Develop cubic splines for the data in Prob. 18.6 and (a) predict f (4) and f (2.5) and (b) verify that
(a)
To calculate: The value of
Answer to Problem 14P
Solution:
Explanation of Solution
Given Information:
Write the data points mentioned in the Prob. 18.6.
1 | 2 | 3 | 5 | 7 | 8 | |
3 | 6 | 19 | 99 | 291 | 444 |
Formula Used:
A third order polynomial is derived for each interval between data points in cubic splines.
Write the expression of polynomial for each interval from Eq.18.36.
Here, 4n conditions are required to evaluate 4n unknowns.
Write the equations to calculate the unknowns,
Calculation:
Recall the table mentioned in the Prob. 18.6.
1 | 2 | 3 | 5 | 7 | 8 | |
3 | 6 | 19 | 99 | 291 | 444 |
From the table the value is,
Recall the equations to calculate the unknowns,
Write the equation(1) for
Substitute all the above values.
Because of natural spline condition
Similarly, equation(1) can be write for all interior knot
Write all the above equations in matrix form.
Solve the equations by using the MATLAB,
Write the following codes in MATLAB.
%
The values are,
Recall the polynomial expression for each interval,
Write the equation (2) for
Substitute the all values.
Again, substitute all the values.
Similarly solve for the
Calculate the value of
Substitute
Calculate the value of
Substitute
Therefore, the value
(b)
To show: The value of
Answer to Problem 14P
Solution:
Explanation of Solution
Given Information:
Calculation:
Calculate the value of
Recall the expression of
Substitute
Calculate the value of
Substitute
Thus,
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