unbiased recursive least squares estimation utilizing dichotomous coordinate descent iterations we propose an unbiased recursive least squares algorithm for errors in variables system identification the proposed algorithm called urls removes the noise induced bias when both input and output are contaminated with noise and the input noise is colored and correlated with the output noise to develop the algorithm we define an exponentially weighted least squares optimization problem that yields an unbiased estimate then we solve the system of linear equations of the associated normal equations utilizing the dichotomous coordinate descent iterations the urls algorithm features significantly reduced computational complexity as well as improved numerical stability compared with a previously proposed bias compensated recursive least squares algorithm while having similar estimation performance we show that the urls algorithm is asymptotically unbiased and convergent in the mean square sense we also calculate its steady state mean square deviation simulation results corroborate the efficacy of the urls algorithm and the accuracy of the theoretical findings

computation of optimal feedforward and feedback control by a modified reduced gradient method a computational scheme for solving problems of optimal open loop control policy and optimal state feedback control was introduced the problems were first converted into a nonlinear programming problem the conjugate reduced gradient method was then applied to find the minimum solution the sensitivity matrices of the state trajectories with respect to the coefficients of control variables could be recursively calculated so that the reduced gradients could be easily evaluated in this computational algorithm the h orthogonal property for linear optimal control problems was demonstrated such that the global minimum solution of such problems was thus achieved in finite iterations using this algorithm the computational strategy was easily and efficiently extended to determine the optimal state feedback gain matrices

a chebyshev expansion method for solving nonlinear optimal control problems this paper presents a numerical technique for solving the controlled duffing oscillator the control and state variables are approximated by chebyshev series a technique is provided for approximating the system dynamics boundary conditions and performance index this technique is based on using an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives the system dynamics and the performance index are converted into some algebraic equations then the optimal control problem is reduced to constrained optimization problem results and comparisons are given at the end of this paper

lossless convexification of non convex optimal control problems for state constrained linear systems this paper analyzes a class of finite horizon optimal control problems with mixed non convex and convex control constraints and linear state constraints a convex relaxation of the problem is proposed and it is proved that a solution of the relaxed problem is also a solution of the original problem this process is called lossless convexification and its generalization for problems with state constraints is the primary contribution of the paper doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non convex problem the approach is also demonstrated on example problems

a fast recursive total least squares algorithm for adaptive fir filtering this paper proposes a new fast recursive total least squares n rtls algorithm to recursively compute the tls solution for adaptive finite impulse response fir filtering the n rtls algorithm is based on the minimization of the constrained rayleigh quotient c rq in which the last entry of the parameter vector is constrained to the negative one as analysis results on the convergence of the proposed algorithm we study the properties of the stationary points of the c rq the high computational efficiency of the new algorithm depends on the efficient computation of the fast gain vector fgv and the adaptation of the c rq since the last entry of the parameter vector in the c rq has been fixed as the negative one a minimum point of the c rq is searched only along the input data vector and a more efficient n rtls algorithm is obtained by using the fgv as compared with davila s rtls algorithms the n rtls algorithm saves the 6m number of multiplies divides and square roots mads the global convergence of the new algorithm is studied by lasalle s invariance principle the performances of the relevant algorithms are compared via simulations and the long term numerical stability of the n rtls algorithm is verified

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

brief numerical treatment of multiobjective optimal control problems we present a numerical procedure for solving optimal control problems with both linear terminal constraints and multiple criteria using a chebyshev spectral procedure the problem reduces to a constrained optimization problem which can be solved using hybrid penalty partial quadratic interpolation hppqi technique the proposed procedure compares quite favorably with other methods on a sample of well known examples

low complexity rls algorithms using dichotomous coordinate descent iterations in this paper we derive low complexity recursive least squares rls adaptive filtering algorithms we express the rls problem in terms of auxiliary normal equations with respect to increments of the filter weights and apply this approach to the exponentially weighted and sliding window cases to derive new rls techniques for solving the auxiliary equations line search methods are used we first consider conjugate gradient iterations with a complexity of operations per sample being the number of the filter weights to reduce the complexity and make the algorithms more suitable for finite precision implementation we propose a new dichotomous coordinate descent dcd algorithm and apply it to the auxiliary equations this results in a transversal rls adaptive filter with complexity as low as multiplications per sample which is only slightly higher than the complexity of the least mean squares lms algorithm multiplications simulations are used to compare the performance of the proposed algorithms against the classical rls and known advanced adaptive algorithms fixed point fpga implementation of the proposed dcd based rls algorithm is also discussed and results of such implementation are presented

a chebyshev technique for solving nonlinear optimal control problems a numerical technique for solving nonlinear optimal control problems is introduced the state and control variables are expanded in the chebyshev series and an algorithm is provided for approximating the system dynamics boundary conditions and performance index application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the chebyshev coefficients the optimum condition is obtained by applying the method of constrained extremum for linear quadratic optimal control problems the state and control variables are determined by solving a set of linear equations in the chebyshev coefficients applicability is illustrated with the minimum time and maximum radius orbit transfer problems

an efficient chebyshev algorithm for the solution of optimal control problems in this paper we derive an efficient chebyshev algorithm for solving optimal control problems the chebyshev expansions are employed to approximate both the control and the state functions the discretizing process and the related techniques are unique compared to existing methods the optimal control problems are transformed into the resulting mathematical programming problems theoretical analysis is given to support the method further numerical examples and comparisons are presented to illustrate the efficiency of the method

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

the parameterization method in optimal control problems and differential algebraic equations the paper is devoted to the explanation of the numerical parameterization method pm for optimal control oc problems with intermediate phase constraint and to its circumstantiation for classical calculus of variation cv problems that arise in connection with singular odes or daes especially in cases of their essential degeneracy the pm is based on a finite parameterization of control functions and on derivation of the problem functional with respect to control parameters the first and the second derivatives are calculated with the help of adjoint vector and matrix impulses results of the solution to one phase constrained oc and two degenerate cv problems connected with singular daes nonreducible to the normal form are presented

bias compensation for the bearings only pseudolinear target track estimator the bearings only pseudolinear target track estimator is known to suffer from severe bias problems this paper presents a bias analysis for the pseudolinear estimator and develops a method of bias compensation resulting in a closed form reduced bias pseudolinear estimator the reduced bias estimator is then incorporated into an instrumental variable estimator to produce asymptotically unbiased target motion parameter estimates unlike batch iterative estimators the proposed instrumental variable estimator has a closed from solution and therefore avoids the convergence problems associated with iterative estimators the performance of the proposed instrumental variable estimator is illustrated by way of simulation examples and is shown to be almost identical to that of the computationally more demanding iterative maximum likelihood estimator

fast approximate inverse power iteration algorithm for adaptive total least squares fir filtering the presence of contaminating noises at both the input and the output of an finite impulse response fir system constitutes a major impediment to unbiased parameter estimation the total least squares tls method is known to be effective in achieving unbiased estimation in this correspondence we develop a fast recursive algorithm with a view to finding the tls solution for adaptive fir filtering given the fact that the tls solution is obtainable via inverse power iteration we introduce a novel but approximate inverse power iteration in combination with galerkin method so that the tls solution can be updated adaptively at a lower computational cost we also take advantage of the regular form of the tls solution to constrain the last element of the filter parameter vector to the negative one we further reduce the computational complexity of the developed algorithm by making efficient computation of the fast gain vector defined in and using rank one update of the augmented autocorrelation matrix the developed algorithm saves seven m mad s number of multiplies divides and square roots when compared with the recursive tls algorithm in moreover unlike the algorithms given in and the developed algorithm does not deal with the solution to a one variable quadratic equation and it avoids square root operation therefore it has the simpler structure and may be more easily implemented we then make a careful investigation into global convergence of the developed algorithm simulation results are provided that clearly illustrate appealing performance of the developed algorithm including its good long term numerical stability

numerical method for a class of optimal control problems subject to nonsmooth functional constraints in this paper we consider a class of optimal control problems which is governed by nonsmooth functional inequality constraints involving convolution first we transform it into an equivalent optimal control problem with smooth functional inequality constraints at the expense of doubling the dimension of the control variables then using the chebyshev polynomial approximation of the control variables we obtain an semi infinite quadratic programming problem at last we use the dual parametrization technique to solve the problem

the signature of γ 0 n abstract the signature of 0 en2 where e is square free is completely determined if e 3 or n is odd or e where is the set of all square free integers e n such that i if is odd then admits no divisors of the form 8 3 ii if is even then admits no divisors of the form 8 7 in particular v2 number of elliptic classes of period 2 of 0 n is expressed as a linear combination of multiplicative functions

on the distribution of eigenvalues of maass forms on certain moonshine groups in this paper we study both analytically and numerically questions involving the distribution of eigen values of maass forms on the moonshine groups 0 n where n 1 is a square free integer after we prove that 0 n has one cusp we compute the constant term of the associated non holomorphic eisenstein series we then derive an average weyl s law for the distribution of eigenvalues of maass forms from which we prove the classical weyl s law as a special case the groups corresponding to n 5 and n 6 have the same signature however our analysis shows that asymptotically there are infinitely more cusp forms for 0 5 than for 0 6 we view this result as being consistent with the phillips sarnak philosophy since we have shown unconditionally the existence of two groups which have different weyl s laws in addition we employ hejhal s algorithm together with recently developed refinements from 32 and numerically determine the first 3557 of 0 5 and the first 12474 eigenvalues of 0 6 with this information we empirically verify some conjectured distributional properties of the eigenvalues

qr based tls and mixed ls tls algorithms with applications to adaptive iir filtering the least squares ls total least squares tls and mixed ls tls approaches are compared as to their properties and performance on several classical filtering problems mixed ls tls is introduced as a qr decomposition based algorithm for unbiased equation error adaptive infinite impulse response iir filtering the algorithm is based on casting adaptive iir filtering into a mixed ls tls framework this formulation is shown to be equivalent to the minimization of the mean square equation error subject to a unit norm constraint on the denominator parameter vector an efficient implementation of the mixed ls tls solution is achieved through the use of back substitution and inverse iteration unbiasedness of the system parameter estimates is established for the mixed ls tls solution in the case of uncorrelated output noise and the algorithm is shown to converge to this solution ls tls and mixed ls tls performance is then compared for the problems of echo cancellation noise reduction and frequency equalization

bias correction in least squares identification abstract the least squares method in system identification leads generally to biased parameter estimates a conceptually simple modification is to estimate the bias and to compute compensated parameter estimates when white output noise is the only disturbance this principle compensated least squares cls can readily be used to obtain consistent estimates the main purpose of the paper is to investigate the accuracy of the parameter estimates obtained when the cls method is used it is proved that the estimates are asymptotically gaussian distributed an explicit expression for the covariance matrix is given it is also shown that a commonly used instrumental variable method gives asymptotically better accuracy than the compensated least squares method

optimality condition of the nonlinear impulsive system in fed batch fermentation abstract based on the nonlinear impulsive system of fed batch fermentation which is used in the process of bio dissimilation of glycerol to 1 3 propanediol by klebsiella pneumoniae this paper is concerned with the optimal control of the volumes of infused glycerol at impulsive moments the optimal control model is developed the authors study the properties of both the nonlinear impulsive system and the solution of it and ascertain the existence of the optimal control then the optimality condition is obtained which can be used as a necessary condition that the optimal control should satisfy

a pmgop method of parameter identification for a class of continuous nonlinear systems a series of piecewise multiple general orthogonal polynomials pmgops is introduced and applied to the parameter identication problem for a class of continuous nonlinear systems after introducing pmgops and discussing their main properties an effective procedure for the parameter identification of a large class of continuous systems called parameter separable systems is proposed some relevant algorithms are presented the procedure given in the paper has the following advantages compared with other methods the identification algorithm ia obtained is computationally fast and accurate the ia can be implemented in a recursive fashion the ia is effective for a small number of data points the ia does not require a priori knowledge of the estimated parameters the ia is tolerant to choices of expanding orders when pmgops are applied some numerical examples together with an experimental example taken from a biochemical fermentation process are given for illustrative purposes

brief paper a unified framework for the numerical solution of optimal control problems using pseudospectral methods a unified framework is presented for the numerical solution of optimal control problems using collocation at legendre gauss lg legendre gauss radau lgr and legendre gauss lobatto lgl points it is shown that the lg and lgr differentiation matrices are rectangular and full rank whereas the lgl differentiation matrix is square and singular consequently the lg and lgr schemes can be expressed equivalently in either differential or integral form while the lgl differential and integral forms are not equivalent transformations are developed that relate the lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem the lg and lgr discrete costate systems are full rank while the lgl discrete costate system is rank deficient the lgl costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the lgl discrete costate system an example is considered to assess the accuracy and features of each collocation scheme

reduced complexity constrained recursive least squares adaptive filtering algorithm a linearly constrained recursive least squares adaptive filtering algorithm based on the method of weighting and the dichotomous coordinate descent dcd iterations is proposed the method of weighting is employed to incorporate the linear constraints into the least squares problem the normal equations of the resultant unconstrained least squares problem are then solved using the dcd iterations the proposed algorithm has a significantly smaller computational complexity than the previously proposed constrained recursive least square crls algorithm while delivering convergence performance on par with crls the effectiveness of the proposed algorithm is demonstrated by simulation examples

brief paper lossless convexification of a class of optimal control problems with non convex control constraints we consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost convex state constraints and non convex control constraints we propose a convex relaxation of the non convex control constraints and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem which is referred to as the lossless convexification of the optimal control problem the lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non convex optimal control problem the solution approach is demonstrated on a number of planetary soft landing optimal control problems

state control spectral chebyshev parameterization for linearly constrained quadratic optimal control problems in this paper we propose a computationally attractive numerical method for determining the optimal control of constrained linear dynamic systems with a quadratic performance the method is based upon constructing the mth degree interpolating polynomials using chebyshev nodes to approximate the control and the state vectors the system dynamics are collocated at chebyshev nodes the performance index is discretized by a cell averaging method the state and control inequality constraints are converted into algebraic inequalities through collocation at the nodes the linear quadratic optimal control problem is thereby transformed into a quadratic programming one simulation studies demonstrate computational advantages relative to a standard riccati method a classical chebyshev based method fourier based method and other methods in the literature

optimal control for nonlinear dynamical system of microbial fed batch culture in fed batch culture of glycerol bio dissimilation to 1 3 propanediol 1 3 pd the aim of adding glycerol is to obtain as much 1 3 pd as possible so a proper feeding rate is required during the process taking the concentration of 1 3 pd at the terminal time as the performance index and the feeding rate of glycerol as the control function we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non stationary control a computational approach is constructed to seek the solution of the above model in two aspects on the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space on the other hand by approximating the control function with simple functions we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems which can be solved using gradient based optimization techniques the convergence analysis of this approximation is also investigated numerical results show that by employing the optimal control policy the concentration of 1 3 pd at the terminal time can be increased considerably

groups of units of zero ternary quadratic forms the groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to fuchsian groups and which for zero forms are commensurable with the classical modular group this is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms the signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms this last result should be compared with the corresponding result by mennicke on non zero forms

analysis of the gradient descent total least squares adaptive filtering algorithm the gradient descent total least squares gd tls algorithm is a stochastic gradient adaptive filtering algorithm that compensates for error in both input and output data we study the local convergence of the gd tls algoritlun and find bounds for its step size that ensure its stability we also analyze the steady state performance of the gd tls algorithm and calculate its steady state mean square deviation our steady state analysis is inspired by the energy conservation based approach to the performance analysis of adaptive filters the results predicted by the analysis show good agreement with the simulation experiments

cell averaging chebyshev methods for hyperbolic problems this paper describes a cell averaging method for the chebyshev approximations of first order hyperbolic equations in conservation form we present formulas for transforming between pointwise data at the collocation points and cell averaged quantities and vice versa this step trivial for the finite difference and fourier methods is nontrivial for the global polynomials used in spectral methods we then prove that the cell averaging methods presented are stable for linear scalar hyperbolic equations and present numerical simulations of shock density wave interaction using the new cell averaging chebyshev methods

analysis of gradient algorithms for tls based adaptive iir filters steepest descent gradient algorithms for unbiased equation error adaptive infinite impulse response iir filtering are analyzed collectively for both the total least squares and mixed least squares total least squares framework these algorithms have a monic normalization that allows for a direct filtering implementation we show that the algorithms converge to the desired filter coefficient vector we achieve the convergence result by analyzing the stability of the equilibrium points and demonstrate that only the desired solution is locally stable additionally we describe a region of initialization under which the algorithm converges to the desired solution we derive the results using interlacing relationships between the eigenvalues of the data correlation matrices and their respective schur complements finally we illustrate the performance of these new approaches through simulation

recursive total least squares algorithm based on inverse power method and dichotomous coordinate descent iterations we develop a recursive total least squares rtls algorithm for errors in variables system identification utilizing the inverse power method and the dichotomous coordinate descent dcd iterations the proposed algorithm called dcd rtls outperforms the previously proposed rtls algorithms which are based on the line search method with reduced computational complexity we perform a comprehensive analysis of the dcd rtls algorithm and show that it is asymptotically unbiased as well as being stable in the mean we also find a lower bound for the forgetting factor that ensures mean square stability of the algorithm and calculate the theoretical steady state mean square deviation msd we verify the effectiveness of the proposed algorithm and the accuracy of the predicted steady state msd via simulations

learning continuous trajectories in recurrent neural networks with time dependent weights the paper is concerned with a general learning with arbitrary criterion and state dependent constraints of continuous trajectories by means of recurrent neural networks with time varying weights the learning process is transformed into an optimal control framework where the weights to be found are treated as controls a learning algorithm based on a variational formulation of pontryagin s maximum principle is proposed this algorithm is shown to converge under reasonable conditions to an optimal solution the neural networks with time dependent weights make it possible to efficiently find an admissible solution i e initial weights satisfying state constraints which then serves as an initial guess to carry out a proper minimization of a given criterion the proposed methodology may be directly applicable to both classification of temporal sequences and to optimal tracking of nonlinear dynamic systems numerical examples are also given which demonstrate the efficiency of the approach presented

data smoothing and interpolation using eighth order algebraic splines a new type of algebraic spline is used to derive a filter for smoothing or interpolating discrete data points the spline is dependent on control parameters that specify the relative importance of data fitting and the derivatives of the spline a general spline of arbitrary order is first formulated using matrix equations we then focus on eighth order splines because of the continuity of their first three derivatives desirable for motor and robotics applications the spline s matrix equations are rewritten to give a recursive filter that can be implemented in real time for lengthy data sequences the filter is lowpass with a bandwidth that is dependent on the spline s control parameters numerical results including a simple image processing application show the tradeoffs that can be achieved using the algebraic splines

numerical methods based on extended one step methods for solving optimal control problems in this article we develop extended one step methods for solving the dynamic system of optimal control problems governed by ordinary differential equations the control variables are approximated by polynomial functions the proposed problem is reduced to either a constrained or unconstrained minimization problem according to the nature of the dynamic system and the given conditions numerical results and comparisons with other methods are presented

legendre series approach to identification and analysis of linear systems in this paper the legendre operational matrix of integration is introduced and it is subsequently used for the identification and analysis of linear systems the algorithms proposed are easily implementable in a digital computer and they are analogous to those already derived for other types of orthogonal functions

a chebyshev approximation for solving optimal control problems abstract this paper presents a numerical solution for solving optimal control problems and the controlled duffing oscillator a new chebyshev spectral procedure is introduced control variables and state variables are approximated by chebyshev series then the system dynamics is transformed into systems of algebraic equations the optimal control problem is reduced to a constrained optimization problem results and comparisons are given at the end of the paper

the pseudospectral legendre method for discretizing optimal control problems this paper presents a computational technique for optimal control problems including state and control inequality constraints the technique is based on spectral collocation methods used in the solution of differential equations the system dynamics are collocated at legendre gauss lobatto points the derivative x spl dot t of the state x t is approximated by the analytic derivative of the corresponding interpolating polynomial state and control inequality constraints are collocated at legendre gauss lobatto nodes the integral involved in the definition of the performance index is discretized based on the gauss lobatto quadrature rule the optimal control problem is thereby converted into a mathematical programming program thus existing well developed optimization algorithms may be used to solve the transformed problem the method is easy to implement capable of handling various types of constraints and yields very accurate results illustrative examples are included to demonstrate the capability of the proposed method and a comparison is made with existing methods in the literature

on rademacher s conjecture congruence subgroups of genus zero of the modular group we list all genus zero congruence subgroups of psl2 z there are altogether 132 of them up to conjugation in psl2 z geometrical invariants genus v2 v3 number of cusps index in psl2 z fundamental polygons farey symbol and independent generators of all such groups are determined generators of the function fields associated to such groups are also determined http www math nus edu sg matlml

a numerical solution of problems in calculus of variation using direct method and nonclassical parameterization in this paper a direct method for solving variational problems using nonclassical parameterization is presented a nonclassical parameterization based on nonclassical orthogonal polynomials is first introduced to reduce a variational problem to a nonlinear mathematical programming problem then using the lagrange multiplier technique the problem is converted to that of solving a system of algebraic equations illustrative examples are given to demonstrate the validity and applicability of the technique

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods