basics of numerical analysis

basics of numerical analysis - Basics of Numerical Analysis...

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Basics of Numerical Analysis FM 5012 Sandra Paterlini
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Jacobi, Gauss-Seidel and SOR Methods
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3 Iterative Methods Jacobi x ( k +1) = D -1 (-( L + U ) x ( k ) + b) Iteration matrix: J = - D -1 ( L + U )=- D -1 ( A-D ) Gauss-Seidel x ( k +1) = D -1 (- Lx ( k +1) Ux ( k ) + b ) Iteration matrix: -( L + D ) -1 U SOR x ( k +1) = x ( k ) + w { D -1 (- Lx ( k +1) - Ux ( k ) + b ) - x ( k ) } Iteration matrix: ( w L + D ) -1 [(1- w ) D - w U ]
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4 Iterative Methods SOR Method For a broad class of matrix, we can speed up the SOR method by considering where ρ J =max| μ | is the spectral radius of the Jacobi iteration matrix If ρ J =max| μ |, then ρ GS = ρ 2 J 2 2 *, * * 1 11 J   
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5 Iterative Methods Projected SOR SOR can be easily adapted to other type of problems Suppose we wish to solve the linear complementarity problem Ax ³ b, x ³ f ( Ax-b )( x-f )= 0 where the inequalities hold component by component We modify SOR such that at each iteration x ( k ) ³ f x ( k +1) =max( x ( k ) + w { D -1 (- Lx ( k +1) - Ux ( k ) + b ) - x ( k ) }, f ) where max() applies component by component to the vector quantities
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6 Iterative Methods Projected SOR Projected SOR can help us with American options Projected SOR can be used Crank-Nicolson time stepping for the American put
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The Conjugate Gradient Method
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8 Eigenvectors Eigenvectors An eigenvector x of a matrix A is a nonzero vector that does not rotate when A is applied x Ax A 2 x A 3 x x Ax A 2 x A 3 x | λ |>1 | λ |<1
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9 Quadratic Form Quadratic form 1 () 2 TT fx  xAx bx c Source: http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf a) Positive definite matrix b) Negative definite matrix c) Singular and (positive indefinite matrix) d) Indefinite matrix
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basics of numerical analysis - Basics of Numerical Analysis...

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