La interpolación es la técnica de estimar el valor de una función para cualquier valor intermedio de la variable independiente, mientras que el proceso de calcular el valor de la función fuera del rango dado se llama extrapolación .
Diferencias centrales : El operador de diferencia central d está definido por las relaciones:
De manera similar, las diferencias centrales de alto orden se definen como:
Nota – Las diferencias centrales en la misma línea horizontal tienen el mismo sufijo
Fórmula de interpolación de Bessel –
Es muy útil cuando u = 1/2 . Da una mejor estimación cuando 1/4 < u < 3/4
Aquí f(0) es el punto de origen que generalmente se toma como punto medio, ya que Bessel se usa para interpolar cerca del centro.
h se llama intervalo de diferencia y u = ( x – f(0) ) / h, aquí f(0) es el término en el origen elegido.
Ejemplos –
Entrada: ¿Valor en 27,4?
Producción :
El valor a 27,4 es 3,64968
Implementación de la Interpolación de Bessel –
C++
// CPP Program to interpolate using Bessel's interpolation
#include <bits/stdc++.h>
using namespace std;
// calculating u mentioned in the formula
float ucal(float u, int n)
{
if (n == 0)
return 1;
float temp = u;
for (int i = 1; i <= n / 2; i++)
temp = temp * (u - i);
for (int i = 1; i < n / 2; i++)
temp = temp * (u + i);
return temp;
}
// calculating factorial of given number n
int fact(int n)
{
int f = 1;
for (int i = 2; i <= n; i++)
f *= i;
return f;
}
int main()
{
// Number of values given
int n = 6;
float x[] = { 25, 26, 27, 28, 29, 30 };
// y[][] is used for difference table
// with y[][0] used for input
float y[n][n];
y[0][0] = 4.000;
y[1][0] = 3.846;
y[2][0] = 3.704;
y[3][0] = 3.571;
y[4][0] = 3.448;
y[5][0] = 3.333;
// Calculating the central difference table
for (int i = 1; i < n; i++)
for (int j = 0; j < n - i; j++)
y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
// Displaying the central difference table
for (int i = 0; i < n; i++) {
for (int j = 0; j < n - i; j++)
cout << setw(4) << y[i][j] << "\t";
cout << endl;
}
// value to interpolate at
float value = 27.4;
// Initializing u and sum
float sum = (y[2][0] + y[3][0]) / 2;
// k is origin thats is f(0)
int k;
if (n % 2) // origin for odd
k = n / 2;
else
k = n / 2 - 1; // origin for even
float u = (value - x[k]) / (x[1] - x[0]);
// Solving using bessel's formula
for (int i = 1; i < n; i++) {
if (i % 2)
sum = sum + ((u - 0.5) *
ucal(u, i - 1) * y[k][i]) / fact(i);
else
sum = sum + (ucal(u, i) *
(y[k][i] + y[--k][i]) / (fact(i) * 2));
}
cout << "Value at " << value << " is " << sum << endl;
return 0;
}
Java
// Java Program to interpolate using Bessel's interpolation
import java.text.*;
class GFG{
// calculating u mentioned in the formula
static double ucal(double u, int n)
{
if (n == 0)
return 1;
double temp = u;
for (int i = 1; i <= n / 2; i++)
temp = temp * (u - i);
for (int i = 1; i < n / 2; i++)
temp = temp * (u + i);
return temp;
}
// calculating factorial of given number n
static int fact(int n)
{
int f = 1;
for (int i = 2; i <= n; i++)
f *= i;
return f;
}
public static void main(String[] args)
{
// Number of values given
int n = 6;
double x[] = { 25, 26, 27, 28, 29, 30 };
// y[][] is used for difference table
// with y[][0] used for input
double[][] y=new double[n][n];
y[0][0] = 4.000;
y[1][0] = 3.846;
y[2][0] = 3.704;
y[3][0] = 3.571;
y[4][0] = 3.448;
y[5][0] = 3.333;
// Calculating the central difference table
for (int i = 1; i < n; i++)
for (int j = 0; j < n - i; j++)
y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
// Displaying the central difference table
DecimalFormat df = new DecimalFormat("#.########");
for (int i = 0; i < n; i++) {
for (int j = 0; j < n - i; j++)
System.out.print(y[i][j]+"\t");
System.out.println("");
}
// value to interpolate at
double value = 27.4;
// Initializing u and sum
double sum = (y[2][0] + y[3][0]) / 2;
// k is origin thats is f(0)
int k;
if ((n % 2)>0) // origin for odd
k = n / 2;
else
k = n / 2 - 1; // origin for even
double u = (value - x[k]) / (x[1] - x[0]);
// Solving using bessel's formula
for (int i = 1; i < n; i++) {
if ((i % 2)>0)
sum = sum + ((u - 0.5) *
ucal(u, i - 1) * y[k][i]) / fact(i);
else
sum = sum + (ucal(u, i) *
(y[k][i] + y[--k][i]) / (fact(i) * 2));
}
System.out.printf("Value at "+value+" is %.5f",sum);
}
}
// This code is contributed by mits
Python3
# Python3 Program to interpolate
# using Bessel's interpolation
# calculating u mentioned in the
# formula
def ucal(u, n):
if (n == 0):
return 1;
temp = u;
for i in range(1, int(n / 2 + 1)):
temp = temp * (u - i);
for i in range(1, int(n / 2)):
temp = temp * (u + i);
return temp;
# calculating factorial of
# given number n
def fact(n):
f = 1;
for i in range(2, n + 1):
f *= i;
return f;
# Number of values given
n = 6;
x = [25, 26, 27, 28, 29, 30];
# y[][] is used for difference
# table with y[][0] used for input
y = [[0 for i in range(n)]
for j in range(n)];
y[0][0] = 4.000;
y[1][0] = 3.846;
y[2][0] = 3.704;
y[3][0] = 3.571;
y[4][0] = 3.448;
y[5][0] = 3.333;
# Calculating the central
# difference table
for i in range(1, n):
for j in range(n - i):
y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
# Displaying the central
# difference table
for i in range(n):
for j in range(n - i):
print(y[i][j], "\t", end = " ");
print("");
# value to interpolate at
value = 27.4;
# Initializing u and sum
sum = (y[2][0] + y[3][0]) / 2;
# k is origin thats is f(0)
k = 0;
if ((n % 2) > 0): # origin for odd
k = int(n / 2);
else:
k = int(n / 2 - 1); # origin for even
u = (value - x[k]) / (x[1] - x[0]);
# Solving using bessel's formula
for i in range(1, n):
if (i % 2):
sum = sum + ((u - 0.5) *
ucal(u, i - 1) *
y[k][i]) / fact(i);
else:
sum = sum + (ucal(u, i) * (y[k][i] +
y[k - 1][i]) / (fact(i) * 2));
k -= 1;
print("Value at", value, "is", round(sum, 5));
# This code is contributed by mits
C#
// C# Program to interpolate using Bessel's interpolation
class GFG{
// calculating u mentioned in the formula
static double ucal(double u, int n)
{
if (n == 0)
return 1;
double temp = u;
for (int i = 1; i <= n / 2; i++)
temp = temp * (u - i);
for (int i = 1; i < n / 2; i++)
temp = temp * (u + i);
return temp;
}
// calculating factorial of given number n
static int fact(int n)
{
int f = 1;
for (int i = 2; i <= n; i++)
f *= i;
return f;
}
public static void Main()
{
// Number of values given
int n = 6;
double []x = { 25, 26, 27, 28, 29, 30 };
// y[,] is used for difference table
// with y[,0] used for input
double[,] y=new double[n,n];
y[0,0] = 4.000;
y[1,0] = 3.846;
y[2,0] = 3.704;
y[3,0] = 3.571;
y[4,0] = 3.448;
y[5,0] = 3.333;
// Calculating the central difference table
for (int i = 1; i < n; i++)
for (int j = 0; j < n - i; j++)
y[j,i] = y[j + 1,i - 1] - y[j,i - 1];
// Displaying the central difference table
for (int i = 0; i < n; i++) {
for (int j = 0; j < n - i; j++)
System.Console.Write(y[i,j]+"\t");
System.Console.WriteLine("");
}
// value to interpolate at
double value = 27.4;
// Initializing u and sum
double sum = (y[2,0] + y[3,0]) / 2;
// k is origin thats is f(0)
int k;
if ((n % 2)>0) // origin for odd
k = n / 2;
else
k = n / 2 - 1; // origin for even
double u = (value - x[k]) / (x[1] - x[0]);
// Solving using bessel's formula
for (int i = 1; i < n; i++) {
if ((i % 2)>0)
sum = sum + ((u - 0.5) *
ucal(u, i - 1) * y[k,i]) / fact(i);
else
sum = sum + (ucal(u, i) *
(y[k,i] + y[--k,i]) / (fact(i) * 2));
}
System.Console.WriteLine("Value at "+value+" is "+System.Math.Round(sum,5));
}
}
// This code is contributed by mits
PHP
<?php
// PHP Program to interpolate
// using Bessel's interpolation
// calculating u mentioned
// in the formula
function ucal($u, $n)
{
if ($n == 0)
return 1;
$temp = $u;
for ($i = 1;
$i <= (int)($n / 2); $i++)
$temp = $temp *
($u - $i);
for ($i = 1;
$i < (int)($n / 2); $i++)
$temp = $temp * ($u + $i);
return $temp;
}
// calculating factorial
// of given number n
function fact($n)
{
$f = 1;
for ($i = 2; $i <= $n; $i++)
$f *= $i;
return $f;
}
// Number of values given
$n = 6;
$x = array(25, 26, 27,
28, 29, 30);
// y[][] is used for difference
// table with y[][0] used for input
$y;
for($i = 0; $i < $n; $i++)
for($j = 0; $j < $n; $j++)
$y[$i][$j] = 0.0;
$y[0][0] = 4.000;
$y[1][0] = 3.846;
$y[2][0] = 3.704;
$y[3][0] = 3.571;
$y[4][0] = 3.448;
$y[5][0] = 3.333;
// Calculating the central
// difference table
for ($i = 1; $i < $n; $i++)
for ($j = 0; $j < $n - $i; $j++)
$y[$j][$i] = $y[$j + 1][$i - 1] -
$y[$j][$i - 1];
// Displaying the central
// difference table
for ($i = 0; $i < $n; $i++)
{
for ($j = 0; $j < $n - $i; $j++)
echo str_pad($y[$i][$j], 4) . "\t";
echo "\n";
}
// value to interpolate at
$value = 27.4;
// Initializing u and sum
$sum = ($y[2][0] +
$y[3][0]) / 2;
// k is origin thats is f(0)
$k;
if ($n % 2) // origin for odd
$k = $n / 2;
else
$k = $n / 2 - 1; // origin for even
$u = ($value - $x[$k]) /
($x[1] - $x[0]);
// Solving using
// bessel's formula
for ($i = 1; $i < $n; $i++)
{
if ($i % 2)
$sum = $sum + (($u - 0.5) *
ucal($u, $i - 1) *
$y[$k][$i]) / fact($i);
else
$sum = $sum + (ucal($u, $i) *
($y[$k][$i] +
$y[--$k][$i]) /
(fact($i) * 2));
}
echo "Value at " . $value .
" is " . $sum . "\n";
// This code is contributed by mits
?>
Javascript
<script>
// Javascript Program to interpolate
// using Bessel's interpolation
// Calculating u mentioned in the formula
function ucal(u, n)
{
if (n == 0)
return 1;
var temp = u;
for(var i = 1; i <= n / 2; i++)
temp = temp * (u - i);
for(var i = 1; i < n / 2; i++)
temp = temp * (u + i);
return temp;
}
// Calculating factorial of given number n
function fact(n)
{
var f = 1;
for(var i = 2; i <= n; i++)
f *= i;
return f;
}
// Driver code
// Number of values given
var n = 6;
var x = [ 25, 26, 27, 28, 29, 30 ];
// y is used for difference table
// with y[0] used for input
var y = Array(n).fill(0.0).map(x => Array(n).fill(0.0));;
y[0][0] = 4.000;
y[1][0] = 3.846;
y[2][0] = 3.704;
y[3][0] = 3.571;
y[4][0] = 3.448;
y[5][0] = 3.333;
// Calculating the central difference table
for(var i = 1; i < n; i++)
for(var j = 0; j < n - i; j++)
y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
// Displaying the central difference table
for(var i = 0; i < n; i++)
{
for(var j = 0; j < n - i; j++)
document.write(y[i][j].toFixed(6) +
" ");
document.write('<br>');
}
// Value to interpolate at
var value = 27.4;
// Initializing u and sum
var sum = (y[2][0] + y[3][0]) / 2;
// k is origin thats is f(0)
var k;
// Origin for odd
if ((n % 2) > 0)
k = n / 2;
else
// Origin for even
k = n / 2 - 1;
var u = (value - x[k]) / (x[1] - x[0]);
// Solving using bessel's formula
for(var i = 1; i < n; i++)
{
if ((i % 2) > 0)
sum = sum + ((u - 0.5) *
ucal(u, i - 1) * y[k][i]) / fact(i);
else
sum = sum + (ucal(u, i) *
(y[k][i] + y[--k][i]) / (fact(i) * 2));
}
document.write("Value at " + value.toFixed(6) +
" is " + sum.toFixed(6));
// This code is contributed by Princi Singh
</script>
Producción:
4 -0.154 0.0120001 -0.00300002 0.00399971 -0.00699902 3.846 -0.142 0.00900006 0.000999689 -0.00299931 3.704 -0.133 0.00999975 -0.00199962 3.571 -0.123 0.00800014 3.448 -0.115 3.333 Value at 27.4 is 3.64968
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Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA