Least-squares Approximation
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whose pseudoinverse is
![](https://raw.githubusercontent.com/xibsked/menka/master/books/design-analysis-of-algorithm/c9b99e00c63a24dd3c387cdce19f098a1.png)
Multiplying y by A , we obtain the coefficient vector
![](https://raw.githubusercontent.com/xibsked/menka/master/books/design-analysis-of-algorithm/9a1c1302fd9e8167a7d5cbc4e777a7b01.png)
which corresponds to the quadratic polynomial
F(x) = 1.200 - 0.757x 0.214x2
as the closest-fitting quadratic to the given data, in a least-squares sense.
As a practical matter, we solve the normal equation (28.33) by multiplying y by AT and then finding an LU decomposition of AT A. If A has full rank, the matrix AT A is guaranteed to be nonsingular, because it is symmetric and positive-definite.