the role of difference equations in numerical analysis abstract the central role of difference equations and principally the role of their stability properties in numerical analysis is discussed by analyzing the special problem of approximating the solutions of ordinary differential equations we show how a deeper knowledge of the stability properties of difference equations may be useful in designing numerical methods

mammographic feature enhancement by multiscale analysis introduces a novel approach for accomplishing mammographic feature analysis by overcomplete multiresolution representations the authors show that efficient representations may be identified within a continuum of scale space and used to enhance features of importance to mammography methods of contrast enhancement are described based on three overcomplete multiscale representations 1 the dyadic wavelet transform separable 2 the spl phi transform nonseparable nonorthogonal and 3 the hexagonal wavelet transform nonseparable multiscale edges identified within distinct levels of transform space provide local support for image enhancement mammograms are reconstructed from wavelet coefficients modified at one or more levels by local and global nonlinear operators in each case edges and gain parameters are identified adaptively by a measure of energy within each level of scale space the authors show quantitatively that transform coefficients modified by adaptive nonlinear operators can make more obvious unseen or barely seen features of mammography without requiring additional radiation the authors results are compared with traditional image enhancement techniques by measuring the local contrast of known mammographic features they demonstrate that features extracted from multiresolution representations can provide an adaptive mechanism for accomplishing local contrast enhancement by improving the visualization of breast pathology one can improve chances of early detection while requiring less time to evaluate mammograms for most patients

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

sub hexagonal phase correlation for motion estimation we present a novel frequency domain motion estimation technique which operates on hexagonal images and employs the hexagonal fourier transform our method involves image sampling on a hexagonal lattice followed by a normalised hexagonal cross correlation in the frequency domain the term subpixel or subcell is defined on a hexagonal grid in order to achieve floating point registration experiments using both artificially induced motion and actual motion demonstrate that the proposed method outperforms the state of the art in frequency domain motion estimation operating on a square lattice in the shape of phase correlation in terms of subpixel accuracy for a range of test material and motion scenarios

multiresolution moment filters theory and applications we introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales when the window function satisfies a two scale relation we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet like algorithm we show that b splines are well suited window functions because in addition to being refinable they are positive symmetric separable and very nearly isotropic gaussian shape we present three applications of these multiscale local moments the first is a feature extraction method for detecting and characterizing elongated structures in images the second is a noise reduction method which can be viewed as a multiscale extension of savitzky golay filtering the third is a multiscale optical flow algorithm that uses a local affine model for the motion field extending the lucas kanade optical flow method the results obtained in all cases are promising

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

texture enhanced histogram equalization using tv rm l 1 image decomposition histogram transformation defines a class of image processing operations that are widely applied in the implementation of data normalization algorithms in this paper we present a new variational approach for image enhancement that is constructed to alleviate the intensity saturation effects that are introduced by standard contrast enhancement ce methods based on histogram equalization in this paper we initially apply total variation tv minimization with a l1 fidelity term to decompose the input image with respect to cartoon and texture components contrary to previous papers that rely solely on the information encompassed in the distribution of the intensity information in this paper the texture information is also employed to emphasize the contribution of the local textural features in the ce process this is achieved by implementing a nonlinear histogram warping ce strategy that is able to maximize the information content in the transformed image our experimental study addresses the ce of a wide variety of image data and comparative evaluations are provided to illustrate that our method produces better results than conventional ce strategies

recognizing direct products from their conjugate type vectors see 5 for results of a similar nature there is astrong relation between information about character degrees and sizes ofconjugacy classes if the multiplication constants for the conjugacy classesare known then the character table can be reconstructed and similarly inreverse however if one knows only the sizes then there is less completeinformation and it is not possible to obtain a complete translation toillustrate this point if the conjugacy classes have sizes 1 4

on techniques for detecting circumscribed masses in mammograms a method for detecting one type of breast tumor circumscribed masses in mammograms is presented it relies on a combination of criteria used by experts including the shape brightness contrast and uniform density of tumor areas the method uses modified median filtering to enhance mammogram images and template matching to detect the tumors in the template matching step suspicious areas are identified by thresholding the cross correlation values and a percentile method is used to determine a threshold for each film in addition two tests are used to remove false alarms from the resulting candidates the results obtained by applying these techniques to a set of test images are described they are judged encouraging

fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms the objective of this research is to model the mammographic parenchymal and ductal patterns and enhance the microcalcifications using a deterministic fractal approach according to the theory of deterministic fractal geometry images can be modeled by deterministic fractal objects which are attractors of sets of two dimensional 2 d affine transformations the iterated functions systems and the collage theorem are the mathematical foundations of fractal image modeling here a methodology based on fractal image modeling is developed to analyze and model breast background structures the authors show that general mammographic parenchymal and ductal patterns can be well modeled by a set of parameters of affine transformations therefore microcalcifications can be enhanced by taking the difference between the original image and the modeled image the authors results are compared with those of the partial wavelet reconstruction and morphological operation approaches the results demonstrate that the fractal modeling method is an effective way to enhance microcalcifications it may also be able to improve the detection and classification of microcalcifications in a computer aided diagnosis system

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

tridiagonal matrices invertibility and conditioning abstract tridiagonal matrices arise in a large variety of applications most of the time they are diagonally dominant and this is indeed the case most extensively studied in this paper we study in a unified approach the invertibility and the conditioning of such matrices the results presented provide practical criteria for a tridiagonal and irreducible matrix to be both invertible and well conditioned an application to a singular perturbation boundary value problem is then presented

complexity regularized image denoising we study a new approach to image denoising based on complexity regularization this technique presents a flexible alternative to the more conventional l sup 2 l sup 1 and besov regularization methods different complexity measures are considered in particular those induced by state of the art image coders we focus on a gaussian denoising problem and derive a connection between complexity regularized denoising and operational rate distortion optimization this connection suggests the use of efficient algorithms for computing complexity regularized estimates bounds on denoising performance are derived in terms of an index of resolvability that characterizes the compressibility of the true image comparisons with state of the art denoising algorithms are given

bounding fitting heights of character degree graphs ofg written g isthegraphwhosevertexsetis g twoverticespandqin g areadjacentifthereis some degree a cd g where pqdivides a these graphs have beenstudiedinseveralplaces thebestsourcesforgeneralinformationaboutthese graphs are theorem 14 of 2 chapter 30 of 3 and sections 18and19of 10 in 8 adegreegraph wassaidtohave

high order finite difference schemes for the solution of second order bvps we introduce new methods in the class of boundary value methods bvms to solve boundary value problems bvps for a second order ode these formulae correspond to the high order generalizations of classical finite difference schemes for the first and second derivatives in this research we carry out the analysis of the conditioning and of the time reversal symmetry of the discrete solution for a linear convection diffusion ode problem we present numerical examples emphasizing the good convergence behavior of the new schemes finally we show how these methods can be applied in several space dimensions on a uniform mesh

orthogonal and biorthogonal sqrt 3 refinement wavelets for hexagonal data processing the hexagonal lattice was proposed as an alternative method for image sampling the hexagonal sampling has certain advantages over the conventionally used square sampling hence the hexagonal lattice has been used in many areas a hexagonal lattice allows radic3 dyadic and radic7 refinements which makes it possible to use the multiresolution multiscale analysis method to process hexagonally sampled data the radic3 refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale and hence it provides more resolution levels from which one can choose this fact is the main motivation for the study of radic3 refinement surface subdivision and it is also the main reason for the recommendation to use the radic3 refinement for discrete global grid systems however there is little work on compactly supported radic3 refinement wavelets in this paper we study the construction of compactly supported orthogonal and biorthogonal radic3 refinement wavelets in particular we present a block structure of orthogonal fir filter banks with twofold symmetry and construct the associated orthogonal radic3 refinement wavelets we study the sixfold axial symmetry of perfect reconstruction biorthogonal fir filter banks in addition we obtain a block structure of sixfold symmetric radic3 refinement filter banks and construct the associated biorthogonal wavelets

speckle reduction and contrast enhancement of echocardiograms via multiscale nonlinear processing this paper presents an algorithm for speckle reduction and contrast enhancement of echocardiographic images within a framework of multiscale wavelet analysis the authors apply wavelet shrinkage techniques to eliminate noise while preserving the sharpness of salient features in addition nonlinear processing of feature energy is carried out to enhance contrast within local structures and along object boundaries the authors show that the algorithm is capable of not only reducing speckle but also enhancing features of diagnostic importance such as myocardial walls in two dimensional echocardiograms obtained from the parasternal short axis view shrinkage of wavelet coefficients via soft thresholding within finer levels of scale is carried out on coefficients of logarithmically transformed echocardiograms enhancement of echocardiographic features is accomplished via nonlinear stretching followed by hard thresholding of wavelet coefficients within selected midrange spatial frequency levels of analysis the authors formulate the denoising and enhancement problem introduce a class of dyadic wavelets and describe their implementation of a dyadic wavelet transform their approach for speckle reduction and contrast enhancement was shown to be less affected by pseudo gibbs phenomena the authors show experimentally that this technique produced superior results both qualitatively and quantitatively when compared to results obtained from existing denoising methods alone a study using a database of clinical echocardiographic images suggests that such denoising and enhancement may improve the overall consistency of expert observers to manually defined borders

numerical methods for second order singular pertubation problems abstract by using sufficient conditions which ensure the well conditioning of tridiagonal matrices we derive numerical methods for second order singular pertubations boundary values problems under hypotheses similar to the ones usually used in the continuous theory methods are derived which use constant step sizes h o 1 2 in the more general case methods with variable step sizes are defined which solve the problem with n 1 steps examples on standard test problems are shown

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed

texture classification by wavelet packet signatures this correspondence introduces a new approach to characterize textures at multiple scales the performance of wavelet packet spaces are measured in terms of sensitivity and selectivity for the classification of twenty five natural textures both energy and entropy metrics were computed for each wavelet packet and incorporated into distinct scale space representations where each wavelet packet channel reflected a specific scale and orientation sensitivity wavelet packet representations for twenty five natural textures were classified without error by a simple two layer network classifier an analyzing function of large regularity d sub 20 was shown to be slightly more efficient in representation and discrimination than a similar function with fewer vanishing moments d sub 6 in addition energy representations computed from the standard wavelet decomposition alone 17 features provided classification without error for the twenty five textures included in our study the reliability exhibited by texture signatures based on wavelet packets analysis suggest that the multiresolution properties of such transforms are beneficial for accomplishing segmentation classification and subtle discrimination of texture

fir filter banks for hexagonal data processing images are conventionally sampled on a rectangular lattice thus traditional image processing is carried out on the rectangular lattice the hexagonal lattice was proposed more than four decades ago as an alternative method for sampling compared with the rectangular lattice the hexagonal lattice has certain advantages which include that it needs less sampling points it has better consistent connectivity and higher symmetry the hexagonal structure is also pertinent to the vision process in this paper we investigate the construction of symmetric fir hexagonal filter banks for multiresolution hexagonal image processing we obtain block structures of fir hexagonal filter banks with 3 fold rotational symmetry and 3 fold axial symmetry these block structures yield families of orthogonal and biorthogonal fir hexagonal filter banks with 3 fold rotational symmetry and 3 fold axial symmetry in this paper we also discuss the construction of orthogonal and biorthogonal fir filter banks with scaling functions and wavelets having optimal smoothness in addition we present a few of such orthogonal and biorthogonal fir filters banks

multiresolution detection of spiculated lesions in digital mammograms we present a novel multiresolution scheme for the detection of spiculated lesions in digital mammograms first a multiresolution representation of the original mammogram is obtained using a linear phase nonseparable two dimensional 2 d wavelet transform a set of features is then extracted at each resolution in the wavelet pyramid for every pixel this approach addresses the difficulty of predetermining the neighborhood size for feature extraction to characterize objects that may appear in different sizes detection is performed from the coarsest resolution to the finest resolution using a binary tree classifier this top down approach requires less computation by starting with the least amount of data and propagating detection results to finer resolutions experimental results using the mias image database have shown that this algorithm is capable of detecting spiculated lesions of very different sizes at low false positive rates

boundary value problems and dichotomic stability since the conditioning of a boundary value problem bvp is closely related to the existence of a dichotomic fundamental solution i e where one set of modes is increasing and a complementary set is decreasing it is important to have discretization methods that conserve this dichotomy property the conditions this imposes on such a method are investigated in this paper they are worked out in more detail for scalar second order equations the central difference scheme and for linear first order systems as well for the latter type both one step methods including collocation and multistep methods those that may be used in multiple shooting are examined

multiresolution statistical analysis of high resolution digital mammograms a multiresolution statistical method for identifying clinically normal tissue in digitized mammograms is used to construct an algorithm for separating normal regions from potentially abnormal regions that is small regions that may contain isolated calcifications this is the initial phase of the development of a general method for the automatic recognition of normal mammograms the first step is to decompose the image with a wavelet expansion that yields a sum of independent images each containing different levels of image detail when calcifications are present there is strong empirical evidence that only some of the image components are necessary for the purpose of detecting a deviation from normal the underlying statistic for each of the selected expansion components can be modeled with a simple parametric probability distribution function this function serves as an instrument for the development of a statistical test that allows for the recognition of normal tissue regions the distribution function depends on only one parameter and this parameter itself has an underlying statistical distribution the values of this parameter define a summary statistic that can be used to set detection error rates once the summary statistic is determined spatial filters that are matched to resolution are applied independently to each selected expansion image regions of the image that correlate with the normal statistical model are discarded and regions in disagreement suspicious areas are flagged these results are combined to produce a detection output image consisting only of suspicious areas this type of detection output is amenable to further processing that may ultimately lead to a fully automated algorithm for the identification of normal mammograms

comparison of multiwavelet wavelet haralick and shape features for microcalcification classification in mammograms we present an evaluation and comparison of the performance of four different texture and shape feature extraction methods for classification of benign and malignant microcalcifications in mammograms for 103 regions containing microcalcification clusters texture and shape features were extracted using four approaches conventional shape quantifiers co occurrence based method of haralick wavelet transformations and multi wavelet transformations for each set of features most discriminating features and their optimal weights were found using real valued and binary genetic algorithms ga utilizing a k nearest neighbor classifier and a malignancy criterion for generating roc curves for measuring the performance the best set of features generated areas under the roc curve ranging from 0 84 to 0 89 when using real valued ga and from 0 83 to 0 88 when using binary ga the multi wavelet method outperformed the other three methods and the conventional shape features were superior to the wavelet and haralick features

lagrange wavelets for signal processing this paper deals with the design of interpolating wavelets based on a variety of lagrange functions combined with novel signal processing techniques for digital imaging halfband lagrange wavelets b spline lagrange wavelets and gaussian lagrange lagrange distributed approximating functional daf wavelets are presented as specific examples of the generalized lagrange wavelets our approach combines the perceptually dependent visual group normalization vgn technique and a softer logic masking slm method these are utilized to rescale the wavelet coefficients remove perceptual redundancy and obtain good visual performance for digital image processing

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

the effect of median filtering on edge estimation and detection in this paper we consider the effect of median prefiltering on the subsequent estimation and detection of edges in digital images where possible a quantitative statistical comparison is made for a number of filters defined with two dimensional geometries in some cases one dimensional analyses are required to illustrate certain points noise images prefiltered by median filters defined with a variety of windowing geometries are used to support the analysis and it is found that median prefiltering improves the performance of both thresholding and zero crossing based edge detectors

backward error analysis of cyclic reduction for the solution of tridiagonal systems tridiagonal systems play a fundamental role in matrix computation in particular in recent years parallel algorithms for the solution of tridiagonal systems have been developed among these the cyclic reduction algorithm is particularly interesting here the stability of the cyclic reduction method is studied under the assumption of diagonal dominance a backward error analysis is made yielding a representation of the error matrix for the factorization and for the solution of the linear system the results are compared with those for lu factorization

square character degree graphs yield direct products abstract if g is a solvable group we take g to be the character degree graph for g with primes as vertices we prove that if g is a square then g must be a direct product

irreducible character degree sets of solvable groups in this paper we examine the following general question given a set x of positive integers that includes 1 is there a finite group g whose 4 character degree set cd g s x 1 x g irr g equals x technically we should call cd g the set of irreducible character degrees but these are the only character degrees that we will look at in this paper thus when we say the character degrees of g we mean the irreducible character degrees while this is a very interesting question in order to answer it one must restrict the focus of the question we restrict p to be a prime and m and n to be coprime integers that are greater than 1 and are not divisible 4 by p we look at sets x that have the form x s 1 j p j m j n where 4 4 p is a set of powers of p m m mp with m g 1 and n n np with n g 1 we would like to know under what conditions x s cd g for some solvable group g in this paper we are able to give the following partial answer

determining group structure from the sets of character degrees let cd g be the set of all irreducible complex characters of a finite group g in 4 lewis proved that if p q r and s are distinct primes and cd g 1 p q r pq pr or cd g 1 p q r s pr ps qr qs then g is the direct product of two normal non abelian subgroups of g we generalize lewis results by loosening the primeness hypothesis of cd g

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

steerable pyramids and tight wavelet frames in l_ 2 bbr d we present a functional framework for the design of tight steerable wavelet frames in any number of dimensions the 2 d version of the method can be viewed as a generalization of simoncelli s steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization the backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal the steerable wavelets are obtained by applying a one to many mapping n th order generalized riesz transform to the primal ones the shaping of the steerable wavelets is controlled by an m m unitary matrix where m is the number of wavelet channels that can be selected arbitrarily this allows for a much wider range of solutions than the traditional equiangular configuration steerable pyramid we give a complete functional description of these generalized wavelet transforms and derive their steering equations we describe some concrete examples of transforms including some built around a mallat type multiresolution analysis of l2 rd and provide a fast fourier transform based decomposition algorithm we also propose a principal component based method for signal adapted wavelet design finally we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms the results are in favor of an optimized wavelet design equalized principal component analysis which consistently performs best

a wavelet perspective on variational perceptually inspired color enhancement the issue of perceptually inspired correction of color and contrast in digital images has been recently analyzed with the help of variational principles these techniques allowed building a general framework in which the action of many already existing algorithms can be more easily understood and compared in terms of intensification of local contrast and control of dispersion around the average intensity value in this paper we analyze this issue from the dual perspective of wavelet theory showing that it is possible to build energy functionals of wavelet coefficients that lead to a multilevel perceptually inspired color correction by computing the euler lagrange equations associated to the wavelet based functionals we were able to find an analytical formula for the modification of wavelet detail coefficients that overcomes the problem of an ad hoc selection based on empirical considerations besides these theoretical results the wavelet perspective provides the computational advantage of generating much faster algorithms in comparison with the spatial variational framework

groups with two extreme character degrees and their normal subgroups we study the finite groups g for which the set cd g of irreducible complex character degrees consists of the two most extreme possible values that is 1 and g z g 1 2 we are easily reduced to finite p groups for which we derive the following group theoretical characterization they are the p groups such that ig z g l is a square and whose only normal subgroups are those containing g or contained in z g by analogy we also deal with pgroups such that g z g i p2n 1 is not a square and we prove that cd g 1 pn if and only if a similar property holds for any n g either g n or nz g z g p the proof of these results requires a detailed analysis of the structure of the p groups with any of the conditions above on normal subgroups which is interesting for its own sake it is especially remarkable that these groups have small nilpotency class and that if the nilpotency class is greater than 2 then the index of the centre is small and in some cases we may even bound the order of g

finite groups with real valued irreducible characters of prime degree abstract in this paper we describe the structure of finite groups whose real valued nonlinear irreducible characters have all prime degree the more general situation in which the real valued irreducible characters of a finite group have all squarefree degree is also considered

finite groups whose all irreducible character degrees are hall numbers abstract we call m is a hall number for g if m is the order of a hall subgroup of g that is gcd g m m 1 the aim of this paper is to investigate the structure of the finite group g whose all irreducible character degrees are hall numbers for g

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough