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

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

forwardsub generates a solution vector x to the equation $Lx=b$.

If b is omitted, or b is 'false' then L is assumed to be an augmented matrix and the last column of L 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 L is the result of applying Gaussian elimination to the augmented matrix of a system of linear equations, as might be obtained from LUdecomp, forwardsub completes the solution by forward 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 the top. Such a matrix is produced by obtaining the LU decomposition.

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

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

$A≔\mathrm{array}\left(1..3\,1..4\,\left[\left[1\,-2\,3\,1\right]\,\left[2\,k\,6\,6\right]\,\left[-1\,3\,k-3\,0\right]\right]\right)\:$

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

$\left[\begin{array}{cccc}1& \mathrm{-2}& 3& 1\\ 0& k+4& 0& 4\\ 0& 0& k& \frac{k}{k+4}\end{array}\right]$

$b≔\mathrm{vector}\left(\left[1\,2\,3\right]\right)\:$

$\mathrm{v1}≔\mathrm{forwardsub}\left(l\,b\right)$

$\mathrm{v1}≔\left[\begin{array}{ccc}1& 0& 4\end{array}\right]$

$\mathrm{v2}≔\mathrm{backsub}\left(u\,\mathrm{v1}\right)$

$\mathrm{v2}≔\left[\begin{array}{cccc}-\frac{k^2{\mathrm{\_t}}_1-k^2+9k{\mathrm{\_t}}_1+8k+48}{\left(k+4\right)k}& -\frac{4{\mathrm{\_t}}_1}{k+4}& -\frac{k{\mathrm{\_t}}_1-4k-16}{\left(k+4\right)k}& {\mathrm{\_t}}_1\end{array}\right]$

$\mathrm{map}\left(\mathrm{normal}\,\mathrm{evalm}\left(b-A\&*\mathrm{v2}\right)\right)$

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