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Difference between revisions of "Stability of a computational algorithm"

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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
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A partially resolving operator $L_m^h$, uniformly bounded in $h$ and $m$, 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>
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$$L^hu^h=f^h,$$
  
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]]]).
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for example, a grid equation with step $h$ (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 $p(h)=\sup\|L_m^h\|$ grows comparatively slowly and a corresponding strengthening of the influence of computational errors for $h\to0$ 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====
 
====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>
 
<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>

Latest revision as of 14:06, 14 August 2014

A partially resolving operator $L_m^h$, uniformly bounded in $h$ and $m$, describing the succession of steps in the computational algorithm for solving the equation

$$L^hu^h=f^h,$$

for example, a grid equation with step $h$ (cf. 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 $p(h)=\sup\|L_m^h\|$ grows comparatively slowly and a corresponding strengthening of the influence of computational errors for $h\to0$ remains practically admissible. The concept of stability of a computational algorithm has been made concrete in applications to grid-projection methods (cf. [4]) and in applications to iterative methods (cf. [6]). There are also other definitions of the stability of a computational algorithm (cf. e.g. [1], [3]).

References

[1] I. [I. Babushka] Babuška, M. Práger, E. Vitásek, "Numerical processes in differential equations" , Wiley (1966)
[2] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[3] M.K. Gavurin, "Lectures on computing methods" , Moscow (1971) (In Russian)
[4] G.I. Marchuk, V.I. Agoshkov, "Introduction to grid-projection methods" , Moscow (1981) (In Russian)
[5] A.A. Samarskii, A.V. Gulin, "Stability of difference schemes" , Moscow (1973) (In Russian)
[6] A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)
How to Cite This Entry:
Stability of a computational algorithm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_of_a_computational_algorithm&oldid=32911
This article was adapted from an original article by A.F. Shapkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article