{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lecture01 - Lecture 1 Introduction to Numerical Methods W...

This preview shows pages 1–11. Sign up to view the full content.

Lecture 1 Introduction to Numerical Methods W. Gropp Guest Lecturer Stephen Bond Department of Computer Science University of Illinois at Urbana-Champaign January 19, 2010 W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 1 / 34

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Numerical Methods? Numerical Analysis? W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 2 / 34
Numerical Calculation vs. Symbolic Calculation Numerical Calculation: involve numbers directly I manipulate numbers to produce a numerical result Symbolic Calculation: symbols represent numbers I manipulate symbols according to mathematical rules to produce a symbolic result Example (numerical) (17 . 36) 2 - 1 17 . 36 + 1 = 16 . 36 Example (symbolic) x 2 - 1 x + 1 = x - 1 W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 3 / 34

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Analytic Solution vs. Numerical Solution Analytic Solution (a.k.a. symbolic): The exact numerical or symbolic representation of the solution I may use special characters such as π , e , or tan (83) Numerical Solution: The computational representation of the solution I entirely numerical Example (analytic) 1 4 1 3 π tan (83) Example (numerical) 0 . 25 0 . 33333 . . . (?) 3 . 14159 . . . (?) 0 . 88472 . . . (?) W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 4 / 34
Numerical Computation and Approximation Numerical Approximation is needed to carry out the steps in the numerical calculation. The overall process is a numerical computation. Example (symbolic computation, numerical solution) 1 2 + 1 3 + 1 4 - 1 = 1 12 = 0 . 083333333 . . . Example (numerical computation, numerical approximation) 0 . 500 + 0 . 333 + 0 . 250 - 1 . 000 = 0 . 083 W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 5 / 34

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Method vs. Algorithm vs. Implementation Method: a general (mathematical) framework describing the solution process Algorithm: a detailed description of executing the method Implementation: a particular instantiation of the algorithm Is it a “good” method? Is it a robust algorithm? Is it a fast implementation? W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 6 / 34
The Big Theme Accuracy Cost W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 7 / 34

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
History: Numerical Algorithms date to 1650 BC: The Rhind Papyrus of ancient Egypt contains 85 problems; many use numerical algorithms (T. Chartier, Davidson) Approximates π with (8 / 9) 2 * 4 3 . 1605 W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 8 / 34
History: Archimedes 287-212BC developed the ”Method of Exhaustion” Method for determining π I find the length of the permieter of a polygon inscribed inside a circle of radius 1 / 2 I find the permiter of a polygon circumscribed outside a circle of radius 1 / 2 I the value of π is between these two lengths W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 9 / 34

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
History: Method of Exhaustion A circle is not a polygon A circle is a polygon with an infinite number of sides C n = circumference of an n-sided polygon inscribed in a circle of radius 1 / 2 lim n →∞ = π Archimedes deterimined 223 71 <π < 22 7 3 . 1408 <π < 3 . 1429 two places of accuracy ....
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern