Matrix Functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix Functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.

Modified Normalized Least Mean Square (MNLMS) algorithm, which is a sign form of NLMS based on set-membership (SM) theory in the class of Optimal Bounding ellipsoid (OBE) algorithms, requires a priori knowledge of error bounds that is unknown in most applications. In a special but popular case of measurement noise, a simple algorithm has been proposed. With some simulation examples the performance of algorithm is compared with MNLMS.

Background and Objectives: Since exact manner of BKZ algorithm for higher block sizes cannot be studied by practical running, therefore simulation of BKZ is used to predict the total cost of BKZ and quality of output basis. This paper revises some main components of BKZ-simulation for better predictions.Methods: At first, by definition of full-enumeration success probability, the Optimal enumeration radius is formally defined. Next, this paper defines three more pruning types, besides the well-known pruning by Bounding Function in GNR-enumerations, and consequently uses these four pruning types collectively in revision of success probability estimation. Also, by using these four pruning types and the process of updating-radius, this paper revises the estimation of enumeration cost. Finally, this paper introduces a simple technique to generate partially better Bounding Functions. Results: For block sizes of 50≤β≤240, better domains of radius parameters in GNR enumeration are formally introduced. Also, our revised estimation of success probability (for GNR Bounding Function) in our test results shows non-negligible gap from former estimations in some main former studies. Moreover, our results show that the cost results by our proposed estimator of GNR-enumeration cost are closer to the cost results determined in experimental running of enumeration, than the cost results by Chen-Nguyen estimator.Conclusion: This paper revises the estimators of cost and success probability for GNR-Enumeration, and justifies the value of these revised estimators by sufficient test results (in actual running and simulation of BKZ). Also our novel definition of Optimal enumeration radius can be used effectively in actual running and simulation of BKZ.

This paper studies the effect of flexible linear torso on the dynamics of passive quadruped Bounding. A reduced-order passive and conservative model with linear flexible torso and springy legs is introduced. The model features extensive spine deformation during high-speed Bounding, resembling those observed in a cheetah. Fixed points corresponding to cyclic Bounding motions are found and calculated using numerical return map methodologies. Results show that the corresponding robot gaits and the associated performance resemble those of its natural counterparts.

The exact manner of BKZ algorithm for higher block sizes cannot be studied by practical running, so simulation of BKZ can be used to predict the total cost and output quality of BKZ algorithm. Sampling method of enumeration solution vector v is one of the main components of designing BKZ-simulation and can be divided into two phases: sampling norm of solution vector v and sampling corresponding coefficient vectors. This paper introduces a simple and efficient idea for sampling the norm of enumeration solution v for any success probability of enumeration Bounding Functions, while to the best of our knowledge, no such sampling method for norm of enumeration solution is proposed in former studies. Next, this paper analyzes the structure and probability distribution of coefficient vectors (corresponding with enumeration solution v), and consequently introduces the sampling methods for these coefficient vectors which are verified by our test results, while no such a deep analysis for sampling coefficient vectors is considered in design of former BKZ-simulations. Moreover, this paper proposes an approximation for cost of enumerations pruned by Optimal Bounding Functions.