Backward consistency concept and round-off error propagation dynamics in recursive least-squares algorithms

Slock, Dirk

doi:10.1117/12.57673

Cited by 44 publications

(61 citation statements)

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“…For the conventional RLS algorithms, the round-off error propagation is the best studied of the three aforementioned subproblems [3]- [5]. Such studies typically examine the linearized round-off error propagation system and focus on the derivation of its exponential stability; this, in turn, implies local exponential stability of the corresponding nonlinear round-off error propagation system [6, pp.…”

Section: Introductionmentioning

confidence: 99%

On the numerical stability and accuracy of the conventional recursive least squares algorithm

Liavas¹,

Regalia²

1999

IEEE Trans. Signal Process.

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Abstract-We study the nonlinear round-off error accumulation system of the conventional recursive least squares algorithm, and we derive bounds for the relative precision of the computations in terms of the conditioning of the problem and the exponential forgetting factor, which guarantee the numerical stability of the finite-precision implementation of the algorithm; the positive definiteness of the finite-precision inverse data covariance matrix is also guaranteed. Bounds for the accumulated round-off errors in the inverse data covariance matrix are also derived. In our simulations, the measured accumulated roundoffs satisfied, in steady state, the analytically predicted bounds. We consider the phenomenon of explosive divergence using a simplified approach; we identify the situations that are likely to lead to this phenomenon; simulations confirm our findings.

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Section: Introductionmentioning

confidence: 99%

On the numerical stability and accuracy of the conventional recursive least squares algorithm

Liavas¹,

Regalia²

1999

IEEE Trans. Signal Process.

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“…If so, how large, and under what circumstances will perturbations be allowed? The question of consistency has been studied by Regalia and Slock (see References [2,4], for example). If a ÿlter can only withstand slight perturbations of its parameters before the resultant solution is rendered meaningless, there should be guidelines which would guarantee that the perturbations are within the tolerance to maintain consistency.…”

Section: Notationmentioning

confidence: 99%

“…Although numerous variations of the adaptive ÿlters have been, and are being, designed and implemented in ÿnite precision, the question of algorithmic stability continues to initiate further research to ÿnd better and more robust algorithmic designs [2][3][4]. For example, Ling and Manolakis [5] demonstrated through their experiments that an algebraically equivalent re-expression of the a priori and a posteriori RLSL algorithm greatly enhances the performance of the algorithm when the computations were performed using computer arithmetic.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the direct and indirect a posteriori RLSL algorithm

Bunch

Borne

Proudler³

2001

Numerical Linear Algebra App

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SUMMARYThe numerical properties of two implementations of the a posteriori recursive least squares algorithm, the direct and the indirect algorithms, are analysed in order to qualify their numerical trustworthiness. Although both implementations include update recursions that have the potential for large relative errors, the manner in which they are propagated into the update recursions for the re ection coe cients is shown to di er. It is this di erence that allows for the direct implementation to be more numerically trustworthy with respect to the indirect implementation.

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“…The derivation starts by considering the partially triangularized input matrix (6) where , and a similar definition holds for The element vectors and denote the residual vectors of forward and backward prediction operations. Thus, the input signal matrix is triangularized, except for its first and last columns and rows.…”

Section: Sliding Window Fast Qr-lattice Algorithmmentioning

confidence: 99%

“…The most two well-known alternatives to plane hyperbolic rotations are the Chambers' method, which is also known as stabilized hyperbolic rotations [3], and the LINPACK method [4]. It is shown in [5] that these two methods satisfy the "relational stability" condition, which is claimed to help maintain better numerical accuracy, although backward stability [6]- [8] is not guaranteed. We show in the simulations that relational stability may worsen the numerical conditioning of the fast algorithms for short wordlengths.…”

mentioning

confidence: 99%

Sliding window adaptive fast QR and QR-lattice algorithms

Baykal

Constantinides

1998

IEEE Trans. Signal Process.

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Abstract-Sliding window formulations of the fast QR and fast QR-lattice algorithms are presented. The derivations are based on the partial triangularization of raw data matrices. Three methods for window downdating are discussed: the method of plane hyperbolic rotations, the Chambers' method, and the LINPACK algorithm. A numerically ill-conditioned stationary signal and a speech signal are used in finite wordlength simulations of the full QR (nonfast), fast QR, and QR-lattice algorithms. All algorithms are observed to be numerically stable over billions of iterations for double-precision mantissas (53 bits), but as the number of bits is decreased in the mantissa, the algorithms exhibit divergent behavior. Hence, practically, the algorithms can be regarded as numerically stable for long wordlengths.

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Backward consistency concept and round-off error propagation dynamics in recursive least-squares algorithms

Cited by 44 publications

References 0 publications

On the numerical stability and accuracy of the conventional recursive least squares algorithm

On the numerical stability and accuracy of the conventional recursive least squares algorithm

Analysis of the direct and indirect a posteriori RLSL algorithm

Sliding window adaptive fast QR and QR-lattice algorithms

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