Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[LeastSquares], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The call leastsqrs(A, b) returns the vector that best satisfies $Ax=b$ in the least-squares sense. The result returned is the vector x which minimizes $\mathrm{norm}\left(Ax-b,2\right)$.

The call leastsqrs(S, v) finds the values for the variables in v which minimize the equations or expressions in S in the least-squares sense. The result returned is a set of equations whose left-hand sides are from v.

For the linear case, if the third optional argument is 'optimize', the routine will find the optimal least square solution (i.e. the vector x with $\mathrm{norm}\left(x\,2\right)$ being the smallest). At present, the matrix entries must be rationals.

The command with(linalg,leastsqrs) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$A≔\mathrm{array}\left(\left[\left[1\,1\,1\right]\,\left[1\,2\,4\right]\,\left[1\,0\,0\right]\,\left[1\,-1\,1\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 4\\ 1& 0& 0\\ 1& \mathrm{-1}& 1\end{array}\right]$

$b≔\mathrm{array}\left(\left[3\,10\,3\,\frac{9}{10}\right]\right)$

$b≔\left[\begin{array}{cccc}3& 10& 3& \frac{9}{10}\end{array}\right]$

$\mathrm{leastsqrs}\left(A\,b\right)$

$\left[\begin{array}{ccc}\frac{327}{200}& \frac{301}{200}& \frac{49}{40}\end{array}\right]$

$S≔\left\{\mathrm{c0}-\frac{9}{10}\,\mathrm{c0}+\mathrm{c1}+\mathrm{c2}-3\,\mathrm{c0}+2\mathrm{c1}+4\mathrm{c2}-10\,\mathrm{c0}-\mathrm{c1}+\mathrm{c2}-3\right\}$

$S≔\left\{\mathrm{c0}-\frac{9}{10}\,\mathrm{c0}-\mathrm{c1}+\mathrm{c2}-3\,\mathrm{c0}+\mathrm{c1}+\mathrm{c2}-3\,\mathrm{c0}+2\mathrm{c1}+4\mathrm{c2}-10\right\}$

$\mathrm{leastsqrs}\left(S\,\left\{\mathrm{c0}\,\mathrm{c1}\,\mathrm{c2}\right\}\right)$

$\left\{\mathrm{c0}=\frac{159}{200}\,\mathrm{c1}=\frac{7}{200}\,\mathrm{c2}=\frac{91}{40}\right\}$

$A≔\mathrm{array}\left(\left[\left[1\,-1\,1\right]\,\left[1\,1\,-2\right]\,\left[2\,0\,-1\right]\right]\right)\:$

$b≔\mathrm{vector}\left(\left[1\,2\,4\right]\right)\:$

$\mathrm{leastsqrs}\left(A\,b\right)$

$\left[\begin{array}{ccc}{\mathrm{\_t}}_1& -5+3{\mathrm{\_t}}_1& 2{\mathrm{\_t}}_1-\frac{11}{3}\end{array}\right]$

$\mathrm{leastsqrs}\left(A\,b\,'\mathrm{optimize}'\right)$

$\left[\begin{array}{ccc}\frac{67}{42}& -\frac{3}{14}& -\frac{10}{21}\end{array}\right]$