symbolic and numeric computations, and hybrid matrices

eigenvalues and eigenvectors, both classical and generalized.

linear algebra over finite fields.

matrix factorizations and system solvers

numerical methods for dense and sparse systems with a high degree of user control

hardware float and arbitrary precision software float data

numeric routines from CLAPACK and optimized vendor BLAS (ATLAS and MKL) libraries, called automatically when appropriate.

automatically parallelized numeric computation, when appropriate

$\mathrm{B1}≔\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& \mathrm{d\_\_i}\\ 0& 0& 0& 1\end{array}\right]\.\left[\begin{array}{cccc}\cos \left(\mathrm{\θ\_\_i}\right)& -\sin \left(\mathrm{\θ\_\_i}\right)& 0& 0\\ \sin \left(\mathrm{\θ\_\_i}\right)& \cos \left(\mathrm{\θ\_\_i}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\:$

$\mathrm{B2}≔\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \left(\mathrm{\α\_\_i}\right)& -\sin \left(\mathrm{\α\_\_i}\right)& 0\\ 0& \sin \left(\mathrm{\α\_\_i}\right)& \cos \left(\mathrm{\α\_\_i}\right)& 0\\ 0& 0& 0& 1\end{array}\right]\.\left[\begin{array}{cccc}1& 0& 0& \mathrm{a\_\_i}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\:$

$\mathrm{H}≔\mathrm{B1}\.\mathrm{B2}$

$H≔\left[\begin{array}{cccc}\cos \left(\mathrm{\θ\_\_i}\right)& -\sin \left(\mathrm{\θ\_\_i}\right)\cos \left(\mathrm{\α\_\_i}\right)& \sin \left(\mathrm{\θ\_\_i}\right)\sin \left(\mathrm{\α\_\_i}\right)& \cos \left(\mathrm{\θ\_\_i}\right)\mathrm{a\_\_i}\\ \sin \left(\mathrm{\θ\_\_i}\right)& \cos \left(\mathrm{\θ\_\_i}\right)\cos \left(\mathrm{\α\_\_i}\right)& -\cos \left(\mathrm{\θ\_\_i}\right)\sin \left(\mathrm{\α\_\_i}\right)& \sin \left(\mathrm{\θ\_\_i}\right)\mathrm{a\_\_i}\\ 0& \sin \left(\mathrm{\α\_\_i}\right)& \cos \left(\mathrm{\α\_\_i}\right)& \mathrm{d\_\_i}\\ 0& 0& 0& 1\end{array}\right]$

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

$M≔\mathrm{RandomMatrix}\left(1000\,1000\,\mathrm{density}\=0.001\,\mathrm{datatype}\=\mathrm{float}\left[8\right]\right)\;$

$v≔\mathrm{RandomVector}\left(1000\,\mathrm{density}\=0.001\,\mathrm{datatype}\=\mathrm{float}\left[8\right]\right)$

$\mathbf{for} i \mathbf{from} 1 \mathbf{to} 1000 \mathbf{do} M\left[i\,i\right]≔i\:\mathbf{end} \mathbf{do}\:$

$x≔\mathrm{LinearSolve}\left(M\,v\right)$

$\mathrm{Norm}\left(M\.x-v\right)$

$0.$