牛顿除数差插值公式

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📜 牛顿除数差插值公式

📅 最后修改于: 2021-04-27 18:21:09 🧑 作者: Mango

插值是对值序列中两个已知值中的一个值的估计。

牛顿除数差分插值公式是一种在所有值序列的间隔差都不相同时使用的插值技术。

假设f(x ₀ )，f(x ₁ )，f(x ₂ )………f(x _n )是函数y = f(x)对应于参数x = x的(n + 1)个值₀ ，x ₁ ，x ₂ …x _n ，其中间隔差异不相同
然后，第一个除法差为

$f[x_0, x_1]=\frac{f(x_1)-f(x_0)}{x_1-x_0}$

第二个分开的差异由下式给出

$f[x_0, x_1, x_2]=\frac{f[x_1, x_2]-f[x_0, x_1]}{x_2-x_0}$

等等…
分开的差异相对于参数是对称的，即独立于参数的顺序。
所以，
f [x ₀ ，x ₁ ] = f [x ₁ ，x ₀ ]
f [x ₀ ，x ₁ ，x ₂ ] = f [x ₂ ，x ₁ ，x ₀ ] = f [x ₁ ，x ₂ ，x ₀ ]
通过使用第一个除法差，第二个除法差等。形成一个表，称为除法差表。

差异表：

牛顿除法差分插值公式

f(x)=f(x_0)+(x-x_0)f[x_0, x_1]+(x-x_0)(x-x_1)f[x_0, x_1, x_2]+..........................+(x-x_0)(x-x_1)...(x-x_k_-_1)f[x_0, x_1, x_2...x_k]

为什么编程需要懂一点英语

例子：

输入：7处的值输出： 7的值为13.47

以下是牛顿除数差分插值方法的实现。

C++

// CPP program for implementing
// Newton divided difference formula
#include 
using namespace std;
  
// Function to find the product term
float proterm(int i, float value, float x[])
{
    float pro = 1;
    for (int j = 0; j < i; j++) {
        pro = pro * (value - x[j]);
    }
    return pro;
}
  
// Function for calculating
// divided difference table
void dividedDiffTable(float x[], float y[][10], int n)
{
    for (int i = 1; i < n; i++) {
        for (int j = 0; j < n - i; j++) {
            y[j][i] = (y[j][i - 1] - y[j + 1]
                         [i - 1]) / (x[j] - x[i + j]);
        }
    }
}
  
// Function for applying Newton's
// divided difference formula
float applyFormula(float value, float x[],
                   float y[][10], int n)
{
    float sum = y[0][0];
  
    for (int i = 1; i < n; i++) {
      sum = sum + (proterm(i, value, x) * y[0][i]);
    }
    return sum;
}
  
// Function for displaying 
// divided difference table
void printDiffTable(float y[][10],int n)
{
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n - i; j++) {
            cout << setprecision(4) << 
                                 y[i][j] << "\t ";
        }
        cout << "\n";
    }
}
  
// Driver Function
int main()
{
    // number of inputs given
    int n = 4;
    float value, sum, y[10][10];
    float x[] = { 5, 6, 9, 11 };
  
    // y[][] is used for divided difference
    // table where y[][0] is used for input
    y[0][0] = 12;
    y[1][0] = 13;
    y[2][0] = 14;
    y[3][0] = 16;
  
    // calculating divided difference table
    dividedDiffTable(x, y, n);
  
    // displaying divided difference table
    printDiffTable(y,n);
  
    // value to be interpolated
    value = 7;
  
    // printing the value
    cout << "\nValue at " << value << " is "
               << applyFormula(value, x, y, n) << endl;
    return 0;
}

Java

// Java program for implementing
// Newton divided difference formula
import java.text.*;
import java.math.*;
  
class GFG{
// Function to find the product term
static float proterm(int i, float value, float x[])
{
    float pro = 1;
    for (int j = 0; j < i; j++) {
        pro = pro * (value - x[j]);
    }
    return pro;
}
  
// Function for calculating
// divided difference table
static void dividedDiffTable(float x[], float y[][], int n)
{
    for (int i = 1; i < n; i++) {
        for (int j = 0; j < n - i; j++) {
            y[j][i] = (y[j][i - 1] - y[j + 1]
                        [i - 1]) / (x[j] - x[i + j]);
        }
    }
}
  
// Function for applying Newton's
// divided difference formula
static float applyFormula(float value, float x[],
                float y[][], int n)
{
    float sum = y[0][0];
  
    for (int i = 1; i < n; i++) {
    sum = sum + (proterm(i, value, x) * y[0][i]);
    }
    return sum;
}
  
// Function for displaying 
// divided difference table
static void printDiffTable(float y[][],int n)
{
    DecimalFormat df = new DecimalFormat("#.####");
    df.setRoundingMode(RoundingMode.HALF_UP);
      
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n - i; j++) {
            String str1 = df.format(y[i][j]);
            System.out.print(str1+"\t ");
        }
        System.out.println("");
    }
}
  
// Driver Function
public static void main(String[] args)
{
    // number of inputs given
    int n = 4;
    float value, sum;
    float y[][]=new float[10][10];
    float x[] = { 5, 6, 9, 11 };
  
    // y[][] is used for divided difference
    // table where y[][0] is used for input
    y[0][0] = 12;
    y[1][0] = 13;
    y[2][0] = 14;
    y[3][0] = 16;
  
    // calculating divided difference table
    dividedDiffTable(x, y, n);
  
    // displaying divided difference table
    printDiffTable(y,n);
  
    // value to be interpolated
    value = 7;
  
    // printing the value
    DecimalFormat df = new DecimalFormat("#.##");
    df.setRoundingMode(RoundingMode.HALF_UP);
      
    System.out.println("\nValue at "+df.format(value)+" is "
            +df.format(applyFormula(value, x, y, n)));
}
}
// This code is contributed by mits

Python3

# Python3 program for implementing 
# Newton divided difference formula 
  
# Function to find the product term 
def proterm(i, value, x): 
    pro = 1; 
    for j in range(i): 
        pro = pro * (value - x[j]); 
    return pro; 
  
# Function for calculating 
# divided difference table 
def dividedDiffTable(x, y, n):
  
    for i in range(1, n): 
        for j in range(n - i): 
            y[j][i] = ((y[j][i - 1] - y[j + 1][i - 1]) /
                                     (x[j] - x[i + j]));
    return y;
  
# Function for applying Newton's 
# divided difference formula 
def applyFormula(value, x, y, n): 
  
    sum = y[0][0]; 
  
    for i in range(1, n):
        sum = sum + (proterm(i, value, x) * y[0][i]); 
      
    return sum; 
  
# Function for displaying divided 
# difference table 
def printDiffTable(y, n): 
  
    for i in range(n): 
        for j in range(n - i): 
            print(round(y[i][j], 4), "\t", 
                               end = " "); 
  
        print(""); 
  
# Driver Code
  
# number of inputs given 
n = 4; 
y = [[0 for i in range(10)] 
        for j in range(10)]; 
x = [ 5, 6, 9, 11 ]; 
  
# y[][] is used for divided difference 
# table where y[][0] is used for input 
y[0][0] = 12; 
y[1][0] = 13; 
y[2][0] = 14; 
y[3][0] = 16; 
  
# calculating divided difference table 
y=dividedDiffTable(x, y, n); 
  
# displaying divided difference table 
printDiffTable(y, n); 
  
# value to be interpolated 
value = 7; 
  
# printing the value 
print("\nValue at", value, "is",
        round(applyFormula(value, x, y, n), 2))
  
# This code is contributed by mits

C#

// C# program for implementing 
// Newton divided difference formula 
using System;
  
class GFG{ 
// Function to find the product term 
static float proterm(int i, float value, float[] x) 
{ 
    float pro = 1; 
    for (int j = 0; j < i; j++) { 
        pro = pro * (value - x[j]); 
    } 
    return pro; 
} 
  
// Function for calculating 
// divided difference table 
static void dividedDiffTable(float[] x, float[,] y, int n) 
{ 
    for (int i = 1; i < n; i++) { 
        for (int j = 0; j < n - i; j++) { 
            y[j,i] = (y[j,i - 1] - y[j + 1,i - 1]) / (x[j] - x[i + j]); 
        } 
    } 
} 
  
// Function for applying Newton's 
// divided difference formula 
static float applyFormula(float value, float[] x, 
                float[,] y, int n) 
{ 
    float sum = y[0,0]; 
  
    for (int i = 1; i < n; i++) { 
    sum = sum + (proterm(i, value, x) * y[0,i]); 
    } 
    return sum; 
} 
  
// Function for displaying 
// divided difference table 
static void printDiffTable(float[,] y,int n) 
{ 
    for (int i = 0; i < n; i++) { 
        for (int j = 0; j < n - i; j++) { 
            Console.Write(Math.Round(y[i,j],4)+"\t "); 
        } 
        Console.WriteLine(""); 
    } 
} 
  
// Driver Function 
public static void Main() 
{ 
    // number of inputs given 
    int n = 4; 
    float value; 
    float[,] y=new float[10,10]; 
    float[] x = { 5, 6, 9, 11 }; 
  
    // y[][] is used for divided difference 
    // table where y[][0] is used for input 
    y[0,0] = 12; 
    y[1,0] = 13; 
    y[2,0] = 14; 
    y[3,0] = 16; 
  
    // calculating divided difference table 
    dividedDiffTable(x, y, n); 
  
    // displaying divided difference table 
    printDiffTable(y,n); 
  
    // value to be interpolated 
    value = 7; 
  
    // printing the value 
      
    Console.WriteLine("\nValue at "+(value)+" is "
            +Math.Round(applyFormula(value, x, y, n),2)); 
} 
} 
// This code is contributed by mits

PHP

输出：

12     1     -0.1667     0.05     
13     0.3333     0.1333     
14     1     
16     

Value at 7 is 13.47