Generalizing from a straight line (i.e., first degree polynomial) to a th degree polynomial
(1)
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the residual is given by
(2)
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The partial derivatives (again dropping superscripts) are
(3)
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(4)
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(5)
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These lead to the equations
(6)
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(7)
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(8)
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or, in matrix form
(9)
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This is a Vandermonde matrix. We can also obtain the matrix for a least squares fit by writing
(10)
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Premultiplying both sides by the transpose of the first matrix then gives
(11)
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so
(12)
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As before, given points and fitting with polynomial coefficients , ..., gives
(13)
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In matrix notation, the equation for a polynomial fit is given by
(14)
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This can be solved by premultiplying by the transpose ,
(15)
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This matrix equation can be solved numerically, or can be inverted directly if it is well formed, to yield the solution vector
(16)
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Setting in the above equations reproduces the linear solution.