lecture19 - CMPSC/MATH 451 Numerical Computations Lecture...

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CMPSC/MATH 451 Numerical Computations Lecture 19 October 5, 2011 Prof. Kamesh Madduri
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Class Overview Newton’s method convergence theorems Exercises Secant method 2
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Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Bisection Method Fixed-Point Iteration and Newton’s Method Additional Methods Convergence of Newton’s Method Newton’s method transforms nonlinear equation f ( x ) = 0 into fixed-point problem x = g ( x ) , where g ( x ) = x - f ( x ) /f 0 ( x ) and hence g 0 ( x ) = f ( x ) f 00 ( x ) / ( f 0 ( x )) 2 If x * is simple root (i.e., f ( x * ) = 0 and f 0 ( x * ) 6 = 0 ), then g 0 ( x * ) = 0 Convergence rate of Newton’s method for simple root is therefore quadratic ( r = 2 ) But iterations must start close enough to root to converge < interactive example > Michael T. Heath Scientific Computing 27 / 55
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Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Bisection Method Fixed-Point Iteration and Newton’s Method
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lecture19 - CMPSC/MATH 451 Numerical Computations Lecture...

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