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# 关于zju1007.奇怪！

jwd_1_cool 2003-08-01 08:58:03

y(x)=(1-x)((2-x)((3-x)((4-x)((5-x)((6-x)((7-x)((8-x)R+1/40320)+1/35280)+1/4320)+1/600)+1/96)+1/18)+1/4)+1

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boodweb 2003-08-12

f(x)=phi(x)-phi(m)，其中m=floor(x)

g(x)=f(x)-(x-m)f(m+1)=(m-x)(m+1-x)∑[1/k/(k+m)/(k+m+1)/(k+x)]
phi(x)=f(x)+phi(m)=g(x)+(x-m)f(m+1)+phi(m)

#include <stdio.h>
#include <iostream>
#include <cmath>

using namespace std;

const double ESP=1e-13;
const double E=1e-3;
const int r=4;
double phi_m=1.0; //phi(m)
double myfloor(double x)
{
double t=floor(x);
if(x-t>E+0.9) return t+1.0;
return t;
}
double f(int m)
{
double ret=0.0;
int k;
for(k=1;k<=m-1;k++) ret+=1.0/k;
ret*=-1.0/(m-1)/m;
ret+=1.0/m/m;
return ret;
}
//phi(x)=Σ(1/k/(k+x)),k=1..infinity
double phi(double x,int n)//r=4
{
double ret=0.0;
int k;
int m;
double t=myfloor(x);
if(x<=2.0+E) m=1;
else if(x-t<=E)
m=t-1;
else
m=t;
for(k=1;k<=n;k++)
{
ret+=1.0/k/(k+m)/(k+m+1)/(k+x);
}
ret*=(m-x)*(m+1-x);
ret+=phi_m-(m-x)*f(m+1);
if(x>=2.0-E && x-t<E)
phi_m=ret;
return ret;
}
int main()
{
double x;
int n;
n=(int)pow(1.0/(r-1.0)/ESP,1.0/(r-1.0))+2;
for(x=0.0;x<=300.0+E;x+=0.1)
{
printf("%6.2f %16.12f\n", x, phi(x,n) );
}
return 0;
}

boodweb 2003-08-11

btw: uva真夸张,zju只到2.0阿...

mmmcd 2003-08-10
1/(k(k+x)) - 1/(k(k+1)) = (1-x)/(k(k+1)(k+x))

sigma 1/(k(k+x)) - sigma 1/(k(k+1)) = (1-x) * sigma 1/(k(k+1)(k+x))

sigma 1/(k(k+x)) = (1-x) sigma 1/(k(k+1)(k+x)) + sigma 1/(k(k+1))
= (1-x) sigma 1/(k(k+1)(k+x)) + 1/1 - 1/2 + 1/2 - 1/3 + ...
= (1-x) sigma 1/(k(k+1)(k+x)) + 1/1

1/(k(k+1)(k+x)) - 1/(k(k+1)(k+2)) = (2-x)/(k(k+1)(k+2)(k+x))

sigma 1/(k(k+1)(k+x)) - sigma 1/(k(k+1)(k+2))
= (2-x) * sigma 1/(k(k+1)(k+2)(k+x))

sigma 1/(k(k+1)(k+x))
= (2-x) * sigma 1/(k(k+1)(k+2)(k+x)) + sigma 1/(k(k+1)(k+2))
= (2-x) * sigma 1/(k(k+1)(k+2)(k+x))+1/2/(1*2)-1/2/(2*3)+1/2/(2*3)-1/2/(3*4)+...
= (2-x) * sigma 1/(k(k+1)(k+2)(k+x))+1/2/(1*2)

sigma 1/(k(k+1)(k+2)(k+x)) - sigma 1/(k(k+1)(k+2)(k+3)) =
sigma (3-x)/(k(k+1)(k+2)(k+3)(k+x)) = (3-x)*sigma 1/(k(k+1)(k+2)(k+3)(k+x))

...

sigma 1/(k(k+1)(k+2)...(k+n-1)(k+x)) - sigma 1/(k(k+1)(k+2)...(k+n)) =
sigma (n-x)/(k(k+1)(k+2)...(k+n)(k+x))
=(n-x)*sigma 1/(k(k+1)(k+2)...(k+n)(k+x))

sigma 1/(k(k+1)(k+2)...(k+n-1)(k+x))=
(n-x)*sigma 1/(k(k+1)(k+2)...(k+n)(k+x)) + 1/n/n!

jwd_1_cool 2003-08-10

mmmcd 2003-08-09

boodweb 2003-08-08

jwd_1_cool 2003-08-05

mmmcd 2003-08-03

As an example, the sample output below shows 4 acceptable lines out of 3001, which might appear in the output file.

The values of x should start at 0.00 and increase by 0.1 until the line with x=300.00 is output.

jwd_1_cool 2003-08-03

jwd_1_cool 2003-08-03

f(3)=-(1/18)
g(x) = (1-x) * (2-x) * (3-x)*∑[1/(k * (k + 1) * (k + 2) * (k+3) * (k + x) )

phi(x) = f(x) + phi(1) = g(x) + (x-1)*f(2) + (x-2)*f(3) + phi(1)

boodweb 2003-08-03

/*
Algorithm:
We know that phi(1) = 1.
Let f(x) = phi(x) - phi(1),
then we get f(x) = (1-x)* ∑[ 1/(k * (k+1) * (k + x)) ],
and f(1) = 0, f(2) = -0.25.
Set g(x) = f(x) - (x-1)*f(2),
and we can get g(x) = (1-x) * (2-x) * ∑[1/(k * (k + 1) * (k + 2) * (k + x) )
].
So phi(x) = f(x) + phi(1) = g(x) + (x-1)*f(2) + phi(1)

More over, we use the inequation given by the problem,
and we can see that ∑1/(k^r) < 1/ [(r-1)* ((n-1)^(r-1))], for k = n-1, ...,∞
.
Let ∑1/(k^r) < 1/ [(r-1)* ((n-1)^(r-1))] < ESP,
where ESP = 1e-12 given by the problem,
and we can calculate the max k we should count.

*/

jwd_1_cool 2003-08-02
#include<stdio.h>
int main()
{
long k; /* dummy summation variable */
long n; /* number of terms to sum */
double kd; /* double version of k */
double x; /* the variable */
double sum; /* used for computation summations */
double px, p1; /* psi(x) and psi(1) */
double dpx; /* value of the first accelerated series at x */
double dp2,dp3,dp4,dp5,dp6,dp7,dp8; /* value of the first accelerated series at 2 */
double ddpx,dddpx,ddddpx,dddddpx,ddddddpx,dddddddpx,ddddddddpx; /* value of the second accelerates series at x */
long i; /* counter for elements of table */

p1 = 1.0;
dp2 = 0.25;
dp3=1.0/18.0;
dp4=1.0/96.0;
dp5=1.0/600.0;
dp6=1.0/4320.0;
dp7=1.0/35280.0;
dp8=1.0/40320.0;
n = 12000;

/* loop for the 2001 table entries */
for (i=0; i<2001; i++ )
{
x = i * 0.001;
sum = 0.0;
for( k=1; k<n; k++ ){
kd = k;
sum += 1.0/(kd*(kd+1)*(kd+2)*(kd+x) );
}
//ddddddddpx=(8.0-x)*sum;
//dddddddpx=(7.0-x)*(ddddddddpx+dp8);
//dddddddpx=(7.0-x)*sum;
//ddddddpx=(6.0-x)*(dddddddpx+dp7);
//dddddpx = (5.0-x)*(dp6+ddddddpx);
//ddddpx = (4.0-x)*(dp5+dddddpx);
//dddpx = (3.0-x)*(ddddpx+dp4);
//dddpx= (3.0-x)*sum;
//ddpx = (2.0-x)*(dp3+dddpx);
ddpx=(2.0-x)*sum;
dpx = (1.0-x)*(dp2+ddpx);
px = p1+dpx;
printf("%5.3f %16.12f\n", x, px );
}
return 1;
}

jwd_1_cool 2003-08-02

http://online-judge.uva.es/p/v7/731.html

mmmcd 2003-08-02

18=3*3！
96=4*4！

mmmcd 2003-08-02

4=2*2！
16=3*3！
18=4*4！
...
35280=7*7！
...

BlueSky2008 2003-08-01

jwd_1_cool 2003-08-01

3.2w+

2003-08-01 08:58
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