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A contradiction between built-in functions lpc and filter

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zero
zero on 20 Oct 2017
I am using built-in function * lpc* to return the coefficients of a forward linear predictor by minimizing the predictor error; for example:
x = [1,2,4,8]
[a, e] = lpc(x,1)
% print results
a = 1.000000 -0.4941176
e = 16.0617
I thought it should have output `a=1.00, -2`, because I was using a 1-order linear predictor, it was easy to estimate that x(t) = 2*x(t-1).
Something might be wrong because of my superficial understanding. Then I read the `lpc.m` source code, and its comment says:
% Xp(n) = -A(2)*X(n-1) - A(3)*X(n-2) - ... - A(N+1)*X(n-N)
So, a(2) should be -2;
Then I went on reading lpc help document and its example codes, such as:
noise = randn(50000,1);
x = filter(1,[1 1/2 1/3 1/4],noise);
x = x(45904:50000);
a = lpc(x,3)
it printed the result of a with [1.000000000000000 0.488260222978665 0.337341624006294 0.255574741469446] it was exactly equal to the coefficients in filter function.
However, I read the source code of filter.m, it was using another AR model:
% a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-nb)
% - a(2)*y(n-1) - ... - a(na+1)*y(n-na)
but lpc's output result agreed with filter's, how could that happen?
Is there something wrong with me or built-in functions? You known, I need your help. Thanks a lot.

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