3. Independently set unequally spaced interpolation nodes.

   1. Construct interpolation polynomials using Lagrange's interpolation formula, Newton's interpolation formula for unequal interpolation nodes.

• Lagrange's interpolation formula:

1. Construct a table of 4 unequal interpolation nodes.

x
y = f(x)

2. Substitute the table values into the formula.

3. The obtained function is

• Newton's interpolation formula for unequal interpolation nodes:

1. The formula:

+ … +

2. Write the formula for our case of 4 nodes:

3. Substitute the values into the formula:

x
y = f(x)

4. The obtained function is

3.2 Calculate the value of the given function y = f(x), Lagrange polynomial and Newton polynomials at interpolation nodes x = 1, 3, 7, 13 and intermediate points x = 2, 5, 9 on each partial segment of interpolation.

1. The value of function :

1. The value of Lagrange’s polynomial

2. The value of Lagrange’s polynomial

3. Table for each obtained value.

x	Function	Lagrange's	Newton's
1	1.60944	1.57	1.6
2	4.96981	4.92	5.03
3	9.13357	8.87	9.1
5	19.03331	18.33	18.92
7	30.40664	29.47	30.58
9	42.85957	41.81	43.6
13	70.17612	68.17	71.8

3.3 Calculate the absolute errors. Let Function = , Lagrange’s polynomial = and Newton’s polynomial = , then Lagrange polynomial errors at the given points are

and Newton’s polynomial’s errors

f(x)	L(x)	∆L(x)	N(x)	∆N(x)
1.609438	1.57	0.03944	1.6	0.009438
4.969813	4.92	0.04981	5.03	0.060187
9.133567	8.87	0.26357	9.1	0.033567
19.03331	18.33	0.70331	18.92	0.113312
30.40664	29.47	0.93664	30.58	0.173362
42.85957	41.81	1.04957	43.6	0.740435
70.17612	68.17	2.00612	71.8	1.623885

4. Plot the graphs of the given function , the Lagrange polynomial and Newton polynomial in the same coordinate system.

Conclusion: During the work we performed a lot of calculations to interpolation the given function. We first constructed them at equally spaced nodes, using Lagrange’s, Newton’s first and second and spline formulas and then set unequally spaced nodes. All of them gave us appropriate approximate result. But, by finding errors of these polynomials we can conclude several things: 1. Taking equally spaced nodes is surely more precise than taking unequally spaced because the function values begin to range greater and errors between calculation rise, 2. The rounding errors dominate when working with very small values and when taking equally spaced nodes you don’t have to round a lot of values so in the end you obtain that Lagrange’s polynomial equals to Newton’s, they have different formulas but same result, so the errors are as well the same. 3. Finally, considering all the obtained errors and graphs we can say that for equally spaced nodes spline is the most precise option, but as for unequally spaced nodes the Newton’s interpolation is better – the errors are smaller the values are better.