Errors and arithmetic operations of floating points numbers

1. Numerical Methods and
Computer Programming
Prof. A. P.Phatale
Asst. Prof.
ECT Dept.

2. Error & their computations
• Two kinds of numbers
• Exact and approximate numbers.
• Exact Example: 1,2,3 ….1/2,3/2…..pi etc.
• Approximate Example: Number that represent to be a certain degree
of accuracy.
• Example 3.1416

3. Significant digit or significant figures
• The digits that are used to express a numbers are called significant
digits.
• Numbers are 3.1416,0.66667,4.0687 contain 5 significant digit each.
• 0.00023= 2 significant digit 2 and 3.
• Zero serve only to fix the position of decimal point.
• 0.00145, 0.000145 and 0.0000145 all have 3 significant 25600 is
having number of significant figures is uncertain
• 2.56*104, 2.560*104, 2.5600*104 have 3,4 & 5 significant digits resp.

4. Arithmetic operation with Normalized floating
point nos.
• Examples
• 1) 0.4546E3 + 0.5454E8
• 2) 0.4546E5 + 0.5433E7
• 3) 0.9432E-4 +0.6353E-5
• 4) 0.5452E-3 + 0.9432E-4

5. • 0.5454E8 0.5433E7 0.9432E-4
• 0.0000E8 0.0045E7 0.0635E-4
• ---------------- ----------------- -----------------
• 0.5454E8 0.5478E7 1.0067E-4--0.1006E-3

6. Subtraction
• Adding a negative number and remaining thing is same
• 0.9432E-4 - 0.5452 E-3
• 0.5452E-3
• - 0.0943E-3
• ------------------
• 0.4509E-3

7. Multiplication
• Two numbers are multiplied in the normalized floating point mode by
multiplying the mantissa and adding the exponent.After the
multiplication of the mantissa the result mantissa is normalized as in
addition.subtraction operation and exponenr appropriately adjusted.
• 0.5543E12 *0.4111E-15
• 0.5543E12
• *0.4111E-15
• ---------------
• 0.22787273E-3

8. • 0.1111E10 * 0.1234E15 = 0.013709E25= 0.1370E24
• 0.1111E51 * 0.4444E50 = 0.04937284E101 = 0.4937E100 Answer
overflow
• 0.1234E-49 * 0.1111E -54 = 0.1370 E -104 Result underflow

9. Division
• Mantissa of the numerator is divided by that of denominator.
• Denominator Exponent is subtracted from the numerator exponent.
• Quotient mantissa is normalized to make the Most significant digit
non zero and the exponent appropriately adjusted.
• Mantissa of the result is chopped down to occupy 4 digits

10. • 1) 0.9998 E1 / 0.1000 E -99
• = 9.9980E100
• 0.9998E101  Result overflow
• 2) 0.9998E-5 / 0.1000 E98
• = 0.9998E -104 Result underflow
• 3) 0.1000E5 / 0.9999E3 = 0.1000E2

11. ERROR ANALYSIS
• Defination:
• The error of a quantity is the difference between it’s true value and approximated
value.

12. Source: Open source free Internet

13. Source: Open source free Internet