MATH 248--Computers and Numerical Algorithms--Spring 1999/Pruett

 Week 16--Final Exam (3 Hours): Monday, May 3, 1999, 6-9pm, Burruss 126

Week 15 "We shall not cease from exploration, and the end of all our exploring will be to arrive where we started and to know the place for the first time." T.S. Eliot

 Date Topics Reading HW Assignments 54 Apr. 26 M Numerical Integration (Quadrature) NM 4.1 PA7 due 55 Apr. 27 T Simple Quadrature Rules NM 4.2 PA8 (Optional) 56 Apr. 28 W Composite Quadrature Rules NM 4.3 57 Apr. 29 Th Exam Review/Course Evaluation Celebrate!

Week 14 "God sleeps in minerals, awakens in plants, walks in animals, and thinks in humans" The Sanskrit

 Date Topics Reading HW Assignments 50 Apr. 19 M Numerical Differentiation NM 4.9 HW3 (ungraded) 51 Apr. 20 T Discretization Error NM 4.9 again 52 Apr. 21 W Higher Order Methods 53 Apr. 22 Th Fortran Odds and Ends Laboratory

Week 13 "We have met the enemy, and he is us!" Pogo

 Date Topics Reading HW Assignments 46 Apr. 12 M Interpolation Error Analysis I NM 3.2 again PA6 Due 47 Apr. 13 T Interpolation Error Analysis II Probs. 1, 2 48 Apr. 14 W Taylor Polynomial NM 1.2 & 3.1 PA7 / Probs. 3, 4 49 Apr. 15 Th Polynomial Laboratory

1. Consider quadratic interpolation on equally spaced intervals. Find a spacing h between node points such that P2(x) interpolates sin(x) correctly to at least two decimal places.
2. Explain mathematically why extrapolation is usually a bad idea, particularly on equally spaced intervals.
3. Approximate the function f(x) = sqrt(x + 1) by a third-order Taylor polynomial T3(x) about a=0.
4. Use T3(x) above to approximate sqrt(0.75). Bound the error using the remainder term of Taylor's theorem and then check to make sure the actual error is less than or equal to your bound.

Week 12 "The present is passed over in the race for the future." Anne Morrow Lindbergh

 Date Topics Reading HW Assignments 42 Apr. 05 M Lagrange Interpolation NM 3.1-3.2 Prob. 1, 2 43 Apr. 06 T Newton's Divided Differences I NM 3.3 Prob. 3 44 Apr. 07 W Newton's Divided Differences II NM 3.3 again 45 Apr. 08 Th 2D-Array Laboratory

1. Write the Vandermonde matrix that defines the quadratic polynomial passing through the points (0,3), (1,4), and (2,3). Solve the resulting matrix system. In general, is this a computationally good approach to finding interpolating polynomials? Why or why not?
2. Find the polynomial that interpolates the points above by the Lagrange approach. Show that this is indeed the same polynomial as found in Prob. 1.
3. (Text Problem 3.2/16) The following table lists the population of the US from 1950 to 1990.

 Year 1950 1960 1970 1980 1990 Population (1000s) 151,326 179,323 203,302 226,542 249,633

a) Construct Newton's divided difference table for this data. b) Find the 4th-order polynomial that interpolates this data. c) Use the polynomial to approximate what the population was in 1965 and what it is now (1999). Which value is a "safer" bet?

Week 11 "Two roads diverged in a yellow wood…and I, I took the one less traveled by, and that has made all the difference." Robert Frost

 Date Topics Reading HW Assignments 38 Mar. 29 M Banded Matrices/Lin. Alg. Packages NM 6.6 Prob. 1,2 39 Mar. 30 T Dynamic Allocation Laboratory F90 8.2 40 Mar. 31 W Interp. Poly./Existence & Uniqueness 41 Apr. 01 Th ************* Test II **************

Week 10 "Security is mostly a superstition. It does not exist in nature… . Life is either a daring adventure or nothing." Helen Keller

 Date Topics Reading HW Assignments 34 Mar. 22 M Elementary Row Ops./Back Subs. NM 6.2-6.4 again Prob. 1, 2, 3 35 Mar. 23 T Forward Elimination/Pivoting Prob. 4 36 Mar. 24 W Computing the Matrix Inverse PA5 due/Prob. 5 37 Mar. 25 Th Subroutine Laboratory F90 Chapter 7

Week 9 "Sit loosely in the saddle of life." Emerson

 Date Topics Reading HW Assignments --- Mar. 15 M SNOW DAY!---CLASS CANX 31 Mar. 16 T Dot Product & Matrix Multiplication NM 6.1-6.2 again Probs. 1,2,3 32 Mar. 17 W Existence and Uniqueness NM 6.3-6.4 33 Mar. 18 Th 1D-Array Laboratory F90 Chapter 8

1. Find two square matrices A and B for which AB is not equal to BA.
2. Find two square matrices A and B such that AB is the zero matrix, but neither A nor B is the zero matrix.
3. If A and B are m x p and p x n matrices, respectively, how many machine operations (adds or multiplies) are required to compute the product AB?

SPRING BREAK: March 8-12

Week 8 "How can it be that mathematics, after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?" Einstein

 Date Topics Reading HW Assignments 27 Mar. 01 M Function Subprograms in Fortran 90 F90 6.1, 6.3 PA4 Due 28 Mar. 02 T Function Laboratory 29 Mar. 03 W Introduction to Numerical Linear Algebra NM 6.1-6.2 HW2 Due/ Prob. 1 30 Mar. 04 Th Vectors & Matrices: Basic Notions

1. Set up and solve a linear system of equations to answer the following riddle: How many of each type of coin (nickels, dimes, quarters) do I have if 1) the total value is \$0.90, 2) there are 10 coins, and 3) the total weight of the coins is 17 grams? (Assume that nickels, dimes, and quarters weigh 2, 1, and 3 grams, respectively.)

Week 7 "One who would know the secrets of nature must practice more humanity than others." Thoreau

 Date Topics Reading HW Assignments 24 Feb. 22 M Rootfinding/Fixed-Point Iteration Method NM 2.1-2.2 Prob. 1 --- Feb. 23 T ASSESSENT DAY--NO CLASS ----- ----- 25 Feb. 24 W Exist. & Uniqueness of Roots/Bisection NM 2.3-2.4 Prob. 2,3 26 Feb. 25 Th Newton's Method/Secant Method

1. Let g(x) = 1/2 cos(x) and find the fixed point p of the iteration xi+1 = g(xi) to 5 decimal places. To get an initial value x0 , sketch y = x and y = 1/2 cos(x) on the same axes. Also estimate g'(x) near the fixed point. Should the iteration converge? How fast?
2. Consider the function f(x) = x2 - 3. Find a monotonic interval in which the positive root is trapped.
3. Compute the square root of 3 to 3 decimal places using only the function above, the bisection algorithm, and a 4-function calculator.

Week 6 "Every mystery solved brings us to the threshold of an even greater one." Rachel Carson

 Date Topics Reading HW Assignments 20 Feb. 15 M Test I Post Mortem 21 Feb. 16 T Fixed-Point Iteration Laboratory 22 Feb. 17 W Orbits of Fixed-Point Iterations On Res. (Mathews) PA3 Due/Prob. 1/PA4 23 Feb. 18 Th Fixed-Point Iteration Theorem On Res. (Mathews) Prob. 2

1. Consider the iteration function g(x) = (x+2)(1/2) . Show that p=2 is a fixed point of the iteration xi+1 = g(xi). Perform the iteration on a hand calculator starting from x0 = 1.0 and verify that the iteration appears to be converging to p.
2. Consider the iteration function g(x) = (x2-2). Show that p=2 is a fixed point of the iteration xi+1 = g(xi). Perform the iteration on a hand calculator starting from x0 = 1.0 and verify that the iteration fails to converge. Tell why.

Week 5 "Simplicity is the greatest sophistication." Thoreau

 Date Topics Reading HW Assignments 16 Feb. 08 M Floating-Point Numbers/Unit Roundoff Read NM 1.3 PA3/Prob. 1 17 Feb. 09 T Roundoff Error/Precision Read NM 1.4 NM1.3/2a,b + 1.4/7 18 Feb. 10 W Error Propagation/Loss of Significance Prob. 2 19 Feb. 11 Th ************* Test I **************

1. Find the precision of real numbers on a Cray supercomputer with a 64-bit word, of which 15 bits are reserved for the exponent.
2. Find the exact binary representation of x = 1/6. Now find fl(x) on a machine with a 10-bit mantissa if a) chopping is used b) rounding is used. Also find the exact absolute and relative roundoff errors.

Week 4 "One machine can do the work of fifty ordinary men [women]. No machine can do the work of one extraordinary man [woman]," Elbert Hubbard

 Date Topics Reading HW Assignments 12 Feb. 01 M Positional Number Systems Books on Reserve Prob. 1 13 Feb. 02 T Base Conversions Books on Reserve Probs. 2,3 14 Feb. 03 W Machine Integers & KIND Number Read F90 1.1 PA2 Due/Probs. 4,5 15 Feb. 04 Th Integer Laboratory

1. Convert (BC123)16 to a decimal integer two ways: following Ex. 1 and by Algorithm 1.1
2. Convert (375)10 to a binary integer.
3. Convert (0.1)10 to a binary decimal
4. Convert the binary number 101.011011 to a decimal number.
5. Compute 12 x 9 by converting each operand to an integer and performing the multiplication in binary arithmetic. Check your result by converting the binary result back to decimal notation.

Week 3 "There is geometry in the humming of the strings; there is music in the motions of the spheres," Pythagoras

 Date Topics Reading HW Assignments 08 Jan. 25 M Relational Operators & Logical Exp. Read F90/3.1-3.2 PA1 Due/HW1 09 Jan. 26 T If-Then-Else Construct Read F90/3.4 HW1 Due/Lab3/PA2 10 Jan. 27 W Programming for Repetition/Loops Read F90/4.1-4.3 11 Jan. 28 Th Loop Constructs

Week 2 "Mathematics is the alphabet with which God has written the universe," Galileo

 Date Topics Reading HW Assignments 04 Jan. 18 M Program Structure/Building Blocks Read F90/2.1-2.2 Prog. Add 05 Jan. 19 T Variables & Type Declaration Read F90/2.3-2.4 06 Jan. 20 W I-O/Assignment/Intrinsic Functions Read F90/2.5-2.6 07 Jan. 21 Th Real, Integer, & Mixed-Mode Arithmetic

Week 1 "Never in the history of mankind has it been possible to produce so many wrong answers so quickly,'' Carl-Erik Froeberg

 Date Topics Reading HW Assignments 01 Jan. 12 T Introduction/Expectations Read F90/1.1-1.2 02 Jan. 13 W Historical Perspectives (Video) Read F90/1.3-1.4 03 Jan. 14 Th Basic Terminology ("Valleyspeak") Lab0