CGandPCG - The Landscape of Ax=b Solvers Direct A = LU...

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The Landscape of Ax=b Solvers The Landscape of Ax=b Solvers Pivoting LU GMRES, BiCGSTAB, Cholesky Conjugate gradient Direct A = LU Iterative y’ = Ay Non- symmetric Symmetric positive definite More Robust Less Storage (if sparse) More Robust More General D
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Conjugate gradient iteration for Ax = b Conjugate gradient iteration for Ax = b x 0 = 0 approx solution r 0 = b residual = b - Ax d 0 = r 0 search direction for k = 1, 2, 3, . . . x k = x k-1 + … new approx solution r k = new residual d k = new search direction
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Conjugate gradient iteration for Ax = b Conjugate gradient iteration for Ax = b x 0 = 0 approx solution r 0 = b residual = b - Ax d 0 = r 0 search direction for k = 1, 2, 3, . . . α k = step length x k = x k-1 + α k d k-1 new approx solution r k = new residual d k = new search direction
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Conjugate gradient iteration for Ax = b Conjugate gradient iteration for Ax = b x 0 = 0 approx solution r 0 = b residual = b - Ax d 0 = r 0 search direction for k = 1, 2, 3, . . . α k = (r T k-1 r k-1 ) / (d T k-1 Ad k-1 ) step length x k = x k-1 + α k d k-1 new approx solution r k = new residual d k = new search direction
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Conjugate gradient iteration for Ax = b Conjugate gradient iteration for Ax = b x 0 = 0 approx solution r 0 = b residual = b - Ax d 0 = r 0 search direction for k = 1, 2, 3, . . . α
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CGandPCG - The Landscape of Ax=b Solvers Direct A = LU...

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