NUMERICAL STABILITY; IMPLICIT METHODS
If a numerical method has no restrictions on in order to have y n!0 as n !1, we say the numerical method is A-stable. THE BACKWARD EULER METHOD Expand the function Y(x) as a linear Taylor polynomial about x n+1: Y(x) = Y(x n+1) + (x x n+1)Y0(x n+1) + 1 2 (x x n+1) 2 Y00( n) with n between x and x n+1. Let x = x
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THE SECANT METHOD - University of Iowa
homepage.math.uiowa.eduTHE SECANT METHOD Newton’s method was based on using the line tangent to the curve of y = f(x), with the point of tangency (x 0;f(x 0)).When x 0 ˇ , the graph of the tangent line is approximately the same as the
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homepage.math.uiowa.eduFor f(x) = log 10 x, with 1 x 0 x x 2 10; this leads to jlog 10 x P 2(x)j h3 9 p 3 max x0 x x2 2log 10 e x3:05572h3 x3 0 For the case of h = :01, we have jlog 10 x P 2(x)j 5:57 10 8 x3 0 5:57 10 8 For comparison, jlog 10 x P 1(x)j 5:43 10 6
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