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''method of successive approximation, method of repeated substitution, method of simple iteration''
 
''method of successive approximation, method of repeated substitution, method of simple iteration''
  
 
One of the general methods for the approximate solution of equations. In many cases the good convergence properties of the approximations constructed by this method allow one to apply it to practical computations.
 
One of the general methods for the approximate solution of equations. In many cases the good convergence properties of the approximations constructed by this method allow one to apply it to practical computations.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846101.png" /> be some set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846102.png" /> an operator (not necessarily linear) on this set mapping it into itself. Suppose one has to find a fixed point of this mapping, i.e. a solution of the equation
+
Let $  E $
 +
be some set and $  A $
 +
an operator (not necessarily linear) on this set mapping it into itself. Suppose one has to find a fixed point of this mapping, i.e. a solution of 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/s084/s084610/s0846103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= A ( x) ,\ \
 +
x \in E .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846104.png" /> be a solution of (1) and let its first approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846105.png" /> be given by some method. Then all other approximations in the method of successive approximation are constructed by the formula
+
Let $  x  ^ {*} $
 +
be a solution of (1) and let its first approximation $  x _ {0} \in E $
 +
be given by some method. Then all other approximations in the method of successive approximation are constructed by the formula
  
<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/s084/s084610/s0846106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
x _ {n+ 1}  = A x _ {n} ,\ \
 +
n = 0 , 1 ,\dots .
 +
$$
  
 
This process is called a simple one-step iteration.
 
This process is called a simple one-step iteration.
Line 15: Line 37:
 
To study the convergence properties of the sequence (2) and to prove the existence of a solution to (1), the contracting-mapping principle formulated below is widely used (cf. also [[Contracting-mapping principle|Contracting-mapping principle]]).
 
To study the convergence properties of the sequence (2) and to prove the existence of a solution to (1), the contracting-mapping principle formulated below is widely used (cf. also [[Contracting-mapping principle|Contracting-mapping principle]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846107.png" /> be a complete metric space with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846108.png" />; let the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s0846109.png" /> be defined on a closed ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461010.png" /> with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461011.png" /> and centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461012.png" />:
+
Let $  E $
 +
be a complete metric space with metric $  \rho $;  
 +
let the operator $  A $
 +
be defined on a closed ball $  S $
 +
with radius $  \delta $
 +
and centre at the point $  x _ {0} $:
  
<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/s084/s084610/s08461013.png" /></td> </tr></table>
+
$$
 +
= \{ {x \in E } : {\rho ( x , x _ {0} ) \leq  \delta } \}
 +
;
 +
$$
  
let for any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461015.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461016.png" /> the following relation hold:
+
let for any elements $  x $
 +
and $  y $
 +
from $  S $
 +
the following relation hold:
  
<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/s084/s084610/s08461017.png" /></td> </tr></table>
+
$$
 +
\rho ( A x , A y )  \leq  \alpha \rho ( x , y ) ,\ \
 +
0 < \alpha = \textrm{ const } < 1 ;
 +
$$
  
let for the initial approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461018.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461019.png" /> be valid, and let for the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461020.png" /> the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461021.png" /> be valid.
+
let for the initial approximation $  x _ {0} $
 +
the inequality $  \rho ( A x _ {0} , x _ {0} ) \leq  m $
 +
be valid, and let for the numbers $  \alpha , \delta , m $
 +
the condition $  m / ( 1 - \alpha ) \leq  \delta $
 +
be valid.
  
Then: 1) the successive approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461022.png" /> calculated by the rule (2) can be found for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461023.png" /> and they all belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461024.png" />; 2) the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461025.png" /> converges to some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461026.png" />; 3) the limit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461027.png" /> is a solution of (1); and 4) for the approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461028.png" /> the following estimate of the distance to the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461029.png" /> holds:
+
Then: 1) the successive approximations $  x _ {n} $
 +
calculated by the rule (2) can be found for any $  n $
 +
and they all belong to $  S $;  
 +
2) the sequence $  x _ {n} $
 +
converges to some point $  x _ {*} \in S $;  
 +
3) the limit element $  x _ {*} $
 +
is a solution of (1); and 4) for the approximation $  x _ {n} $
 +
the following estimate of the distance to the solution $  x _ {*} $
 +
holds:
  
<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/s084/s084610/s08461030.png" /></td> </tr></table>
+
$$
 +
\rho ( x _ {n} , x _ {*} )  \leq 
 +
\frac{m}{1 - \alpha }
 +
\alpha  ^ {n} .
 +
$$
  
Further, on any subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461031.png" /> on which for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461032.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461033.png" /> holds, (1) cannot have more than one solution.
+
Further, on any subset in $  E $
 +
on which for any two points $  x , y $
 +
the inequality $  \rho ( A x , A y ) < \rho ( x , y ) $
 +
holds, (1) cannot have more than one solution.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461034.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461035.png" />-dimensional real vector space, and let the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461036.png" /> in (1) have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461038.png" /> is a square matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461039.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461040.png" /> be given and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461041.png" /> be the unknown vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461042.png" />. If in this space the metric is defined by the formula
+
Let $  E = \mathbf R  ^ {n} $
 +
be the $  n $-dimensional real vector space, and let the operator $  A $
 +
in (1) have the form $  A x = B x + b $,  
 +
where $  B = \| a _ {ik} \| $
 +
is a square matrix of order $  n $;  
 +
let $  b = ( b _ {1}, \dots, b _ {n} ) $
 +
be given and let $  x = ( x _ {1}, \dots, x _ {n} ) $
 +
be the unknown vector in $  \mathbf R  ^ {n} $.  
 +
If in this space the metric is defined by the formula
  
<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/s084/s084610/s08461043.png" /></td> </tr></table>
+
$$
 +
\rho ( x , y )  = \max _ { i }  | x _ {i} - y _ {i} |
 +
$$
  
and if the entries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461044.png" /> satisfy the condition
+
and if the entries of $  B $
 +
satisfy the condition
  
<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/s084/s084610/s08461045.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= 1 }^ { n }
 +
| a _ {ik} |  < 1
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461047.png" />, then the contracting-mapping principle implies that the system of algebraic equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461048.png" /> has a unique solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461049.png" /> which can be obtained by the method of successive approximation starting from an arbitrary initial approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461050.png" />.
+
for all $  i $,
 +
$  i = 1, \dots, n $,  
 +
then the contracting-mapping principle implies that the system of algebraic equations $  x = A x $
 +
has a unique solution in $  \mathbf R  ^ {n} $
 +
which can be obtained by the method of successive approximation starting from an arbitrary initial approximation $  x _ {0} \in \mathbf R  ^ {n} $.
  
If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461051.png" /> the Euclidean metric
+
If in $  \mathbf R  ^ {n} $
 +
the Euclidean metric
  
<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/s084/s084610/s08461052.png" /></td> </tr></table>
+
$$
 +
\rho ( x , y )  = \sqrt {\sum _ { i= 1} ^ { n }
 +
( x _ {i} - y _ {i} )  ^ {2} }
 +
$$
  
 
is given, then one obtains another condition of convergence for the successive approximations:
 
is given, then one obtains another condition of convergence for the successive approximations:
  
<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/s084/s084610/s08461053.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= 1} ^ { n }  \sum _ { k= 1} ^ { n }  a _ {ik}  ^ {2}  < 1 .
 +
$$
  
 
Let (1) be an integral equation in which
 
Let (1) be an integral equation in which
  
<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/s084/s084610/s08461054.png" /></td> </tr></table>
+
$$
 +
A x ( t)  = f ( t) + \lambda \int\limits _ { a } ^ { b }
 +
K ( t , s ) x ( s)  d s ,
 +
$$
  
where the given functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461056.png" /> are square integrable on the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461058.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461059.png" /> is a numerical parameter. Then the contracting-mapping principle implies that if
+
where the given functions $  f $,  
 +
$  K $
 +
are square integrable on the sets $  [ a , b ] $
 +
and $  [ a , b ] \times [ a , b ] $,  
 +
respectively, and $  \lambda $
 +
is a numerical parameter. Then the contracting-mapping principle implies that if
  
<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/s084/s084610/s08461060.png" /></td> </tr></table>
+
$$
 +
| \lambda |  < \left ( \int\limits _ { a } ^ { b }
 +
\int\limits _ { a } ^ { b }  K  ^ {2} ( s , t )  dt  ds \right )  ^ {- 1/2} ,
 +
$$
  
then the considered integral equation has a unique solution in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084610/s08461061.png" /> which can be constructed by the method of successive approximation.
+
then the considered integral equation has a unique solution in the space $  L _ {2} ( [ a , b ] ) $
 +
which can be constructed by the method of successive approximation.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.K. Faddeev,  V.N. Faddeeva,  "Computational methods of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Krylov,  V.V. Bobkov,  P.I. Monastyrnyi,  "Numerical methods" , '''1–2''' , Moscow  (1976–1977)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Collatz,  "Funktionalanalysis und numerische Mathematik" , Springer  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.K. Faddeev,  V.N. Faddeeva,  "Computational methods of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Krylov,  V.V. Bobkov,  P.I. Monastyrnyi,  "Numerical methods" , '''1–2''' , Moscow  (1976–1977)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Collatz,  "Funktionalanalysis und numerische Mathematik" , Springer  (1964)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 23:24, 7 May 2022


method of successive approximation, method of repeated substitution, method of simple iteration

One of the general methods for the approximate solution of equations. In many cases the good convergence properties of the approximations constructed by this method allow one to apply it to practical computations.

Let $ E $ be some set and $ A $ an operator (not necessarily linear) on this set mapping it into itself. Suppose one has to find a fixed point of this mapping, i.e. a solution of the equation

$$ \tag{1 } x = A ( x) ,\ \ x \in E . $$

Let $ x ^ {*} $ be a solution of (1) and let its first approximation $ x _ {0} \in E $ be given by some method. Then all other approximations in the method of successive approximation are constructed by the formula

$$ \tag{2 } x _ {n+ 1} = A x _ {n} ,\ \ n = 0 , 1 ,\dots . $$

This process is called a simple one-step iteration.

To study the convergence properties of the sequence (2) and to prove the existence of a solution to (1), the contracting-mapping principle formulated below is widely used (cf. also Contracting-mapping principle).

Let $ E $ be a complete metric space with metric $ \rho $; let the operator $ A $ be defined on a closed ball $ S $ with radius $ \delta $ and centre at the point $ x _ {0} $:

$$ S = \{ {x \in E } : {\rho ( x , x _ {0} ) \leq \delta } \} ; $$

let for any elements $ x $ and $ y $ from $ S $ the following relation hold:

$$ \rho ( A x , A y ) \leq \alpha \rho ( x , y ) ,\ \ 0 < \alpha = \textrm{ const } < 1 ; $$

let for the initial approximation $ x _ {0} $ the inequality $ \rho ( A x _ {0} , x _ {0} ) \leq m $ be valid, and let for the numbers $ \alpha , \delta , m $ the condition $ m / ( 1 - \alpha ) \leq \delta $ be valid.

Then: 1) the successive approximations $ x _ {n} $ calculated by the rule (2) can be found for any $ n $ and they all belong to $ S $; 2) the sequence $ x _ {n} $ converges to some point $ x _ {*} \in S $; 3) the limit element $ x _ {*} $ is a solution of (1); and 4) for the approximation $ x _ {n} $ the following estimate of the distance to the solution $ x _ {*} $ holds:

$$ \rho ( x _ {n} , x _ {*} ) \leq \frac{m}{1 - \alpha } \alpha ^ {n} . $$

Further, on any subset in $ E $ on which for any two points $ x , y $ the inequality $ \rho ( A x , A y ) < \rho ( x , y ) $ holds, (1) cannot have more than one solution.

Let $ E = \mathbf R ^ {n} $ be the $ n $-dimensional real vector space, and let the operator $ A $ in (1) have the form $ A x = B x + b $, where $ B = \| a _ {ik} \| $ is a square matrix of order $ n $; let $ b = ( b _ {1}, \dots, b _ {n} ) $ be given and let $ x = ( x _ {1}, \dots, x _ {n} ) $ be the unknown vector in $ \mathbf R ^ {n} $. If in this space the metric is defined by the formula

$$ \rho ( x , y ) = \max _ { i } | x _ {i} - y _ {i} | $$

and if the entries of $ B $ satisfy the condition

$$ \sum _ { k= 1 }^ { n } | a _ {ik} | < 1 $$

for all $ i $, $ i = 1, \dots, n $, then the contracting-mapping principle implies that the system of algebraic equations $ x = A x $ has a unique solution in $ \mathbf R ^ {n} $ which can be obtained by the method of successive approximation starting from an arbitrary initial approximation $ x _ {0} \in \mathbf R ^ {n} $.

If in $ \mathbf R ^ {n} $ the Euclidean metric

$$ \rho ( x , y ) = \sqrt {\sum _ { i= 1} ^ { n } ( x _ {i} - y _ {i} ) ^ {2} } $$

is given, then one obtains another condition of convergence for the successive approximations:

$$ \sum _ { i= 1} ^ { n } \sum _ { k= 1} ^ { n } a _ {ik} ^ {2} < 1 . $$

Let (1) be an integral equation in which

$$ A x ( t) = f ( t) + \lambda \int\limits _ { a } ^ { b } K ( t , s ) x ( s) d s , $$

where the given functions $ f $, $ K $ are square integrable on the sets $ [ a , b ] $ and $ [ a , b ] \times [ a , b ] $, respectively, and $ \lambda $ is a numerical parameter. Then the contracting-mapping principle implies that if

$$ | \lambda | < \left ( \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K ^ {2} ( s , t ) dt ds \right ) ^ {- 1/2} , $$

then the considered integral equation has a unique solution in the space $ L _ {2} ( [ a , b ] ) $ which can be constructed by the method of successive approximation.

References

[1] D.K. Faddeev, V.N. Faddeeva, "Computational methods of linear algebra" , Freeman (1963) (Translated from Russian)
[2] V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Numerical methods" , 1–2 , Moscow (1976–1977) (In Russian)
[3] L. Collatz, "Funktionalanalysis und numerische Mathematik" , Springer (1964)

Comments

The contracting-mapping principle is of special interest for non-linear equations; see, e.g., [a1], [a2].

References

[a1] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955)
[a2] J. Cronin, "Fixed points and topological degree in nonlinear analysis" , Amer. Math. Soc. (1964)
How to Cite This Entry:
Sequential approximation, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sequential_approximation,_method_of&oldid=14953
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article