Find cube root of 20 using regula falsi method
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One approach is to use Newton's Method, which provides a convergent numerical approximation (with the caveat that you choose an initial guess that is 'close enough' to the real solution; in this sense, doing so for large numbers gets more difficult).
Newton's Method is based upon finding roots of a function f(x)f(x). To see how this applies to square or cube roots, suppose that y=n−−√y=n for some fixed nn. Well, then this yy would be a root of the equation f(x)=x2−nf(x)=x2−n. Similarly, f(x)=x3−nf(x)=x3−n would provide us with a way to calculate the cube root of nn.
Newton's Method works as follows: start with an initial guess x0x0, and then, for each nn, define xn+1:=xn−f(xn)f′(xn)xn+1:=xn−f(xn)f′(xn).
Take the simple example of calculating 10−−√10. We use f(x)=x2−10f(x)=x2−10 with Newton's Method to find this. In this case, f′(x)=2xf′(x)=2x. Letting out initial guess be x0=3x0=3 (32=932=9 which is close to 1010), we have x1=3−32−102(3)=3+1/6=19/6x1=3−32−102(3)=3+1/6=19/6. Then x2=19/6−(19/6)2−1019/3≈3.16228x2=19/6−(19/6)2−1019/3≈3.16228. For comparison, 10−−√≈3.16227710≈3.162277, which is already very close to 3.16228.
Newton's Method is based upon finding roots of a function f(x)f(x). To see how this applies to square or cube roots, suppose that y=n−−√y=n for some fixed nn. Well, then this yy would be a root of the equation f(x)=x2−nf(x)=x2−n. Similarly, f(x)=x3−nf(x)=x3−n would provide us with a way to calculate the cube root of nn.
Newton's Method works as follows: start with an initial guess x0x0, and then, for each nn, define xn+1:=xn−f(xn)f′(xn)xn+1:=xn−f(xn)f′(xn).
Take the simple example of calculating 10−−√10. We use f(x)=x2−10f(x)=x2−10 with Newton's Method to find this. In this case, f′(x)=2xf′(x)=2x. Letting out initial guess be x0=3x0=3 (32=932=9 which is close to 1010), we have x1=3−32−102(3)=3+1/6=19/6x1=3−32−102(3)=3+1/6=19/6. Then x2=19/6−(19/6)2−1019/3≈3.16228x2=19/6−(19/6)2−1019/3≈3.16228. For comparison, 10−−√≈3.16227710≈3.162277, which is already very close to 3.16228.
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