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lecture16

# lecture16 - Michael T Heath Scientiﬁc Computing 10 55...

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CMPSC/MATH 451 Numerical Computations Lecture 16 September 28, 2011 Prof. Kamesh Madduri

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This class I handed out graded midterm exams I went over exam solutions in class (I will post the solutions in the “exam” folder on Angel) Non-linear equations Conditioning, absolute condition number 2
Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Sensitivity and Conditioning Conditioning of root finding problem is opposite to that for evaluating function Absolute condition number of root finding problem for root x * of f : R R is 1 / | f ( x * ) | Root is ill-conditioned if tangent line is nearly horizontal In particular, multiple root ( m > 1 ) is ill-conditioned Absolute condition number of root finding problem for root x * of f : R n R n is J - 1 f ( x * ) , where J f is Jacobian matrix of f , { J f ( x ) } ij = ∂f i ( x ) /∂x j Root is ill-conditioned if Jacobian matrix is nearly singular

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Unformatted text preview: Michael T. Heath Scientiﬁc Computing 10 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Sensitivity and Conditioning Michael T. Heath Scientiﬁc Computing 11 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Sensitivity and Conditioning What do we mean by approximate solution ˆ x to nonlinear system, k f (ˆ x ) k ≈ or k ˆ x-x * k ≈ 0 ? First corresponds to “small residual,” second measures closeness to (usually unknown) true solution x * Solution criteria are not necessarily “small” simultaneously Small residual implies accurate solution only if problem is well-conditioned Michael T. Heath Scientiﬁc Computing 12 / 55...
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