PowerSeriesInversion

PowerSeriesInversion may refer to the case of two quantities, say, \(x\) and \(y\) related through

\( y=a_1x+a_2x^2+a_3x^3+... \)

It is supposed that

\( x=A_1y+A_2y^2+A_3y^3+.... \)

The PowerSeriesInversion may refer also the set of formulas that express the coefficients \(A\) through the coefficients \(a\).

Some of these formulas are collected below.

Explicit formula by Wolfram MathWorld

The Wolfram MathWorld suggests the following relations:

\( A_1 = a_1^{-1} \\ A_2 = -a_1^{-3}a_2 \\ A_3 = a_1^{-5}(2a_2^2-a_1a_3) \\ A_4 = a_1^{-7}(5a_1a_2a_3-a_1^2a_4-5a_2^3) \\ A_5 = a_1^{-9}(6a_1^2a_2a_4+3a_1^2a_3^2+14a_2^4-a_1^3a_5-21a_1a_2^2a_3) \\ A_6 = a_1^{-11}(7a_1^3a_2a_5+7a_1^3a_3a_4+84a_1a_2^3a_3-a_1^4a_6-28a_1^2a_2a_3^2-42a_2^5-28a_1^2a_2^2a_4) \\ A_7 = a_1^{-13}(8a_1^4a_2a_6+8a_1^4a_3a_5+4a_1^4a_4^2+120a_1^2a_2^3a_4+180a_1^2a_2^2a_3^2+132a_2^6-a_1^5a_7-36a_1^3a_2^2a_5-72a_1^3a_2a_3a_4-12a_1^3a_3^3-330a_1a_2^4a_3) \)

Case of a_1=1

\( A_1 = a_1 =1 \\ A_2 = - a_2 \\ A_3 = 2a_2^2-a_3 \\ A_4 = 5a_2a_3-a_4-5a_2^3 \\ A_5 = 6a_2a_4+3a_3^2+14a_2^4-a_5-21a_2^2a_3 \\ A_6 = 7a_2a_5+7a_3a_4+84a_2^3a_3-a_6-28a_2a_3^2-42a_2^5-28a_2^2a_4 \\ A_7 = 8a_2a_6+8a_3a_5+4a_4^2+120a_2^3a_4+180a_2^2a_3^2+132a_2^6-a_7-36a_2^2a_5-72a_2a_3a_4-12a_3^3-330a_2^4a_3 \)

ChatGPT

//ChatGPT suggests the following C++ implementation:

//

void PowerSeriesInversion(double *a, double *b, int N)
{
    // b[1..N] given
    // compute a[1..N]

    a[1] = 1.0 / b[1];

    for(int n=2; n<=N; n++)
    {
        double sum = 0.0;

        // compute sum_{k=2}^n b_k * C_{n,k}
        for(int k=2; k<=n; k++)
        {
            // compute coefficient of y^n in (A(y))^k
            // by convolution

            // temporary polynomial for A(y)
            double power[N+1] = {0};
            double next[N+1] = {0};

            // initialize power = A(y)
            for(int i=1; i<=n; i++)
                power[i] = a[i];

            // raise to k-th power
            for(int p=2; p<=k; p++)
            {
                for(int i=0;i<=n;i++) next[i]=0.0;

                for(int i=1;i<=n;i++)
                    for(int j=1;j<=n;j++)
                        if(i+j<=n)
                            next[i+j] += power[i]*a[j];

                for(int i=0;i<=n;i++)
                    power[i] = next[i];
            }

            sum += b[k] * power[n];
        }

        a[n] = - sum / b[1];
    }
}
//

Example 1

//

#include <math.h>
#include  <stdio.h>
#include  "PowerSeriesInversion.cin"

int main(){ int n,k; int N=4; int N1=N+1;
double a[N1]; double b[N1];
double f=1; for(n=1;n<N1;n++) {f/=n; b[n]=f;}
printf("before:\n");
for(n=0;n<N1;n++)  printf("%20.16lf",a[n]);
printf("\n");
for(n=0;n<N1;n++)  printf("%20.16lf",b[n]);
printf("\nPowerSeriesInversion(a,b,N):\n");
PowerSeriesInversion(a,b,N);
for(n=0;n<N1;n++)  printf("%20.16lf",a[n]);
printf("\n");
}
//

output:

before:
  0.0000000000000000  0.0000000000000000  0.0000000000000000  0.0000000000000000  0.0000000000000000
  0.0000000000000000  1.0000000000000000  0.5000000000000000  0.1666666666666667  0.0416666666666667
PowerSeriesInversion(a,b,N):
  0.0000000000000000  1.0000000000000000 -0.5000000000000000  0.3333333333333334 -0.2500000000000001

References

Keywords

«Asymptotic», «ChatGPT», «Power series», «PowerSeriesInversion», «Wolfram MathWorld»,