An iterative procedure for the solution of linear and nonlinear equations

Abstract

The proposed procedure applies to the general linear equation

$$XoA = B$$

where B denotes a given L₂ function (or vector) and X_oA the product of the unknown function (or vector) X by a given linear operator (or matrix) A which satisfies the inequality condition

$$||UoA|| \leqslant M||U||,Mfinite$$

for any L₂ function (or vector) U (∥U∥ is defined by ∥U∥=mrŪ², with

$$UV = \int\limits_a^b {U(x)V(x)dxor\sum {U_i V_i .} }$$

The procedure is iterative and defined by the following formulae, where

$$\begin{gathered}\hat X = X/||X||^2 , \hfill \\Xo = Uo = O,V_1 = B \hfill \\Up = Up - 1 + \hat VpoA* \hfill \\Xp = Xp - 1 + \hat Up \hfill \\Vp + 1 = B - XpoA \hfill \\\end{gathered}$$

Provided the equation has at least one L₂ solution X_p converges in the quadratic mean towards the solution with the smallest norm. If the equation is Fredholm's second kind with a square summable kernel, the rate of convergence is factorial. The calculation requires only three main subprograms:

• -NORM (α) to calculate -X² for any X

• -AR (X) to calculate product X_oA (right)

• -AL (X) to calculate product X_oA* (left)

where A* is defined by Green's identity.

$\overline {U(VoA)} = \overline {(UoA*)V}$ for any L₂ functions U and V. (2)