the data in Table 2.8 to computeusing and .20 Use the data in Table 2.7 to computeusing and .Solution: Using we getusing , we get
21 Show that the trapezoid rule is exact for all linear polynomials.
22 Prove Theorem 2.4.Solution: Let ; then we can apply Theorem 2.2 to get thatwhere . Therefore,
23 Extend the discussion on stability to include changes in the interval of integration instead of changes in the function. State and prove a theorem that bounds the change in the trapezoid rule approximation towhen the upper limit of integration changes from to , but remains the same.
24 Consider a function which is the floating point representation of ; thus is the rounding error in computing . If we assume that for all , show thatWhat does this say about the effects of rounding error on the trapezoid rule?Solution:sowhere . The result follows by simple manipulation.
25 We will end this section with a few exercises designed to illustrate the trapz command. Strictly speaking, trapz simply does a specialized summation. If S = trapz(Y) , thenwhich is the trapezoid rule with . To do an actual trapezoid rule for a uniform discretization, just multiply by ; for , we havewhere f is the vector of values.For each of the following integrals, use trapz to compute an approximate value.Apply the trapezoid rule with to approximate the integralCompare what you got here to what you got in Problem 6.Solution: The following script will do the job. I got trapy = 0.385643909952095 ; in Problem 6 we got 0.3856439099.N = 8; h = 1/N; x = [0:N]/N; y = log(1 + x); S = trapz(y); trapy = h*SRepeat the above for .Solution: If we set N = 100 in the above script, we getand the error is now much less than before.Apply the trapezoid rule with to approximate the integralCompare what you got here to what you got in Problem 7.Repeat the above for .Apply the trapezoid rule with to approximate the integralCompare what you got here to what you got in Problem 8.Repeat the above for .
2.6 SOLUTION OF TRIDIAGONAL LINEAR SYSTEMS
Exercises:
1 Use the tridiagonal algorithm in this section to compute the solution to the following system of equations:Solution: After the elimination step is completed, the triangular system isand the solution is
2 Write a computer code to solve the previous problemSolution: The following is a MATLAB script that produced the previous solutionfunction [delta, f, x] = trisol(l,d,u,b) n = length(d); x = zeros(n,1); for k=2:n d(k) = d(k) - u(k-1)*l(k)/d(k-1); b(k) = b(k) - b(k-1)*l(k)/d(k-1); end x(n) = b(n)/d(n); for k=(n-1):(-1):1 x(k) = (b(k) - u(k)*x(k+1))/d(k); end delta = d; f = b;
3 Use the algorithm of this section to solve the following system of equations:You should get the solution .
4 Write a computer code to solve the previous problem.
5 The diagonal dominance condition is an example of a sufficient but not necessary condition. That is, the algorithm will often work for systems that are not diagonally dominant. Show that the following system is not diagonally dominant, but then use the tridiagonal algorithm in this section to compute the solution to it:You should get the solution .Solution: The matrix fails to be diagonally dominant because of the values in the second, third, and fourth rows. However, if we apply the algorithm, we get the triangular systemand then the correct values for the solution, . (It always helps to type in the correct right‐side vector when checking these things.)
6 Use the tridiagonal algorithm in this section to compute the solution to the following system of equations:Note that this is a very small change from the previous problem, since the only difference is that has changed by only . How much has the answer changed?Solution: The solution is nowConsidering the small change in the problem, this is a substantial change in the solution.
7 Write a computer code to do the previous two problems.
8 Verify that the following system is diagonally dominant and use the algorithm of this section to find the solution.
9 Use the algorithm of this section to find the solution to this system:Note that the right side here is different from that in the previous problem by only a small amount in the component. Comment on your results here as compared to those in the previous problem.Solution: The solution is nowAgain, this is a very large change in the solution for so modest a change in the problem.
10 Write a computer code to do the previous two problems.
11 Write a code that carries out the tridiagonal solution algorithm, and test it on the following system of equations,where is withand for all . Check your results by computing the residual . What is the largest component (in absolute value) of ? (You could also check your results by using MATLAB's backslash operator to solve the system.)
12 Extend the tridiagonal algorithm to a pentadiagonal matrix, i.e., one with five non‐zero diagonals. Write a program to carry out this solution algorithm, and apply it to the systemCheck your results by again computing the residual vector, or by using MATLAB's backslash operation.Solution: A MATLAB script for doing this is given below. On the example in the exercise it returns the (exact) solution .function [delta, f, x] = pentasol(k,l,d,u,v,b) n = length(d); x = zeros(n,1); d(2) = d(2) - l(2)*u(1)/d(1); u(2) = u(2) - l(2)*v(1)/d(1); b(2) = b(2) - l(2)*b(1)/d(1); for j=3:n l(j) = l(j) - k(3)*u(j-2)/d(j-2); d(j) = d(j) - k(j)*v(j-2)/d(j-2); b(j) = b(j) - k(j)*b(j-2)/d(j-2); d(j) = d(j) - l(j)*u(j-1)/d(j-1); if j < n u(j) = u(j) - l(j)*v(j-1)/d(j-1); end b(j) = b(j) - l(j)*b(j-1)/d(j-1); end delta = d; f = b; x = f; % x(n) = f(n)/d(n); x(n-1) = (b(n-1) - u(n-1)*x(n))/d(n-1); for j=(n-2):(-1):1 x(j) = (b(j) - u(j)*x(j+1) - v(j)*x(j+2))/d(j) end
13 Consider the family of tridiagonal problems defined by the matrix , withand a randomly defined right‐hand‐side vector. (Use rand to generate the random vectors.) Solve the system over the range ; use the appropriate timing functions to estimate the CPU time required for each case, and plot the result as a function of . Comment on your results. (You will probably want to use a semilog plot.)Solution: I got the plot in Fig. 2.5. While this is a “noisy” plot, it is very much generally a logarithmic curve, which suggests the actual timing numbers are a straight line, which is what we would expect, given that the cost of doing the solution is linear in the size of the matrix.Figure 2.5 Estimated timing cost for Problem 13, semilog scale.
2.7 APPLICATION: SIMPLE TWO‐POINT BOUNDARY VALUE PROBLEMS
Exercises:
1 Solve, by hand, the two‐point BVP (2.28) and (2.29) when , using . Write out the linear system explicitly prior to solution. You should get the following system:Solution: The approximate solution is
2 Repeat the above, this time using . What is the system now?
3 Repeat it again, this time using . What is the system now? (For this problem, you probably will want to use a computer code to actually solve the system.)Solution: The approximate solution is
4 Write a program which solves the two‐point BVP (2.28) and (2.29) where is as given below:;;;.The exact solutions are as given. Using , do we get the same kind of accuracy as in Table 2.10? Explain why or why not.Solution: The approximate solutions will display the kind of accuracy suggested in the text, except for (d); in this case, the singularity in the logarithm function in the solution affects the accuracy of the approximation.
5 Try to apply the ideas of this section to approximating the solution of the two point boundary value problemCan we get
