NUMERICAL METHODS

ROOTS OF EQUATIONS

DEFINITION

REAL ROOT COMPLEX ROOTS

ALGEBRAIC EQUATIONS

Solving a first-degree algebraic equation

solution is:

Solution of a second degree algebraic equation

solution is:

Solution of a transcendental equation:

solution is:

SEARCH OF ROOT

'Finding Roots of an Equation'
Suppose you have the equation y=h(x) and want to know the values of x for which y=0, i.e. the roots of h(x). To find the roots using KaleidaGraph:
Generate data for the equation and create a plot.
Define the curve fit equation as (h(a) +1)*x;a=(initial guess for root value), where h(a) is simply h(x) with each x replaced by the parameter a.
Apply the curve fit. The resulting value of a is the value of x for which h(x) is zero.
This routine works because KaleidaGraph is fitting (h(a) +1)*x to the curve 1*x, hence a is found such that h(a)=0 and thus h(a)+1=1.
Example: Let h(x)=(sin(x+2)/(cos(x)+3))-1.1.
Create a data set and plot the data for this function. From looking at the plot of this function, it has root values near 110 and 170.
Define the curve fit as: (((sin(a)+2)/(cos(a)+3))-1.1+1)*x;a=200
The upper root value is x=166.743. To find the lower root value, an initial guess of 100 can be used which yields a result of x=108.709.

Bisection
False position

FIXED POINT
Newton Raphson
SECANT

GRAPHICAL METHODS
As visual aids in the understanding of numerical methods open many closed as to identify the number of possible roots and identifying cases in which the open methods will not work.
MISCELLANEOUS ESTATE SEARCH

ROOTS OF POLYNOMIALS

EXAMPLES OF APPLICATION IN ENGINEERING

Bisection method

-Is to consider an interval (xi, xs) which ensures that the function root.

-The segment bisects,taking xr bisection point as a proxy for the desired root.

-It then identifies which of the two intervals is the root

-The process is repeated n times, until the point of bisection xr practically coincides with the exact value of the root.

METHOD OF RULE FALSE -Is to consider an interval (xi, xs) which ensures that the function root.

-Draw a line through the points (xi, f (xi)), (xs, f (xs)).
-We get the point of intersection of this line with the x-axis: (xr, 0); xr taken as an approximation of the desired root. -It then identifies which of the two intervals is the root.
- The process is repeated n times, until the point of intersection xr practically coincides with the exact value of the root.
Newton Raphson METHOD
- It consists in choosing any one initial point x1 to approximate the root.
- It consists in choosing any one initial point x1 as a proxy for the root and get the value of the function at that point.Draw a line tangent to the function at that point.
- The point of intersection of this line with the x-axis (xr, 0), represents a second approximation of the root. - The process is repeated n times until the twelfth intersection point practically coincides with the exact value of the root. - Although the method works well, there is no guarantee of convergence.

SECANT METHOD It consists in choosing any two initial points x0, x1 for which assesses the function values:
f(x0) = f(x1) f (x0) = f (x1)
A secant line is drawn to the role of these two points.
The point of intersection of this line with the x-axis (x2, 0) represents a second approximation of the root.
The process is repeated n times until the twelfth intersection point practically coincides with the exact value of the root.
FIXED POINT METHOD Consider the decomposition of the function f (x) in a difference of two functions: the first g (x) and the second, provided the function x.
The root of the function f (x) is when f (x) = 0, ie when g (x) - x = 0, so that g (x) = x.
The point of intersection of the two functions, then gives the exact value of the root.
The method is to consider an initial value x0, as an approximation to the root, to assess the value of this function g (x0), considering it as an approximation of the root.
The process is repeated n times until g (x) is roughly equal to x.
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*********SYSTEM OF NONLINEAR EQUATIONS.****************

FIXED POINT METHODS IN NONLINEAR SYSTEMS OF EQUATIONS
-Consider the intersection of two nonlinear functions u (x, y) v (x, y).

-The intersection of the curves u (x, y) v (x, y) occurs when u (x, y) - v (x, y) = 0, so that u (x, y) = v (x, y).

-The point of intersection of the two functions, then gives the exact value of the root.The method is to consider an initial value x0, as an approximation to the root, to assess the value of this function g (x0), considering it as a second approximation of the root.
The process is repeated n times until g (x) is roughly equal to x..

-However, fixed point method, the convergence depends on how they formulate the equations of recurrence and chosen initial values close enough to the solution. In the two formulations following the method diverges.

Newton Raphson METHOD IN NONLINEAR SYSTEMS OF EQUATIONS

- It consists in choosing the coordinates of a point (x1, y1) as an approximation of the intersection point of the functions u (x, y) v (x, y) that make them declared void.

-Obtain the values of the functions u (x, y), v (x, y) valued with the coordinates (x1, y1) and locate the four points u (x1, y), v (x1, y), u (x , y1) v (x, y1).

-Draw a tangent line parallel to the secant joining the points u (x1, y) u (x, y1) and a tangent parallel to the secant joining the points v (x1, y) v (x, y1).).

- The point of intersection of these two tangents is a second approach (x2, y2) the point of intersection of the two functions.)

-The process is repeated n times until the intersection point coordinates (xn, yn) practically coincides with the exact value of the intersection between the two curves.

-This procedure corresponds, analytically, to extend the use of the derivative, now to calculate the intersection between two nonlinear functions.
As for a single equation, the calculation is based on the expansion of Taylor series of first order, now many variables to consider the contribution of more than one independent variable in determining the root.
For two variables, the number of first-order Taylor writes, for each nonlinear equation: