Find the polynomial which takes the following values given f(0) = -1 , f(1)=1 and f(2)=4 using the Newton's interpolating formula
1/2 x(x-1) + 2(x-1) + 1
Writing
To find the polynomial using Newton's interpolating formula, we need to first find the divided difference table.
The divided difference table is a table that helps us calculate the coefficients of the polynomial. Here's how to calculate it:
Step 1: Write down the given values of x and f(x) in a table:
x | f(x)
--------------
0 | -1
1 | 1
2 | 4
Step 2: Calculate the divided differences. The first divided difference is simply the f(x) values in the table. The second divided differences can be calculated using the formula:
f[x1,x2] = (f(x2) - f(x1)) / (x2 - x1)
Using this formula, we have:
f[0,1] = (1 - (-1)) / (1 - 0) = 2
f[1,2] = (4 - 1) / (2 - 1) = 3
Step 3: Calculate the higher-order divided differences using the formula:
f[x1,x2,x3] = (f[x2,x3] - f[x1,x2]) / (x3 - x1)
Using this formula, we have:
f[0,1,2] = (3 - 2) / (2 - 0) = 0.5
Step 4: Arrange the divided differences in a table:
x | f(x) | f[x1,x2] | f[x1,x2,x3]
-----------------------------------
0 | -1 |
1 | 1 | 2
2 | 4 | 3 | 0.5
Step 5: Write down the polynomial using Newton's interpolating formula:
f(x) = f[x0] + f[x0,x1] * (x-x0) + f[x0,x1,x2] * (x-x0)*(x-x1) + ...
Substituting the values from the table, we get:
f(x) = -1 + 2 * (x-0) + 0.5 * (x-0)*(x-1)
Simplifying, we find the polynomial:
f(x) = -1 + 2x + 0.5x^2
Therefore, the polynomial that takes the given values is f(x) = -1 + 2x + 0.5x^2.
To find the polynomial using Newton's interpolating formula, follow these steps:
Step 1: Write down the Newton's interpolating formula:
f(x) = f(x0) + (x - x0)f[x0, x1] + (x - x0)(x - x1)f[x0, x1, x2] + ...
Step 2: Fill in the given values:
Using the given values f(0) = -1, f(1) = 1, f(2) = 4, we have:
f(x) = -1 + (x - 0)f[0, 1] + (x - 0)(x - 1)f[0, 1, 2] + ...
Step 3: Determine the divided differences:
The divided differences, denoted as f[x0, x1], f[x0, x1, x2], etc., can be found using the following formula:
f[x0, x1] = (f(x1) - f(x0)) / (x1 - x0)
f[x0, x1, x2] = (f[x1, x2] - f[x0, x1]) / (x2 - x0)
and so on.
For our problem, we can calculate:
f[0, 1] = (f(1) - f(0)) / (1 - 0) = (1 - (-1)) / (1 - 0) = 2/1 = 2
f[0, 1, 2] = (f[1, 2] - f[0, 1]) / (2 - 0) = (4 - 2) / (2 - 0) = 2/2 = 1
Step 4: Substitute the divided differences into the Newton's interpolating formula:
Now we can substitute the calculated divided differences into the formula from Step 2:
f(x) = -1 + (x - 0) * 2 + (x - 0)(x - 1) * 1 + ...
Simplifying and expanding, we get:
f(x) = -1 + 2x + x(x - 1) + ...
Step 5: Simplify the polynomial:
To simplify further, expand and collect like terms:
f(x) = -1 + 2x + x^2 - x + ...
Combining like terms, we have:
f(x) = x^2 + x - 1
Therefore, the polynomial that takes the given values f(0) = -1, f(1) = 1, and f(2) = 4 is f(x) = x^2 + x - 1.