I LOVE METHODS

28 julio 2010

What is a Mathematical Model ???

Types of modeling

Representation of reality by means of abstractions. The models focus on important parts of a system (at least, that she is interested in a specific model type), downplaying others.Models are created using modeling tools.

Components of a Mathematical Model

Dependent Variables

Independent Variables

Parameters

Functions of Force

Operators

How to Round-Off

Round-off

Round-off errors arise because it is impossible to represent all real numbersexactly on a finite-state machine (which is what all practical digital computers are).

All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).

CASE A:
In rounding off numbers, the last figure kept should beunchanged if the first figure dropped is less than 5.
For example, if only one decimal is to be kept, then 6.422 becomes 6.4.

CASE B:
In rounding off numbers, the last figure kept should beincreased by 1 if the first figure dropped is greater than 5.
For example, if only two decimals are to be kept, then 6.4872 becomes 6.49. Similarly, 6.997 becomes 7.00.

CASE C:
In rounding off numbers, if the first figure dropped is 5, and all the figures following the five are zero or if there are no figures after the 5, then the last figure kept should be unchanged if that last figure is even.
For example, if only one decimal is to be kept, then 6.6500 becomes 6.6.
For example, if only two decimals are to be kept, then 7.485 becomes 7.48.

CASE D:
In rounding off numbers, if the first figure dropped is 5, and all the figures following the five are zero or if there are no figures after the 5, then the last figure kept should be increased by 1 if that last figure is odd.
For example, if only two decimals are to be kept, then 6.755000 becomes 6.76.
For example, if only two decimals are to be kept, 8.995 becomes 9.00.

CASE E:
In rounding off numbers, if the first figure dropped is 5, and there are any figures following the five that arenotzero, then the last figure kept should be increased by 1.
For example, if only one decimal is to be kept, then 6.6501 becomes 6.7.
For example, if only two decimals are to be kept, then 7.4852007 becomes 7.49.

27 julio 2010

Truncation and Discretization Error

Truncations errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution. Similarly, discretization induces adiscretization errorbecause the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of 3x³ + 4 = 28, after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.

Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.

Root-Finding Equations

There are three kinds of methods to solve equations:

Graphics Methods
Open methods
Methods Closed

Graphics Methods

Systems of equations or simultaneous equations can also be solved by the graphical method.

To do that, we draw the graph for each of the equation and look for a point of intersection between the two graphs. The coordinates of the point of intersection would be the solution to the system of equations.

If the two graphs do not intersect - which means that they are parallel - then there is no solution.

Example :
Using the graphical method, find the solution of the systems of equations

y + x = 3 y = 4x - 2

Solution :

Draw the two lines graphically and determine the point of intersection from the graph.

Features:
• Calculations are not accurate. • They have limited practical value.
• Allows initial estimate values.
• Allows understanding of the properties of
functions.
• They can help prevent failures in the methods.
• in general can be considered closed as ¨ closed¨

CASE A: In rounding off numbers, the last figure kept should beunchanged if the first figure dropped is less than 5.	For example, if only one decimal is to be kept, then 6.422 becomes 6.4.
CASE B: In rounding off numbers, the last figure kept should beincreased by 1 if the first figure dropped is greater than 5.	For example, if only two decimals are to be kept, then 6.4872 becomes 6.49. Similarly, 6.997 becomes 7.00.
CASE C: In rounding off numbers, if the first figure dropped is 5, and all the figures following the five are zero or if there are no figures after the 5, then the last figure kept should be unchanged if that last figure is even.	For example, if only one decimal is to be kept, then 6.6500 becomes 6.6. For example, if only two decimals are to be kept, then 7.485 becomes 7.48.
CASE D: In rounding off numbers, if the first figure dropped is 5, and all the figures following the five are zero or if there are no figures after the 5, then the last figure kept should be increased by 1 if that last figure is odd.	For example, if only two decimals are to be kept, then 6.755000 becomes 6.76. For example, if only two decimals are to be kept, 8.995 becomes 9.00.
CASE E: In rounding off numbers, if the first figure dropped is 5, and there are any figures following the five that arenotzero, then the last figure kept should be increased by 1.	For example, if only one decimal is to be kept, then 6.6501 becomes 6.7. For example, if only two decimals are to be kept, then 7.4852007 becomes 7.49.