Node:Numerical Precision, Next:, Previous:Linearity, Up:CLPQR

### Numerical Precision and Rationals

The fact that you can switch between clp(R) and clp(Q) should solve most of your numerical problems regarding precision. Within clp(Q), floating point constants will be coerced into rational numbers automatically. Transcendental functions will be approximated with rationals. The precision of the approximation is limited by the floating point precision. These two provisions allow you to switch between clp(R) and clp(Q) without having to change your programs.

What is to be kept in mind however is the fact that it may take quite big rationals to accommodate the required precision. High levels of precision are for example required if your linear program is ill-conditioned, i.e. in a full rank system the determinant of the coefficient matrix is close to zero. Another situation that may call for elevated levels of precision is when a linear optimization problem requires exceedingly many pivot steps before the optimum is reached.

If your application approximates irrational numbers, you may be out of space particularly soon.

The following program implements N steps of Newton's approximation for the square root function at point 2.

```           % library('clpqr/examples/root')

root(N, R) :-
root(N, 1, R).

root(0, S, R) :- !, S=R.
root(N, S, R) :-
N1 is N-1,
{ S1 = S/2 + 1/S },
root(N1, S1, R).
```

It is known that this approximation converges quadratically, which means that the number of correct digits in the decimal expansion roughly doubles with each iteration. Therefore the numerator and denominator of the rational approximation have to grow likewise:

```clp(q) ?- use_module(library('clpqr/examples/root')).
clp(q) ?- root(3,R),print_decimal(R,70).
1.4142156862 7450980392 1568627450 9803921568 6274509803 9215686274
5098039215

R = 577/408

clp(q) ?- root(4,R),print_decimal(R,70).
1.4142135623 7468991062 6295578890 1349101165 5962211574 4044584905
0192000543

R = 665857/470832

clp(q) ?- root(5,R),print_decimal(R,70).
1.4142135623 7309504880 1689623502 5302436149 8192577619 7428498289
4986231958

R = 886731088897/627013566048

clp(q) ?- root(6,R),print_decimal(R,70).
1.4142135623 7309504880 1688724209 6980785696 7187537723 4001561013
1331132652

R = 1572584048032918633353217/1111984844349868137938112

clp(q) ?- root(7,R),print_decimal(R,70).
1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679
7379907324

R = 4946041176255201878775086487573351061418968498177 /
3497379255757941172020851852070562919437964212608
```

Iterating for 8 steps produces no further change in the first 70 decimal digits of `sqrt(2)`. After 15 steps the approximating rational number has a numerator and a denominator with 12543 digits each, and the next step runs out of memory.

Another irrational number that is easily computed is e. The following program implements an alternating series for `1/e`, where the absolute value of last term is an upper bound on the error.

```           % library('clpqr/examples/root')

e(N, E) :-
{ Err =:= exp(10,-(N+2)), Half =:= 1/2 },
inv_e_series(Half, Half, 3, Err, Inv_E),
{ E =:= 1/Inv_E }.

inv_e_series(Term, S0, _, Err, Sum) :-
{ abs(Term) =< Err }, !,
S0 = Sum.
inv_e_series(Term, S0, N, Err, Sum) :-
N1 is N+1,
{ Term1 =:= -Term/N, S1 =:= Term1+S0 },
inv_e_series(Term1, S1, N1, Err, Sum).
```

The computation of the rational number E that approximates e up to at least 1000 digits in its decimal expansion requires the evaluation of 450 terms of the series, i.e. 450 calls of `inv_e_series/5`.

```clp(q) ?- e(1000,E).

E = 7149056228932760213666809592072842334290744221392610955845565494
3708750229467761730471738895197792271346693089326102132000338192
0131874187833985420922688804220167840319199699494193852403223700
5853832741544191628747052136402176941963825543565900589161585723
4023097417605004829991929283045372355639145644588174733401360176
9953973706537274133283614740902771561159913069917833820285608440
3104966899999651928637634656418969027076699082888742481392304807
9484725489080844360397606199771786024695620205344042765860581379
3538290451208322129898069978107971226873160872046731879753034549
3130492167474809196348846916421782850086985668680640425192038155
4902863298351349469211627292865440876581064873866786120098602898
8799130098877372097360065934827751120659213470528793143805903554
7928682131082164366007016698761961066948371407368962539467994627
1374858249110795976398595034606994740186040425117101588480000000
0000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000
/
2629990810403002651095959155503002285441272170673105334466808931
6863103901346024240326549035084528682487048064823380723787110941
6809235187356318780972302796570251102928552003708556939314795678
1978390674393498540663747334079841518303636625888963910391440709
0887345797303470959207883316838346973393937778363411195624313553
8835644822353659840936818391050630360633734935381528275392050975
7271468992840907541350345459011192466892177866882264242860412188
0652112744642450404625763019639086944558899249788084559753723892
1643188991444945360726899532023542969572584363761073528841147012
2634218045463494055807073778490814692996517359952229262198396182
1838930043528583109973872348193806830382584040536394640895148751
0766256738740729894909630785260101721285704616818889741995949666
6303289703199393801976334974240815397920213059799071915067856758
6716458821062645562512745336709063396510021681900076680696945309
3660590933279867736747926648678738515702777431353845466199680991
73361873421152165477774911660108200059
```

The decimal expansion itself looks like this:

```clp(q) ?- e(1000, E), print_decimal(E, 1000).
2.
7182818284 5904523536 0287471352 6624977572 4709369995 9574966967
6277240766 3035354759 4571382178 5251664274 2746639193 2003059921
8174135966 2904357290 0334295260 5956307381 3232862794 3490763233
8298807531 9525101901 1573834187 9307021540 8914993488 4167509244
7614606680 8226480016 8477411853 7423454424 3710753907 7744992069
5517027618 3860626133 1384583000 7520449338 2656029760 6737113200
7093287091 2744374704 7230696977 2093101416 9283681902 5515108657
4637721112 5238978442 5056953696 7707854499 6996794686 4454905987
9316368892 3009879312 7736178215 4249992295 7635148220 8269895193
6680331825 2886939849 6465105820 9392398294 8879332036 2509443117
3012381970 6841614039 7019837679 3206832823 7646480429 5311802328
7825098194 5581530175 6717361332 0698112509 9618188159 3041690351
5988885193 4580727386 6738589422 8792284998 9208680582 5749279610
4841984443 6346324496 8487560233 6248270419 7862320900 2160990235
3043699418 4914631409 3431738143 6405462531 5209618369 0888707016
7683964243 7814059271 4563549061 3031072085 1038375051 0115747704
1718986106 8739696552 1267154688 9570350354
```