Math. Math. 455-01, Spring, 1997 Assignments

After assignments are handed back, copies of the answers are placed on reserve in Dupré Library.

Lesson 1: (Due Tuesday, January 21)

• page 9: 1c, 2a, 3ac, 9

Lessons 1--2: (Due Thursday, January 23)

• page 10: 12
• page 14: 1a, 2a, 3abc, 4

Lesson 2: (Due Tuesday, January 28)

• p. 19: 1ab, 3
• Refer to the course handout An Introduction to Interval Arithmetic . Evaluate f as in problem 3b on page 19, but using four digit interval arithmetic, with x = [2.278,2.280]. Click here for an answer and an analysis.

Lesson 3: (Due Thursday, January 30)

• p. 19: 10b
• Supplementary lesson
  1. Study the indicated Fortran 90 program.
  2. Describe what the Fortran 90 program does.
  3. Describe the bit-structure of each of the printed floating point numbers, indicating what the sign bit, exponent number, exponent bias, and mantissa are.

Lesson 5: (Due Tuesday, February 4, 1997)

These may be done by hand, by using the programs that come with the book, or by writing your own Fortran 90 programs. (See the example Fortran 90 program for a simple Fortran 90 program for Newton's method that you may use.

1. page 29: 4, 8, 13
2. Do 4 on page 29, but using the secant method instead of the method of bisection.
3. Do 4 on page 29, but using Newton's method instead of the method of bisection.

Lesson 10: (Due Thursday, February 20)

• page 61: 1a, 5a, 6

Lesson 11: (Due Thursday, March 6)

• page 83:
  1. No. 12. Do using the program ncubsp34.for from the book, or other natural cubic spline program.
  2. Also do the same problem with a polynomial interpolant obtained with neville31.for or other polynomial interpolation program.
  3. Extra credit (15 points added to your last exam score): Draw graphs of both the polynomial interpolant and the spline interpolant over the interval of interest. Analyze these graphs carefully, stating both your reasoning and your conclusions.
• page 89: 3a (Do it by hand, and show all of your work.)

For Tuesday, March 11:

• p. 98: 1a, 2a, 3a, 4a, 5a, 6a. Also compute an enclosure to the integral in 1b using the Simpson's rule and bounding the error term with interval arithmetic.
• p. 105, no. 6.
• Compute the integral from 0 to 0.1 of cos(x) with a 2-point Gaussian quadrature rule. Bound the error in the formula. Also compute an enclosure to the solution by evaluating the error term with interval arithmetic. Carry out eight digits in these computations.

For Tuesday, March 18:

• Evaluate the integral from 0 to 1 of cos(x) to within an error of .0001, by hand. To do this, step through the adaptive quadrature procedure with Simpson's rule. You may use either the heuristic error estimate or the rigorous bound based on interval arithmetic.
• p. 124, no. 7. (You may use adapq43.for from the routines supplied with the text, quanc8.for from the Forsythe / Malcolm / Moler routines, or other adaptive quadrature routine, including one you write yourself.)

For Tuesday, March 25:

• Use the handout from March 18 (or the routines from the Examples given in class page) to complete a module for automatic differentiation that has +, -, *, /, **, and sin.
• Use the module to complete problems 2 and 3 of the handout Elements of Automatic Differentiation.

For Tuesday, April 1:

• pp. 159-160, nos. 3a and 4a.
• pp. 159-160, nos. 3a and 4a, except use a fourth order Runge Kutta method.
• p. 167, no. 14.

For Tuesday, April 15:

• p. 167, no. 14, but use the Runge-Kutta-Fehlberg 4-5 method with TOL=1E-2 and TOL=1E-4. Graph the iterates.
• pp. 193-194, nos. 2a and 6.
• Try solving #14, p. 167, but with k=6.22E-10, using an implicit Euler method, with 20, 100, and 1000 steps. Graph the approximate solutions.

For Thursday, April 17:

• p. 209, nos. 2cf, 4cf, 6cf

For Thursday, April 24:

• Do two iterations, by hand, of the multivariate Newton method, with the system: x(1)**2 - x(2)**2 = 0, 2*x(1)*x(2) - 2 = 0 and starting values x(1)=0.9, x(2)=1.1. (Note that the first iteration was done in class.)

For Thursday, April 24:

• p. 215, nos. 2cf, 4cf, 6cf

For Tuesday, April 29:

• Compute the condition number in the infinity norm of a 3 by 3 Hilbert matrix.
• Do five iterations of the power method for the matrix A = [1,2; 3 4], starting with initial vector v = [1;1]. (MATLAB notation is used for A and v.)

For a later date:

Solve problem 3c on page 9 with an interval Newton method