Numerisk analys

FMNF05

"This is probably the hardest course you will take..." - Andreas Langer

It is possible to get extra points for the final exam by completing homework assignments.

Its possible to gain 10% of exam points (6p) from assignments, and we are allowed to do them in groups.

Learn by trying.

Theorem

Let beta >= 2 be a fixed bases.

Then we have that:

For all epsilon larger than zero and for all x_beta in R there exists an representation y_b such that |y_b - x_b| < epsilon.

Whereas y_b has a finite representation.

Any number can be representated by a finite number of digits.

Let N in N be the memory positions of a computer, x_b = (-1)^s d_n-2 d_n-3 .. d_k .. d_0

s in {0, 1} has one memory position.

N-k-1 memory positions for the integer digits.

Fixed-point system.

Remedy:

Scaling => Floating point

x_b = (-1)^s d_0 ... d_k-1 * b^e (e is an exponent, a number not eulers)

k in N is the precision.

m = d_0 ... d_k-1 (is called mantissa)

+- 1.bbbbbbb*2^e Normalized

Example:

9 base 10 = 1001 base 2

IEEE floating point rep. -> +1.001 * 2^3 [Move the left most one to the correct location]

epsilon_mach ? 2 ^-52 [machine precision]

The relative error is almost less or equal to the machine precision.

9.4 to binary

9: 1001

0.4: 0.4 * 2 -> 0

0.8 * 2 = 1.6 -> 1

0.6 * 2 = 1.2 -> 1

0,2 * 2 = 0.4 -> 0

This will keep on going 011001100110 etc

9.4 == 1001.0110 = 1.0010110 * 2^3

Into IEEE -> +1.00101100110...01100*2 ^3

Chopping:

Simple but biased towards zero

Rounding: