Properties of Limits

Error Control Inequalities

Here suppose C is an approximation to c and D is an approximation to d. Further suppose |C-c| £ e₁ and |D-d| £ e₂. You could imagine that e₁ and e₂ are both numbers of the form [1/2]10^-k. Alternatively, you could imagine that C and D are estimates (from measurements perhaps) and that the true values of c and d lie in the intervals [C-e₁,C+e₁] and [D-e₂,D+e₂], respectively. Here the values of the errors would depend on circumstances beyond the control of a mathematics book. Now if the uncertain approximations C and D are used in computations in place of the true values c and d, there will be an error. Thus error control in computations will be of interest.

The following theorem describes the maximum possible error in the approximations C+D, C·D, [(C)/(D)] and [1/(D)] to the true values of c+d, c·d, [(c)/(d)] and [1/(d)].

Theorem E.1 [Error Control Inequalities] Assume |c-C| £ e₁ and |d-D| £ e₂. Suppose k is a real number. Then

|(C+D)- (c+d)| £ e1+e2 |CD-cd| £ |C|e2+|D|e1+e1e2 |kC-kc| £ |k|e1

Moreover if D^* = |D|₂ > 0 then 

ê ê ê C D - c d ê ê ê £ |C| D* e1+ |C| |D|D* e2

and

ê ê ê 1 D - 1 d ê ê ê £ |C| |D| |D*| e2.

Corollary:  If  c and d agree with C and D to k+1 decimal places  then c+d agrees with C+D to k decimal places.

These inequalities or similar ones can be used to control the absolute error in sums and the relative error in products and quotients. That being said, the principle of error control is easier to understand than the practice. In practice, numerical experimentation may be attempted, or not, to empirically estimate variations in results due to errors in the approximation of numbers. But the principle guides theoretical development of approximations and limit-based calculations.

The values of the approximations C and D will be available unless c and/or d are known exactly. Because of this, the above maximum error estimates stated above involve the two estimates C and D of c and d, and not the exact values of the latter. But the derivation of these inequalities, show that the error estimates also hold with c and d in place of C and D. The conclusions would still hold if c and d were regarded as approximations to C and D.

The conclusion indicates the maximum possible absolute error in the calculation of sums, products, quotients and divisors. Roughly speaking, if the error was known precisely then the approximation could be improved to eliminate the error in them. Thus with the given information about the approximations, the above inequalities cannot be improved.

Learn More: The above inequalities and a different perspective on them can be found along with proofs in the first chapter in the book Calculus by L. Bers (Holt, Rinehart and Winston 1969, SBN 03-065240-5). The chapter in question provides further background information on the decimal and decimal-free representation of real numbers.

In the direct use of the above inequalities or error estimates, the maximum possible errors e₁ and e₂ in the approximations are specified first. In the indirect use of the above inequalities, or error estimates, we may specify a maximum allowable (target) error e > 0 for a computation first, and then try to choose e₁ and e₂ second, so that the maximum allowable error is not exceeded. For example in the estimation C+D of the sum c+d, the error bounds e₁ ³ 0 and e₂ ³ 0 should satisfy

e ³ e1+e2

so that the actual error |(c+d)-(C+D)| £ e. The proof and the next pages may further clarify this error control effort.

Proof of the Inequalities - optional readings.

For the sum, observe

|(C+D)- (c+d)| = |(C-c)+(D-d)| £ |(C-c)|+|(D-d)| £ e1+e2

For the product, observe

|cd- CD| = |(C+(c-C))(D+(d-D))-CD| = |[CD+(c-C)D+(d-D)D+(c-C)(d-D)]-CD|    = |(c-C)D+(d-D)D+(c-C)(d-D)]| £ |(c-C)D|+|(d-D)D|+|(c-C)(d-D)| £ |c-C|·|D|+|d-D|·|D|+|c-C|·||d-D| £ e1·|D|+e2·|D|+e1e2

For the reciprocal, observe |d-D| £ e₂ implies

|d| > |D|-e2 = D* > 0

and therefore [1/(|d|)] > [1/(|D|-e₂)] = [1/(D^*)]. Therefore

ê ê ê 1 d - 1 D ê ê ê = ê ê ê (d-D) Dd ê ê ê £ e2 |D|·|d| £ e2 |D|D*

For the quotient observe [1/(|d|)] > [1/(|D|-e₂)] = [1/(D^*)] implies

ê ê ê c d - C D ê ê ê = ê ê ê cD-Cd dD ê ê ê = ê ê ê [C+(c-C)]D-C[D+(d-D)] dD ê ê ê = |(c-C)D-C(d-D)| |Dd| £ |(c-C)D|+|C(d-D)| |Dd| £ |c-C|·|D|+|C|·|d-D| |Dd| £ e1|D|+|C|e2 |Dd| £ e1|D|+|C|e2 |D|D*

 

Calculus Appetizers
& Lessons

YOU are better than YOU think. Show yourself  how:

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and writing

   |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
|      | |     |
   \             /   
\    =   /

 -/[]\- 
||
  _ / \     
 |||||||||||||||||||||||||||| .

www.whyslopes.com   [Top of this Page]

Custom www Search

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb Please report by email,  errors in mathematics or grammar or terminology to site author If an arithmetic topic you need is not covered in site pages,  report that as well. Topics in most demand will be covered first in site growth.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby (email form) All Rights Reserved.