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By their nature, steepest descent and hill climbing methods use only local information. This isbecause the update from a point x [ k ] depends only on the value of x [ k ] and on the value of its derivative evaluated at that point. This can be a problem,since if the objective function has many minima, the steepest descent algorithm may become “trapped” at a minimum that is not (globally)the smallest. These are called local minima. To see how this can happen, consider the problem of finding the value of x that minimizes the function

J ( x ) = e - 0 . 1 | x | sin ( x ) .

Applying the chain rule, the derivative is

e - 0 . 1 | x | cos ( x ) - 0 . 1 e - 0 . 1 | x | sin ( x ) sign ( x ) ,

where

sign ( x ) = 1 x > 0 - 1 x < 0

is the formal derivative of | x | . Solving directly for the minimum point is nontrivial (try it!). Yet implementing a steepest descentsearch for the minimum can be done in a straightforward manner using the iteration

x [ k + 1 ] = x [ k ] - μ e - 0 . 1 | x [ k ] | · ( cos ( x [ k ] ) - 0 . 1 sin ( x [ k ] ) sign ( x ) ) .

To be concrete, replace the update equation in polyconverge.m with

x(k+1)=x(k)-mu*exp(-0.1*abs(x(k)))*(cos(x(k))...            -0.1* sin(x(k))*sign(x(k)));

Implement the steepest descent strategy to find the minimum of J ( x ) in [link] , modeling the program after polyconverge.m . Run the program for different values of mu , N , and x(1) , and answer the same questions as in Exercise [link] .

One way to understand the behavior of steepest descent algorithms is to plot the error surface , which is basically a plot of the objective as a function of the variablethat is being optimized. [link] (a) displays clearly the single global minimum of the objective function [link] while [link] (b) shows the many minima of the objective function defined by [link] . As will be clear to anyone who has attempted Exercise  [link] , initializing within any one of the valleys causes the algorithmto descend to the bottom of that valley. Although true steepest descent algorithms can never climb over a peak to enter another valley(even if the minimum there is lower) it can sometimes happen in practice when there is a significant amount of noise inthe measurement of the downhill direction.

Essentially, the algorithm gradually descends the error surface by moving in the (locally)downhill direction, and different initial estimates may lead to different minima. Thisunderscores one of the limitations of steepest descent methods—if there are many minima, then it is important to initialize near an acceptable one. In someproblems such prior information may easily be obtained, while in others it may be truly unknown.

The examples of this section are somewhat simple because they involve static functions. Most applications incommunication systems deal with signals that evolve over time, and the next section applies thesteepest descent idea in a dynamic setting to the problem of Automatic Gain Control (AGC). The AGC provides a simple settingin which all three of the major issues in optimization must be addressed: setting the goal, choosing a method of solution, andverifying that the method is successful.

Error surfaces corresponding to (a) the objective function Equation 13 and (b) the objective function Equation 18.
Error surfaces corresponding to (a) the objective function [link] and (b) the objective function [link] .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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