternating direction implicit adi algorithm with the concept of preconditioning and apply it to linear systems discretized from the 2d steady state diffusion equations with orthotropic heterogeneous coefficients by the finite element method assuming tensor product basis functions specifically we adopt the compound iteration idea and use adi iterations as the preconditioner for the outside krylov subspace method that is used to solve the preconditioned linear system an efficient algorithm to perform each adi iteration is crucial to the efficiency of the overall iterative scheme we exploit the kronecker product structure in the matrices inherited from the tensor product basis functions to achieve high efficiency in each adi iteration meanwhile in order to reduce the number of krylov subspace iterations we incorporate partially the coefficient information into the preconditioner by exploiting the local support property of the finite element basis functions numerical results demonstrated the efficiency and quality of the proposed preconditioner

a proof of crouzeix s conjecture for a class of matrices abstract crouzeix s conjecture is that for any square matrix a and any polynomial p we have p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for the matrices of the form 1 n 1 n where c and 1 n c n

an error analysis for rational galerkin projection applied to the sylvester equation in this paper we suggest a new formula for the residual of galerkin projection onto rational krylov spaces applied to a sylvester equation and establish a relation to three different underlying extremal problems for rational functions these extremal problems enable us to compare the size of the residual for the above method with that obtained by adi in addition we deduce several new a priori error estimates for galerkin projection onto rational krylov spaces both for the sylvester and for the lyapunov equation

transform domain lms algorithm the concept of transform domain adaptive filtering is introduced in certain applications filtering in the transform domain results in great improvements in convergence rate over the conventional time domain adaptive filtering the relationship between several existing frequency domain adaptive filtering algorithms is established applications of the discrete fourier transform dft and the discrete cosine transform dct domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed

m adic polyadic invariance and its applications the concept of m adic invariance allows approximation of a linear time invariant lti system by a linear m adic invariant lmi system or equivalently approximation of a circulant matrix by a super circulant matrix the approximation reduces the number of multiplies required for computing n point cyclic convolution to 2 log m n 1 n where n mn the error introduced by the approximation can be removed if desired by subsequent processing in one concrete case determination of a small number of noncontiguous frequencies this approach approximation and subsequent correction can effect substantial savings in a number of multiplies compared to both fast fourier transform fft algorithm and direct discrete fourier transform ddft these applications are preceded by a tutorial presentation of concepts which are basic to m adic invariant systems

improved parallel qr method for large least squares problems involving kronecker products abstract a new algorithm is presented for the efficient solution of large least squares problems in which the coefficient matrix of the linear system is a kronecker product of two smaller dimension matrices the solution algorithm is based on qr factorizations of the smaller dimension matrices near perfect load balancing is achieved by exploiting a commutativity property of the kronecker product and communication requirements are minimized by employing a binary exchange algorithm for matrix transposition the parallel algorithm is presented and timing results are shown from test runs on an intel i 860 computer

on the convergence of krylov subspace methods for matrix mittag leffler functions in this paper we analyze the convergence of some commonly used krylov subspace methods for computing the action of matrix mittag leffler functions as is well known such functions find application in the solution of fractional differential equations we illustrate the theoretical results by some numerical experiments

computational methods for modifying seemingly unrelated regressions models computational efficient methods for updating seemingly unrelated regressions models with new observations are proposed a recursive algorithm to solve a series of updating problems is developed the algorithm is based on orthogonal transformations and has as main computational tool the updated generalized qr decomposition ugqrd strategies to compute the orthogonal factorizations by exploiting the block sparse structure of the matrices are designed the problems of adding and deleting exogenous variables from the seemingly unrelated regressions model have also been investigated the solution of these problems utilize the strategies for computing the ugqrd

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

a modified gautschi s method without order reduction when integrating boundary value nonlinear wave problems abstract in this paper we analyse the order reduction which turns up when integrating nonlinear wave problems with non homogeneous and time dependent boundary conditions with the well known gautschi s method moreover a technique is suggested to avoid that order reduction so that the classical local order 4 and global order 2 are recovered on the other hand the usual approximation for the derivative which is used together with this method is also analysed and a substantial improvement is suggested some numerical results are shown which corroborate the performed analysis

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

on hermitian and skew hermitian splitting iteration methods for the linear matrix equation axb c in this paper a hermitian and skew hermitian splitting hss iteration method for solving the linear matrix equation axb c is proposed this method is formed by extending the corresponding hss iteration method for solving ax b the analysis shows that the hss iteration method will converge under certain assumptions moreover the optimal parameter of this iteration method is presented numerical results show that this new method is more efficient and robust than existing methods

a rational arnoldi approach for ill conditioned linear systems for the solution of full rank ill posed linear systems a new approach based on the arnoldi algorithm is presented working with regularized systems the method theoretically reconstructs the true solution by means of the computation of a suitable function of matrix in this sense the method can be referred to as an iterative refinement process numerical experiments arising from integral equations and interpolation theory are presented finally the method is extended to work in connection with the standard tikhonov regularization with the right hand side contaminated by noise

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

characterization of half radial matrices abstract numerical radius r a is the radius of the smallest ball with the center at zero containing the field of values of a given square matrix a it is well known that r a a 2 r a where is the matrix 2 norm matrices attaining the lower bound are called radial and have been analyzed thoroughly this is not the case for matrices attaining the upper bound where only partial results are available in this paper we consider matrices satisfying r a a 2 and call them half radial we summarize the existing results and formulate new ones in particular we investigate their singular value decomposition and algebraic structure and provide other necessary and sufficient conditions for a matrix to be half radial based on that we study the extreme case of the attainable constant 2 in crouzeix s conjecture the presented results support the conjecture of greenbaum and overton that the crabb choi crouzeix matrix always plays an important role in this extreme case

parameterized preconditioned hermitian and skew hermitian splitting iteration method for a class of linear matrix equations recently wang et al 2013 proposed an efficient hermitian and skew hermitian splitting hss iteration method to approximate the solution of linear matrix equation to further improve the efficiency of the hss method we in this paper establish a parameterized preconditioned hss pphss iteration method by introducing two hermitian positive definite preconditioners and two more different iteration parameters for the hss method the convergence properties of the pphss method used for solving linear matrix equation a x b c are strictly analyzed and the quasi optimal values of the iteration parameters for the pphss method are also derived numerical results demonstrate the feasibility and the efficiency of the pphss iteration method

stationary splitting iterative methods for the matrix equation axb c abstract stationary splitting iterative methods for solving a x b c are considered in this paper the main tool to derive our new method is the induced splitting of a given nonsingular matrix a m n by a matrix h such that i h 1 exists convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants in particular for certain surface fitting applications our method is much more efficient than the progressive iterative approximation pia a conventional iterative method often used in computer aided geometric design cagd

variational analysis of the crouzeix ratio let w a denote the field of values numerical range of a matrix a for any polynomial p and matrix a define the crouzeix ratio to have numerator max left p zeta zeta in w a right max p w a and denominator vert p a vert 2 p a 2 crouzeix s 2004 conjecture postulates that the globally minimal value of the crouzeix ratio is 1 2 over all polynomials p of any degree and matrices a of any order we derive the subdifferential of this ratio at pairs p a for which the largest singular value of p a is simple in particular we show that at certain candidate minimizers p a the crouzeix ratio is clarke regular and satisfies a first order nonsmooth optimality condition and hence that its directional derivative is nonnegative there in every direction in polynomial matrix space we also show that pairs p a exist at which the crouzeix ratio is not regular

the jacobi and gauss seidel type iteration methods for the matrix equation a x b c in this paper the jacobi and gauss seidel type iteration methods are proposed for solving the matrix equation a x b c which are based on the splitting schemes of the matrices a and b the convergence and computational cost of these iteration methods are discussed furthermore we give the preconditioned jacobi and gauss seidel type iteration methods numerical examples are given to demonstrate the efficiency of these methods proposed in this paper

derivatives of fourth order kronecker power systems with applications in nonlinear elasticity abstract a natural way to describe systems with polynomial nonlinearities is using the kronecker product particularly third order kronecker power systems can express a wide range of systems from electronic engineering to nonlinear elasticity but such development e g equations of motions of elastic structures from nonlinear strain energy requires standard formulation for the derivative of the kronecker power of vectors with respect to the same vector such standard way cannot be found in literature this paper presents a method to obtain the derivatives of kronecker powers of vectors with respect to itself up to a power of four and also third order kronecker power systems containing those terms in a concise matrix form the matrix expression of these systems provides new approach for efficient numerical implementation organized analysis and linearization to demonstrate the strength of this method an example of application for a finite element nonlinear euler beam is also presented

numerical investigation of crouzeix s conjecture abstract crouzeix s conjecture states that for all polynomials p and matrices a the inequality p a 2 p w a holds where the quantity on the left is the 2 norm of the matrix p a and the norm on the right is the maximum modulus of the polynomial p on w a the field of values of a we report on some extensive numerical experiments investigating the conjecture via nonsmooth minimization of the crouzeix ratio f p w a p a using chebfun to evaluate this quantity accurately and efficiently and the bfgs method to search for its minimal value which is 0 5 if crouzeix s conjecture is true almost all of our optimization searches deliver final polynomial matrix pairs that are very close to nonsmooth stationary points of f with stationary value 0 5 for which w a is a disk or smooth stationary points of f with stationary value 1 for which w a has a corner our observations have led us to some additional conjectures as well as some new theorems we hope that these give insight into crouzeix s conjecture which is strongly supported by our results

tests for complete k spectral sets abstract let be a family of functions analytic in some neighborhood of a complex domain and let t be a hilbert space operator whose spectrum is contained in our typical result shows that under some extra conditions if the closed unit disc is complete k spectral for t for every then is complete k spectral for t for some constant k in particular we prove that under a geometric transversality condition the intersection of finitely many k spectral sets for t is again k spectral for some k k these theorems generalize and complement results by mascioni stessin stampfli badea beckermann crouzeix and others we also extend to non convex domains a result by putinar and sandberg on the existence of a skew dilation of t to a normal operator with spectrum in as a key tool we use the results from our previous paper 11 on traces of analytic uniform algebras

noniterative map reconstruction using sparse matrix representations we present a method for noniterative maximum a posteriori map tomographic reconstruction which is based on the use of sparse matrix representations our approach is to precompute and store the inverse matrix required for map reconstruction this approach has generally not been used in the past because the inverse matrix is typically large and fully populated i e not sparse in order to overcome this problem we introduce two new ideas the first idea is a novel theory for the lossy source coding of matrix transformations which we refer to as matrix source coding this theory is based on a distortion metric that reflects the distortions produced in the final matrix vector product rather than the distortions in the coded matrix itself the resulting algorithms are shown to require orthonormal transformations of both the measurement data and the matrix rows and columns before quantization and coding the second idea is a method for efficiently storing and computing the required orthonormal transformations which we call a sparse matrix transform smt the smt is a generalization of the classical fft in that it uses butterflies to compute an orthonormal transform but unlike an fft the smt uses the butterflies in an irregular pattern and is numerically designed to best approximate the desired transforms we demonstrate the potential of the noniterative map reconstruction with examples from optical tomography the method requires offline computation to encode the inverse transform however once these offline computations are completed the noniterative map algorithm is shown to reduce both storage and computation by well over two orders of magnitude as compared to a linear iterative reconstruction methods

a new construction framework for polyphase complete complementary codes with various lengths complete complementary codes ccc with equal sequence lengths have found numerous applications in wireless communications owing to their ideal correlation properties in modern multi user multi rate communication systems cccs with various sequence lengths for different complementary codes are desirable in this paper we propose a new construction of polyphase ccc with various lengths by using unimodular paraunitary pu matrices our key idea is to decompose a large square pu matrix into direct sum of matrices with smaller sizes which is equivalent to block diagonalization of matrices in order to construct ccc with various lengths we show that our proposed constructions not only generalize several previously known methods but also lead us to new cccs with both equal and various lengths