How to obtain a convergent solution iteratively for a linear system of equations?
I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is:
$$\underbrace{\begin{bmatrix} r_{11} & r_{12} & r_{13} & \cdots & r_{1j} \ r_{21} & r_{22} & r_{23} & \cdots & r_{2j} \ r_{31} & r_{32} & r_{33} & \cdots & r_{3j} \ \vdots & \vdots & \vdots & \ddots & \vdots \ r_{i1} & r_{i2} & r_{i3} & \cdots & r_{ij} \end{bmatrix}}\textit{R} \underbrace{\begin{bmatrix} a{1} & 0 & 0 & \cdots & 0 \ 0 & a_{2} & 0 & \cdots & 0 \ 0 & 0 & a_{3} & \cdots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & a_{j} \end{bmatrix}}\textit{A} + \underbrace{\begin{bmatrix} b{1} & b_{2} & b_{3} & \cdots & b_{j} \ b_{1} & b_{2} & b_{3} & \cdots & b_{j} \ b_{1} & b_{2} & b_{3} & \cdots & b_{j} \ \vdots & \vdots & \vdots & \ddots & \vdots \ b_{1} & b_{2} & b_{3} & \cdots & b_{j} \end{bmatrix}}\textit{B} = \underbrace{\begin{bmatrix} c{1} & c_{1} & c_{1} & \cdots & c_{1} \ c_{2} & c_{2} & c_{2} & \cdots & c_{2} \ c_{3} & c_{3} & c_{3} & \cdots & c_{3} \ \vdots & \vdots & \vdots & \ddots & \vdots \ c_{i} & c_{i} & c_{i} & \cdots & c_{i} \end{bmatrix}}\textit{C}\ R{i, j}A_{j, j} + B_{i, j} = C_{i, j} $$ Now matrix R is fully known (input), matrices A, B, C are unknown. I am working on an iterative procedure in which I provide initial guesses for A and B, then calculate C. 490
The iteration is to be carried out till I get a converged value for C (output I require). The code is being developed in Python. So far no luck in choosing the initial guess too. 416