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jiri 2009-11-05 10:57:43 +00:00
parent e090cc5712
commit e5351499fa
1 changed files with 75 additions and 6 deletions
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81
matexp.h
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@ -49,6 +49,7 @@ return y;
}
//algorithm which minimazes number of multiplications, at the cost of storage
template<class T,class R>
const T polynom(const T &x, const NRVec<R> &c)
@ -174,12 +175,9 @@ if(n<=1.) return 1;
return int(ceil(log(n)/log2-log(.75)));
}
template<class T, class C>
NRVec<C> exp_aux(const T &x, int &power,int maxpower= -1, int maxtaylor= -1, typename LA_traits<T>::elementtype prescale=1.)
{
//should better be computed by mathematica to have accurate last digits, chebyshev instead, see exp in glibc
static double exptaylor[]={
//should better be computed by mathematica to have accurate last digits, perhaps chebyshev instead, see exp in glibc
//is shared also for sine and cosine now
static const double exptaylor[]={
1.,
1.,
0.5,
@ -201,7 +199,14 @@ static double exptaylor[]={
1.5619206968586225271e-16,
8.2206352466243294955e-18,
4.1103176233121648441e-19,
1.9572941063391262595e-20,
0.};
template<class T, class C>
NRVec<C> exp_aux(const T &x, int &power,int maxpower= -1, int maxtaylor= -1, typename LA_traits<T>::elementtype prescale=1.)
{
double mnorm= x.norm() * abs(prescale);
power=nextpow2(mnorm);
if(maxpower>=0 && power>maxpower) power=maxpower;
@ -237,6 +242,44 @@ return taylor2;
template<class T, class C>
void sincos_aux(NRVec<C> &si, NRVec<C> &co, const T &x, int &power,int maxpower= -1, int maxtaylor= -1, typename LA_traits<T>::elementtype prescale=1.)
{
double mnorm= x.norm() * abs(prescale);
power=nextpow2(mnorm);
if(maxpower>=0 && power>maxpower) power=maxpower;
double scale=exp(-log(2.)*power);
//find how long taylor expansion will be necessary
const double precision=1e-14; //further decreasing brings nothing
double s,t;
s=mnorm*scale;
int n=0;
t=1.;
do {
n++;
t*=s;
}
while(t*exptaylor[n]>precision);//taylor 0 will terminate in any case
if(maxtaylor>=0 && n>maxtaylor) n=maxtaylor; //useful e.g. if the matrix is nilpotent in order n+1 as the CC T operator for n electrons
if((n&1)==0) ++n; //force it to be odd to have same length in sine and cosine
si.resize((n+1)/2);
co.resize((n+1)/2);
int i; //adjust the coefficients in order to avoid scaling the argument
for(i=0,t=1.;i<=n;i++)
{
if(i&1) si[i>>1] = exptaylor[i]* (i&2?-t:t);
else co[i>>1] = exptaylor[i]* (i&2?-t:t);
t*=scale;
}
cout <<"TEST sin "<<si<<endl;
cout <<"TEST cos "<<co<<endl;
}
//it seems that we do not gain anything by polynom vs polynom0, check the m-optimization!
template<class T>
const T exp(const T &x, bool horner=true, int maxpower= -1, int maxtaylor= -1 )
@ -256,6 +299,32 @@ return r;
}
//make exp(iH) with real H in real arithmetics
template<class T>
void sincos(T &s, T &c, const T &x, bool horner=true, int maxpower= -1, int maxtaylor= -1 )
{
int power;
NRVec<typename LA_traits<T>::normtype> taylors,taylorc;
sincos_aux<T,typename LA_traits<T>::normtype>(taylors,taylorc,x,power,maxpower,maxtaylor);
//could we save something by computing both polynoms simultaneously?
{
T x2 = x*x;
s = horner?polynom0(x2,taylors):polynom(x2,taylors);
c = horner?polynom0(x2,taylorc):polynom(x2,taylorc);
}
s = s * x;
//power the results back
for(int i=0; i<power; i++)
{
T tmp = c*c - s*s;
s = s*c; s *= 2.;
c=tmp;
}
}
//this simple implementation seems not to be numerically stable enough