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Physical Mathematics by Kevin Cahill

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1.22 Linear least squares

Suppose we have a system of M > N equations in N unknowns xk

image

(1.216)

This problem is over-determined and, in general, has no solution, but it does have an approximate solution due to Carl Gauss (1777–1855).

If the matrix A and the vector y are real, then Gauss’s solution is the N values xk that minimize the sum E of the squares of the errors

image

(1.217)

The minimizing values xk make the N derivatives of E vanish

(1.218)

or in matrix notation Ay = AAx. Since A is real, the matrix AA is nonnegative ...

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