Numerical method errors analysis examines the difference between true and approximate values. Absolute error is the difference between true and approximate values, while relative error is the ratio of absolute error to true value. Percentage error is calculated by taking the absolute difference between true and approximate values, dividing by the absolute true value, and multiplying by 100. Examples are provided to demonstrate calculating absolute, relative, and percentage errors.

1. 📖
Numerical method
errors analysis

2. What is Error Analysis in
Numerical method?

3. Errors Analysis
The error of a quantity is the difference
between it’s true value and
approximate.

4. Absolute Error
+ The absolute error of a quantity is the absolute
value of the difference between the true value X
and the approximate value x. It is denoted by

5. An approximate value of ᴫ is 3.1428571 and true
value is 3.1415926.
Find the absolute Error
𝐸𝐴 = 𝑋 − 𝑥 = 3.1415926 − 3.1428571
= − 0 .0 0 1 2 6 4 5
= 0 .0 0 1 2 6 4 5

6. Find the absolute errors of the number 8.6 if both of its
digits are correct.
• If the number X is rounded to N decimal places, then 𝐸𝐴 =
1
2
(10−𝑁
)

7. Evaluate the sum S = 2 + 3 + 5 to 4 significant
digits and find its absolute errors.

8. The relative error of a
quantity is the ratio of it’s
absolute error to it’s true
value. It is denoted by ER
Relative Error
ER =
EA
𝑋

9.  Find the absolute, relative and percentage errors of the number 8.6 if both of
its digits are correct.
+Solution:
The given number is X =
8.6
Since both digits are
correct so N = 1
The absolute error is,
EA =
1
2
( 10−1
)
= 0.05
The relative error is,
ER =
EA
𝑋
=
0.05
8.6
= 0.0058
The percentage
error is,
EP = 100 ER
= 100 × 0.0058
= o.58

10. Percentage Error
+Formula
PE =
|𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒−𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒|
|𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒|
x 100

11. How to Calculate
In order to calculate percentage error in some experiment, one needs to follow
the steps written below:
Step 1: Obtain the true value and approximated value.
Step 2: Find the difference between them and take absolute value, ignore if there is a negative
sign. This is known as error.
Step 3: Find the absolute value of true of exact value as well.
Step 4: Divide the absolute error by absolute true value.
Step 5: Multiply the outcome by 100 to convert it to the percent value and add a "%" sign at the
end

12. Example 1: It was assumed that around 1,00,000 people would reach at a certain hill station in a month
summer. But the exact number of people counted was 88,000. Calculate the percentage error.
Solution: Assumed or approximated value = 100000
true value = 88000
The formula for the percentage error is:
PE =
|𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒− 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒|
|𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒|
x 100
PE =
|88000−100000|
|88000|
x 100
PE =
|22000|
|88000|
x 100
PE= 25%

13. Example 2: The approximated time of a ball to reach at ground when dropped from a 4-
meter height is 3 seconds. But during the experiment, it was found that it took 2.1 seconds.
Solution: Approximated time = 3 sec
true time = 2.1 sec
The formula for the percentage error is given by:
PE =
|𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒−𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒|
|𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒|
x 100
PE =
|2.1−3|
|2.1|
x 100
PE =
|0.9|
|2.1|
x 100
PE = 42.86%

14. Thanks!
Any questions?