Richardson's algorithm

 

Our first illustration of an iterative method for solving the linear system A x = b, we describe Richardson’s algorithm. The input to the algorithm is a matrix A ∈ ℂ^n×n, a column vector b ∈ ℂn×1, and a vector of unknowns x ∈ ℂn×1. An initial guess to the true solution is a vector x(0) ∈ ℂn×1. We may take x(0) = 0 if we have no initial information about x. The algorithm proceeds by forming a sequence of approximate solutions according to the formula:
\[ {\bf x}^{(k+1)} = {\bf x}^{(k)} + \left( {\bf b} - {\bf A}\,{\bf x}^{(k)} \right) , \qquad k=0,1,2,\ldots . \]
   
Example 1: Let S be a set of two vectors in ℝ³ Loehr page 215    ■
End of Example 1
   
Example 13: Let S be a set of two vectors in ℝ³    ■
End of Example 13

  2. Loehr, Advanced Linear Algebra,

 

  2. Loehr, N., Advanced Linear Algebra, CRC Press, 2014.