Variable step‐size implementation of sixth‐order Numerov‐type methods

2019 ◽  
Vol 43 (3) ◽  
pp. 1204-1215 ◽  
Author(s):  
Marina A. Medvedeva ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Sixth Order ◽  
Variable Step Size ◽  
Step Size ◽  
Variable Step

scholarly journals Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 387
Author(s):  
Faieza Samat ◽  
Eddie Shahril Ismail
Keyword(s):  
Hybrid Method ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Sixth Order ◽  
Variable Step Size ◽  
Local Error ◽  
Step Size ◽  
Oscillatory Solutions ◽  
Variable Step ◽  
Exponentially Fitted

For the numerical integration of differential equations with oscillatory solutions an exponentially fitted explicit sixth-order hybrid method with four stages is presented. This method is implemented using variable step-size while its derivation is accomplished by imposing each stage of the formula to integrate exactly { 1 , t , t 2 , … , t k , exp ( ± μ t ) } where the frequency μ is imaginary. The local error that is employed in the step-size selection procedure is approximated using an exponentially fitted explicit fourth-order hybrid method. Numerical comparisons of the new and existing hybrid methods for the spring-mass and other oscillatory problems are tabulated and discussed. The results show that the variable step exponentially fitted explicit sixth-order hybrid method outperforms the existing hybrid methods with variable coefficients for solving several problems with oscillatory solutions.

A Variable Step-Size for Sparse Nonlinear Adaptive Filters

2021 ◽  
Author(s):  
Alberto Carini ◽  
Markus V. S. Lima ◽  
Hamed Yazdanpanah ◽  
Simone Orcioni ◽  
Stefania Cecchi
Keyword(s):  
Adaptive Filters ◽  
Variable Step Size ◽  
Step Size ◽  
Variable Step

Active control for vehicle interior noise using the improved iterative variable step-size and variable tap-length LMS algorithms

2019 ◽  
Vol 67 (6) ◽  
pp. 405-414 ◽  
Author(s):  
Ningning Liu ◽  
Yuedong Sun ◽  
Yansong Wang ◽  
Hui Guo ◽  
Bin Gao ◽  
...  
Keyword(s):  
Steady State ◽  
Noise Control ◽  
Convergence Speed ◽  
Interior Noise ◽  
Lms Algorithm ◽  
Variable Step Size ◽  
Step Size ◽  
Vehicle Interior ◽  
Vehicle Interior Noise ◽  
Variable Step

Active noise control (ANC) is used to reduce undesirable noise, particularly at low frequencies. There are many algorithms based on the least mean square (LMS) algorithm, such as the filtered-x LMS (FxLMS) algorithm, which have been widely used for ANC systems. However, the LMS algorithm cannot balance convergence speed and steady-state error due to the fixed step size and tap length. Accordingly, in this article, two improved LMS algorithms, namely, the iterative variable step-size LMS (IVS-LMS) and the variable tap-length LMS (VT-LMS), are proposed for active vehicle interior noise control. The interior noises of a sample vehicle are measured and thereby their frequency characteristics. Results show that the sound energy of noise is concentrated within a low-frequency range below 1000 Hz. The classical LMS, IVS-LMS and VT-LMS algorithms are applied to the measured noise signals. Results further suggest that the IVS-LMS and VT-LMS algorithms can better improve algorithmic performance for convergence speed and steady-state error compared with the classical LMS. The proposed algorithms could potentially be incorporated into other LMS-based algorithms (like the FxLMS) used in ANC systems for improving the ride comfort of a vehicle.

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