Mathematical Errors

              Mathematical Errors

(1) Inherent Errors :

            Errors which are already present in the statement of a problem before its solution , are called inherent errors . Such errors arise either due to the given data being mathematical tables , calculators or the digital computer . Inherent errors can be minimized by taking better data or by using high precision computing aids .

(2) Rounding Errors :

               It arises from the process of rounding off the numbers during the computation . Such errors are unavoidable in most of the calculations due to the limitations of the computing aids . Rounding errors can however be reduced by retaining at least one more significant figure at each step than that given in the data and rounding off at the last step . 

(3) Truncation Errors :

               These are used by using approximate results or on replacing an infinite process by a finite one . If we are using a decimal computer having a fixed word length of 4 digits , rounding off of 13.658 gives 13.66 whereas truncation gives 13.65 .

    For example ,

   if eˣ = 1+x + x²/2! +x³/3! +.....∞ = X (say) is replaced by 1+x+x²/2! + x³/3! = X' (say)

then the truncation error is X-X'

Truncation error is a type of algorithm error .

(4) Absolute , Relative,Percentage              Errors : 

     If X is the true value of a quantity and X' is its approximate value , then |X-X'| is called the absolute error Eₐ .

 The relative error is defined by 

                   Eᵣ = |X-X' / X |

and the percentage error is 

                   Eₚ = 100Eᵣ = 100 × |X-X' /X|

If X̅ be such a number that 

                     |X-X'| ≤ X̅ 

 then X̅ is an upper limit on the magnitude of absolute error and measures the absolute accuracy  .