Transforming from Maclaurin to Taylor

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Angus Wong
Angus Wong 2020-5-15
编辑： Angus Wong 2020-5-15
I have a polynomial function in terms of x, in which the highest power never exceeds 6. This function will look like a Maclaurin Series with finite terms. How to tell Matlab to rewrite this function into a Taylor one, which will express the function in terms of (x-a), where a is a real, positive number?
E.g. Original function: f(x)=x^3+x^2+x+1
Transformed functions:
If a=1, g(x)=(x-1)^3+4*(x-1)^2+6*(x-1)+4
if a=2, g(x)=(x-2)^3+7*(x-2)^2+17*(x-2)+15
and etc.
I tried using rem but it returned an error message, saying that f(x) must be a real value in order to use rem.

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William Alberg
William Alberg 2020-5-15
If i understand correctly, your code looks something like this:
syms x
f(x) = x^3 + x^2 + x +1
f(x-1)
f(x-2)
And you want to expand the expressions. This can be done with the expand command:
expand(f(x-1))
>> x^3 - 2*x^2 + 2*x
expand(f(x-2))
>> x^3 - 5*x^2 + 9*x - 5
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William Alberg
William Alberg 2020-5-15
Ahh, I completely misunderstood that!
I dont know how to achieve that, since matlab dont like the "k*(x-a)^1" part, and will rewrite it. I can however get the correct "k*(x-a)^n" for n > 1
My (not complete) solution is this:
f(x) = x^3 + x^2 + x +1;
a = 1;
temp = expand(subs(f(x),x,y + a))
g(x) = subs(temp,y,x-a)
>>6*x + 4*(x - 1)^2 + (x - 1)^3 - 2
Angus Wong
Angus Wong 2020-5-15
编辑：Angus Wong 2020-5-15
I appologise again for my vague wordings - I have no idea what I was typing earlier. Your new solution do help, thanks a lot!

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