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Memory Cards — Chapter 7
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QR solution
QR factorization avoids forming \(A^TA\) and is usually more stable.
Normal equations
\(A^TAx=A^Tb\) are simple but square the condition number.
Least squares
Least squares solves \(\min_x\|Ax-b\|_2\), especially for overdetermined inconsistent data.
Rank deficiency
If \(A\) is rank deficient, solutions may be nonunique; use SVD for minimum-norm solutions.
Regularization
Regularization stabilizes ill-conditioned inverse or fitting problems by penalizing undesirable solutions.
Projection
The residual at the solution is orthogonal to the column space of \(A\).