Newton Raphson Method

NEWTON RAPHSON

If we want to find r, the root of a function f(x), we start with X₀ an estimate of r. From X₀ we produce an estimate X₁ , this is a improve value of the root. 

 From X₁ we can produce an estimate X₂ which is a improve value of the root. 

In this way we are going to reach an approximation close to r. 

Next dias gram shows steps to obtain an approximation using Newton Raphson Method:

Source: http://www.polyvore.com/cgi/img-set?.out=jpg&id=CiLA-pZS3hGf6J_BzGWFag&size=l

Bisection Method

 Bisection method

Consider the equation f(x) continuous in [x₁ , x₂ ], we evaluate Xm, the midpoint of the interval.

Xm = (X1+X2)/ 2

For example: 

Source: http://www.ita.uni-heidelberg.de/~banerjee/teaching/UKNum/Lecture_03.pdf

 We have two options:

• 1.       If f(x₁) and f(x_m) have opposite signs then we can reduce the range, now instead of  [x₁ , x₂ ] new interval is  [x₁ , x_m ].
• 2.       If f(x_m) and f(x₂) have opposite signs then we can reduce the range, now instead of  [x₁ , x₂ ] new interval is  [x_m , x₂ ].

We can be sure that within these values is the real root because if the function is continuous it must be zero between the selected values.

We have different possible cases working with bisection methods:

Examples:

Case 1

Case 2

 Case 3

Case 4

Source: http://www.ita.uni-heidelberg.de/~banerjee/teaching/UKNum/Lecture_03.pdf

Advantages & disadvantages of bisection method!

Chris Madden Cartoon

Roots of equations

Roots of equations

If we have the function:

Y=f(x)

The values that makes y=0 are called roots of equation. In case of real estate it correspond to x values where the function had cut the x axis.

Fundamental theorem of algebra states that a polynomial of degree n has n roots, for example, a cubic polynomial has three distinct roots:

1.         A real root.2.       A simple real root and one real root.
3.       Oner real root and a complex conjugate pair.

The methods  for finding the real roots are divides into intervals and open methods.

Methods of intervals: or closed. This methods needs two initial values to restrict the root, this values establish a interval, which will be gradually reduced. In this way we can reach an approximation of the real value, for this reason these methods are CONVERGENTS.

Open methods: this methods needs a single initial value, x, this is a initial approach to the root. In some cases this methods don´t reach a good approximation, instead of this methods are DIVERGENTS. 

1.  

GLOSSARY

GLOSSARY

Accuracy: truthfulness and certainly of a value.


Approximation: Estimation or inexact representation of a value. An approximation is a quantity that represents a desider value.


Analytical solution: Mathematical exppression that represent a system. Gives a real result. 

Mathematical Model: formulation that expresses a physical phenomena or system in mathematical terms. 


Method:technic for doing something.

Numerical method: “techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations.” Steven Chapra


Numerical solution: Approximation  to the result of a system.
  

Parameters: compounds of a model.

Significant digits: digit(s) that gives confidence to a number.

Total numerical error: is the summation of the truncation and round-off error.

Truncation error: “is the discrepancy introduced but the fact that numerical methods may employ approximations to represent exact mathematical operations and quantities” Steven Chapra