Numerical Method Test number 1

Numerical methods involve approximating roots of equations. So all of these methods are code vba autocad toward that goal.

1.Bracketting Method

This is the genral idea of graphical, bisection and false-position method.

We start with two value x1 and code vba autocad x2 that are approximation of the real root xr where f(x1) * f(x2) < 0. Since the code vba autocad function usually changes sign in the vincinity of a root. We then code vba autocad through calculations reduce the size of this bracket. If all things works out (let’s hope it should?) we will be closing in on the real root.

a. Graphical Method:

Graph the code vba autocad function and zoom in to where it intercepts the x-axis (obvious, but not literally)

b. Bisection method: this is better than the last one.

We start with x1 and x2 as our bracket. Then we calculate x3 = (x1+x2)/2. Then our next bracket Would be xi, xj with i,j in {1,2,3} where f(xi) * f(xj) < 0. (since we already have f(x1) and f(x2) of opposite sign, anything between it code vba autocad can only evaluate to the same size with EITHER f(x1) and f(x2) not both. We just reduce our search region in half. We keep doing this code vba autocad until xn and x_n-1 is within a desired error margin.

Error margin = |[x_n - x_(n-1) ] / x_n| * 100%

(absolute value of: newest estimation minus the code vba autocad one prior divided by the newest times 100%. This should yield something like 10% or 1% etc. If it’s 1% we know that we are getting there.)

c. False-position method

The idea of this code vba autocad method is we treat the function over the interval [x1, x2] as a code vba autocad straight line. and wherever this line intercepts the x-axis is code vba autocad our new estimation for the root x. 7th grade mathematic (thank you co Tu’) yields this code vba autocad function for the x-intercept of a line passing thru x1 and code vba autocad x2 of function f(x):

                 f(x2)*(x1-x2)
xr = x2 - __________
                 f(x1) - f(x2)

of course through several iteration we will be code vba autocad closing on the real roots.

2. Open Methods

Open methods differs from code vba autocad Bracketting method because it does not involve two estimation around the code vba autocad roots. For Open Method we start with 1 value and make our way toward the code vba autocad real roots. We just need to have a very good starting point (guess work based on knowledge of the code vba autocad physical system and stuff, perhaps graphical method)

 a. Simple fixed-point iteration

For this method we simple rearrange our equation

f(x) = 0

into:

g(x) = x

Then our next guess for x_n is g(x_(n-1));

It’s very mind boggling simple

Example:

f(x) = e^(-x) - x

we rearrange:

x = e^(-x);

Starting with x0 = 0 we evaluate e^(-0) and this will be our x1 (for the next iteration).. (e^0 = 1 so x1 = 1)
For the 2nd iteration to evaluate x2 = we do g(x1) or  1/e = 0.36787 so x2 = .36787

and code vba autocad so forth and so on. each round we evaluate the code vba autocad estimated error and the true error

b. Newton-Raphson

This one of the code vba autocad open method, that is we start out with a single approximated value x_0. We will then code vba autocad find the tangient of the graph at x0. This line intercepts x-axis at our new approximation x1. We repeat this code vba autocad by finding the tangient at x1 which intercepts x-axis at our new approximation x2. Ideally as we progress our approximated error declines because we are code vba autocad approaching the true value of the root.

formula:

                       f(x_i) 
x_(i+1) =     ——-
                       f’(x_i)
c. The Secant method

Like Newton Raphson, but with a code vba autocad twist. When the derivative is hard to evaluate we estimate the code vba autocad derivative at x_i by:

                      f(x_(i-1)) - f(xi)
f ‘(xi)=         —————-
                       x_(i-1) - xi

and code vba autocad just plug this back to the original Newton-Raphson formula above. Pretty straightforward.

3. Müller’s Method

This is code vba autocad a method to find roots of polynomials. We start with 3 values x0, x1 and code vba autocad x2. We then find the corresponding 3 points on the code vba autocad graph. We then We “approximate” a code vba autocad parabola passing through these 3 points and find where this code vba autocad parabola intercepts the x-axis. That would be our x4 and code vba autocad we repeat the process with x_1-thru-4 (dropping x0 out).

 for parabola is in the form:
ax^2 + bx + c = 0

(lots of formula, but simple for values of a, b and c).

After we have code vba autocad the forumla we solve the quadratic equation for the new estimation x_new and repeat the whole process.