Numerical Approach

Numerical Approach

It is a systematic technique whose results are approximations of the true value that assumes the variable of interest, the consistent repetition of the art, to which the called "iterations" is what you closer and closer to the desired value.

Examples:

• 3.1416 is a numerical approximation of π
• 2.7183 is a numerical approximation of e
• 1.4142 is a numerical approximation of the square root of 2
• 0.333333 is a numerical approximation of 1 / 3

Significant Figures

The measurements are normally done through instruments, such as a speedometer to measure the speed of a car, or an odometer to measure the distance covered.

The number of significant figures is the number of digits t, which can be used with confidence to measure a variable, for example: 3 significant figures on the speedometer and 7 on the odometer.

The management of significant figures can develop criteria to detect how accurate are the results and to assess levels of accuracy and precision with which they are expressed some numbers such as pi, or root of 2.

Accuracy And Precition

Precision:
Refers to the number of significant figures representing a quantity.

Accuracy:
It refers to the approach of a number or measure the numerical value is supposed to represent.

The numerical methods should provide sufficiently accurate and precise solutions. The term "error" is used to represent both the inaccuracy and to measure the uncertainty in the predictions.

Covergence And Stability

Convergence:

Convergence is defined as a numerical method ensuring that, when making a "good number" of iterations, the approximations obtained eventually move closer and closer to the true value sought.

To the extent that a numerical method requires fewer iterations to get close to another desired value, is said to have a faster convergence.

Stability:

Stability means of a numerical method the level of assurance of convergence. Some never numerical methods and instead converge, diverge, that is, away from the more desired result.

To the extent that a numerical method, to a very wide range of possibilities of mathematical modeling, cnverja is safer than another, is said to have greater stability.

It is common to find methods that converge quickly, but they are very unstable and, in contrast, very stable models, but slow convergence.

Error In Numerical Methods

There are two types of error that are common in numerical calcula

tions:

Roundoff error:

Is due to the fact that floating

point numbers are represented by finite precision.

Truncation error:

Are related to the method of approach to be used because they generally face an infinite series of terms, will tend to cut the number of terms, introducing an error at that time, not to use the series complete (assumed to be exact). In one itation is understood as the error by not following iteration and continue moving towards the solution.

Source: Error in numerical methods, Brian D. Storey

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