inverse simplicial number function
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78
miscfunc.cc
78
miscfunc.cc
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@ -16,12 +16,13 @@
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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*/
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#include "miscfunc.h"
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#include "laerror.h"
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#include <iostream>
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#include <stdint.h>
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#include <math.h>
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#include "miscfunc.h"
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#include "laerror.h"
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#include "polynomial.h"
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namespace LA {
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@ -64,11 +65,11 @@ double logz,cor,nn,z;
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cor=0.0;
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z=n+1.0;
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logz=log(z);
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logz= std::log(z);
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if(z>=2.0) /*does not have sense for smaller anyway and could make troubles*/
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{
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int i,ii,m;
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m=(int)ceil(0.5-log(EPS/10000.)/logz/2.0);
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m=(int)std::ceil(0.5-std::log(EPS/10000.)/logz/2.0);
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if(m<2)m=2; if(m>16)m=16;
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nn=z*z;
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for(i=m;i>0;i--) {ii=2*i;cor = cor/nn + bernoulli_number(ii)/((double)ii*(ii-1));}
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@ -77,7 +78,7 @@ if(z>=2.0) /*does not have sense for smaller anyway and could make troubles*/
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/*0.5*ln(2*M_PI)=0.91893853320467275836*/
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return((z-0.5)*logz-z+0.918938533204672741780329736406+cor);
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/*cor=log((1.0/(24.0*n)+1.0)/(12.0*n)+1.0)*/
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/*cor=std::log((1.0/(24.0*n)+1.0)/(12.0*n)+1.0)*/
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}
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@ -125,14 +126,14 @@ if(n < 2) return(0.0);
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if(n <= LIMITFACT) /*supposed LIMITFACT > MAXFACT*/
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if(a[n]) return(a[n]); else
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{
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if(n<=MAXFACT) return(a[n]= log(factorial(n)));
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if(n<=MAXFACT) return(a[n]= std::log(factorial(n)));
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{
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int j;
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if(!a[top]) a[top]= log(factorial(top));
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if(!a[top]) a[top]= std::log(factorial(top));
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while (top<n) {
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j=top++;
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a[top]=a[j]+log((double)top);
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a[top]=a[j]+std::log((double)top);
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}
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return a[n];
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}
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@ -619,7 +620,7 @@ double hlp;
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if(x>0.96 && x<1.04 && x!=1.0) /*use laurent series -1 and 0 term and spline */
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{
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double l,e;
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l= -log10(fabs(x-1.0));
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l= -std::log10(std::fabs(x-1.0));
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if(l>=6.5) std::cerr <<"Argument of zeta too near to the pole, result will be very imprecise!\n";
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if(l>12.0) e=0.0;
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else
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@ -630,7 +631,7 @@ if(x>0.96 && x<1.04 && x!=1.0) /*use laurent series -1 and 0 term and spline */
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return(c_euler+e+1.0/(x-1.0));
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}
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if(x==floor(x) && fabs(x)<=(((unsigned int) 1)<<(4*sizeof(int)-1)) )
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if(x==std::floor(x) && std::fabs(x)<=(((unsigned int) 1)<<(4*sizeof(int)-1)) )
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{
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int n;
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n=(int)x;
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@ -642,7 +643,7 @@ if(x==floor(x) && fabs(x)<=(((unsigned int) 1)<<(4*sizeof(int)-1)) )
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else return(0.0);
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}
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if(n==2||n==4||n==6||n==8) /*precalculated few bernoulli numbers */
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return(fabs(bernoulli_number(n))*pow(2*M_PI,(double)n)/factorial(n)/2.0);
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return(std::fabs(bernoulli_number(n))*pow(2*M_PI,(double)n)/factorial(n)/2.0);
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/*otherwise continue like if general real argument*/
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}
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if(x<-1.02 || (x>0.58 && x<1.94)) return (2.0*pow(2.0*M_PI,x-1.0)*sin(M_PI*x/2.0)*gamma(1.0-x)*zeta(1.0-x));
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@ -653,7 +654,7 @@ if(x>=9.1) /* in this region the summation per definition is best and quite exac
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{
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double s;
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int i,n;
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n=(int)ceil(pow(EPS/1000.0,-1.0/x));
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n=(int)std::ceil(pow(EPS/1000.0,-1.0/x));
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s=0.0;for(i=n;i>1;i--) s+= pow((double)i,-x);
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return(1.0+s);
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}
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@ -672,17 +673,17 @@ bool f;
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double p;
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if(x>MAXFACT) laerror("too big argument in gamma");
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f= (fabs(x-floor(x+0.5))<EPS);
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f= (std::fabs(x-std::floor(x+0.5))<EPS);
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if(x<=0.0 && f) laerror("nonpositive integer argument in gamma");
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if(f) return factorial(floor(x+0.5)-1);
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if(f) return factorial(std::floor(x+0.5)-1);
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if(x<8.0)
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{
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p=1.0;
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while(x<8.0) {p /= x; x += 1.0;}
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return(p*exp(stirfact(x-1.0)));
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return(p* std::exp(stirfact(x-1.0)));
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}
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return(exp(stirfact(x-1.0)));
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return( std::exp(stirfact(x-1.0)));
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}
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@ -697,14 +698,14 @@ double gammln(double x)
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{
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bool f;
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f= (fabs(x-floor(x+0.5))<EPS);
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f= (std::fabs(x-std::floor(x+0.5))<EPS);
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if(x<=0.0 && f) laerror("nonpositive integer argument in gamma");
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if(fabs(x-floor(x+0.5))<EPS) return factln(x-1.0);
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if(std::fabs(x-std::floor(x+0.5))<EPS) return factln(x-1.0);
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if(x<8.0)
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{
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double p;
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p=0.0;
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while(x<8.0) {p -= log(fabs(x)); x += 1.0;}
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while(x<8.0) {p -= std::log(std::fabs(x)); x += 1.0;}
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return(p+stirfact(x-1.0));
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}
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return(stirfact(x-1.0));
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@ -759,7 +760,8 @@ return(bern[n]);
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//there is a closed form s(d,n) = binom(n+d-1,d)
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//but that would be inefficient
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//we could also pre-generate these polynomials for small d
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//but that would be inefficient, better is to buffer all values
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//we use recurrence s(d,n) = s(d,n-1) + s(d-1,n)
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//we cannot also use the above efficient binom() since here we need laerge n while d is small
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//
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@ -810,4 +812,40 @@ return stored[d-2][n];
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int inverse_simplicial(int d, unsigned long long s)
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{
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if(s==0) return 0;
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if(d==0 || s==1) return 1;
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if(d==1) return (int)s;
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if(d>=SIMPLICIAL_MAXD) laerror("storage overflow in inverse_simplicial");
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static int maxd=0;
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static Polynomial<double> polynomials[SIMPLICIAL_MAXD];
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static Polynomial<double> polynomials_der[SIMPLICIAL_MAXD];
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for(int i=maxd+1; i<=d; ++i) //generate as needed
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{
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NRVec<double> roots(i);
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for(int j=0; j<i;++j) roots[j] = -j;
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polynomials[i-1] = polyfromroots(roots) * (1./factorial(i));
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polynomials_der[i-1] = polynomials[i-1].derivative(1);
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maxd=i;
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}
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//solve polynomial equation by newton
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double x=std::pow(factorial(d)*(double)s,1./d); //good init guess
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double dx;
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do
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{
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double residual = value(polynomials[d-1],x)-s;
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dx = residual/value(polynomials_der[d-1],x);
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x -= dx;
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}
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while(std::fabs(dx)>.5); //so usually 2 iterations are enough
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return std::floor(x);
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}
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#undef SIMPLICIAL_MAXD
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#undef SIMPLICIAL_MAXN
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}//namespace
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@ -36,6 +36,7 @@ extern double gamma(double x);
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extern double gammln(double x);
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extern double bernoulli_number(int n);
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extern unsigned long long simplicial(int d, int n); //simplicial numbers in a precomputed efficient way
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extern int inverse_simplicial(int d, unsigned long long s);
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9
t.cc
9
t.cc
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@ -3176,11 +3176,14 @@ for(int l=1; l<k; ++l)
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cout <<count<<" "<<binom(n,4)<<endl;
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}
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if(0)
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if(1)
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{
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int d,n;
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cin>>d>>n;
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cout <<simplicial(d,n)<<" "<<binom(n+d-1,d)<<endl;
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unsigned long long s;
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s=simplicial(d,n);
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cout <<s<<" "<<binom(n+d-1,d)<<endl;
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cout <<inverse_simplicial(d,s)<<endl;
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}
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if(0)
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@ -3192,7 +3195,7 @@ cout <<d;
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}
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if(1)
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if(0)
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{
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INDEXGROUP g;
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g.number=3;
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