{{{id=1|
def gaussred(u,v):
    u0=v
    v0=u
    uv=u0*v0
    qu2=u0*u0
    x=floor(uv/qu2+1/2)
    r=v0-x*u0
    v0=u0
    u0=r
    while norm(u0)<norm(v0):
         uv=u0*v0
         qu2=u0*u0
         x=floor(uv/qu2+1/2)
         r=v0-x*u0
         v0=u0
         u0=r
    return (v0,u0)
///
}}}

{{{id=2|
gaussred(vector([5,7]),vector([2,1]))
///
((2, 1), (-1, 4))
}}}

{{{id=3|
a=matrix([[5,7],[2,1]])
///
}}}

{{{id=4|
a.LLL()
///
[ 2  1]
[-1  4]
}}}

{{{id=5|
def sizered(M):
    Mk=M
    n=M.ncols()
    x=vector(range(0,n))
    R,mu=M.gram_schmidt()
    for i in range(1,n):
        for j in range(i-1,-1,-1):
            x[j]=floor(mu[i,j]+1/2)
            Mk[i]=Mk[i]-x[j]*Mk[j]
            mu[i,j]=mu[i,j]-x[j]
            for k in range(j-1):
                mu[i,k]=mu[i,k]-x[j]*mu[j,k]
    return Mk,R,mu
///
}}}

{{{id=6|
def sizered2(M,t):
    Mk=M
    n=M.ncols()
    R,mu=M.gram_schmidt()
    i=t
    for j in range(i-1,-1,-1):
        x=floor(mu[i,j]+1/2)
        Mk[i]=Mk[i]-x*Mk[j]
        mu[i,j]=mu[i,j]-x
        for k in range(j-1):
            mu[i,k]=mu[i,k]-x*mu[j,k]
    return Mk,R,mu
///
}}}

{{{id=7|
M=matrix([[1,2,3],[2,3,4],[4,3,3]])
M1=sizered2(M,1)[0]
M1
sizered2(M1,2)
///
(
[ 1  2  3]  [   1    2    3]  [   1    0    0]
[ 1  1  1]  [ 4/7  1/7 -2/7]  [ 3/7    1    0]
[-1 -3 -4], [ 1/6 -1/3  1/6], [5/14  1/3    1]
)
}}}

{{{id=8|
def myLLL(M):
    k=1
    n=M.ncols()
    Mk=M
    while k <= n-1:
        Mk,R,mu=sizered(Mk)
        if k>0 and norm(R[k-1])>2*norm(R[k]):
            r=Mk[k]
            Mk.set_row(k,Mk[k-1])
            Mk.set_row(k-1,r)
            k=k-1
        else:
            k=k+1
        print k,Mk
    return Mk
///
}}}

{{{id=9|
myLLL(M)
///
0 [ 1  1  1]
[ 1  2  3]
[ 0 -1 -1]
1 [ 1  1  1]
[-1  0  1]
[ 1  0  0]
2 [ 1  1  1]
[-1  0  1]
[ 1  0  0]
1 [ 1  1  1]
[ 1  0  0]
[-1  0  1]
0 [1 0 0]
[1 1 1]
[0 0 1]
1 [ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
2 [ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
3 [ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
[ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
}}}

{{{id=10|
def myLLL2(M):
    k=1
    n=M.ncols()
    Mk=M
    while k <= n-1:
        Mk,R,mu=sizered2(Mk,k)
        if k>0 and norm(R[k-1])>2*norm(R[k]):
            r=Mk[k]
            Mk.set_row(k,Mk[k-1])
            Mk.set_row(k-1,r)
            k=k-1
        else:
            k=k+1
        print k,Mk
    return Mk
///
}}}

{{{id=11|
myLLL2(M)
///
2 [ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
3 [ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
[ 1  0  0]
[ 0  1  1]
[ 0 -1  0]
}}}

{{{id=12|
n=400
R=RealField(n)
C=10^(100)
///
}}}

{{{id=13|
a=sqrt(2)+3^(1/3)+sqrt(3)
///
}}}

{{{id=14|
d=13
col1=[]
for i in range(d):
    col1=col1 + [ZZ(floor(R(C*a^i+1/2)))]
N=Matrix(d,1,col1)
///
}}}

{{{id=15|
M=identity_matrix(ZZ,d)
L=N.augment(M)
///
}}}

{{{id=16|
L.LLL()
///
[      -140      -8648        144       2940       5172       1113      -1656      -1006          0        303        -12        -30          0          1]
[  27628362  104708870  164422286  142362905  115686623 -270080627  -55711656  -41264428   50430737  110838389  192863002 -146433243   -1520620    5022135]
[ 149179087  -18321879  146649981 -216335775  200828125 -130801871  150887421  113254308  174491019 -163989930  160210180    8525884  134151890  -31030760]
[  27201244  -51816322  218405964  267223973 -305311665   -6261693 -262632910   85563736   18618078  -80496572   70066951  -35140090  -24829870    6524749]
[ -24031235  112991991  167647010   -3404539   30975274   -3088335 -439229279  -65082943  176298608   13574835 -209138766 -112699775  -45037985   17232857]
[  -5752450  -53977430   50406237  113995925 -202643814   21925506   22483635 -318538855 -307821801 -232382402  123188621  -76669395  -96321691   24067509]
[   6482137   -3180120   -5825811  -70948835  102608245  124690336   83848350  351053393  113305383   28545778  248505924    9909212 -230575488   47047238]
[-246036581  135105915  150291276  227186173  136999480 -112915045 -127641517  290551525  146126368 -193547570  -49015531  -16006319  -24613894    6968820]
[ 339381268 -112350076 -269066309    9370985 -191660231  202316155  -25670575  144873092 -169664780 -147548240  189295096 -113107808  -73262404   19779765]
[ -84663375 -124721243  -16030321 -308251989   72770063  159692694  305318054  308628206 -162432857  261114821  123301525  -82514548   34785963   -5488488]
[   8750884   10456754  100986650 -111209371  129281164  -26684271  -99075594  373268060 -174638857  -57036243 -176625892  272894711  218290987  -58529606]
[ 139405336   33579849   25968988   29953017   27526066   84863017   35502952  -37425725 -205241731  -34885707 -356299854 -287528858 -222855393   66094915]
[ 225756324  160449307    -322392  463547545  -27157721  186133450   26899316  -76603628 -170916393 -125821758   19391387  148721349  197632361  -49961063]
}}}

{{{id=17|
R=RealField(2000);
Pi =R(pi)
C=10^600
d=9
///
}}}

{{{id=18|
col1=[0 for i in range(8)]
for i in range(8):
  for k in range(1000):
        if k+i<>0:
            col1[i]=col1[i]+R(1/(16^k*(8*k+i)))
col2=[floor(col1[i]*C) for i in range(8)] + [floor(C*Pi)]
///
}}}

{{{id=19|
N=Matrix(d,1,col2)
M=identity_matrix(ZZ,d)
L=N.augment(M)
///
}}}

{{{id=20|
L.LLL()[1]
///
(5, 0, 0, -8, -4, -4, 0, 0, 1, 2)
}}}

{{{id=21|
log(abs(Pi-4*col1[1]+2*col1[4]+col1[5]+col1[6]))/log(10.)
///
-600.764424228000
}}}

{{{id=22|
log(abs(2*Pi-8*col1[2]-4*col1[3]-4*col1[4]+col1[7]))/log(10.)
///
-599.898623325727
}}}

{{{id=23|

///
}}}