Finish chapter 2

README.md

@@ -8,7 +8,7 @@ If you see an issue, just [submit one](https://github.com/cjmlgrto/fit3139-notes
 ## Contents
 
 1. [Errors and Approximation](https://github.com/cjmlgrto/fit3139-notes/blob/master/notes/01-errors_and_approximation.md)
-2. Computer Arithmetic
+2. [Computer Arithmetic](https://github.com/cjmlgrto/fit3139-notes/blob/master/notes/02-computer_arithmetic.md)
 3. Matrices
 4. Eigenvalues & Eigenvectors
 5. Dynamical Systems

notes/02-computer_arithmetic.md

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+[← Return to Index](https://github.com/cjmlgrto/fit3139-notes/)
+
+# Computer Arithmetic
+
+## Floating Point Numbers
+
+* In a digital computer, real numbers are represented using floating points
+* e.g. `-0.0007396`
+	* _Scientific notation_: `7.396 × 10^-4`
+	* _Sign_: `-`
+	* _Mantissa_: `7.396`
+	* _Exponent_: `-4`
+	* _Base_: decimal (base-10)
+* Any floating-point number `x` has the form:
+	* `x = ±( [d_0÷b^0] + [d_1÷b^1] + [d_2÷b^2] + ... + [d_(p-1)÷b^(p-1)] ) × b^E
+	* `b` is the base or _radix_
+	* `p` is the precision
+	* `E ∈ [L,U]` is an exponent
+	* Note that this is the general form for an _unnormalized_ floating point representation
+
+## Normalization
+
+* A floating-point system is said to be **normalized** if the leading digit `d_0` is always nonzero (unless the represented number is 0)
+* Thus, in a normalized system, the mantissa `m` always satisfies `1 ≤ m ≤ base`
+* Floating-point systems are usually normalized because
+	* The representation of each number is then unique
+	* No digits are wasted on leading zeroes, therby maximising precision
+	* In binary (base = 2) system, the leading bit is always 1, thus need not be stored
+
+## Properties of Floating-Point Systems
+
+* The number of normalized floating-point numbers in a given floating-point system is:
+	* `2(b-1)b^{p-1}(U - L + 1) + 1`
+* The smallest positive normalized floating-point number is:
+	* `Underflow level = UFL = b^L`
+* The largest positive normalized floating-point number is:
+	* `Overflow level = OFL = (1-b^{-p})b^{U+1}`
+
+### Single Precision (32-bit)
+
+| 1 bit | 8 bits | 23 bits |
+| --- | --- | --- |
+| Sign (S) | Exponent (E) | Mantissa or Fraction (F) |
+| 31 | 30 ... 23 | 22 ... 0 |
+
+* Decimal value of floating point
+	* `(-1)^S × 2^{E_10 - 127} × (F)_10`
+* `+0`
+	* `S = 0`, `E = 0`, `F = 0`
+* `-0`
+	* `S - 1`, `E = 0`, `F = 0`
+* `+∞`
+	* `S = 0`, `E = 255_10`, `F = 0`
+* `-∞`
+	* `S = 1`, `E = 255_10`, `F = 0`
+* `NaN`
+	* `E = 255_10`, `S × F < 0 or > 0`
+
+### Double Precision (64-bit)
+
+| 1 bit | 11 bits | 52 bits |
+| --- | --- | --- |
+| Sign (S) | Exponent (E) | Mantissa or Fraction (F) |
+| 63 | 62 ... 52 | 51 ... 0 |
+
+* Decimal value of floating point
+	* `(-1)^S × 2^{E_10 - 1023} × (F)_10`
+* `+0`
+	* `S = 0`, `E = 0`, `F = 0`
+* `-0`
+	* `S - 1`, `E = 0`, `F = 0`
+* `+∞`
+	* `S = 0`, `E = 2047_10`, `F = 0`
+* `-∞`
+	* `S = 1`, `E = 2047_10`, `F = 0`
+* `NaN`
+	* `E = 2047_10`, `S × F < 0 or > 0`
+
+## Rounding
+
+* **Rounding by Chopping** — base-b expansion is truncated after the (precision - 1)th digit
+* **Rounding to Nearest** — real number is rounded to the nearest floating-point number
+
+## Machine Epsilon
+
+* **Machine Epsilon** gives an _upper bound on the relative error_ due to rounding in floating-point arithmetic
+	* Rounding by chopping: `ε_mach = b^{1 - p}`
+	* Rounding to nearest: `ε_mach = 0.5b^{1 - p}`
+* The unit roundoff is important because it bounds the relative error in representing the real number `x`
+	* `ε_mach ≥ |(inexact x - exact x) ÷ exact x|`