SLIDE 1
Integers mod M
- Whole numbers 0 .. M-1
- Integers mod 17:
Minimal Fractional Representations of Integers mod M David Greve - - PowerPoint PPT Presentation
Minimal Fractional Representations of Integers mod M David Greve - - PowerPoint PPT Presentation
Minimal Fractional Representations of Integers mod M David Greve ACL2 Workshop May, 2020 Integers mod M Whole numbers 0 .. M-1 Integers mod 17: Signed Representations Negative numbers: (- x) The number added to x to make zero
SLIDE 2
SLIDE 3
Signed Representations
- Negative numbers: (- x)
– The number added to x to make zero (0) (mod M) – The number added to 7 to make zero (0) (mod 17)
- 7 + 10 == 0 % 17
- 10 == -7
- 7 == -10
Is there a “best” signed representation for each number?
SLIDE 4
Signed Representations
- Negative numbers: (- x)
– The number added to x to make zero (0) (mod M) – The number added to 7 to make zero (0) (mod 17)
- 7 + 10 == 0 % 17
- 10 == -7
- 7 == -10
Is there a “best” signed representation for each number?
SLIDE 5
Reciprocals
- Reciprocals: (1/x)
– The number multiplied by x to make one (1) (mod M)? – The number multiplied by 7 to make one (1) (mod 17)?
- 7 * 5 = 1 % 17
- 5 = 1/7
- 7 = 1/5
– Not every x has a reciprocal mod M
SLIDE 6
Fractional Representations
- Fractional Representation (x ~= N/D) mod M
– x * D == N % M – Fractional Representations for 7 mod 17:
Is there a “best” fractional representation for each number?
SLIDE 7
Farey Sequences
N1 + N2 D1 + D2 F3 =
N2*D1 – N1*D2 = 1 F1 = N1/D1 F2 = N2/D2 “Neighbors”
SLIDE 8
Fractional Representations
- Fractional Representation (x ~= N/D) mod M
– x * D == N % M – Fractional Representations for 7 mod 17:
Is there a “best” fractional representation for each number?
SLIDE 9
Fractional Representations
- Fractional Representation (x ~= N/D) mod M
– x * D == N % M – Fractional Representations for 7 mod 17:
Is there a “best” fractional representation for each number?
SLIDE 10
Fractional Representations
- Fractional Representation (x ~= N/D) mod M
– x * D == N % M – Fractional Representations for 7 mod 17:
Is there a “best” fractional representation for each number?
SLIDE 11
Fractional Representations
- Fractional Representation (x ~= N/D) mod M
– x * D == N % M – Fractional Representations for 7 mod 17:
Is there a “best” fractional representation for each number?
SLIDE 12
Minimal Fractional Representation
- A Fractional Representation is “minimal” if :
– No smaller denominator results in a smaller numerator
- Sadly: Not sufficiently general
SLIDE 13
Minimal Fractions Pair
X ~= X ~= N K P M (N < 0)
SLIDE 14
Minimal Fractions Pair
X ~= X ~= N K P M (N < 0) No smaller denominator results in a numerator whose magnitude is smaller than the sum of the two numerator magnitudes
SLIDE 15
Step Minimal Fractions Pair
X ~= X ~= N K P M X ~= N + P K + M
SLIDE 16
Step Minimal Fractions Pair
X ~= X ~= N K P M X ~= N + P K + M
SLIDE 17
Step Minimal Fractions Pair
X ~= X ~= N K P M X ~= N + P K + M
SLIDE 18
Minimal Coefficient Bound
2/20/2020 9:30 pm 2/20/2020 6:30 pm
SLIDE 19
Minimal Fractions mod 17
- smallest maximum coefficient
- smallest denominator
SLIDE 20
Also a Floor Wax ..
- Computes Modular Inverses/GCDs ..
– Generates Same “coefficients” as Euclidean Algorithm
- If you use “division” instead of ”repeated subtraction”
– Like “Extended GCD”
- All computations are in “M”
SLIDE 21
.. and A Dessert Topping!
- Performs Modular “Long Division”
– Traditionally “Division” means multiply by reciprocal
SLIDE 22
Conclusion
- Algorithm for finding minimal fractional representations
– Also performs long division!
- Verified sqrt(M) bound on numerator/denominator
- Culmination of Several Years of .. Contemplation