Numerical methods for linear least squares entails the numerical analysis of linear least squares problems.
Introduction A general approach to the least squares problem m i n ‖ y − X β ‖ 2 {\displaystyle \operatorname {\,min} \,{\big \|}\mathbf {y} -X{\boldsymbol {\beta }}{\big \|}^{2}} can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by
β = S y {\displaystyle {\boldsymbol {\beta }}=S\mathbf {y} }
simply because
X β = X ( S y ) = ( X S ) y {\displaystyle X{\boldsymbol {\beta }}=X(S\mathbf {y} )=(XS)\mathbf {y} }
is exactly a sought for orthogonal projection of y {\displaystyle \mathbf {y} } onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X). A few popular ways to find such a matrix S are described below.
Inverting the matrix of the normal equations The equation
( X T X ) β = X T y {\displaystyle (\mathbf {X} ^{\rm {T}}\mathbf {X} )\beta =\mathbf {X} ^{\rm {T}}y} is known as the normal equation. The algebraic solution of the normal equations with a full-rank matrix XTX can be written as
β ^ = ( X T X ) − 1 X T y = X + y {\displaystyle {\hat {\boldsymbol {\beta }}}=(\mathbf {X} ^{\rm {T}}\mathbf {X} )^{-1}\mathbf {X} ^{\rm {T}}\mathbf {y} =\mathbf {X} ^{+}\mathbf {y} }
where X+ is the Moore–Penrose pseudoinverse of X. Although this equation is correct and can work in many applications, it is not computationally efficient to invert the normal-equations matrix (the Gramian matrix). An exception occurs in numerical smoothing and differentiation where an analytical expression is required. If the matrix XTX is well-conditioned and positive definite, implying that it has full rank, the normal equations can be solved directly by using the Cholesky decomposition RTR, where R is an upper triangular matrix, giving:
R T R β ^ = X T y . {\displaystyle R^{\rm {T}}R{\hat {\boldsymbol {\beta }}}=X^{\rm {T}}\mathbf {y} .}
The solution is obtained in two stages, a forward substitution step, solving for z:
R T z = X T y , {\displaystyle R^{\rm {T}}\mathbf {z} =X^{\rm {T}}\mathbf {y} ,}
followed by a backward substitution, solving for β ^ {\displaystyle {\hat {\boldsymbol {\beta }}}} :
R β ^ = z . {\displaystyle R{\hat {\boldsymbol {\beta }}}=\mathbf {z} .}
Both substitutions are facilitated by the triangular nature of R.
Orthogonal decomposition methods Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product XTX. The residuals are written in matrix notation as
r = y − X β ^ . {\displaystyle \mathbf {r} =\mathbf {y} -X{\hat {\boldsymbol {\beta }}}.}
The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows.
X = Q ( R 0 ) {\displaystyle X=Q{\begin{pmatrix}R\\0\end{pmatrix}}\ } , where Q is an m×m orthogonal matrix (QTQ=I) and R is an n×n upper triangular matrix with r i i > 0 {\displaystyle r_{ii}>0} . The residual vector is left-multiplied by QT.
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