Cimmino Method
Using Cimmino method, this application solves the system of linear equations $Ax=b$, where \[ A = \left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right],\qquad x = \left[\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array}\right],\qquad b = \left[\begin{array}{c} b_1 \\ b_2 \\ \vdots \\ b_n \end{array}\right] \] The updated $x^{k+1}$ is calculated from the relation \[ x^{k+1} = x^k + A^TM(b-Ax^k),\quad\text{where}\ M = \frac{1}{n}\mathrm{diag}\left(\frac{1}{\left\Vert A_1\right\Vert^2},\ldots,\frac{1}{\left\Vert A_n\right\Vert^2}\right) \] The calculation stops when $k=1000$ or $\|b-Ax_k\| < 10^{-8}$ which in the latter case $x_k$ is the solution of the problem.
References
- G. Cimmino, Approximate computation of the solutions of systems of linear equations, Rend. Accad. Sci. Fis. Mat. Napoli (4), 89 (2022), pp. 65–72; English translation by M. Benzi (Sc. Norm. Sup. Pisa, 2015) of Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca Scientifica, II, 9 (1938), pp. 326–333.
- Per Christian Hansen, Tutorial: Row Action Methods, accessed Nov 7, 2024.
Input Data
See Input Data for how to enter data to a matrix and a vector of real numbers.