What is Error in Numerical Analysis

In Numerical techniques, the error was caused by the finite precision of computations involving floating-point and integer values.

The error occurs due to the difference between the exact mathematical solution and the approximate solution obtained when simplifications are made to the mathematical equations to make them more manageable to calculate.

The error comes from the fact that either simplification usually involves the truncation of an infinite series expansion to make the computation possible and practical, or because the least important bits of an arithmetic operation is thrown away.

Types of error

ROUND OFF ERROR

Round-off is also known as rounding error. It is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite precision.

Rounding errors are due to inconsistency in the representation of real numbers and the arithmetic operations which are done with them.

This form is called a quantization error.

 Using approximation values or algorithms, especially when using many digits to represent real numbers, one of the main goals of numerical analysis is to calculate computation errors.

Computation errors are also known as numerical errors, They include both truncation errors and roundoff errors.

When a part of calculations with an input requires any roundoff errors are made, errors may gather, and sometimes control the calculation.

In nonstructural problems, significant errors may gather.

INHERITANCE ERROR

When performing computations with algebraic operations among

approximate values, we carry to some extent the errors of the original

data into the final result. Such errors are called inherent errors of the operation.

 For example, let x=0.4444 and y=2.1516 be two approximate numbers for the

exact number 1/3 and If we perform an algebraic operation between

these two approximate numbers,  that’s why the error will introduce in the final result.

TRUNCATION ERROR

These errors occur due to the use of approximation formula in computation or by truncating the infinite series to some approximation terms. The study of this type of error is usually associated with the problem of convergence of infinite series

LOCAL TRUNCATION ERROR

The local truncation error is the error in numerical techniques which is used to estimate the error,

introduced in a single iteration of the method, assuming that everything fed into the method

was perfectly accurate.lets assume that y1, y2, . . . , yN refer to the numerically computed values

and y(t1), y(t2), . . . , y(tN ) refer to the corresponding exact values (so that yn ≈ y(tn)). To

determine the local truncation error, we have to analyze a general iteration of a method in which the value

yn+1 is computed.

Local truncation error = y(t_n+1) – y_n+1

let’s assume that yn+1 is determined from exact information. That is if we have
a method of the form
                                    y_n+1 = φ(t_n, y_n, f, h)
where φ is considered as the formula for the numerical method, then assume that
                                    y_n = y(tn), i.e.
                                    y_n+1 = φ(t_n, y(t_n), f, h)
and, we have assumed that we are going to examine the difference y(t_n+1) – y_n+1
which we call the local truncation error.

GLOBAL TRUNCATION ERROR

The global truncation error is the collection of the local truncation error over all of the iterations, assuming perfect knowledge of the original solution at the initial time step.

More formally, the global truncation error En, at time of tn

E_n = y(t_n)-y_n                                                           

=y(t_n) – (y₀+hA(t₀,y₀,h,f)+ hA (t₁ , y₁ , h,f) +…+hA(t_n-1, y_n-1 ,h,f))

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