例題1-8 已知5點的座標值 使用牛頓內插法求 差除表

多項式內插法

例題1-8 已知5點的座標值 使用牛頓內插法求 差除表


已知5點的座標值

n=5

  x         f(x)
=============
0.0         0.0
1.1        -3.0
2.0         0.0
3.0        15.0
4.0        48.0
=============

'''
/* ex1-8.py is for developing the divided-defference
 * table for Newton Interpolation polynomial.
 */
'''
n=5
x= [0.0 for i in range(0,n)]     #x [n] 矩陣
f=  [ [0.0 for i in range(n)]  for j in range(n) ]       #[n] * [n] 矩陣
x= [0.0 , 1.0 , 2.0 ,3.0 ,4.0]
f= [[0.0 ,   -3.0   ,0.0  ,15.0 ,48.0  ,0.0  ],
      [0.0 ,     0.0   ,0.0  ,0.0 ,0.0 ,0.0 ],
      [0.0 ,     0.0   ,0.0  ,0.0 ,0.0 ,0.0 ],
      [0.0 ,     0.0   ,0.0  ,0.0 ,0.0 ,0.0 ],
      [0.0 ,     0.0   ,0.0  ,0.0 ,0.0 ,0.0 ],
      [0.0 ,     0.0   ,0.0  ,0.0 ,0.0 ,0.0 ]]

print("\nNewton's Divided Interpolation : ")
print("\nEnter the value of n : ",n)
print("\nEnter the value of x[i] and f[i][0] : \n");
print("\tx[i]\t\tf[i][0]");
for i  in range (len(x)) :   
    print(  '\t',round(x[i],4),"\t\t",round( f[0][i],4))
print()

print("\n        Divided Difference Table:")
print(" =================================="  )

for i in range (1 ,n):
    for j in range (0 , n-i):
        f[i][j]=(f[i-1][j+1]-f[i-1][j]) / (x[i+j]-x[j])

print("i\tx(i)\tf(i)\tf(i,i+1)\tf(i,i+1.i+2),  ......\n");
for i in range (0 ,n):
    print ("{%d}\t{%2.1f}" %(i,x[i]), end='')
    for j in range (0 , n-i):
        print("\t{%2.1f}" %(f[j][i]) ,end='')
    print("\n")

===== RESTART: F:/2018-09勤益科大數值分析/數值分析/PYTHON/EX1-8-1.py ============

Newton's Divided Interpolation : 

Enter the value of n :  5

Enter the value of x[i] and f[i][0] : 

x[i] f[i][0]
0.0 0.0
1.0 -3.0
2.0 0.0
3.0 15.0
4.0 48.0


        Divided Difference Table:
 ==================================
   i   x(i) f(i)         f(i,i+1) f(i,i+1.i+2),  ......

{0} {0.0} {0.0} {-3.0} {3.0} {1.0} {0.0}

{1} {1.0} {-3.0} {3.0} {6.0} {1.0}

{2} {2.0} {0.0} {15.0} {9.0}

{3} {3.0} {15.0} {33.0}

{4} {4.0} {48.0}

>>> 

P(x) = 0.0 + -3.0 (x-0.0)  + 3 (x-0.0) (x-1.0) + 1.0 (x-0.0) (x-1.0)(x-2.) 
           + 0.0  (x-0.0) (x-1.0)(x-2.0) (x-3.0)

P(x)= x^3 + 4x 

P(1.5) = -2.625
P(2.5) = 5.625