Question
. Let f(x)=x~ln(x) be defined at x_{0}=2 x_{1}=2.5 , x_{2}=3.3 , ,x_{3}=4
a). Construct the divided difference table for f(x) with respect to these points
a). Construct the divided difference table for f(x) with respect to these points
Answers
GPT 3.5
To construct a divided difference table for f(x) with respect to these points, we need to calculate the divided differences for each combination of points.
First, let's list the given points and their corresponding function values:
x0 = 2, f(x0) = 2 * ln(2)
x1 = 2.5, f(x1) = 2.5 * ln(2.5)
x2 = 3.3, f(x2) = 3.3 * ln(3.3)
x3 = 4, f(x3) = 4 * ln(4)
Now, let's calculate the divided differences using the following formula:
f[xi] = f(xi) for i = 0, 1, 2, 3
f[xi, xi+1] = (f(xi+1) - f(xi)) / (xi+1 - xi) for i = 0, 1, 2
f[xi, xi+1, xi+2] = (f[xi+1, xi+2] - f[xi, xi+1]) / (xi+2 - xi) for i = 0, 1
f[xi, xi+1, xi+2, xi+3] = (f[xi+1, xi+2, xi+3] - f[xi, xi+1, xi+2]) / (xi+3 - xi) for i = 0
Now, let's calculate the divided differences:
f[x0] = f(2) = 2 * ln(2)
f[x1] = f(2.5) = 2.5 * ln(2.5)
f[x2] = f(3.3) = 3.3 * ln(3.3)
f[x3] = f(4) = 4 * ln(4)
f[x0, x1] = (f[x1] - f[x0]) / (x1 - x0)
= (f(2.5) - f(2)) / (2.5 - 2)
= (2.5 * ln(2.5) - 2 * ln(2)) / 0.5
f[x1, x2] = (f[x2] - f[x1]) / (x2 - x1)
= (f(3.3) - f(2.5)) / (3.3 - 2.5)
= (3.3 * ln(3.3) - 2.5 * ln(2.5)) / 0.8
f[x2, x3] = (f[x3] - f[x2]) / (x3 - x2)
= (f(4) - f(3.3)) / (4 - 3.3)
= (4 * ln(4) - 3.3 * ln(3.3)) / 0.7
f[x0, x1, x2] = (f[x1, x2] - f[x0, x1]) / (x2 - x0)
= (f[x1, x2] - f[x0, x1]) / (2.5 - 2)
= ((3.3 * ln(3.3) - 2.5 * ln(2.5)) / 0.8 - (2.5 * ln(2.5) - 2 * ln(2))) / 0.5
f[x1, x2, x3] = (f[x2, x3] - f[x1, x2]) / (x3 - x1)
= (f[x2, x3] - f[x1, x2]) / (3.3 - 2.5)
= ((4 * ln(4) - 3.3 * ln(3.3)) / 0.7 - (3.3 * ln(3.3) - 2.5 * ln(2.5))) / 0.8
f[x0, x1, x2, x3] = (f[x1, x2, x3] - f[x0, x1, x2]) / (x3 - x0)
= (f[x1, x2, x3] - f[x0, x1, x2]) / (4 - 2)
= ((f[x2, x3] - f[x1, x2]) / (3.3 - 2.5) - ((f[x1, x2] - f[x0, x1]) / (2.5 - 2))) / (4 - 2)
Now, using the above calculations, we can fill out the divided difference table:
--------------------------------
| x | f(x) | Divided Differences |
----------------------------------------------
|x0 = 2 |2ln(2) | - |
|x1 =2.5|2.5ln(2.5) |0.4990 |
|x2 =3.3|3.3ln(3.3) |1.1057 | 0.8264 |
|x3 = 4 |4ln(4) |1.3863 | 0.7623 | 0.3699
First, let's list the given points and their corresponding function values:
x0 = 2, f(x0) = 2 * ln(2)
x1 = 2.5, f(x1) = 2.5 * ln(2.5)
x2 = 3.3, f(x2) = 3.3 * ln(3.3)
x3 = 4, f(x3) = 4 * ln(4)
Now, let's calculate the divided differences using the following formula:
f[xi] = f(xi) for i = 0, 1, 2, 3
f[xi, xi+1] = (f(xi+1) - f(xi)) / (xi+1 - xi) for i = 0, 1, 2
f[xi, xi+1, xi+2] = (f[xi+1, xi+2] - f[xi, xi+1]) / (xi+2 - xi) for i = 0, 1
f[xi, xi+1, xi+2, xi+3] = (f[xi+1, xi+2, xi+3] - f[xi, xi+1, xi+2]) / (xi+3 - xi) for i = 0
Now, let's calculate the divided differences:
f[x0] = f(2) = 2 * ln(2)
f[x1] = f(2.5) = 2.5 * ln(2.5)
f[x2] = f(3.3) = 3.3 * ln(3.3)
f[x3] = f(4) = 4 * ln(4)
f[x0, x1] = (f[x1] - f[x0]) / (x1 - x0)
= (f(2.5) - f(2)) / (2.5 - 2)
= (2.5 * ln(2.5) - 2 * ln(2)) / 0.5
f[x1, x2] = (f[x2] - f[x1]) / (x2 - x1)
= (f(3.3) - f(2.5)) / (3.3 - 2.5)
= (3.3 * ln(3.3) - 2.5 * ln(2.5)) / 0.8
f[x2, x3] = (f[x3] - f[x2]) / (x3 - x2)
= (f(4) - f(3.3)) / (4 - 3.3)
= (4 * ln(4) - 3.3 * ln(3.3)) / 0.7
f[x0, x1, x2] = (f[x1, x2] - f[x0, x1]) / (x2 - x0)
= (f[x1, x2] - f[x0, x1]) / (2.5 - 2)
= ((3.3 * ln(3.3) - 2.5 * ln(2.5)) / 0.8 - (2.5 * ln(2.5) - 2 * ln(2))) / 0.5
f[x1, x2, x3] = (f[x2, x3] - f[x1, x2]) / (x3 - x1)
= (f[x2, x3] - f[x1, x2]) / (3.3 - 2.5)
= ((4 * ln(4) - 3.3 * ln(3.3)) / 0.7 - (3.3 * ln(3.3) - 2.5 * ln(2.5))) / 0.8
f[x0, x1, x2, x3] = (f[x1, x2, x3] - f[x0, x1, x2]) / (x3 - x0)
= (f[x1, x2, x3] - f[x0, x1, x2]) / (4 - 2)
= ((f[x2, x3] - f[x1, x2]) / (3.3 - 2.5) - ((f[x1, x2] - f[x0, x1]) / (2.5 - 2))) / (4 - 2)
Now, using the above calculations, we can fill out the divided difference table:
--------------------------------
| x | f(x) | Divided Differences |
----------------------------------------------
|x0 = 2 |2ln(2) | - |
|x1 =2.5|2.5ln(2.5) |0.4990 |
|x2 =3.3|3.3ln(3.3) |1.1057 | 0.8264 |
|x3 = 4 |4ln(4) |1.3863 | 0.7623 | 0.3699