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Error analysis is absolutely critical to a successful numerical method. This is because, often, the method involves discrete numerical iteration, such as when solving a problem in integral calculus. Computers have errors in floating point representation, such as truncation and round-off. These errors can accumulate, and actually overwhelm the result.

For example, if your floating point format has 24 bit resolution (which is the size for a typical 32 bit float), adding 1 to 1×1025 will not change the result. If your program involves a loop, it could fail in this case. It is important to add and subtract numbers of comparable magnitude.

Another example is Taylor series, used for generating trignonometric functions such as sin(x). These series are most accurate between -pi/2 and +pi/2. If you were calculating sin(x) for large values of x, you would want to normalize x to be within that range by adding or subtracting 2 pi and then finally pi as needed. Problem is, that, at large values of x, 2 pi might only represent 1 or 2 bits of resolution, and your answer will be way off.

A third example is the solution of 3 equations in 3 unknowns. This represents three planes in 3-D space, which should, if not parallel, resolve to a single point. Error is measurement, and error in floating point representation, could easily (if two of the planes are nearly parallel) result in large error in the result.

The art is in balancing the accumulation of error against the increase in resolution as things become smaller and smaller, such as delta x when doing integration, a fourth example.

Q: Significance of error analysis in numerical method?

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Numerical Analysis - an area of mathematics that uses various numerical methods to find numerical approximations to mathematical problems, while also analysing those methods to see if there is any way to reduce the numerical error involved in using them, thus resulting in more reliable numerical methods that give more accurate approximations than previously.

Error propagation in numerical analysis is just calculating the uncertainty or error of an approximation against the actual value it is trying to approximate. This error is usually shown as either an absolute error, which shows how far away the approximation is as a number value, or as a relative error, which shows how far away the approximation is as a percentage value.

in trpezoidal rule for numerical integration how you can find error

the precentage of error in data or an experiment

The main disadvantage of the bisection method for finding the root of an equation is that, compared to methods like the Newton-Raphson method and the Secant method, it requires a lot of work and a lot of iterations to get an answer with very small error, whilst a quarter of the same amount of work on the N-R method would give an answer with an error just as small.In other words compared to other methods, the bisection method takes a long time to get to a decent answer and this is it's biggest disadvantage.

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Numerical Analysis - an area of mathematics that uses various numerical methods to find numerical approximations to mathematical problems, while also analysing those methods to see if there is any way to reduce the numerical error involved in using them, thus resulting in more reliable numerical methods that give more accurate approximations than previously.

Error propagation in numerical analysis is just calculating the uncertainty or error of an approximation against the actual value it is trying to approximate. This error is usually shown as either an absolute error, which shows how far away the approximation is as a number value, or as a relative error, which shows how far away the approximation is as a percentage value.

in trpezoidal rule for numerical integration how you can find error

J. E. Akin has written: 'Finite element analysis with error estimators' -- subject(s): Error analysis (Mathematics), Finite element method, Structural analysis (Engineering) 'Finite Elements for Analysis and Design' 'Finite Elements for Analysis and Design' 'Application and implementation of finite element methods' -- subject(s): Data processing, Finite element method

Daniele Funaro has written: 'A new method of imposing boundary conditions for hyperbolic equations' -- subject(s): Numerical analysis 'Convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment' -- subject(s): Approximation, Boundary value problems, Convergence, Error analysis, Hyperbolic functions, Penalty function, Spectral methods 'Computational aspects of pseudospectral Laguerre approximations' -- subject(s): Numerical analysis, Laguerre polynomials

Ross N. Hoffman has written: 'Distortion representation of forecast errors for model skill assessment and objective analysis' -- subject(s): Errors, Error analysis, Numerical weather forecasting

the precentage of error in data or an experiment

Numerical analysis has numerous applications in all fields of science and some fields of engineering, and essentially any type of work that requires calculations to give very precise solutions. The point of numerical analysis is to analyse methods that are used to give approximate number solutions to situations where it is unlikely to find the real solution quickly, and to try and improve upon these methods so as to reduce the amount of error generated by computer calculation. It is essential in work that requires precise numbers to get very good approximations with very little error in them, if approximations with just even 1 or 2% error are used in another calculation, and the answer of that calculation used in another, and so on, the errors will build up and you end up with very unreliable numbers. This is why it is a good idea to study numerical anlysis if you intend to go into any area of work requiring precise calculations, so as to be able to identify if there are areas you can improve so as to better your methods in finding solutions and reducing error.

The main disadvantage of the bisection method for finding the root of an equation is that, compared to methods like the Newton-Raphson method and the Secant method, it requires a lot of work and a lot of iterations to get an answer with very small error, whilst a quarter of the same amount of work on the N-R method would give an answer with an error just as small.In other words compared to other methods, the bisection method takes a long time to get to a decent answer and this is it's biggest disadvantage.

Experiments are often likely to contain errors. Quantitative error analysis means determining uncertainty, precision and error in quantitative measurements.

If the estimated value is given, then calculating the numerical error from the percentage error, or the other way around, is a trivial exercise. If the estimated value is not known then it is impossible to tell which of the two is clearer.

Robert J. Melosh has written: 'Manipulation errors in finite element analysis of structures' -- subject(s): Data processing, Error analysis (Mathematics), Structural analysis (Engineering) 'Structural engineering analysis by finite elements' -- subject(s): Finite element method, Structural analysis (Engineering)