View source for Stability of a computational algorithm

A partially resolving operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869901.png" />, uniformly bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869903.png" />, describing the succession of steps in the computational algorithm for solving the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869904.png" /></td> </tr></table>

for example, a grid equation with step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869905.png" /> (cf. [[Closure of a computational algorithm|Closure of a computational algorithm]]). Stability of a computational algorithm guarantees weak influence of computational errors on the result of the calculation. However, the possibility is not excluded that the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869906.png" /> grows comparatively slowly and a corresponding strengthening of the influence of computational errors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086990/s0869907.png" /> remains practically admissible. The concept of stability of a computational algorithm has been made concrete in applications to grid-projection methods (cf. [[#References|[4]]]) and in applications to iterative methods (cf. [[#References|[6]]]). There are also other definitions of the stability of a computational algorithm (cf. e.g. [[#References|[1]]], [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. [I. Babushka] Babuška,   M. Práger,   E. Vitásek,   "Numerical processes in differential equations" , Wiley  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.K. Gavurin,   "Lectures on computing methods" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.I. Marchuk,   V.I. Agoshkov,   "Introduction to grid-projection methods" , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Samarskii,   A.V. Gulin,   "Stability of difference schemes" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Samarskii,   E.S. Nikolaev,   "Numerical methods for grid equations" , '''1–2''' , Birkhäuser  (1989)  (Translated from Russian)</TD></TR></table>