Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.

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

backsub generates a solution vector x to the equation $Ux=b$.

If b is omitted, or b is 'false' then U is assumed to be an augmented matrix and the last column of U is used in place of b.

If b is a matrix, then x (the solution) will also be a matrix with the same number of columns.

If U is the result of applying forward Gaussian elimination to the augmented matrix of a system of linear equations, as might be obtained from gausselim or gaussjord, backsub completes the solution by back substitution.  If a solution exists, it is returned as a vector.  If no solution exists, an error will be generated.

If the solution is not unique, it will be parameterized in terms of the symbols v[1], v[2], ..., etc. or v[1,k],v[2,k], ... as in the case where b is a matrix. If the third argument v is not specified, the global variable _t will be used.

The input matrix must be in row-echelon form with all zero rows grouped at bottom. Such a matrix is produced by applying gausselim or gaussjord to the augmented matrix of a system of linear equations or by obtaining the LU decomposition.

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

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

$A≔\mathrm{randmatrix}\left(3\,4\right)\:$

$F≔\mathrm{gausselim}\left(A\right)$

$F≔\left[\begin{array}{cccc}\mathrm{-7}& 22& \mathrm{-55}& \mathrm{-94}\\ 0& \frac{1522}{7}& -\frac{4785}{7}& -\frac{8612}{7}\\ 0& 0& -\frac{56633}{1522}& -\frac{56043}{761}\end{array}\right]$

$\mathrm{backsub}\left(F\right)$

$\left[\begin{array}{ccc}-\frac{19802}{56633}& \frac{31937}{56633}& \frac{112086}{56633}\end{array}\right]$

$H≔\mathrm{matrix}\left(\left[\left[1\,2\,3\right]\,\left[2\,1\,3\right]\,\left[1\,-1\,0\right]\right]\right)\:$

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

$A≔\mathrm{augment}\left(H\,v\right)$

$A≔\left[\begin{array}{cccc}1& 2& 3& 1\\ 2& 1& 3& 2\\ 1& \mathrm{-1}& 0& 1\end{array}\right]$

$F≔\mathrm{gaussjord}\left(A\right)$

$F≔\left[\begin{array}{cccc}1& 0& 1& 1\\ 0& 1& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

$\mathrm{backsub}\left(F\right)$

$\left[\begin{array}{ccc}1-{\mathrm{\_t}}_1& -{\mathrm{\_t}}_1& {\mathrm{\_t}}_1\end{array}\right]$

$\mathrm{backsub}\left(F\,\mathrm{false}\,x\right)$

$\left[\begin{array}{ccc}1-x_1& -x_1& x_1\end{array}\right]$

$u≔\mathrm{LUdecomp}\left(H\,L='l'\right)$

$u≔\left[\begin{array}{ccc}1& 2& 3\\ 0& \mathrm{-3}& \mathrm{-3}\\ 0& 0& 0\end{array}\right]$

$e≔\mathrm{forwardsub}\left(l\,v\right)$

$e≔\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$

$f≔\mathrm{backsub}\left(u\,e\,'s'\right)$

$f≔\left[\begin{array}{ccc}1-s_1& -s_1& s_1\end{array}\right]$

$\mathrm{evalm}\left(\left(l\&*u\right)\&*f-v\right)$

$\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$