Rank: 1715 based on 35 downloads (last 30 days) and 11 files submitted
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Feng Cheng Chang

E-mail
Company/University
Allwave Corporation
Lat/Long
33.853737, -118.36573

Personal Profile:

Professor,
Senior Scientist,
Ph. D. in E.E., University of Alabama,
IEEE Life member,
Retired

Professional Interests:
mathematics, EM waves, Antennas

 

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Files Posted by Feng Cheng View all
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10 Oct 2011 Polynomial division - derived form covolution Polynomial division is derived directly from convolution matrix. Author: Feng Cheng Chang long polynomial divis..., synthetic polynomial ..., convolution polynomia... 1 0
07 Jul 2011 Polynomial division by convolution -- up to finite terms Division of two polynomials by convolution to get up to K terms. Author: Feng Cheng Chang convolution matrix, longhand division, polynomial division, linear algebra 2 0
07 Jul 2011 Polynomial division by convolution - quotient and reminder Division of two polynomials to get quotient and reminder using convolution matrix. Author: Feng Cheng Chang polynomial division, longhand division, linear algebra, synthetic division 2 3
  • 3.0
3.0 | 2 ratings
07 Jan 2011 Solving multiple-root polynomials Find roots and multiplicities of given polynomials using this short compact routine. Author: Feng Cheng Chang polynomial solutions, roots and multiplicit..., rational functions, poles and residues, polynomial divisions, greatest common divis... 7 0
09 Mar 2010 Polynomials with multiple roots solved Find roots of a polynomial with very high degree and multiplicity by this compact routine. Author: Feng Cheng Chang polynomial roots poly... 3 1
Comments and Ratings by Feng Cheng View all
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29 Nov 2009 GCD of Polynomials Find polynomial GCD by "Leading-coefficient Elinimation" Author: Feng Cheng Chang

Pozos:
To answer your question, first find r the GCD of two given polynomials p and q, then, u=p/r, v=q/r and w=u*v.
The partial fraction expansion (PFE) of 1/w will give 1/w=y/u+x/v,
where the two desired polynomials x and y can thus be determined after performing reverse PFE. Thus,
1 = x*u+y*v, or
r=x*p+y*q.
In addition, we may also find the two polynomials t and s, so that
1=u/t+v/s, or
r=p/t+q/s.
It is interesting to note that x and y are expected to be, respectively, equal to 1/t and 1/s. However, it isn't so !
Please let me know if you want find out more detail derivation about these relations.
FC Chang

14 Nov 2009 GCD of Polynomials Find polynomial GCD by "Leading-coefficient Elinimation" Author: Feng Cheng Chang

Pozos:
I do not understand your question. But if you means that you want make a gui from my article, you may go ahead to do so.
Please see my other articles, such as "Solve multiple-root polynomials' that may refer computation of polynomial GCD.
FC Chang
23 Oct 2009 Polynomials with multiple roots solved Find roots of a polynomial with very high degree and multiplicity by this compact routine. Author: Feng Cheng Chang

Self comment:

Amazingly, the revised routine can find the desired roots of test polynomials p(x) -- expanded, such as
       p(x) = (x+987654321)^N,
where N=any positive integers, such as 10, 100, 1000, 10000, ....

For example, find the roots of the following polynomial expanded
       p(x) = (x+987654321)^12345.

>> format long g
>> p = poly([-987654321*ones(1,12345)]);
>> Z = poly_roots(p)
 Z =
          -987654321 12345

It takes about 2 min total time. Most is to create polynomial coefficient vector p, and only 0.5 sec to get the desired root-multiplicity pairs Z.

Of course, any numerical algorithm is not fool prove, neither is mine.

Foe example,it does work for p(x)=(9876x+12345)^70, but fails for p(x)=(9876x+12345)^80.

I hope someone would like to try it for some other existing test polynomials, and give me any valuable comments.

30 Mar 2009 Polynomial division by convolution - quotient and reminder Division of two polynomials to get quotient and reminder using convolution matrix. Author: Feng Cheng Chang

Krishia Prasad:
Please try it again. It should have worked perfectly for both polynomials b(x) and a(x) with trailing zeros.
For the example you gave: b(x)=x^4+1 and a(x)=x^2, yielding the desired results: q(x)=x^2 and r(x)=1.
>> b = [1 0 0 0 1]; a = [1 0 0];
>> [q,r,qc,rc,c] = PolyDiv(b,a)
  q = 1 0 0
  r = 1
  qc = 1 0 0
  rc = 0 1
  c = 1
Also it should be OK for b(x) with leading zero coefficients; however, not for a(x) with trailing zero coefficients.
>> b = [0 0 1 0 0 0 1 0]; a = [1 0 1 0 0];
>> [q,r,qc,rc,c] = PolyDiv(b,a)
 q = 1 0
 r = -1 0 1 0
 qc = 0 0 1 0
 rc = -1 0 1 0
 c = 1
Anyway I thank you for giving me this peculiar case to work with.
I will update the code involving any polynomials with both leading and trailing zero coefficients.
                    Feng Cheng Chang
Comments and Ratings on Feng Cheng's Files View all
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29 Nov 2009 GCD of Polynomials Find polynomial GCD by "Leading-coefficient Elinimation" Author: Feng Cheng Chang Chang, Feng Cheng

Pozos:
To answer your question, first find r the GCD of two given polynomials p and q, then, u=p/r, v=q/r and w=u*v.
The partial fraction expansion (PFE) of 1/w will give 1/w=y/u+x/v,
where the two desired polynomials x and y can thus be determined after performing reverse PFE. Thus,
1 = x*u+y*v, or
r=x*p+y*q.
In addition, we may also find the two polynomials t and s, so that
1=u/t+v/s, or
r=p/t+q/s.
It is interesting to note that x and y are expected to be, respectively, equal to 1/t and 1/s. However, it isn't so !
Please let me know if you want find out more detail derivation about these relations.
FC Chang

24 Nov 2009 GCD of Polynomials Find polynomial GCD by "Leading-coefficient Elinimation" Author: Feng Cheng Chang Pozos, bullpoz

FC Chang:
I would like a favor.I've made a gui that represents the gcd
it is like this: r=poly_gcd(p,q)
i've been asked to make this:
r=xp+ψq
if you understand can you tell me how to show this expression out of the gcd code
Thanks
14 Nov 2009 GCD of Polynomials Find polynomial GCD by "Leading-coefficient Elinimation" Author: Feng Cheng Chang Chang, Feng Cheng

Pozos:
I do not understand your question. But if you means that you want make a gui from my article, you may go ahead to do so.
Please see my other articles, such as "Solve multiple-root polynomials' that may refer computation of polynomial GCD.
FC Chang
09 Nov 2009 GCD of Polynomials Find polynomial GCD by "Leading-coefficient Elinimation" Author: Feng Cheng Chang Pozos, bullpoz

Who can i make a gui of that?please help!
23 Oct 2009 Polynomials with multiple roots solved Find roots of a polynomial with very high degree and multiplicity by this compact routine. Author: Feng Cheng Chang Chang, Feng Cheng

Self comment:

Amazingly, the revised routine can find the desired roots of test polynomials p(x) -- expanded, such as
       p(x) = (x+987654321)^N,
where N=any positive integers, such as 10, 100, 1000, 10000, ....

For example, find the roots of the following polynomial expanded
       p(x) = (x+987654321)^12345.

>> format long g
>> p = poly([-987654321*ones(1,12345)]);
>> Z = poly_roots(p)
 Z =
          -987654321 12345

It takes about 2 min total time. Most is to create polynomial coefficient vector p, and only 0.5 sec to get the desired root-multiplicity pairs Z.

Of course, any numerical algorithm is not fool prove, neither is mine.

Foe example,it does work for p(x)=(9876x+12345)^70, but fails for p(x)=(9876x+12345)^80.

I hope someone would like to try it for some other existing test polynomials, and give me any valuable comments.

Top Tags Applied by Feng Cheng
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linear algebra, polynomial division, polynomial gcd, polynomial roots, control design
Files Tagged by Feng Cheng View all
  • 1 - 5 of
  • 11
Updated   File Tags Downloads
(last 30 days) 		Comments Rating
10 Oct 2011 Polynomial division - derived form covolution Polynomial division is derived directly from convolution matrix. Author: Feng Cheng Chang long polynomial divis..., synthetic polynomial ..., convolution polynomia... 1 0
07 Jul 2011 Polynomial division by convolution -- up to finite terms Division of two polynomials by convolution to get up to K terms. Author: Feng Cheng Chang convolution matrix, longhand division, polynomial division, linear algebra 2 0
07 Jul 2011 Polynomial division by convolution - quotient and reminder Division of two polynomials to get quotient and reminder using convolution matrix. Author: Feng Cheng Chang polynomial division, longhand division, linear algebra, synthetic division 2 3
  • 3.0
3.0 | 2 ratings
07 Jan 2011 Solving multiple-root polynomials Find roots and multiplicities of given polynomials using this short compact routine. Author: Feng Cheng Chang polynomial solutions, roots and multiplicit..., rational functions, poles and residues, polynomial divisions, greatest common divis... 7 0
09 Mar 2010 Polynomials with multiple roots solved Find roots of a polynomial with very high degree and multiplicity by this compact routine. Author: Feng Cheng Chang polynomial roots poly... 3 1

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