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How I can solve a linear system of the form:

$$∑_{k=1}^{j}b_{k}=c_{1}$$

$$∑_{k=1}^{j}b_{k}2^{k}=c_{2}$$

$$\dots$$

$$∑_{k=1}^{j}b_{k}j^{k}=c_{j}$$

Where $c_{1},c_{2},...c_{j}$ are function in a complex variable $s$ and $(b_{k})_{k}$ are the unknowns?

The problem here is I can construct the system matrix, but I have no idea about how I can proceed after.

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  • $\begingroup$ Well, you can always just Lagrange Interpolate on $(1,c_1), (2,c_2 / 2),..., (j, c_j / j)$ $\endgroup$ – dinoboy Jan 25 '13 at 19:36
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Unless I am mistaken, you seem to be solving for the coefficients of a polynomial that you have sampled at the points $(1,c_1)$,...,$(j,c_j)$. The matrix is called a Vandermonde matrix and is easily inverted. Better yet, you can simply use a Lagrange interpolation, which gives an explicit polynomial given your data points.

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