Assigned: Thursday Nov. 7, 2013
Due Date: Thursday Nov. 14, 2013
MATH 373: Homework 7
“Interpolation III”
Fall 2013
NOTE: For each homework assignment observe the following guidelines:
• Include a cover page.
• Always clearly label all plots (title, x-label, y-label, and legend).
• Use the subplot command from MATLAB when comparing 2 or more plots to make
comparisons easier and to save paper.
1. Explicitly find all the uknown coefficients in the following expression to make it a
cubic spline that satisfies the not-a-knot condition:


16 + b1 (x + 2) + c1 (x + 2)2 + d1 (x + 2)3
−2 ≤ x ≤ −1,



1 + b (x + 1) + c (x + 1)2 + d (x + 1)3
−1 ≤ x ≤ 0,
2
2
2
s(x) =
2
3

c3 x + d3 x
0 ≤ x ≤ 1,



1 + b (x − 1) + c (x − 1)2 + d (x − 1)3
1 ≤ x ≤ 2.
4
4
4
2. SOURCE CODE:
Write the following functions:
• x = trisolver(u,v,w,rhs)
Tridiagonal solver, where u is the main diagonal, v is the lower
diagonal, and w is the upper diagonal of the coefficient matrix.
rhs is the right-hand side in the linear system. x is the solution
to the linear system.
• [a,b,c,d] = cubic spline(x,f)
Function to construct the coefficients a, b, c, and d in the cubic
spline approximation.
• sbar = eval cubic spline(x,a,b,c,d,xbar)
Function to evaluate the cubic spline at the points xbar.
In order to debug your code. Let me suggest the following test for each of the
functions:
• x = trisolver(u,v,w,rhs): Use your tri-diagonal solver to solve the following
6 × 6 linear system:

  

   
1.2857142857142856
x1
7
x1
6 −1
x2  0.7142857142857142
−2 5 0
 x2  1
  


   


x3  1.3571428571428572




5
x
−1
4
1
3


   =   =⇒   = 
x4  0.2857142857142857 .

   
0 3 2 
  


 x4  1
x5  0.0714285714285715

1 2 3 x5  3
2 1
x6
1
1
x6
0.8571428571428571
Assigned: Thursday Nov. 7, 2013
Due Date: Thursday Nov. 14, 2013
• [a,b,c,d] = cubic spline(x,f): Consider the following data:
x = [−2; −1; 0; 1; 2];
f = [0; −1; 2; −3; 4];
Your results should be

 

0
−9.666666666666666
−1
 4.333333333333333 
 





a =  2 , b = 
−1.666666666666667 ,
−3
−3.666666666666666
4
0.000000000000000

12
2
 

c=
−8 ,
6
20



−3.333333333333333
−3.333333333333333



d=
 4.666666666666667  .
 4.666666666666667 
0.000000000000000
• sbar = eval cubic spline(x,a,b,c,d,xbar): Use the above values of x, a,
b, c, and d to evaluate the cubic spline at the points:
xbar = transpose(linspace(−2, 2, 21));
Your results should be

0.000000000000000
−1.480000000000000


−2.160000000000000
−2.200000000000000


−1.759999999999999


−1.000000000000000


−0.079999999999999


 0.840000000000001 


 1.600000000000000 


 2.040000000000000 



sbar = 
 2.000000000000000 
 1.383999999999999 


 0.351999999999998 


−0.872000000000000


−2.064000000000001


−3.000000000000000


−3.456000000000000


−3.207999999999998


−2.031999999999999
 0.296000000000005 

4.000000000000001
3. The following table gives the viscosity, in mili-Pascal-seconds (centipoises) of sulfuric
acid as a function of concentration, in mass percent.
Concentration
Viscosity
0
0.89
20
1.40
40
2.51
60
5.37
80
17.4
100
24.2
(a) Use your code to construct the cubic spline interpolant for this data.
(b) Plot the cubic spline interpolant (also add the data points from the table on
this plot).
(c) The viscosity of sulfuric acid with a 5% concentration is 1.01 and with a 10%
concentration is 1.12 Use these values to asses the accuracy of the cubic spline
interpolant.
4. Consider the following values for the sound speed, a (meters/second), in water as a
function of temperature T (Celcius).
T (Celcius)
a (meters/second)
0
1402
10
1447
20
1482
2
30
1509
40
1529
50
1542
60
1551
70
1553
80
1554
90
1550
100
1543
Assigned: Thursday Nov. 7, 2013
Due Date: Thursday Nov. 14, 2013
(a) Use your code to construct the cubic spline interpolant for this data.
(b) Plot the cubic spline interpolant (also add the data points from the table on
this plot).
(c) Use your interpolant to determine the sound speed in water when T = 34,
T = 68, T = 86, and T = 91.
3