How Can I find the Coefficients of a Polynomials by equating Coefficients Method?

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Ethen 2014-2-24

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评论： Ethen 2014-3-1

Hi guys, I have 2 polynomials, one with known coefficients and one with unknown coefficients. Is there any way in Matlab that I can find the unknown coefficients by equating of coefficients of the like power? For instance, P1=2*s^2+1 and P2=a*s^2+b*s+c, to make P1=P2 we will have a=2,b=0 and c=1. Thanks in advance for the help. Cheers,

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Image Analyst 2014-2-24

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If you have a bunch of (s,P1) pairs (at least 3) then you can train a polynomial with polyfit (does a least squares) and it will tell you the best values of a, b, and c to fit your training data. If that what you mean? If not, explain why not.

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John D'Errico 2014-2-25

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编辑：John D'Errico 2014-2-25

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Not too difficult, and I am sure there are various better solutions.

syms s a b c
P1=2*s^2+1;
P2=a*s^2+b*s+c;
C = coeffs(P1 - P2,s);
solve(C(1),c)
ans =
1
solve(C(2),b)
ans =
0
solve(C(3),a)
ans =
2

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John D'Errico 2014-2-26

编辑：John D'Errico 2014-2-26

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Are you positive of that claim?

C = 1 + 2*s + 3*s^2 + 4*s^3;
expand(A*R + B*S - C)
ans =
a*s - 4*s - 2*a + c*s + a*s^2 + b*s^2 + b*s^3 + c*s^2 - 2*s^2 - 3*s^3 - 1
scoeffs = coeffs(expand(A*R + B*S - C),s)
scoeffs =
[ - 2*a - 1, a + c - 4, a + b + c - 2, b - 3]

As we see, a linear system of 4 equations in 3 unknowns, where each of those coefficients must be zero.

I don't even need to use the symbolic TB at this point.

- We must have b = 3 (from the 4th coefficient)
- Likewise, a = -0.5 (from the first)
- So then we know that c = 4 - a = 4.5.

But then a + b + c = 3 - 0.5 + 4.5 = 7, which contradicts what we know from the third coefficient, that the sum must be 2.

So, NO. Sorry, but A and B had no common factor, yet there exists no solution.

Again, the result of this operation is a linear system with 4 equations in 3 unknowns. Linear algebra tells us that there will often be no solution. For SOME SPECIFIC values of C, there may be a solution.

If A,B,C were all given as simple integers, then your statement is correct, we could find integer solutions R and S under the conditions you pose. It seems not to hold for polynomials though.

John D'Errico 2014-2-27

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The point is, there is NO solution to the problem you have posed for most such problems (which contradicts your claims.) I've even shown how to compute a solution IF one exists, despite what seems to be the moving target of your claims.

Suppose I start with your example, but lets look at whether C can ever be a monic polynomial, as you claim?

syms s a b c
A=s^2+s-2;
B=s^2+s;
R=s+a;
S=b*s+c;
C = expand(A*R + B*S)
C =
  a*s - 2*s - 2*a + c*s + a*s^2 + b*s^2 + b*s^3 + c*s^2 + s^2 + s^3
Ccoeffs = coeffs(C,s)
Ccoeffs =
  [ -2*a, a + c - 2, a + b + c + 1, b + 1]

Hmm. Apparently C is never monic UNLESS b = 0. So if a monic polynomial C is somehow important to you, then the problem reduces to a simpler one.

The point is, I think you don't really know what you want here, or what must happen for a solution to exist. You have just been making random claims, then changing them when I've disproved your claims.

Anyway, lets try a simple version of your problem that actually DOES have a solution. There are some rare cases it might seem. Here I'll arbitrarily construct a problem where

a = 2
b = 0
c = 3

Therefore we would have...

R0 = s + 2;
S0 = 3;
C0 = expand(A*R0 + B*S0)
C0 =
  s^3 + 6*s^2 + 3*s - 4

So, for this simple case, can we recover those coefficient for a,b,c? The problem is now well posed.

scoeffs = coeffs(expand(A*R + B*S - C0),s)
abc = solve(scoeffs)
Warning: 4 equations in 3 variables. 
> In /Applications/MATLAB_R2014a.app/toolbox/symbolic/symbolic/symengine.p>symengine at 56
  In mupadengine.mupadengine>mupadengine.evalin at 97
  In mupadengine.mupadengine>mupadengine.feval at 150
  In solve at 170 
abc = 
    a: [1x1 sym]
    b: [1x1 sym]
    c: [1x1 sym]
abc.a
ans =
2
abc.b
ans =
0
abc.c
ans =
3

So IF a solution exists, what I've showed you is how to solve it (and I showed you that long ago.) But my guess is that soon you will have changed the problem. For example, you called this a Diophantine problem before. Did you really mean that?

Ethen 2014-3-1