09-Newtons-Method-4UP

# 09-Newtons-Method-4UP - Newtons method Examples...

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Newton’s method Examples Implementation details Newton’s Method Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Newton’s Method MATH 2070U 1 / 15 Newton’s method Examples Implementation details Newton’s Method 1 Newton’s method 2 Examples 3 Implementation details c D. Aruliah (UOIT) Newton’s Method MATH 2070U 2 / 15 Newton’s method Examples Implementation details Basic geometric idea x ( k + 1 ) found by extending tangent line from ( x ( k ) , f ( x ( k ) )) Next iterate x ( k + 1 ) is the x -intercept of tangent line Given iterate x ( k ) that approximates zero of f , approximate the graph of f by the tangent line L Follow L to P (where it cuts x -axis) x ( k + 1 ) defined as x -coordinate of point P c D. Aruliah (UOIT) Newton’s Method MATH 2070U 4 / 15 Newton’s method Examples Implementation details Basic geometric idea Value x ( k + 1 ) is hoped to be closer to true zero x * of f ( x * ) = 0 than the original iterate x ( k ) Looking at graph, we have (from point-slope formula) f ( x ( k ) ) = f ( x ( k ) ) - 0 x ( k ) - x ( k + 1 ) .

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