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2-4-3 Numerical Processing

Numerical processing is an algorithm that uses numerical calculation for determining an approximate solution.

(1) Newton’s method

When the approximate solution of n order equation is known where n is large, Newton’s method determines the approximate value of a real solution while this approximate solution is modified.

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[Algorithm of Newton’s method] 
1) Let tangent of y=f(x) and point of intersection with x axis in point p1(x1, y1) be 
x2. 
2) Let tangent of y=f(x) and point of intersection with x axis in point p2(x2, y2) be 
x3. 
3) Similarly, determine x4, x5, ···, and when the difference between xn and xn+1 
converges below a certain value, let xn+1 be the solution. 

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Original Numerical Processing source layout. Source labels: [Algorithm of Newton’s method]; 1) Let tangent of y=f(x) and point of intersection with x axis in point p1(x1, y1) be; x2.; 2) Let tangent of y=f(x) and point of intersection with x axis in point p2(x2, y2) be; x3..

Newton’s method cannot be used unless function f(x) is differentiable, because it is necessary to determine the tangent of function f(x). In addition, x coordinate of one point on the curve can be assigned as the initial value. However, a solution may not converge depending on how the initial value is assigned: that is, approximate value of solution might not be obtained.

(2) Simpson’s method

Trapezoidal rule is one of the methods of calculating the approximate value of an area between the x-axis and the curve over the given interval. In the trapezoidal rule, part of the area to be calculated is split into trapezoids, and the total is determined.

p. 442 · PDF 452

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However, the trapezoidal rule approximates a curve with a straight line. Therefore, it suffers from the disadvantage that errors become large. The only way of reducing the error is to increase the number of trapezoids to be split. However, if the number of trapezoids to be split is large, the algorithm becomes complex and computational complexity increases. For this reason, in Simpson’s method, when the area after splitting is determined, the portion that corresponds to the side of y=f(x) is approximated with a parabola. In specific terms, the area of S1 portion that is split as shown below is determined with the expression by considering a parabola that passes through point p0, p1, and p2 instead of connecting p0 and p2 with a straight line.

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Original Numerical Processing source layout. Source labels: .

In the trapezoidal rule, the portion of the area to be determined is divided into n equal parts. However, the characteristic of Simpson’s method is that it is divided into 2n (i.e., even number) equal parts.