Newton Iterative Method: Introduction, Principle and application

Newton Iterative Method: Introduction, Principle and application

Newton Iterative method is a tool that can find 0 points of an arbitrary function. It is much faster than the dichotomy.

The formula is: X=a-f (a)/F ' (a). Where A is the guess value and x is the new guess value. Constantly iterating, F (x) is getting closer to 0.

Principle

We will f (x) do Taylor first-order expansion: F (x) ∼f (a) + (x-a) f ' (a).

Make f (x) = 0,
∴f (a) + (x-a) f ' (a) = 0
∴f (a) +xf ' (a)-af ' (a) = 0
∴xf ' (a) =af ' (a)-f (a)
∴x=a-f (a)/F ' (a)

Example: Newton iterative method for calculating approximate value of √2

∵	X =√2
∴	x2 = 2
∴	x2-2 = 0

f (x) = equation to the left , then f (x) ∼0↔x∼√2.

F ' (x) = 2x. You can then get an iterative formula:

	X
=	A-f (a)/F ' (a)
=	A-(A2-2)/(2a)
=	a-a/2+1/a
=	a/2+1/a

The code is as follows (requires less than 1e-6 error):

#include <stdio.h> #include <math.h>int main (int argc, char const *argv[]) {Double A = 2.0;double Expect_error = 0.000001;double expect_answer = 1.4142135623731;double x;double actual_error;unsigned iteration_count = 0;do {if (a = = 0. 0) A = 0.1; /* Avoid except 0 */x = a/2 + 1/a;actual_error = fabs (expect_answer-x); a = x;++iteration_count;printf ("%d\t%.9f\t%.9f\n", Iteratio N_count, A, actual_error);} while (Actual_error >= expect_error);p rintf ("%d\n", Iteration_count); return 0;}

Output:

11.5000000000.08578643821.4166666670.00245310431.4142156860.00000212441.4142135620.0000000004

Iterated 4 times. In the dichotomy?

#include <stdio.h> #include <math.h>int main (int argc, char const *argv[]) {Double high = 2.0;double Low = 1.0;d ouble Expect_error = 0.000001;double expect_answer = 1.4142135623731;double x;double actual_error;unsigned iteration_ Count = 0;do {x = (High+low)/2;if (x*x-2 > 0) high = X;elselow = X;actual_error = Fabs (expect_answer-x); ++iteration_co unt;printf ("%d\t%.9f\t%.9f\n", Iteration_count, X, Actual_error); while (Actual_error >= expect_error);p rintf ("%d\n", Iteration_count); return 0;}

Output:

11.5000000000.08578643821.2500000000.16421356231.3750000000.03921356241.4375000000.02328643851.4062500000.00796356261.421 8750000.00766143871.4140625000.00015106281.4179687500.00375518891.4160156250.001802063101.4150390620.000825500111.4145507 810.000337219121.4143066410.000093078131.4141845700.000028992141.4142456050.000032043151.4142150880.000001526161.41419982 90.000013733171.4142074580.000006104181.4142112730.000002289191.4142131810.00000038219

iterated 19 times.

Newton Iterative Method: Introduction, Principle and application