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| {{More footnotes|date=April 2009}}
| | Golda is what's written on my beginning certificate even though it is not the name on my birth certification. For a whilst I've been in Alaska but I will have to transfer in a year or two. The favorite hobby for him and his children is to perform lacross and he would never give it up. He is an info officer.<br><br>Also visit my webpage [http://www.Youronlinepublishers.com/authWiki/AdolphvhBladenqq online psychics] |
| In [[mathematics]], the '''discrete Poisson equation''' is the [[finite difference]] analog of the [[Poisson equation]]. In it, the [[discrete Laplace operator]] takes the place of the [[Laplace operator]]. The discrete Poisson equation is frequently used in [[numerical analysis]] as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in [[discrete mathematics]].
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| ==On a two-dimensional rectangular grid==
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| Using the [[finite difference]] numerical method to discretize
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| the 2 dimensional Poisson equation (assuming a uniform spatial discretization, <math>\Delta x=\Delta y</math>) on an ''m'' × ''n'' grid gives the following formula:<ref>{{citation|title=Numerical Methods for Engineers and Scientists|edition=2nd|first=Joe|last=Hoffman|year=2001|chapter=Chapter 9. Elliptic partial differential equations|publisher=McGraw–Hill|isbn=0-8247-0443-6}}.</ref>
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| :<math>
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| ( {\nabla}^2 u )_{ij} = \frac{1}{\Delta x^2} (u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4 u_{ij}) = g_{ij}
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| </math>
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| where <math> 2 \le i \le m-1 </math> and <math> 2 \le j \le n-1 </math>. The preferred arrangement of the solution vector is to use [[natural ordering]] which, prior to removing boundary elements, would look like:
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| :<math>
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| U =
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| \begin{bmatrix} u_{11} , u_{21} , \ldots , u_{m1} , u_{12} , u_{22} , \ldots , u_{m2} , \ldots , u_{mn}
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| \end{bmatrix}^T
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| </math>
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| This will result in an ''mn'' × ''mn'' linear system:
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| :<math> AU = b </math>
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| where
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| :<math>
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| A =
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| \begin{bmatrix}
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| ~D & -I & ~0 & ~0 & ~0 & \ldots & ~0 \\
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| -I & ~D & -I & ~0 & ~0 & \ldots & ~0 \\
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| ~0 & -I & ~D & -I & ~0 & \ldots & ~0 \\
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| \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
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| ~0 & \ldots & ~0 & -I & ~D & -I & ~0 \\
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| ~0 & \ldots & \ldots & ~0 & -I & ~D & -I \\
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| ~0 & \ldots & \ldots & \ldots & ~0 & -I & ~D
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| \end{bmatrix},
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| </math>
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| <math> I </math> is the ''m'' × ''m'' [[identity matrix]], and <math> D </math>, also ''m'' × ''m'', is given by:
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| :<math>
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| D =
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| \begin{bmatrix}
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| ~4 & -1 & ~0 & ~0 & ~0 & \ldots & ~0 \\
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| -1 & ~4 & -1 & ~0 & ~0 & \ldots & ~0 \\
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| ~0 & -1 & ~4 & -1 & ~0 & \ldots & ~0 \\
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| \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
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| ~0 & \ldots & ~0 & -1 & ~4 & -1 & ~0 \\
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| ~0 & \ldots & \ldots & ~0 & -1 & ~4 & -1 \\
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| ~0 & \ldots & \ldots & \ldots & ~0 & -1 & ~4
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| \end{bmatrix},
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| </math>
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| <ref>Golub, Gene H. and C. F. Van Loan, ''Matrix Computations, 3rd Ed.'',
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| The Johns Hopkins University Press, Baltimore, 1996, pages 177–180.</ref>
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| and <math>b</math> is defined by
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| :<math>
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| b =
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| -\Delta x^2\begin{bmatrix} g_{11} , g_{21} , \ldots , g_{m1} , g_{12} , g_{22} , \ldots , g_{m2} , \ldots , g_{mn}
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| \end{bmatrix}^T.
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| </math>
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| For each <math> u_{ij} </math> equation, the columns of <math> D </math> correspond to the <math> u </math> components:
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| :<math>
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| \begin{bmatrix}
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| u_{1j} , & u_{2j} , & \ldots, & u_{i-1,j} , & u_{ij} , & u_{i+1,j} , & \ldots , & u_{mj}
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| \end{bmatrix}^{T}
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| </math>
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| while the columns of <math> I </math> to the left and right of <math> D </math> correspond to the <math> u </math> components:
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| :<math>
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| \begin{bmatrix}
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| u_{1,j-1} , & u_{2,j-1} , & \ldots, & u_{i-1,j-1} , & u_{i,j-1} , & u_{i+1,j-1} , & \ldots , & u_{m,j-1}
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| \end{bmatrix}^{T}
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| </math>
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| and
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| :<math>
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| \begin{bmatrix}
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| u_{1,j+1} , & u_{2,j+1} , & \ldots, & u_{i-1,j+1} , & u_{i,j+1} , & u_{i+1,j+1} , & \ldots , & u_{m,j+1}
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| \end{bmatrix}^{T}
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| </math>
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| respectively.
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| From the above, it can be inferred that there are <math>n</math> block columns of <math> m </math> in <math> A </math>. It is important to note that prescribed values of <math> u </math> (usually lying on the boundary) would have their corresponding elements removed from <math> I </math> and <math> D </math>. For the common case that all the nodes on the boundary are set, we have <math> 2 \le i \le m - 1 </math> and <math> 2 \le j \le n - 1 </math>, and the system would have the dimensions (''m'' − 2)(''n'' − 2) × (''m'' − 2)(''n'' − 2), where <math> D </math> and <math> I </math> would have dimensions (''m'' − 2) × (''m'' − 2).
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| == Example ==
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| For a 5×5 ( <math> m = 5 </math> and <math> n = 5 </math> ) grid with all the boundary nodes prescribed,
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| the system would look like:
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| :<math>
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| \begin{bmatrix} U \end{bmatrix} =
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| \begin{bmatrix} u_{22}, u_{32}, u_{42}, u_{23}, u_{33}, u_{43}, u_{24}, u_{34}, u_{44}
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| \end{bmatrix}^{T}
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| </math>
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| with
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| :<math>
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| A =
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| \begin{bmatrix}
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| ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 & ~0 \\
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| -1 & ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 \\
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| ~0 & -1 & ~4 & ~0 & ~0 & -1 & ~0 & ~0 & ~0 \\
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| -1 & ~0 & ~0 & ~4 & -1 & ~0 & -1 & ~0 & ~0 \\
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| ~0 & -1 & ~0 & -1 & ~4 & -1 & ~0 & -1 & ~0 \\
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| ~0 & ~0 & -1 & ~0 & -1 & ~4 & ~0 & ~0 & -1 \\
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| ~0 & ~0 & ~0 & -1 & ~0 & ~0 & ~4 & -1 & ~0 \\
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| ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4 & -1 \\
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| ~0 & ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4
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| \end{bmatrix}
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| </math>
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| and
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| :<math>
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| b =
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| \left[\begin{array}{l}
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| -\Delta x^2 g_{22} + u_{12} + u_{21} \\
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| -\Delta x^2 g_{32} + u_{31} ~~~~~~~~ \\
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| -\Delta x^2 g_{42} + u_{52} + u_{41} \\
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| -\Delta x^2 g_{23} + u_{13} ~~~~~~~~ \\
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| -\Delta x^2 g_{33} ~~~~~~~~~~~~~~~~ \\
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| -\Delta x^2 g_{43} + u_{53} ~~~~~~~~ \\
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| -\Delta x^2 g_{24} + u_{14} + u_{25} \\
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| -\Delta x^2 g_{34} + u_{35} ~~~~~~~~ \\
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| -\Delta x^2 g_{44} + u_{54} + u_{45}
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| \end{array}\right].
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| </math>
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| As can be seen, the boundary <math> u </math>'s are brought to the right-hand-side
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| of the equation.<ref>Cheny, Ward and David Kincaid, ''Numerical Mathematics and Computing 2nd Ed.'',
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| Brooks/Cole Publishing Company, Pacific Grove, 1985, pages 443–448.</ref> The entire system is 9 × 9 while <math> D </math> and <math> I </math> are 3 × 3 and given by:
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| :<math>
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| D =
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| \begin{bmatrix}
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| ~4 & -1 & ~0 \\
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| -1 & ~4 & -1 \\
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| ~0 & -1 & ~4 \\
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| \end{bmatrix}
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| </math>
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| and
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| :<math>
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| -I =
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| \begin{bmatrix}
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| -1 & ~0 & ~0 \\
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| ~0 & -1 & ~0 \\
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| ~0 & ~0 & -1
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| \end{bmatrix}.
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| </math>
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| == Methods of solution ==
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| Because <math> \begin{bmatrix} A \end{bmatrix} </math> is block tridiagonal and sparse, many methods of solution
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| have been developed to optimally solve this linear system for <math> \begin{bmatrix} U \end{bmatrix} </math>.
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| Among the methods are a generalized [[Thomas algorithm]], [[cyclic reduction]], [[successive overrelaxation]], and [[Fourier transform]]s. A theoretically optimal <math> O(n) </math> solution can also be computed using [[multigrid methods]].
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| == Applications ==
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| In [[computational fluid dynamics]], for the solution of an incompressible flow problem, the incompressibility condition acts as a constraint for the pressure. There is no explicit form available for pressure in this case due to a strong coupling of the velocity and pressure fields. In this condition, by taking the divergence of all terms in the momentum equation, one obtains the pressure poisson equation.
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| For an incompressible flow this constraint is given by:
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| :<math>
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| \frac{ \partial v_x }{ \partial x} + \frac{ \partial v_y }{ \partial y} + \frac{\partial v_z}{\partial z} = 0
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| </math>
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| where <math> v_x </math> is the velocity in the <math> x </math> direction, <math> v_y </math> is
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| velocity in <math> y </math> and <math> v_z </math> is the velocity in the <math> z </math> direction. Taking divergence of the momentum equation and using the incompressibility constraint, the pressure poisson equation is formed given by:
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| :<math>
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| \nabla^2 p = f(\nu,V)
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| </math>
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| where <math> \nu </math> is the kinematic viscosity of the fluid and <math> V </math> is the velocity vector.<ref>
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| Fletcher, Clive A. J., ''Computational Techniques for Fluid Dynamics: Vol I'', 2nd Ed., Springer-Verlag, Berlin, 1991, page 334–339.
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| </ref>
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| The discrete Poisson's equation arises in the theory of
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| [[Markov chain]]s. It appears as the relative value function for the dynamic programming equation in a [[Markov decision process]], and as the ''[[control variate]]'' for application in simulation variance reduction.<ref name=MCSS> S. P. Meyn and R.L. Tweedie, 2005. [http://decision.csl.uiuc.edu/~meyn/pages/book.html Markov Chains and Stochastic Stability].
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| Second edition to appear, Cambridge University Press, 2009.</ref><ref name=CTCN> S. P. Meyn, 2007. [http://decision.csl.uiuc.edu/~meyn/pages/CTCN/CTCN.html Control Techniques for Complex Networks], Cambridge University Press, 2007. </ref><ref name=AG07> Asmussen, Søren, Glynn, Peter W., 2007. "Stochastic Simulation: Algorithms and Analysis". Springer. Series: Stochastic Modelling and Applied Probability, Vol. 57, 2007.</ref>
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| ==Footnotes==
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| <references/>
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| ==References==
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| *Hoffman, Joe D., '' Numerical Methods for Engineers and Scientists, 4th Ed.'', McGraw–Hill Inc., New York, 1992.
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| *Sweet, Roland A., '' SIAM Journal on Numerical Analysis, Vol. 11, No. 3 '', June 1974, 506–520.
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 20.4. Fourier and Cyclic Reduction Methods | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1053}}
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| [[Category:Finite differences]]
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| [[Category:Numerical differential equations]]
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