Lecture-8 - Lecture-8 Conjugate Direction Algorithm...

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1 Lecture-8 Conjugate Direction Algorithm (Solution of Linear System or Minimization of Quadratic Function) Conjugate Gradient • Linear conjugate gradient: for solving linear systems Ax= b with PD matrix, A. • Non-linear conjugate gradient: for solving large-scale non-linear optimization problems. – Fletcher and Reeves, 1960s
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2 Solution of A linear System • Gaussian Elimination, Backward Substitution • Matrix Factorization • Iterative Techniques b Ax = n i b x a n j i j ij , , 2 , 1 for ) ( 1 K = = =
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3 Iterative Methods for Solving Linear Systems • For large sparse system Gaussian Elimination and Backward substitution is not suitable. • Approximate solution using iterative methods Jacobi n i b x a n j i j ij , , 2 , 1 for ) ( 1 K = = = n i a b x a x ii n i j j i k j ij k i , , 2 , 1 for ) ( , 1 1 K = + - = = - C TX X + =
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Gauss-Seidel n i a b x a x a x ii i j n i j i k j ij k j ij k i , , 2 , 1 for ) ( ) ( 1 1 1 1 K = + - - = - = + = - n i b x a n j i j ij , , 2 , 1 for ) ( 1 K = = = ii k ii k i k i ii k ii k i k i a r x x a r x x w + = + = - - 1 1 C TX X + = Interpretation of Gauss-Seidel n i a b x a x a x ii i j n i j i k j ij k j ij k i , , 2 , 1 for ) ( ) ( 1 1 1 1 K = + - - = - = + = - x A b r ~ - = 1 1 1 1 1 ) ( ) ( - - = + = - - - - = k i i j n i j ii k j ij
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Lecture-8 - Lecture-8 Conjugate Direction Algorithm...

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