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

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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
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Numerical Methods? Numerical Analysis? W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 2 / 34
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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
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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
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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
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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
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The Big Theme Accuracy Cost W. GroppGuest Lecturer Stephen Bond (UIUC) CS 357 January 19, 2010 7 / 34
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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
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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
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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 ....
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