update readme with sqrt explication
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hugogogo
163
README.md
163
README.md
@@ -63,19 +63,176 @@ this project uses submodules (maybe recursively), so either :
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-> Δ == 0 -> 1 solution : x = ( -b / 2a )
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-> Δ < 0 -> 2 solutions : x = ( -b / 2a ) +- i( √|Δ| / 2a )
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-> solution :
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- a; // double
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- b; // double
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- c; // double
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- delta_sign; // + or -
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- delta_absolute; // |Δ|
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- delta_sqrt; // √|Δ|
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- first_term_gcd; // gcd(b, 2a)
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- first_term_numerator; // -b / gcd
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- first_term_denominator; // 2a / gcd
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- first_term; // double (-b / 2a)
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- second_term_gcd; // gcd(√|Δ|, 2a)
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- second_term_numerator; // √|Δ| / gcd
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- second_term_denominator; // 2a / gcd
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- second_term; // double (√|Δ| / 2a)
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- double solution1; // first_term + second_term
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- double solution2; // first_term - second_term
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6. print solution
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---
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# sqrt implementation
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finding the square root of x
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## dichotomy method
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```
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solution
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↓
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|------------------------------ 1. define range bound_1 : 0
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|-----------------------------| 2. choose range bound_2 : x
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|--------------+--------------| 3. take mid : (bound_1 + bound_2) / 2
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|--------------|-------ø------| 4. choose range bound : bound_1 or bound_2
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---------------|--------------- 1. range bound_1 = mid
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|--------------|--------------- 2. range bound_2 = chosen previous bound
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|------+-------|--------------- 3. mid
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|--ø---|-------|--------------- 4. choose range
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-------|----------------------- 1. range bound_1 = mid
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-------|-------|--------------- 2. range bound_2 = chosen previous bound
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-------|---+---|--------------- 3. mid
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-------|---|-ø-|--------------- 4. choose range
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-----------|------------------- 1. range bound_1 = mid
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-------|---|------------------- 2. range bound_2 = chosen previous bound
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-------|-+-|------------------- 3. mid
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-------|-|ø|------------------- 4. choose range
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---------|--------------------- 1. range bound_1 = mid
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-------|-|--------------------- 2. range bound_2 = chosen previous bound
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-------|+|--------------------- 3. mid
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-------ø|---------------------- 4. choose range
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--------|---------------------- --> solution
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```
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## Newton–Raphson method
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it's like a self-correcting binary search, we get rid of the step "choose range" :
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```
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solution
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↓
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------------------------------| 1. define range bound_1 : v
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--|---------------------------| 2. choose range bound_2 : x/v
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--|-------------+-------------| 3. take mid : (v + x/v) / 2
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----------------|-------------- 1. range bound_1 : v = mid
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------|---------|-------------- 2. range bound_2 : x/v
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------|----+----|-------------- 3. mid
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-----------|------------------- 1. range bound_1 : v = mid
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-|---------|------------------- 2. range bound_2 : x/v
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-|----+----|------------------- 3. mid
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------|------------------------ 1. range bound_1 : v = mid
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------|-------|---------------- 2. range bound_2 : x/v
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------|---+---|---------------- 3. mid
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----------|-------------------- 1. range bound_1 : v = mid
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------|---|-------------------- 2. range bound_2 : x/v
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------|-+-|-------------------- 3. mid
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--------|---------------------- --> solution
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```
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### mathematical proof that each range is automatically in the right range :
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- if the value was higher than the answer, then new value is below old value, and vice versa
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- how ? :
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- define `x`, solution `s = √x`, and value `v = (old_value + x / old_value) / 2`
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- supposition : if `v < s` , then `new_v > v`, else `new_v < v` :
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- demonstration :
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1. if `v < s` :
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v < s
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<=> v < √x
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<=> v² < x (that's actually how we know that v < s)
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<=> v²/v < x/v
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<=> v < x/v
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-> and is s < x/v ? :
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v < s
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<=> v < √x
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<=> v² < x (as previously)
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<=> v² < x²/x
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<=> v² * x < x²
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<=> (v * √x)² < x²
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<=> v * √x < x
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<=> v * √x < v * x/v
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<=> √x < x/v
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-> so indeed : if v < √x, then v < √x < x/v == v < s < x/v
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-> conclusion, the new range < v , x/v > contains the solution
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2. the same demonstration works for `v > s`
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### here is a more intuitive demonstration, with x = 20 :
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1. **show that if `v² > x` (== `v > s`) then `v > s > x/v`, and if `v² < x` (== `v < s`) then `v < s < x/v` :**
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1.1. **for value too high `v > s` :**
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1.1.1 **why `v > x/v` :**
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- let's take initial value v = 5 :
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- is 5² the solution ? 5² == 25 -> so no, 5 is not the sqrt, it's too high
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```
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v v²
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0 (5) 10 15 20 25
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v : |----|----|----|----|----|
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x/v: |---|---|---|---|---| <----- squiz it, so the previous 5 portions fit the x = 20 size
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0 (4) 8 12 16 20
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x/v x
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```
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- the value of the new portion is 4, and we can visually see that it's lower than the previous portion 5
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- so : `v > x/v`
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1.1.2 **why `s > x/v` :**
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- let's take the value v = 5 :
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- we already showed that it's too high, now we will see that x/v == 20/5 is too low :
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```
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v
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0 (5) 10 15 20 25
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v : |----|----|----|----|----|
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x/v: |---|---|---|---|---| <----- squizz
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0 *1 *2 *3 *4 *5 -> number of portions
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01234 -> portion size
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(x/v)²: |---|---|---|---|
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0 4 8 12 16
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```
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- the portion size is smaller than the number of portions, so it's too small to be the sqrt, indeed we visually see that this portion size `x/v` is a root of a smaller number : `(x/v)² == 16`
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- so : `s > x/v`
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1.1.3. **conclusion :**
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- v > s
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- and v > x/v (<- this proof is not essential)
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- and s > x/v (<- we actually only need this proof)
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- so `v > s > x/v`
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1.2. **for value too high `v < s` :**
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- this is the same demonstration but in other direction, let's just summarize it :
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- let's take initial value v = 4 :
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```
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v v²
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0 (4) 8 12 16
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(1) v : |---|---|---|---|
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(2) x/v: |----|----|----|----| -----> stretch
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(3) 0 (5) 10 15 20
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x/v x
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(4) 0 *1 *2 *3 *4 -> number of portions
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012345 -> portion size
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(5) (x/v)²: |----|----|----|----|----|
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0 5 10 15 20 25
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```
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- (1) : 4 is not the sqrt of x == 20, it's too smalle : (4² == 16) < 20
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- (2) : stretch it, so the previous 4 portions fit the x = 20 size
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- (3) : the new portion x/v == 5, is more than v == 4, so `v < x/v`
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- (4) : portion size is bigger than number of portions, so it's too big to be the root
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- (5) : indeed, we see that the portion² == (x/v)² is bigger than x, so √x < x/v == `s < x/v`
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- so `v < s < x/v`
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