I need your help folks

Hi everyone,

I've managed to boil loads of work down to one simple problem. But my brain's gone offline. For any coprime or irrational interval a/b, and with j = 0,1,2,3,..., there is an integer k that always gives the *smallest* value of |k(a - b) - j(a + b)|. Now I 'know' that this integer is k = Round[j(a + b)/(a - b)]. But I'm at a loss to prove it.

Eg, 81/64 = a/b, j = 1 gives k = 9. j = 2 gives k = 17 etc...

The wave convergents are then easily obtained as p/q = (k + j)/(k - j).

(9 + 1)/(9 - 1) = 5/4, (17 + 2)/(17 - 2) = 19/15 etc...

Is this type of math using the round function familiar to anyone? It's the last remaining step before I can write a proper paper.

Thanks

-Rick

On 15 July 2010 18:08, rick <rick_ballan@...> wrote:
> Hi everyone,
>
> I've managed to boil loads of work down to one simple problem. But my brain's gone offline. For any coprime or irrational interval a/b, and with j = 0,1,2,3,..., there is an integer k that always gives the *smallest* value of |k(a - b) - j(a + b)|. Now I 'know' that this integer is k = Round[j(a + b)/(a - b)]. But I'm at a loss to prove it.

For real numbers, the smallest value of |x| is also the smallest value
of x^2 (x squared). So minimize

[k(a - b) - j(a + b)]^2

= k^2 (a-b)^2 + j^2(a+b)^2 - 2kj(a-b)(a+b)

To find the minimum point of this, differentiate and set to zero.

2k (a-b)^2 - 2j (a-b)(a+b) = 0

Assuming a and be are not equal,

k (a-b) - j (a+b) = 0

k = j (a + b) / (a - b)

You're then taking the nearest integer to this, which is a pretty good bet.

> Is this type of math using the round function familiar to anyone? It's the last remaining step before I can write a proper paper.

I feel safer with floor and ceil, but round is correct here. As it's
a quadratic function, the slope will be the same either side of the
minimum, and the nearest integer to that minimum will be the best one.

Graham

Oh of course. That's fantastic Graham, thanks very much. Now I can move on.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 15 July 2010 18:08, rick <rick_ballan@...> wrote:
> > Hi everyone,
> >
> > I've managed to boil loads of work down to one simple problem. But my brain's gone offline. For any coprime or irrational interval a/b, and with j = 0,1,2,3,..., there is an integer k that always gives the *smallest* value of |k(a - b) - j(a + b)|. Now I 'know' that this integer is k = Round[j(a + b)/(a - b)]. But I'm at a loss to prove it.
>
> For real numbers, the smallest value of |x| is also the smallest value
> of x^2 (x squared). So minimize
>
> [k(a - b) - j(a + b)]^2
>
> = k^2 (a-b)^2 + j^2(a+b)^2 - 2kj(a-b)(a+b)
>
> To find the minimum point of this, differentiate and set to zero.
>
> 2k (a-b)^2 - 2j (a-b)(a+b) = 0
>
> Assuming a and be are not equal,
>
> k (a-b) - j (a+b) = 0
>
> k = j (a + b) / (a - b)
>
> You're then taking the nearest integer to this, which is a pretty good bet.
>
> > Is this type of math using the round function familiar to anyone? It's the last remaining step before I can write a proper paper.
>
> I feel safer with floor and ceil, but round is correct here. As it's
> a quadratic function, the slope will be the same either side of the
> minimum, and the nearest integer to that minimum will be the best one.
>
>
> Graham
>