more golay sequences davis and jedwab s construction of golay sequences as reed muller codewords was believed to generate possibly all golay sequences over z sub 2 sup h of length 2 sup m from exhaustive search 1024 quaternary h 2 golay sequences of length 16 m 4 are found which are not within davis and jedwab s construction a table of 16 characteristic sequences is presented from which all 1024 golay sequences can be reconstructed the pairing information and the papr s of associated ofdm signals are given

comments on a new construction of 16 qam golay complementary sequences and extension for 64 qam golay sequences for original article by c v chong see ibid vol 49 no 11 p 2953 2959 nov 2003 some corrections are given for the sequence pairing descriptions of 16 qam golay complementary sequences in chong venkataramani and tarokh s paper together with a related correction for lee and golomb s 64 qam golay sequence construction lee and golomb obtained 496 808 and 976 offset pairs for length 2m 64 qam golay sequences m 2 3 4 we obtained 724 972 and 1224 offset pairs adding w 1 to case iii in lee and golomb s construction gives some additional offset pairs others are new and exist for m ges 3 only descriptions of new offset pairs and enumeration for all first order offset pairs are proposed as conjectures an example is given to show that there exist other 64 qam golay sequences not within this construction

construction of ofdm m qam sequences with low peak to average power ratio we present a technique to derive m quadrature amplitude modulation qam signals from quaternary phase shift keying qpsk constellations when m 2 sup n and n is an even number by utilizing qpsk golay sequences we have constructed m qam sequences with low peak to mean envelope power ratios several upper bounds for these m qam sequences were derived

relative perturbation bound for invariant subspaces of graded indefinite hermitian matrices abstract we give a bound for the perturbations of invariant subspaces of graded indefinite hermitian matrix h d ad which is perturbed into h h d a a d such relative perturbations include an important case where h is given with an element wise relative error application of our bounds requires only the knowledge of the size of relative perturbation a and not the perturbation a itself this typically occurs when data are given with relative uncertainties when the matrix is being stored into computer memory and when analyzing some numerical algorithms subspace perturbations are measured in terms of perturbations of angles between subspaces and our bound is therefore a relative variant of the well known davis kahan sin theorem our bounds generalize some of the recent relative perturbation results

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

optimal odd length binary z complementary pairs a pair of sequences is called a golay complementary pair gcp if their aperiodic auto correlation sums are zero for all out of phase time shifts existing known binary gcps only have even lengths in the form of 21026 where are non negative integers to fill the gap left by the odd lengths we investigate the optimal oddlength binary pairs which display the closest correlation property to that of gcps our criteria of closeness is that each pair has the maximum possible zero correlation zone zcz width and minimum possible out of zone aperiodic auto correlation sums such optimal pairs are called optimal odd length binary z complementary pairs ob zcp in this paper we show that each optimal ob zcp has maximum zcz width of n 1 2 and minimum out of zone aperiodic sum magnitude of 2 where n denotes the sequence length odd systematic constructions of such optimal op zcps are proposed by insertion and deletion of certain binary gcps which settle the 2011 li fan tang tu open problem positively the proposed optimal ob zcps may serve as a replacement for gcps in many engineering applications where odd sequence lengths are preferred in addition they give rise to a new family of base two almost difference families adf which are useful in studying partially balanced incomplete block design bibd index terms aperiodic correlation almost difference set ads almost difference families adf golay complementary pair gcp zero correlation zone zcz z complementary pair zcp

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

an implicit jacobi like method for computing generalized hyperbolic svd abstract in this paper we introduce a joint hyperbolic orthogonal decomposition of two matrices which we call a generalized hyperbolic singular value decomposition or ghsvd this decomposition can be used for finding the eigenvalues and eigenvectors of a symmetric definite pencil x t x y t y where diag 1 we also present an implicit jacobi like method for computing this ghsvd

perfect gaussian integer sequences of arbitrary composite length a composite number can be factored into either n mp or n 2 n where p is an odd prime and m n geq 2 are integers this paper proposes a method for constructing degree 3 and degree 4 perfect gaussian integer sequences pgiss of an arbitrary composite length utilizing an upsampling technique and the base sequence concept proposed by hu wang and li in constructing the pgiss the degree of the sequence is defined as the number of distinct nonzero elements within one period of the sequence this paper commences by constructing degree 3 pgiss of odd prime length followed by degree 2 pgiss of odd prime length the proposed method is then extended to the construction of degree 3 and degree 4 pgiss of composite length n mp finally degree 3 and degree 4 pgiss of length n 4 are built to facilitate the construction of degree 3 and degree 4 pgiss of length n 2 n where n geq 3

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

a direct and generalized construction of polyphase complementary sets with low pmepr and high code rate for ofdm system a major drawback of orthogonal frequency division multiplexing ofdm systems is their high peak to mean envelope power ratio pmepr the pmepr problem can be solved by adopting large codebooks consisting of complementary sequences with low pmepr in this paper we present a new construction of polyphase complementary sets css using generalized boolean functions gbfs which generalizes schmidt s construction in 2007 paterson s construction in 2000 and golay complementary pairs gcps given by davis and jedwab in 1999 compared with schmidt s approach our proposed css lead to lower pmepr with higher code rate for sequences constructed from higher order 3 gbfs we obtain polyphase complementary sequences with maximum pmepr of 2 k 1 and 2 k 2 2m where k m are non negative integers that can be easily derived from the gbf associated with the cs

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

an asymptotic preserving scheme for the streamer simulation in this work we describe the development of a robust simulator of the streamer phenomenon a very common problem of computational plasma physics is the stability of the numerical scheme when the quasi neutral regime is approached actually the stability condition may imply very tight time step constraints we prove both numerically and analytically the good behavior of our approach when the system approaches the quasi neutral regime in particular the semi implicit method we employ requires mild time step restrictions to guarantee the stability some other properties such as the positivity of the scheme are investigated and a detailed review of the numerical implementation is included

on some mixed finite element methods with numerical integration in this paper a new family of mixed finite volume methods is analyzed for the approximation of a reaction diffusion problem all the methods are obtained starting from the dual mixed formulation of the problem and then employing the lowest order raviart thomas finite element spaces plus a suitable quadrature formula for the mass matrix this allows for the use of different averages of the inverse diffusion coefficient to enforce the constitutive law for the fluxes at the interelement boundary in a finite volume fashion the soundness of the methods is supported by an error analysis which shows optimal cal o h convergence rate with respect to the standard mixed finite element norm in the case of both smooth and piecewise smooth coefficients numerical results on test problems with both smooth and nonsmooth coefficients support the theoretical estimates

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

new complete complementary codes for peak to mean power control in multi carrier cdma owing to the zero non trivial aperiodic correlation sum properties complete complementary codes ccc have been applied to asynchronous multi carrier code division multiple access mc cdma communications in order to provide zero interference performance when each complementary code is arranged to be a matrix the peak to mean envelope power ratio pmepr of the ccc mc cdma system is determined by the column sequences of the complementary matrices the existing ccc have the column sequence pmepr of m where m denotes the number of subcarriers in a ccc mc cdma system in practice m is generally large and a pmepr approaching this value is unacceptable to solve this problem a new class of ccc using generalized boolean functions and with a column sequence pmepr of at most 2 is proposed in this paper

novel 16 qam and 64 qam near complementary sequences with low pmepr in ofdm systems in this paper we first propose a novel construction of 16 quadrature amplitude modulation qam near complementary sequences with low peak to mean envelope power ratio pmepr in orthogonal frequency division multiplexing ofdm systems the proposed 16 qam near complementary sequences can be constructed by utilizing novel nonlinear offsets where the length of the sequences is n 2 m the family size of the newly constructed 16 qam near complementary sequences is 8 times m 2 times 4 m 1 and the pmepr of these sequences is proven to satisfy pmepr 2 4 thus the proposed construction can generate a number of 16 qam near complementary sequences with low pmepr resulting in the improvement of the code rate in ofdm systems furthermore we also propose a novel construction of 64 qam near complementary sequences with low pmepr which is the first proven construction of 64 qam near complementary sequences the pmeprs of two types of the proposed 64 qam near complementary sequences are proven to satisfy that pmepr 3 62 or pmepr 2 48 respectively the family size of the newly constructed 64 qam near complementary sequences is 64 times m 2 times 4 m 1

a construction of ofdm 16 qam sequences having low peak powers using a realization of 16 qam as set addition of two scaled versions of the 4 psk constellation we construct 16 qam sequences having low peak to mean envelope power ratios pmepr from 4 psk golay sequences various upper bounds on the peak envelope powers of these sequences are established transmission using these sequences provides twice as much rate as transmission using complementary golay sequences with a maximum pmepr of 3 6 5 56 db

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

a new relative perturbation theorem for singular subspaces abstract this note addresses the sensitivity of singular subspaces of a matrix under relative perturbations it employs a new technique of separating a multiplicative perturbation d into two components one is the distance of a scalar multiple of d to the nearest unitary matrix q and the other is the distance of q to the identity consequently the new bounds reflect the intrinsic differences in how left and right multiplicative perturbations affect left and right singular subspaces

an analysis of the rayleigh ritz method for approximating eigenspaces this paper concerns the rayleigh ritz method for computing an approximation to an eigenspace x of a general matrix a from a subspace w that contains an approximation to x the method produces a pair n x that purports to approximate a pair l x where x is a basis for x and ax xl in this paper we consider the convergence of n x as the sine e of the angle between x and w approaches zero it is shown that under a natural hypothesis called the uniform separation condition the ritz pairs n x converge to the eigenpair l x when one is concerned with eigenvalues and eigenvectors one can compute certain refined ritz vectors whose convergence is guaranteed even when the uniform separation condition is not satisfied an attractive feature of the analysis is that it does not assume that a has distinct eigenvalues or is diagonalizable

a note on optimal block scaling of matrices after pointing out that two recent results on optimal blockscaling are equivalent a new short and simple proof of both results is given

new constructions of general qam golay complementary sequences a construction of general quadrature amplitude modulation qam golay complementary sequences based on quadrature phase shift keying golay davis jedwab sequences gdj sequences is described existing constructions of 16 and 64 qam golay sequences are extended to 4q qam sequences of length 2m for q 1 m 2 this construction gives m 1 42 q 1 m 1 4 q 1 2q 1 m 2 4 m 1 golay complementary sequences a previous offset pair enumeration conjecture for 64 qam golay sequences is proved as a special case of the enumeration for 4q qam golay sequences when used for orthogonal frequency division multiplexing signals the peak to mean envelope power ratio upper bound is shown to be 6 2q 1 2q 1 approaching 6 as the qam constellation size increases

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

near complementary sequences with low pmepr for peak power control in multicarrier communications new families of near complementary sequences are presented for peak power control in multicarrier communications a framework for near complementary sequences is given by an explicit boolean expression and an equivalent matrix structure the framework transforms seed pairs to near complementary sequences by the aid of golay complementary sequences new families of near complementary sequences of various lengths and pmepr 4 are then presented where the sequences are constructed by the framework employing the seeds of shortened and extended golay complementary pairs the families present in a constructive way a large number of sequences of pmepr 4 for the lengths 100 of 24 28 30 34 36 48 56 60 62 66 68 72 and 96 where no golay pairs have been reported the sequence families can find potential applications for peak power control requiring codewords or sequences of various lengths as well as low peak to mean envelope power ratio pmepr

how do more golay sequences arise in 1999 davis and jedwab gave a direct construction of golay complementary sequences over zopf2 h of length 2 m recently li and chu found 1024 more quaternary golay complementary sequences of length 16 that cannot be obtained by the direct construction using exhaustive computer enumeration it is shown how these sequences arise from interleaving and concatenation of two classes of golay complementary sequences given as an example by davis and jedwab these examples spawn new golay sequences over zopf2 h of length 2m for all h ges 2 and m ges 4

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

a note on the existence of the hyperbolic singular value decomposition abstract in this note we present a new matrix decomposition for a matrix pair a b with a hermitian the hyperbolic svd comes as a special case of the decomposition with a set to be the signature matrix

a new construction of 16 qam golay complementary sequences we present a new construction of 16 qam golay sequences of length n 2 sup m the number of constructed sequences is 14 12m m 2 4 sup m 1 when employed as a code in an orthogonal frequency division multiplexing ofdm system this set of sequences has a peak to mean envelope power ratio pmepr of 3 6 by considering two specific subsets of these sequences we obtain new codes with pmepr bounds of 2 0 and 2 8 and respective code sizes of 2 2m m 2 4 sup m 1 and 4 4m m 2 4 sup m 1 these are larger than previously known codes for the same pmepr bounds

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

a sin2θ theorem for graded indefinite hermitian matrices this paper gives double angle theorems that bound the change in an invariant subspace of an indefinite hermitian matrix in the graded form h d ad subject to a perturbation h h d a a d these theorems extend recent results on a definite hermitian

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

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

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

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

a note on unifying absolute and relative perturbation bounds abstract perturbation bounds for invariant subspaces and eigenvalues of complex matrices are presented that lead to absolute as well as a large class of relative bounds in particular it is shown that absolute bounds such as those by davis and kahan bauer and fike and hoffman and wielandt and some relative bounds are special cases of universal bounds as a consequence we obtain a new relative bound for subspaces of normal matrices which contains a deviation of the matrix from positive definiteness we also investigate how row scaling affects eigenvalues and their sensitivity to perturbations and we illustrate how the departure from normality can affect the condition number with respect to inversion of the scaled eigenvectors

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

simplified p n approximations to the equations of radiative heat transfer and applications simplified pn spn approximations to the equations of radiative heat transfer are derived for optically thick diffusive systems and appropriate boundary conditions are formulated the spn equations are derived by an asymptotic analysis while the boundary conditions are obtained from a variational analysis numerical comparisons of the spn equilibrium diffusion and full transport equations are presented to demonstrate the accuracy and efficiency of the spn approximations

asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi complementary sequence sets a quasi complementary sequence set qcss refers to a set of two dimensional matrices with low nontrivial aperiodic auto and cross correlation sums for multicarrier code division multiple access applications the availability of large qcsss with low correlation sums is desirable the generalized levenshtein bound glb is a lower bound on the maximum aperiodic correlation sum of qcsss the bounding expression of glb is a fractional quadratic function of a weight vector mathbf w and is expressed in terms of three additional parameters associated with qcss the set size k the number of channels m and the sequence length n it is known that a tighter glb compared to the welch bound is possible only if the condition m geq 2 and k geq overline k 1 where overline k is a certain function of m and n is satisfied a challenging research problem is to determine if there exists a weight vector that gives rise to a tighter glb for all not just some k geq overline k 1 and m geq 2 especially for large n i e the condition is asymptotically both necessary and sufficient to achieve this we analytically optimize the glb which is in general nonconvex as the numerator term is an indefinite quadratic function of the weight vector our key idea is to apply the frequency domain decomposition of the circulant matrix in the numerator term to convert the nonconvex problem into a convex one following this optimization approach we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the glb under certain conditions

synthesis of power efficient multitone signals with flat amplitude spectrum it is shown that generally any binary or polyphase sequence belonging to a pair of complementary sequences can be used to construct the initial phases of tones forming a multitone signal with the crest factor less than or equal to 6 db the previously proposed shapiro rudin sequences are representatives of a much larger family of binary golay complementary sequences numerical investigations show that the values of the crest factors obtained by applying complementary sequences binary or polyphase are between 5 and 6 db regardless of the number of tones

paraunitary based boolean generator for qam complementary sequences of length 2 k a boolean generator for a large number of standard complementary qam sequences of length 2 k is proposed this boolean generator is derived from the authors earlier paraunitary generator which is based on matrix multiplications both generators are based on unitary matrices in contrast to previous boolean qam algorithms which represent complementary sequences as a weighted sum our algorithm has a multiplicative form any element of a sequence can be generated efficiently by indexing the entries of unitary matrices with the binary representation of the discrete time index which is easily implemented as a binary counter our 1qum based on one qam unitary matrix and 2qum based on two qam unitary matrices algorithms generate generalized case i iii sequences and generalized case iv and v sequences respectively as specified by liu et al in 2013 in addition to many new 2qum sequences the ratio of the numbers of sequences that are generated by our new construction and the previous construction increases with the constellation size for example for a 1024 qam sequence of length 1024 this ratio is 4 4 however if we compare only 2qum sequences to case iv and v sequences this ratio is 267

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

canonical angles of unitary spaces and perturbations of direct complements abstract let n plane1d 4b2 be a given direct sum decomposition we determine the largest number r i such that all subspaces for which the gap between and plane1d 4b2 satisfies have the property dim plane1d 4b2 the problem involves angles between subspaces i e singular values of products of projections

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

existence of the hyperbolic singular value decomposition abstract the hyperbolic singular value decomposition is defined on a general n m matrix and an m m signature matrix pair it is employed in finding the eigenstructure of any matrix that is expressed as the difference of two matrix outer products such differences arise in signal processing applications in the context of the covariance differencing the hyperbolic svd applies in problems where the conventional svd cannot be employed the existence of the hyperbolic singular value decomposition is here extended to the most general case where neither the general matrix nor the matrix product is assumed full rank

on cosets of the generalized first order reed muller code with low pmepr golay sequences are well suited for use as codewords in orthogonal frequency division multiplexing ofdm since their peak to mean envelope power ratio pmepr in q ary phase shift keying psk modulation is at most 2 it is known that a family of polyphase golay sequences of length 2m organizes in m 2 cosets of a q ary generalization of the first order reed muller code rmq 1 m in this paper a more general construction technique for cosets of rm q 1 m with low pmepr is established these cosets contain so called near complementary sequences the application of this theory is then illustrated by providing some construction examples first it is shown that the m 2 cosets of rmq 1 m comprised of golay sequences just arise as a special case second further families of cosets of rmq 1 m with maximum pmepr between 2 and 4 are presented showing that some previously unexplained phenomena can now be understood within a unified framework a lower bound on the pmepr of cosets of rmq 1 m is proved as well and it is demonstrated that the upper bound on the pmepr is tight in many cases finally it is shown that all upper bounds on the pmepr of cosets of rmq 1 m also hold for the peak to average power ratio papr under the walsh hadamard transform wht

on expansion of search subspaces for large non hermitian eigenproblems abstract how to expand a given subspace efficiently is an important task for large sparse eigenvalue problems in this paper we are interested in the following question which vector from a given subspace after multiplied by the matrix a will give a better expanded search subspace in 2008 33 ye investigated this problem and determined how to choose the optimal expansion vector theoretically unfortunately the result is not computable since it involves some unknown information on the desired eigenvector when a is symmetric it was suggested that the ritz vector may lead to a good candidate for subspace expansion however when a is non hermitian the ritz vector may not be a satisfactory choice the contribution of this paper is twofold first we suggest to use the refined ritz vector to expand the search subspace and propose a residual expansion arnoldi method for subspace expansion theoretical results justify the use of the refined ritz vector second we prove that the elements of the primitive refined ritz vector have a decreasing pattern going to zero we then show that the decreasing pattern exists in an arbitrary primitive approximate eigenvector and derive an inexact residual expansion arnoldi method for subspace expansion numerical examples show the effectiveness of our theoretical results and illustrate the numerical behavior of the new method for subspace expansion

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

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

new 64 qam golay complementary sequences a construction of 64 qam golay complementary sequences was proposed by lee and golomb where three qpsk golay davis jedwab complementary sequences gdj sequences or standard golay sequences related by a pair of offsets were combined to form each 64 qam golay sequence feasible offset pairs were given as first order generalized boolean functions in algebraic normal form new offset pairs were described in a conjecture of li leading to new 64 qam golay complementary sequences the proof of the conjecture is presented

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

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

full block j jacobi method for hermitian matrices abstract the paper considers convergence accuracy and efficiency of a block j jacobi method the method is a proper blas 3 generalization of the known method of veselic for computing the hyperbolic singular value decomposition of rectangular matrices at each step the proposed algorithm diagonalizes the block pivot submatrix the convergence is proved for cyclic strategies which are weakly equivalent to the row cyclic strategy the relative accuracy is proved under the standard conditions numerical tests show improved performance with respect to the block oriented generalization of the original method of veselic combined with the hermitian indefinite factorization the proposed method becomes accurate and efficient eigensolver for hermitian indefinite matrices

a framework for the construction ofgolay sequences in 1999 davis and jedwab gave an explicit algebraic normal form for m 2 2h m 1 ordered golay pairs of length 2mmiddot over z2h involving m 2 2h m 1 golay sequences in 2005 li and chu unexpectedly found an additional 1024 length 16 quaternary golay sequences fiedler and jedwab showed in 2006 that these new golay sequences exist because of a crossover of the aperiodic autocorrelation function of certain quaternary length eight sequences belonging to golay pairs and that they spawn further new quaternary golay sequences and pairs of length 2m for m 4 under budisin s 1990 iterative construction the total number of golay sequences and pairs spawned in this way is counted and their algebraic normal form is given explicitly a framework of constructions is derived in which turyn s 1974 product construction together with several variations plays a key role all previously known golay sequences and pairs of length 2m over z2h can be obtained directly in explicit algebraic normal form from this framework furthermore additional quaternary golay sequences and pairs of length 2m are produced that cannot be obtained from any other known construction the framework generalizes readily to lengths that are not a power of 2 and to alphabets other than z2h

a new construction of 64 qam golay complementary sequences in this correspondence we present a new construction for 64 qam golay sequences of length n 2 sup m for integer m the peak envelope power pep of 64 qam golay sequences is shown to be bounded by 4 66n the new construction of 64 qam golay sequences of length n 2 sup m is based on our earlier construction of new offsets of 16 qam golay sequences which are also presented here the total number of offsets of 64 qam golay sequences is 496 for m 2 808 for m 3 and 976 for m 4 obtained by computer search we also computed the pep distribution for 64 qam golay sequences for m 2 m 3 and m 4

an iterative block arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems a modified m step arnoldi algorithm proposed by jia and elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version in the new method we use new modified approximate eigenvectors obtained from a linear combination of ritz vectors and the wasted m 1 th block basis vector whose residual norms are satisfied with some p 1 dimensional minimization problems the resulting block arnoldi algorithm is not only better than the standard m step one both in theory and in practice but also cheaper than the standard m 1 step one the relationships among the residual norms of ritz pairs and those of new approximate eigenpairs are analyzed theoretical results show that the new method can overcome the drawback of nonconvergence which exists in the conventional one in some extent we carry out some numerical experiments which exhibit that the new algorithm is more efficient and works better than its counterparts

optimal block diagonal scaling of block 2 cyclic matrices abstract we describe a class of optimal block diagonal scalings preconditionings of a symmetric positive definite block 2 cyclic matrix generalizing a result of forsythe and strauss 1 for point 2 cyclic matrices

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 pde based partial discharge simulator abstract partial discharges are the main ageing and failure mechanism of solid insulating materials subjected to alternated current stresses this phenomenon from a simulation point of view has been almost always tackled using semi empirical schemes in this work a fully physically based model based on a set of conservation partial differential equations is introduced a numerical algorithm specifically designed to solve this particular problem is developed and its validation is discussed considering some experimental data acquired in a simple geometry containing an isolated void

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

16 qam almost complementary sequences with low pmepr a pair of sequences is said to be an almost complementary pair acp if they have zero aperiodic autocorrelation sums except at only one position over all the positive negative time shifts having correlation property very close to that of golay complementary pairs gcps acps may be used as an alternative to gcps in many applications in communications and radar for high rate code keying ofdm communication we construct novel 16 qam acps from three new classes of quadratic offsets leading to three large sets of 16 qam almost complementary sequences with maximum peak to mean envelope power ratio pmepr of 2 4

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

singular values doubly stochastic matrices and applications abstract the hadamard square of any square matrix a is bounded above and below by some doubly stochastic matrices times the square of the largest and the smallest singular values of a applications to graphs permanents and eigenvalue perturbations are discussed

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

residual bounds for some or all singular values abstract for a given real or complex matrix we derive perturbation bounds for some or all singular values the bounds are of the same quality as corresponding residual bounds for the hermitian eigenvalue problem they are suitable for numerical computation