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Cholesky-decomposition of a Matrix

Cholesky-decomposition of a Matrix When a coefficient matrix of a linear system of equations is positive definite and symmetric, then Cholesky decomposition can be used. A positive definite symmetric matrix satisfies the condition $x^{T}Ax > 0$ and the elements of the lower $L$ and upper $U$ triangular matrix satisfies the condition $L_{i,j}=U_{j,i}$ i.e. $U = L^{T}$ and hence $A = LL^{T}$ . A symmetric matrix is positive definite if all its diagonal elements are positive and the diagonal element is larger than the sum of all other elements in that row (i.e. diagonally dominant). Example: choleskydeco_example function [X,C] = choleskydecomposition(A,B) % Applicable to only symmetric positive definite matrix % Input : % A : Coefficient matrix % B : right hand vector % Output : % X : Solution vector % C : Upper triangular metrix [n,m] = size(A); % rank of matrix A C = zeros(n,n); for i=1:n
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Backward Substitution in MATLAB

Backward Substitution Backward Substitution Consider a Upper Triangular matrix U of dimension n, U x = b $$\begin{bmatrix} u_{1,1}&u_{1,2}&\dots&u_{1,n}\\ 0&u_{2,2}&\dots&u_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&u_{n,n} \end{bmatrix} \begin{bmatrix} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{bmatrix} = \begin{bmatrix} b_{1}\\ b_{2}\\ \vdots\\ b_{n} \end{bmatrix}$$ Solving Procedure: $$x_{n} = \frac{b_{n}}{u_{n,n}}$$ Moving to ${n-1}^{th}$ row, there are two coefficients related to $x_{n}$(already known) and $x_{n-1}$ (unkown). $$u_{n-1,n-1}x_{n-1}+u_{n-1,n}x_{n}=b_{n-1}$$ Solving for unknown $x_{n}$, we have $$x_{n-1} = \frac{b_{n-1}-u_{n-1,n}x_{n}}{u_{n-1,n-1}}$$ Generalizing for $i^{th}$ row, $$x_{i} = \frac{1}{u_{i,i}}\left( b_{i}-\sum_{j=i+1}^{n}u_{i,j}x_{j} \right)$$ A
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Location: India
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Solve Poissons Equation using 5-Point Stencil with SOR method

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Solution Poisson's Equation In a Rectangular Domain Using 5-Point Stencil Poisson's equation in a square domain $0 \leq x,y \leq 1$ with source term given by $q(x)=100*\sin(\pi x)*\sin(\pi y)$. The boundaries of the domain are maintained at $u=0$. Determine the solution using finite difference method. Solution: Discretizing domain uniformly with step size h and k along x and y direction. Using five point stencil, equation at each node is given by $$\frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^{2}}+\frac{u_{i,j+1}-2u{i,j}+u{i,j-1}}{k^{2}}=-q(i,j)$$ $$u_{i,j}=\frac{1}{\frac{2}{h^{2}}+\frac{2}{k^{2}}}(\frac{u_{i+1,j}+u_{i-1,j}}{h^{2}}+\frac{u_{i,j+1}+u{i,j-1}}{k^{2}}+q(i,j))$$ where $q(i,j)=100*\sin(\pi x_{i})*\sin(\pi y_{j})$ Here a MATLAB code is given below. Choosen $m=n=11$ grid to solve the problem using SOR (successive overrelaxation). Mathematica Code: In this algo
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Solve Poisson's Equation in Matlab

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Solution Poisson's Equation In a Rectangular Domain Using 5-Point Stencil Poisson's equation in a square domain $0 \leq x,y \leq 1$ with source term given by $q(x)=100*\sin(\pi x)*\sin(\pi y)$. The boundaries of the domain are maintained at $u=0$. Determine the solution using finite difference method. Solution: Discretizing domain uniformly with step size h and k along x and y direction. Using five point stencil, equation at each node is given by $$\frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^{2}}+\frac{u_{i,j+1}-2u{i,j}+u{i,j-1}}{k^{2}}=-q(i,j)$$ $$u_{i,j}=\frac{1}{\frac{2}{h^{2}}+\frac{2}{k^{2}}}(\frac{u_{i+1,j}+u_{i-1,j}}{h^{2}}+\frac{u_{i,j+1}+u{i,j-1}}{k^{2}}+q(i,j))$$ where $q(i,j)=100*\sin(\pi x_{i})*\sin(\pi y_{j})$ Here a MATLAB code is given below. Choosen $m=n=81$, these are the grid spacing. cme_assignment8_b clear; close; clc; n = 81; m = 81; x = 0:1/(n-1):1; y = 0:1/(m-1):1; h = 1/(n-1); k = 1/(m-1); [Y,X] = meshgrid(y,x); q = 100*
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Solving Poisson's Equation Using 5-Point Stencil In A Rectangular Domain

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Solving Poisson's Equation Using 5-Point Stencil: Example: Consider the Poisson's equation on a unit square domain with the source term given by $q(x)=100*\sin(\pi x)*\sin(\pi y)$. The boundaries of the domain are maintained at $u=0$. Reactangular region: $0 \leq x,y \leq 1$. Determine the solution using finite difference method. Solution: 5-point stencil and Gauss-Seidel methods are used to solve Poisso's equations. Here a Mathematica code is give to sove this following program. dx = 0.1; x = Range[0, 1, dx]; y = x; n = Length[x];m = Length[y]; xy = Transpose[Join[{x}, {y}]]; (*Print["xy: ",xy//MatrixForm];*) xMeshgrid = ConstantArray[10, {n, n}]; yMeshgrid = ConstantArray[10, {m, m}]; o[Do[xMeshgrid[[i, j]] = x[[i]]; yMeshgrid[[j, i]] = y[[i]], {j, 1, m}], {i, 1, n}]; (*Print["xMeshgrid: ",xMeshgrid//MatrixForm]; Print["yMeshgrid: ",yMeshgrid//MatrixForm];*) X = T
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