1. Introduction to Scientific Computation
Whom do you call…?

2. Copyright, G. A. Tagliarini, PhD
Common Problems
Intractable analytical representations
Model building: sample data are available but no closed form analytical representation for a relation
Need to estimate intermediate behavior (interpolation)
Need to predict/forecast behavior (extrapolation)
Change representation from one domain to another (e.g., DSP)
11/21/2018
Copyright, G. A. Tagliarini, PhD

3. Copyright, G. A. Tagliarini, PhD
Computing Issues
Inherently discrete representation
Finite ranges that vary according to the space allocated to store the representation
Alternative arithmetic systems
Integers (two’s complement representation)
Floating point (IEEE 754)
11/21/2018
Copyright, G. A. Tagliarini, PhD

4. Computer Integer Arithmetic
Representations assume n bits bn-1…b0
Unsigned
Range is [0, 2n-1], there is no sign so all bits can be used to represent magnitude
Example with n=8 and x = =
Typically used to represent storage addresses
Signed magnitude
Two’s complement
11/21/2018
Copyright, G. A. Tagliarini, PhD

5. Computer Integer Arithmetic
Representations assume n bits bn-1…b0
Unsigned
Signed magnitude
Range is [-(2n-1-1), 2n-1-1] including two zeros!
In the representation bn-1…b0, bn-1 represents the sign and bn-2…b0 the magnitude
Example with n=8 and x = =
Typically not used
Two’s complement
11/21/2018
Copyright, G. A. Tagliarini, PhD

6. Computer Integer Arithmetic
Representations assume n bits bn-1…b0
Unsigned
Signed magnitude
Two’s complement
Asymmetric range is [-2n-1, 2n-1-1] includes only one zero!
In the representation bn-1…b0, bn-1 indicates the sign but is also used to determine the magnitude
Example with n=8 and x = = 2810
11/21/2018
Copyright, G. A. Tagliarini, PhD

7. Two’s Complement Representation of a Negative Quantity
Assume x<0
Find y = |x|
Determine the unsigned binary representation of y = bn-1…b0
Complement each bit (replace each 0 with 1 and each 1 with 0) to form y’ = bn-1’…b0’, where bi’=(1-bi), for i e {0, 1,…, n-1}
Add 1 to complete the representation
11/21/2018
Copyright, G. A. Tagliarini, PhD

8. Two’s Complement Representation of a Negative Quantity
Assume n = 8 and x = -28 (note x<0)
Find y = |x| → y = 2810
Determine the unsigned binary representation of y = bn-1…b0 → y =
Complement each bit to form y’ =
Add 1 to form the two’s complement representation of -28 which is y’+1 =
Note: = 0, as expected! (Remember that the field width is fixed.)
11/21/2018
Copyright, G. A. Tagliarini, PhD

9. Two’s Complement Interpretation
If bn-1 = 0, interpret as the corresponding unsigned (non-negative) integer
If bn-1 = 1,
Complement each bit
Add 1
Interpret the result as an unsigned quantity
Attach the “-” sign
11/21/2018
Copyright, G. A. Tagliarini, PhD

10. Two’s Complement Interpretation
Example where bn-1 = 1, x =
Complement each bit →
Add 1 →
Interpret the result as an unsigned quantity → 10010
Attach the “-” sign →
11/21/2018
Copyright, G. A. Tagliarini, PhD

11. Review of Single Precision IEEE 754 Floating Point Notation
Uses 32-bits in three fields:
Sign (a 1-bit field)
0 - nonnegative
1 - negative
Biased exponent (an 8-bit field)
Bias equals (or )
Normalized mantissa (a 23-bit field)
Leading “1” omitted
11/21/2018
Copyright, G. A. Tagliarini, PhD

12. Conversion to IEE 754 Representation
Select the appropriate value for the sign bit
Write the quantity in normalized binary scientific notation as a product of:
A value x such that 12 ≤ x < 102 and a
A power 102
Add the bias to the exponent to obtain the biased exponent field value
Write the mantissa by
Dropping the leading “1” (to the left of the binary point)
Extending or rounding the remaining to 23-bits
11/21/2018
Copyright, G. A. Tagliarini, PhD

13. IEEE 754 Conversion Example
Convert 1 3/16
Sign bit = 0
1 3/16 = = x (102)02
Biased exponent = =
Mantissa = (note that the leading “1” has been omitted and trailing zeros added to create a total of 23 bits)
Completed representation or 3F
11/21/2018
Copyright, G. A. Tagliarini, PhD

14. IEEE 754 Conversion Example 2
Convert
Sign bit = 1
3.110 = …2 = …2 x (102)12
Biased exponent = =
Mantissa = (note that the leading “1” has been omitted, repeating digits used and the representation rounded to a total of 23 bits)
Completed representation or C
11/21/2018
Copyright, G. A. Tagliarini, PhD

15. What Difference Does It Make?
Patriot Missile Failure (28 killed, 100 injured)
Ariane 5 Explosion ($7 Billion)
11/21/2018
Copyright, G. A. Tagliarini, PhD

16. Some IEEE 754 Floating Point Values
Sign
Bit
Exponent
Field
Mantissa
Floating Point Value
1.0
1.0 + eM
2.0
2.0*(1.0 + eM)
4.0
4.0*(1.0 + eM)
8.0
8.0*(1.0 + eM)
11/21/2018
Copyright, G. A. Tagliarini, PhD

17. Some IEEE 754 Floating Point Values
SignBit
Exponent
Field
Mantissa
Interpretation
+0.0 1
-0.0 +∞ -∞
xxxx xxxx xxxx xxxx xxxx xxx
(at least one x≠0)
NaN
eMax=2+127*(2-2-23)
≈2+128
eMin=2-126*2-23=2-149
11/21/2018
Copyright, G. A. Tagliarini, PhD

18. Some Notes About IEEE 754, Floating Point Representation
The density of the distribution of floating point values varies along the number line
Half of the possible floating point values lie in the interval [-1, 1]
For single precision representation
Machine epsilon εM = 2-23
The minimum representable value εMin = 2-149
The maximum representable value εMax ≈ 2128
11/21/2018
Copyright, G. A. Tagliarini, PhD

19. Finding Machine Epsilon, εM
Set εM = 1 Do
εM = εM/2
x = 1 + εM
While (x>1)
εM = 2*εM
11/21/2018
Copyright, G. A. Tagliarini, PhD

20. Copyright, G. A. Tagliarini, PhD
Errors
Round-off
Arise from limitations in the space allocated for representing specific type of datum
Truncation
Arise from abbreviating an analytical representation
11/21/2018
Copyright, G. A. Tagliarini, PhD

21. Copyright, G. A. Tagliarini, PhD
Random Numbers
Uniform random distribution (e.g., Math.random())
For u1 and u2 are chosen from a uniform distribution, then have a Gaussian distribution
11/21/2018
Copyright, G. A. Tagliarini, PhD