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Math Help - Writing Matices... I know it looks daunting...

  1. #1
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    Exclamation Writing Matices... I know it looks daunting...

    ...but any help is much appreciated!

    Our aim is to find a polynomial of minimal degree passing through N
    points Pi. All our constructions should depend on N.
    The basic data is two matrices X:=matrix(1,N,[...]);
    Y:=matrix(1,N,[...]), whose entries X[1,i] and Y[1,i] are the
    coordinates of the point Pi, for 1 <= i <= N.
    Assume that the the N points lie on the graph of the polynomial
    T(X) = A1 + . . .ANX^(N−1). The coefficients Ai satisfy a system of
    linear equations that can be written M.x = b where M and b depend
    on the coordinates of the points Pi.

    A. Write the matrix M as a function of the matrices X and Y .
    B. Write the matrix b as a function of the matrices X and Y .

    I have no idea. Please help!
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    This is called Lagrange interpolation.

    Let p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0 be the polynomial which satisfies p(x_i)=y_i for 0 \leq j \leq n. Then you have

    a_nx_0^n+a_{n-1}x_0^{n-1}+...+a_0=y_0

    a_nx_1^n+a_{n-1}x_1^{n-1}+...+a_0=y_1

    ...

    a_nx_n^n+a_{n-1}x_n^{n-1}+...+a_0=y_n

    The values x_0,...x_n are known, and this is a linear system of equations in the n+1 unknown parameters a_n,...,a_0. Solving for those will give you the polynomial you seek.

    Note that the matrix of coefficients of the system is a Vandermonde matrix, whose inverse is known explicitly. I suspect that with a bit of cleverness, the inverse of the Vandermonde matrix may be retreived from Lagrange's form of the interpolating polynomial.
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