Supp#1 EE 150 Numerical Calculus, Newton's Method-1

Supp#1 EE 150 Numerical Calculus, Newton's Method-1 - EE...

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1 EE 150 Background survey of Calculus concepts applied to numerical computing, Newton’s Method, Euler’s Method Numerical Differentiation In calculus, a field of mathematics that studies functions by looking at small changes in independent variables, the derivative is a measurement of how a function changes when the values of its inputs( independent variables) change. A derivative can be thought of as how much a quantity which is a function of some independent variable , say x , is changing in the neighborhood of some given point , a particular value of x . More exactly it is a ratio formed by the change in the value of a function in the neighborhood of a given point divided the the size of that neighborhood as the size of that neighborhood is allowed to approach zero or become infinitesimal. In numerical computing the size of the neighborhood can never be zero but it can be made small. If it is made small enough the simple ratio ,described without the size of the neighborhood going to zero, becomes an accurate enough description of how the function is changing at a given point as the independent variable varies in the neighborhood of that point. The neighborhood of that point then being finite and occupying some extent in the space. Consider the figure below. The function has a value at the point , x , equal to f(x) . The value of the function at the point x + h is f( x + h ). We can connect the point ( x, f(x)) to the point ( x+h,f(x+h) ) we can form the secant line as shown in figure-1.(If we let x = a , the slope of the secant line is: [equation one]
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2 This is the rate of change of the function in the neighborhood of the region x = a In computing the rate of change of a function using a computer , as long as h is small enough, equation one gives a good numerical approximation of the derivative of the function in the neighborhood of x = a . figure-1 The exact value of the derivative is found by letting the value of h get smaller and smaller the secant line becomes the tangent to the curve at the point of interest x = a . This is shown in figure-2.
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3 Figure-2 Equation one is also sometimes referred to as Newton’s quotient and again it can be used to estimate the value of a derivative numerically. The value of h needs to be made small enough in order for the numerical approximation to give an accurate description of the exact derivative. However if it is made to be too small the numerator in Newton’s quotient, which is the difference of two terms, eventually becomes zero as the two terms become equal. When the two terms become very close for a finite but very small value of h and you reach the limits of the computer’s accuracy for determining the difference between two numbers that are nearly equal, the computer will be forced to make rounding errors and the numerical approximation of the derivative will reflect those errors. For instance if the numerator in equation one gets smaller as h gets smaller. The numerator might be ( 3.1786639 – 3.17865420 ), but if you are in ‘format short’ in Matlab
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Supp#1 EE 150 Numerical Calculus, Newton's Method-1 - EE...

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