MATLAB: Problems in accuracy assessment

accuracy assessmentpisymbolic computation

Dear friends I tried continued fractiona approximation to pi, and got the following rational approximation

1244969988745789040106366155256015114976454553337906033890313/396286255419907262612262286579004636343912658949984351074158

but if I try to find the difference of it with pi, I failed, since

>> sym(1244969988745789040106366155256015114976454553337906033890313/396286255419907262612262286579004636343912658949984351074158)

I got pi directly. How can I assess the difference? Thanks

Best Answer

Using my HPF toolbox, it looks to be correct to about 120 significant digits.

double(log10(abs(hpf('1244969988745789040106366155256015114976454553337906033890313',1000)/hpf('396286255419907262612262286579004636343912658949984351074158',1000) - hpf('pi',1000))))
ans =
    -119.53

Looking back at what you did, this is wrong. Why?

sym(1244969988745789040106366155256015114976454553337906033890313/396286255419907262612262286579004636343912658949984351074158)
ans =
pi

It is wrong because MATLAB will first compute the ratio as a DOUBLE precision number, and only then pass it to sym. sym then recognizes the ratio as a double precision approximation to pi, so it returns that as the result.

If you insist on doing this with the symbolic toolbox, we might try...

log10(abs(vpa(sym('1244969988745789040106366155256015114976454553337906033890313/396286255419907262612262286579004636343912658949984351074158 - pi'),200)))
ans =
-119.52655990941722562288816970381

Of course, if you want a better approximation, say accurate to 250 digits, HPF can provide it too.

[N,D] = rat(hpf('pi',500),hpf('1e-250'))
N =
165213461336854892880967397129207248425612593828416835132438559586109735260365863184040108685528782183687397179966550074377225
D =
52589078074164381557662496030246690228630477463437042437123678996605078304650603819026850043803035007564529614801553827009746

If you need convincing that this ratio is indeed accurate to 250 digits for pi, try this:

double(log10(abs(N/D - hpf('pi',1000))))
ans =
      -250.05

Related Solutions

MATLAB: Displaying numbers explicitly

If you are starting from rational values and manipulating them, and want the results to be in the form you show, then the Symbolic Toolbox will generally do that.

But if you are starting from floating point numbers and want to find some representation in terms of rationals, sqrt() of rational, Pi, and exp() of rationals, then there is a lot of different ways to do that, most of which will be more or less accidental.

For example, sin(1) is about 0.841470984807897 which is not so different from sqrt(2) - pi/2 + 1 which is 0.843417235578199 ... more or less by chance.

sin(1) could also be expressed as being approximately

1 - 1/6 + 1/120 - 1/5040 + 1/362880 - 1/39916800 + 1/6227020800 - 1/1307674368000 + 1/355687428096000 - 1/121645100408832000 + 1/51090942171709440000 - 1/25852016738884976640000 + 1/15511210043330985984000000 - 1/10888869450418352160768000000 + 1/8841761993739701954543616000000 - 1/8222838654177922817725562880000000

which I derive from the taylor series.

but perhaps you would prefer

1 + 1/(-6 + 1/(-3 + 1/(-4 + 1/(-19))))

which is the output of rat(sin(1))...

[GIS] ArcGIS Javascript API with Windows Authentication

A proxy page can also be used to bypass an authenticated service with 10.x versions of the APIs.

A few things to note:

Bypassing the Windows authentication with a proxy page will cause all of your connections to be made by the identity of the application pool that runs the proxy page, or by the identity of the connection you make within the proxy page. This may cause problems for tracking "who does what" since actions made through the proxy will all appear to be the same user.
Bypassing authentication may not be permitted by the publishing authority. Make sure you're allowed to publish the service in such a way before proceeding.
In the ESRI Javascript API, the proxy is set as a global setting, and all connections that would use a proxy will be directed through the same proxy page. So, if you're combining services in your map, make sure your proxy handler is ready to deal with whatever gets thrown at it approriately.

Windows Authentication is normally used on an internal network only, where you may take advantage of the integrated login, and bypass the login prompt that way. If your users are on the internal network, and have already logged on to the domain on their workstation, they shouldn't be challenged by a login prompt (IIS should already have handled that!)

Users that are not logged on to the network already, such as external users being directed to the Windows-secured server, will always be presented with a login prompt. There is no way to apply the login in the background once this prompt appears - its presence indicates that the background login attempt has already failed.

Best Answer

Related Solutions

MATLAB: Displaying numbers explicitly

[GIS] ArcGIS Javascript API with Windows Authentication

Related Question