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

Lab13

 

 

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

Lab12

 

  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

Lab11

 

  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

Lab10/PA6

40

Mar. 31

W

Interp. Poly./Existence & Uniqueness

41

 

Apr. 01

Th

************* Test II **************

 

  1. Problems1-2

 

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

Lab9

 

1. Problems1-5

 

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

Lab8

 

  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

PA5/Lab7

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

HW2

 

  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

Lab6

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

Lab5

 

 

  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

Lab4

 

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

Lab1/PA1

06

Jan. 20

W

I-O/Assignment/Intrinsic Functions

Read F90/2.5-2.6

07

Jan. 21

Th

Real, Integer, & Mixed-Mode Arithmetic

Lab2

 

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