UNIT-II INTERPOLATION & APPROXIMATION
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1 UNIT-II INTERPOLATION & APPROXIMATION LAGRANGE POLYNAMIAL 1. Find the polynomial by using Lagrange s formula and hence find for : : : : Lagrange s interpolation formula, we have To find 2. Using Lagrange s interpolation formula, calculate the profit in the 2000 year from the following data : Given the data s are :
2 Lagrange s interpolation formula, we have To find the profit in the year 2000 Hence the profit in the year 2000 is Using Lagrange s formula find from the following data. : : : : Lagrange s interpolation formula, we have 2
3 To fond 4. Find the third degree polynomial satisfying the following data : : : : Lagrange s interpolation formula, we have 3
4 5. Using Lagrange s interpolation formula find given that. Given the data s are : : Lagrange s interpolation formula, we have 6. Using Lagrange s polynomial fit a polynomial for the following data 4
5 Lagrange s interpolation formula, we have 7. Find the missing term in the following table using Lagrange s interpolation. : : : : Lagrange s interpolation formula, we have 5
6 DIVIDED DIFFERNCES 1. Using Newton s divided difference formula, find given. By Newton s divided difference interpolation formula Here And 2. Find as a polynomial in for the following data by Newton s divided difference formula
7 By Newton s divided difference interpolation formula Here 3. Find by Newton s divided difference formula for the data,
8 By Newton s divided difference interpolation formula Here 4. Find by Newton s divided difference formula for the data, By Newton s divided difference interpolation formula 8
9 Here 5. Using Newton s divided difference formula, find the missing term in the following data By Newton s divided difference interpolation formula Here And NEWTONS FORWARD & BACKWARD INTERPOLATION FORMULA 9
10 Newton forward interpolation formula : Newton backward interpolation formula : 1. Using Newton s forward interpolation formula, find a polynomial satisfying the following data. Hence evaluate We form the difference table There are only four data are given. Hence the polynomial is of order 3 10
11 To find : Verification : Therefore the polynomial is correct. 2. Using Newton s forward formula, find a polynomial satisfying the following data. Hence find We form the difference table 11
12 There are only four data are given. Hence the polynomial is of order 3 To find : 12
13 Verification : 3. A Third degree polynomial passes through the points using Newton s forward interpolation formula find the polynomial. Hence evaluate the value at 1.5 Let us form the difference table There are only four data are given. Hence the polynomial is of order 3 13
14 Verification : 4. Using Newton s forward interpolation formula find the polynomial which takes places the values Evaluate using Newton s backward interpolation formula. Is it the same as obtained from the cubic polynomial found above. Let us form the difference table There are only four data are given. Hence the polynomial is of order 3 The Newton s forward interpolation formula is 14
15 To find : Verification : The Newton s backward interpolation formula is 15
16 Here To find : Therefore the cubic polynomial in both cases are correct. 5. Using Newton s backward formula find the interpolating polynomial of degree 3 for the data.. Hence find. Since. Let us form the difference table 16
17 There are only four data are given. Hence the polynomial is of order 3 The Newton s backward interpolation formula is Here To find : 17
18 Verification : Therefore the polynomial is correct. 6. From the following data, find the number of students whose weight is between 60 to We form the difference table Now let us calculate the no. of students whose The Newton s forward formula is Here Since 18
19 7. The following data are taken from the steam table Find the pressure at temperature. We form the difference table To find the pressure at temperature Let us use the Newton s forward formula : Here Since 19
20 To find the pressure at temperature Let us use the Newton s backward formula : Here 20
Then, Lagrange s interpolation formula is. 2. What is the Lagrange s interpolation formula to find, if three sets of values. are given.
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