# 2.5 Constrained approximation and mixed criteria  (Page 8/10)

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${\stackrel{^}{a}}_{m+1}={\left[{C}^{T}{W}_{m+1}^{T}{W}_{m+1}C\right]}^{-1}{C}^{T}{W}_{m+1}^{T}{W}_{m+1}{A}_{d}.$

The vector of filter coefficients that is actually used is only partially updated using a form of adjustable step size in the following second orderlinearly weighted sum

${a}_{m+1}=\lambda {\stackrel{^}{a}}_{m+1}+\left(1-\lambda \right){a}_{m}$

Using this filter coefficient vector, we solve for the next error vector by going back to [link] and this defines Karlovitz's IRLS algorithm [link] .

In this algorithm, $\lambda$ is a convergence parameter that takes values $0<\lambda \le 1$ . Karlovitz showed that for the proper $\lambda$ , the IRLS algorithm using [link] always converges to the globally optimal ${L}_{p}$ approximation for $p$ an even integer in the range $4\le p<\infty$ . At each iteration the ${L}_{p}$ error has to be minimized over $\lambda$ which requires a line search. In other words, the full Karlovitz method requires a multi-dimensional weighted least squaresminimization and a one-dimensional ${p}^{th}$ power error minimization at each iteration. Extensions of Karlovitz's work [link] show the one-dimensional minimization is not necessary but practice shows thenumber of required iterations increases considerably and robustness in lost.

Fletcher et al. [link] and later Kahng [link] independently derive the same second order iterative algorithm by applying Newton'smethod. That approach gives a formula for $\lambda$ as a function of $p$ and is discussed later in this paper. Although the iteration count for convergence of the Karlovitz method is good, indeed, perhaps the best ofall, the minimization of $\lambda$ at each iteration causes the algorithm to be very slow in execution.

## Newton's methods

Both the new method in section 4.3 and Lawson's method use a second order updating of the weights to obtain convergence of the basic IRLS algorithm.Fletcher et al. [link] and Kahng [link] use a linear summation for the updating similar in form to [link] but apply it to the filter coefficients in the manner of Karlovitz rather thanthe weights as Lawson did. Indeed, using our development of Karlovitz's method, we see that Kahng's method and Fletcher, Grant, andHebden's method are simply a particular choice of $\lambda$ as a function of $p$ in Karlovitz's method. They derive

$\lambda =\frac{1}{p-1}$

by using Newton's method to minimize $\epsilon$ in [link] to give for [link]

${a}_{m}=\left({\stackrel{^}{a}}_{m}+\left(p-2\right){a}_{m-1}\right)/\left(p-1\right).$

This defines Kahng's method which he says always converges [link] . He also notes that the summation methods in the sections Calculation of the Fourier Transform and Fourier Series using the FFT, Sampling Functions-- the Shah Function, and Down-Sampling, Subsampling, or Decimation do not have the possible restarting problem that Lawson's method theoretically does. Because Kahng's algorithm is a form ofNewton's method, its asymptotic convergence is very good but the initial convergence is poor and very sensitive to starting values.

## A new robust irls method

A modification and generalization of an acceleration method suggested independently by Ekblom [link] and by Kahng [link] is developed here and combined with the Newton's method of Fletcher, Grant,and Hebden and of Kahng to give a robust, fast, and accurate IRLS algorithm [link] , [link] . It overcomes the poor initial performance of the Newton's methods and the poor final performance of the RUL algorithms.

Rather than starting the iterations of the IRLS algorithms with the actual desired value of $p$ , after the initial ${L}_{2}$ approximation, the new algorithm starts with $p=K*2$ where $K$ is a parameter between one and approximately two, chosen for the particular problem specifications.After the first iteration, the value of $p$ is increased to $p={K}^{2}*2$ . It is increased by a factor of $K$ at each iteration until it reaches the actual desired value. This keeps the value of $p$ being approximated just ahead of the value achieved. This is similar to a homotopy where we varythe value of $p$ from 2 to its final value. A small value of $K$ gives very reliable convergence because the approximation is achieved at eachiteration but requires a large number of iterations for $p$ to reach its final value. A large value of $K$ gives faster convergence for most filter specifications but fails for some. The rule that is used to choose ${p}_{m}$ at the ${m}^{th}$ iteration is

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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