first_order — Inhomogeneous

Solves the linear inhomogeneous recurrence relation

View versions (2)

Interface

#include <codecogs/maths/discrete/recurrence/linear/inhomogeneous/first_order.h>

using namespace Maths::Discrete::Recurrence::Linear::Inhomogeneous;

double first_order(double u0, double a, double b, int n)

The solution to an inhomogeneous recurrence relationship of the form

$$u_{n+1} = a u_n + b$$

(1)

where $u_0$ is constant is

$$u_n = \left ( u_0 + \frac{b}{a-1} \right ) a^n - \frac{b}{a-1}$$

(2)

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Example 1

The annual birth rate of a small island is 1.1 per capita. There is also a constant influx of 10 new immigrants to the island each year. If the initial population is 580, what is the population after each year for the next 10 years

#include <stdio.h>
#include <codecogs/maths/discrete/recurrence/linear/inhomogeneous/first_order.h>

using namespace Maths::Discrete::Recurrence::Linear::Inhomogeneous;
int main()
{
  double initial_pop=580;  // an initial population
  double birth_rate=1.1;   // average birthrate per capita
  double immigration=10;   // direct immigration
  for(int i=0;i<=10;i++)
    printf("\n Year=%d  Population=%lf",i, first_order(initial_pop, birth_rate, immigration, i));
  return 0;
}

Output

Year=0  Population=580.000000
 Year=1  Population=648.000000
 Year=2  Population=722.800000
 Year=3  Population=805.080000
 Year=4  Population=895.588000
 Year=5  Population=995.146800
 Year=6  Population=1104.661480
 Year=7  Population=1225.127628
 Year=8  Population=1357.640391
 Year=9  Population=1503.404430
 Year=10  Population=1663.744873

Parameters

The first (initial) term of the series.

Homogeneous factor - multiplier for each additional term of the recurrence series.

Inhomogeneous factor - additional constant added at each step of the recurrence.

The number of terms from the recursive series to evaluate.

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Result –