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''minimal residual method''

An iteration method for the solution of an operator equation

$$ \tag{1 }
A u  =  f
$$

with a self-adjoint positive-definite bounded operator  $  A $
acting in a Hilbert space  $  H $, 
and with a given element  $  f \in H $. 
The formula for the minimal discrepancy method takes the form

$$ \tag{2 }
u  ^ {k+} 1  =  u  ^ {k} - \alpha _ {k} ( A u  ^ {k} - f  ) ,\ \ 
k = 0 , 1 \dots
$$

where the parameter

$$ \tag{3 }
\alpha _ {k}  = \ 

\frac{( A \xi  ^ {k} , \xi  ^ {k} ) }{( A \xi  ^ {k} , A \xi  ^ {k} ) }

$$

is chosen in each step  $  k \geq  0 $
from the condition for maximal minimization of the norm of the discrepancy (or residual vector)  $  \xi  ^ {k+} 1 = A u  ^ {k+} 1 - f $; 
that is, it is required that

$$ \tag{4 }
\| \xi  ^ {k} - \alpha _ {k} A \xi  ^ {k} \|  = \ 
\inf _  \alpha  \ 
\| \xi  ^ {k} - \alpha A \xi  ^ {k} \| .
$$

If the spectrum of  $  A $
belongs to an interval  $  [ m , M ] $
on the real line, where  $  m \leq  M $
are positive numbers, then the sequence of approximations  $  \{ u  ^ {k} \} $
in the method (2)–(3) converges to a solution  $  u  ^ {*} $
of (1) with the speed of a geometric progression with multiplier  $  q = ( M - m ) / ( M + m ) < 1 $.

Different ways of defining the scalar product in  $  H $
lead to different iteration methods. In particular, for special scalar products the formulas for the minimal discrepancy method coincide with the formulas for the method of steepest descent (cf. [[Steepest descent, method of|Steepest descent, method of]]) and the method of minimal errors (see [[#References|[2]]]).

The condition for the convergence of the minimal discrepancy method may be weakened in comparison with that given above if it is considered on certain subsets of  $  H $. 
For example, if the minimal discrepancy method is considered only in real spaces, then it is possible to drop the requirement that  $  A $
be self-adjoint (see [[#References|[3]]]–[[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Krasnosel'skii,   S.G. Krein,   "An iteration process with minimal residuals"  ''Mat. Sb.'' , '''31'''  (1952)  pp. 315–334  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Samarskii,   "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.I. Marchuk,   Yu.A. Kuznetsov,   "Iterative methods and quadratic functionals" , Novosibirsk  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.I. Marchuk,   Yu.A. Kuznetsov,   "Sur les méthodes numériques en sciences physique et economique" , Dunod  (1974)  (Translated from Russian)</TD></TR></table>