Lecture 4 Printed Notes

Lecture 4 Printed Notes - ME 218 ENGR COMPUTATIONAL METHODS...

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ME 218: ENGR COMPUTATIONAL METHODS Lecture Notes (Set #4) Secant Method To overcome the need in the Newton-Raphson iterative scheme to evaluate first derivative of the function, or the possibility of the derivative going to zero, the Secant Method can be used. It utilizes the information of a second point to evaluate the derivative, by assuming that the function is linear in the domain of interest. From the Newton-Raphson’s method: x 2 = x 0 – f(x 0 )/f’(x 0 ) But using another point, x 1 , between x 2 and x 0 to determine the derivative, f’(x 0 ) : f’(x 0 ) = (f(x 1 ) – f(x 0 ))/(x 1 – x 0 ) Thus, generalizing the iterative scheme: x N + 1 = x N f(x N ) x N x N 1 f(x N ) f(x N 1 ) Λ Ν Μ Ξ Π Ο The computation of the derivative is avoided by using two starting values, x 0 and x 1 !
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Matrix Algebra A matrix is a rectangular array of numbers in which not only the value of the number is important, but also its position in the array. It is very easy to solve a system of linear equations using matrices. Ω Ψ Χ = = = Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = m j n i for a a a a a a a a a a ij nm n n m m , , 2 , 1 , , 2 , 1 ] [ 2 1 2 22 21 1 12 11 A . ' ' ' ' . ' ' ' ' position column the denotes j position row the denotes i a ij i) Matrix Addition Two matrices of the same size may be added or subtracted.
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Lecture 4 Printed Notes - ME 218 ENGR COMPUTATIONAL METHODS...

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