What is a truncation error?, A truncation is caused by dropping the trailing digits of a figure (number); such as by reducing 99.987 to 99.98 or 99.9. Whereas such errors may be harmless in manual computations, they can become serious mistakes in computer calculations involving thousands or millions of mathematical operations, according to that we go to be careful with the numerical approximations we do and take in account several rules for this process.

So, for avoiding this kind of error you need to considerate how to use your data before you start doing calculations, and there are two factors you may consider for this, they are called Accuracy and Precision.


Accurate means "capable of providing a correct reading or measurement." In physical science it means 'correct'. A measurement is accurate if it correctly reflects the size of the thing being measured.
Precise means "exact, as in performance, execution, or amount. "In physical science it means "repeatable, reliable, getting the same measurement each time."
We can never make a perfect measurement. The best we can do is to come as close as possible within the limitations of the measuring instruments.
Let's use a model to demonstrate the difference.
Suppose you are aiming at a target, trying to hit the bull's eye (the center of the target) with each of five darts. Here are some representative pattern of darts in the target.







Now, you can continue studying about truncation error a here is a link about a nice video about that:

http://www.youtube.com/watch?v=NByHuFBkulw

Reading i have found an interesting subject for numerical methods, and its called Roots of equation, and i am going to talk about polynomial equations. Here are three important theorems relating to the roots of a polynomial:

1.)A polynomial of n-th degree can be factored into n linear factors.
2.)A polynomial equation of degree n has exactly n roots.
3.)If (x − r) is a factor of a polynomial, then x = r is a root of the associated polynomial equation.

The cubic polynomial f(x) = 4x3 − 3x2 − 25x − 6 has degree 3.

This polynomial can be factored and written as

4x3 − 3x2 − 25x − 6 = (x − 3)(4x + 1)(x + 2)

So we see that a 3rd degree polynomial has 3 roots.

The associated polynomial equation is formed by setting the polynomial equal to zero:

f(x) = 4x3 − 3x2 − 25x − 6 = 0

In factored form, this is:

(x − 3)(4x + 1)(x + 2) = 0

We see from the expressions in brackets and using the 3rd theorem from above, that here are 3 roots, x = 3,-1/4,−2.

In this example, all 3 roots of our polynomial equation of degree 3 are real.

Since (x − 3) is a factor, then x = 3 is a root.

Since (4x + 1) is a factor, then x =-1/4 is a root.

Since (x + 2) is a factor, then x = −2 is a root.

The equation x5 − 4x4 − 7x3 + 14x2 − 44x + 120 = 0 can be factored and written as:

(x − 2)(x − 5)(x + 3)(x2 + 4) = 0

We see there are 3 real roots x = 2, 5, -3, and 2 complex roots x = ±2j, (where j = √-1).

So our 5th degree equation has 5 roots altogether.

About complex roots, the following theorem applies :

If the coefficients of the equation f(x) = 0 are real and a + bj is a complex root, then its conjugate a − bj is also a root.

So here its an example:

The factors of the polynomial x3+ 7x2 + 17x + 15 are found using LiveMath:

x3 + 7x2 + 17x + 15 = (x + 3)(x + 2 − j)(x + 2 + j)

So the roots are

x = −3

x = −2 + j and

x = −2 − j

There is one real root and the remaining 2 roots form a complex conjugate pair.

I hope you enjoyed this, and here and interesting link for you to see about newton-raphson method.

http://www.youtube.com/watch?v=lFYzdOemDj8

Many often we do numerical approximations, and we must take care of several problem who comes within, we use numerical approximation when finding an analytical solution to a differential equation is not always a practical option, numerical approximations lead to solutions that are much more readily available.

Nevertheless the are some issues related to this process, the trick to constructing a viable numerical solution of a differential is identifying a reliable approximation of the derivative.

Errors acquired during the construction of a numerical solution come from two sources: roundoff and truncation, roundoff error arises from the limited precision of computer arithmetic. The problem is compounded in that the binary representation of many fractions is irrational, enhancing the effects of the roundoff error.

The second source of error is called truncation error, this error arises when we make discrete approximations of continuous functions. This error can be, to a certain extent, limited by making the step-sizes in the discrete function as small as possible.

Now we now in what we go to be careful when we do numerical approximations, and there are few ways to minimize this errors and we will see them later on this blog.

The nodes of the following diagram represent information to be colected, sorted, evaluated and organized.



the edges of the diagram represent activities of two way communication between the nodes and the corresponding sources of information.

Reading these days about mathematical modeling i found a really interesting sort of things about it, but i asked myself, what is mathematical modeling?, and here is the answer.

Mathematical modeling is an art, because you can translate problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight,answer, and guidance useful for the originating application.

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models.

But there is 1 new question, why is mathematical modeling inportant?, Mathematical modeling is indispensable in many applications, gives precision and direction for problem solution, enables a thorough understanding of the system modeled, prepares the way for better design or control of a system and allows the efficient use of modern computing capabilities. Mathematical modeling have a very importanting application in the petroleum industry and also in geosciences such as prediction of oil or ore deposits, map production and earth quake prediction inter alia.

So, now you can guess that without mathematical modelling the latest triumphs in the petroleum industry would be impossible.

And for saying good bye here is a link about another interesting subject, computer simulation.

http://www.sciencedaily.com/articles/c/computer_simulation.htm