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Oct 18, 2001Talk @ MIT1 Survey and Recent Results: Robust Geometric Computation Chee Yap Department of Computer Science Courant Institute New York University

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Oct 18, 2001Talk @ MIT2 OVERVIEW n Part I: NonRobustness Survey n Part II: Exact Geometric Computation F Core Library F Constructive Root Bounds n Part III: New Directions F Moore’s Law and NonRobustness F Certification Paradigm n Conclusion

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Oct 18, 2001Talk @ MIT3 Numerical Nonrobustness Phenomenon

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Oct 18, 2001Talk @ MIT4 Part I: OVERVIEW n The Phenomenon n What is Geometric? n Taxonomy of Approaches n EGC and relatives

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Oct 18, 2001Talk @ MIT5 Numerical Non-robustness n Non-robustness phenomenon y crash, inconsistent state, intermittent n Round-off errors y benign vs. catastrophic y quantitative vs. qualitative

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Oct 18, 2001Talk @ MIT6 Examples n Intersection of 2 lines y check if intersection point is on line n Mesh Generation y point classification error (dirty meshes) n Trimmed algebraic patches in CAD y bounding curves are approximated leading to topological inconsistencies n Front Tracking Physics Simulation y front surface becomes self-intersecting

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Oct 18, 2001Talk @ MIT7 Responses to Non-robustness n “It is a rare event” n “Use more stable algorithms” n “Avoid ill-conditioned inputs” n “Epsilon-tweaking” n “There is no solution” n “Our competitors couldn’t do it, so we don’t have to bother”

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Oct 18, 2001Talk @ MIT8 Impact of Non-robustness n Acknowledged, seldom demanded n Economic/productivity Impact y barrier to full automation y scientist/programmer productivity y mission critical computation fail F Patriot missile, Ariadne Rocket n E.g. Mesh generation y a preliminary step for simulations y 1 failure/5 million cells [Aftosmis] y tweak data if failure

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Oct 18, 2001Talk @ MIT9 What is Special about Geometry?

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Oct 18, 2001Talk @ MIT10 Geometry is Harder n Geometry = Combinatorics+Numerics n E.g. Voronoi Diagram

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Oct 18, 2001Talk @ MIT11 Example: Convex Hulls 21 21 3 3 44 55 6 6 7 78 8 9 9 InputConvex HullOutput 21 3 4 5 6 7

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Oct 18, 2001Talk @ MIT12 Consistency n Geometric Object Consistency Relation ( P ) E.g. D is convex hull or Voronoi diagram n Qualitative error inconsistency D = (G, L,P) G=graph, L=labeling of G

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Oct 18, 2001Talk @ MIT13 Examples/Nonexamples n Consistency is critical y matrix multiplication y shortest paths in graphs (e.g. Djikstra’s algorithm) y sorting and geometric sorting y Euclidean shortest paths

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Oct 18, 2001Talk @ MIT14 Taxonomy of Approaches

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Oct 18, 2001Talk @ MIT15 Gold Standard n Must understand the dominant mode of numerical computing n “F.P. Mode” : y machine floating point y fixed precision (single/double/quad) y IEEE 754 Standard n What does the IEEE standard do for nonrobustness? Reduces but not eliminate it. Main contribution is cross-platform predictability. n Historical Note

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Oct 18, 2001Talk @ MIT16 Type I Approaches n Basic Philosophy: to make the fast but imprecise (IEEE) arithmetic robust n Taxonomy y arithmetic (FMA, scalar vector, sli, etc) y finite resolution predicates ( -tweaking, - predicates [Guibas-Salesin-Stolfi’89] ) y finite resolution geometry (e.g., grids) y topology oriented approach [Sugihara-Iri’88]

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Oct 18, 2001Talk @ MIT17 Example n Grid Geometry [Greene-Yao’86] n Finite Resolution Geometries

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Oct 18, 2001Talk @ MIT18 Example n What is a Finite Resolution Line? y A suitable set of pixels [graphics] y A fat line [generalized intervals] y A polyline [Yao-Greene, Milenkovic, etc] y A rounded line [Sugihara] fat line: polyline: aX+bY+c=0; (a<2 L, b<2 L, c<2 2L ) rounded line:

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Oct 18, 2001Talk @ MIT19 Example n Topology Oriented Approach of Sugihara-Iri: Voronoi diagram of 1 million points n Priority of topological part over numerical part n Identify relevant and maintainable properties: e.g. planarity n Issue: which properties to choose?

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Oct 18, 2001Talk @ MIT20 Exact Approach n Idea: avoid all errors n Big number packages (big integers, big rationals, big floats, etc) n Only rational problems are exact n Even this is a problem [Yu’92, Karasick- Lieber-Nackman’89]

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Oct 18, 2001Talk @ MIT21 Algebraic Computation n Algebraic number: y y P(x) = x 2 – 2 = 0 n Representation: P(x), 1, 2) n Exact manipulation: y comparison y arithmetic operations, roots, etc. n Most problems in Computational Geometry textbooks requires only +, –, , ,

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Oct 18, 2001Talk @ MIT22 Type II Approach n Basic Philosophy: to make slow but error-free computation more efficient n Taxonomy y Exact Arithmetic (naïve approach) y Expression packages y Compiler techniques y Consistency approach y Exact Geometric Computation (EGC)

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Oct 18, 2001Talk @ MIT23 Consistency Approach n Goal: ensure that no decisions are contradictory n Parsimonious Algorithms [Fortune’89] : only make tests that are independent of previous results n NP-hard but in PSPACE

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Oct 18, 2001Talk @ MIT24 Consistency is Hard n [Hoffmann, Hopcroft, Karasick’88] Geometric Object D = (G, L) G is realizable if there exists L such that (G, L) is consistent Algorithm AL: I G(I), L(I)) AL is geometrically exact if G(I) is the exact structure for input I AL is consistent if G(I) is realizable n Intersecting 3 convex polygons is hard (geometry theorem proving)

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Oct 18, 2001Talk @ MIT25 Exact Geometric Computation

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Oct 18, 2001Talk @ MIT26 Part II: OVERVIEW n Exact Geometric Computing (EGC) n The Core Library n Constructive Root Bounds

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Oct 18, 2001Talk @ MIT27 How to Compute Exactly in the Geometric Sense n Algorithm = sequence of steps y construction steps y conditional or branching steps n Branching based on sign of predicate evaluation Output combinatorial structure G in D=(G,L) is determined by path n Ensure all branches are correct this guarantees that G is exact!

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Oct 18, 2001Talk @ MIT28 Exact Geometric Computation (EGC) n Exactness in the Geometry, NOT the arithmetic (cf.geometric exactness) n Simple but profound implications y We can now use approximate arithmetic [Dube-Yap’94] y EGC tells us exactly how much precision is needed n No unusual geometries y No need to invent new algorithms -- “standard” algorithms apply y no unusual geometries n General solution (algebraic case) y Not algorithm-specific solutions!

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Oct 18, 2001Talk @ MIT29 Constant Expressions n = set of real algebraic operators. n = set of expressions over . n E.g., if , x 1 x 2 then is the integer polynomials over x 1 x 2. n Assume are constant operators ( no variables like x 1 x 2 ). n An expression E is a DAG y E = with and shared R Value function, Val: R where Val(E) may be undefined

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Oct 18, 2001Talk @ MIT30 Fundamental Problem of EGC n Constant Zero Problem y CZP : given E , is Val(E)=0? n Constant Sign Problem y CSP : given E , compute the sign of Val(E). n CSP is reducible to CZP n Potential exponential gap: y sum of square roots y CZP is polynomial time [Bloemer-Yap]

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Oct 18, 2001Talk @ MIT31 Complexity of CSP n Hierarchy: y 0 , y 1 0 y 2 1 y 3 2 Root(P,i) : P(x) Z[x] y 4 0 Sin(x), n Theorem: CSP( 3 ) is decidable. n Theorem: CSP( 1 ) is alternating polynomial time. n Is CSP( 4 ) is decidable?

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Oct 18, 2001Talk @ MIT32 Root Bound A root bound for an expression E is a value such that n E.g., the Cauchy’s bound says that because is a root of the polynomial x 4 0 x 2 1. n Root bit-bound is defined as log(b) b > 0 b =

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Oct 18, 2001Talk @ MIT33 How to use root bounds n Let b be a root bound for n Compute a numerical approximation for. n If then sign sign ( ) Else sign ( E ) = 0. n N.B. root bound is not reached unless sign is really zero! E E. E b/2 E E ( ) = E E

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Oct 18, 2001Talk @ MIT34 Nominally Exact Inputs n EGC Inputs are exact and consistent n Why care about exactness if the input is inexact? Because EGC is the easiest method to ensure consistency.

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Oct 18, 2001Talk @ MIT35 Core Library

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Oct 18, 2001Talk @ MIT36 EGC Libraries n GOAL: any programmer may routinely construct robust programs * n Current Libraries: Real/Expr [Yap-Dube’94] LEDA real [Burnikel et al’99] Core Library [Karamcheti et al’99]

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Oct 18, 2001Talk @ MIT37 Core Library n An EGC Library y C++, compact (200 KB) y Focus on “Numerical Core” of EGC y precision sensitive mechanism y automatically incorporates state of art techniques n Key Design Goal: ease of use y “Regular C++ Program” with preamble: #include “CORE.h” y easy to convert existing programs

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Oct 18, 2001Talk @ MIT38 Core Accuracy API n Four Levels (I) Machine Accuracy (IEEE standard) (II) Arbitrary Accuracy (e.g. Maple ) (III) Guaranteed Accuracy (e.g. Real/Expr ) (IV) Mixed Accuracy (for fine control)

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Oct 18, 2001Talk @ MIT39 Delivery System n No change of programmer behavior y At the flip of a switch! y Benefits: code logic verification, fast debugging #define Level N // N=1,2,3,4 #include “CORE.h” /* ***************************** * any C/C++ Program Here * ***************************** */

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Oct 18, 2001Talk @ MIT40 What is in CORE levels? n Numerical types: y int, long, float, double BigInt, BigFloat, Expr n Promotions (+Demotions): Level II: long BigInt, double BigFloat Level III: long, double Expr

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Oct 18, 2001Talk @ MIT41 What is in Level III? n Fundamental gap between Levels II and III n Need for iteration: consider a = b + c; n Precision sensitive evaluation

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Oct 18, 2001Talk @ MIT42 Relative and Absolute Precision Let real X be an approximation of X. n Composite precision bound [r, a] n If r , then get absolute precision a. n If a , then get relative precision r. Interesting case: [r, a] = [1, ] means we obtain the correct sign of X.

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Oct 18, 2001Talk @ MIT43 Precision-Driven Eval of Expressions n Expr’s are DAGs n Each node stores: an approximate BigFloat value; a precision; a root bound n Down-Up Algorithm: y precision p is propagated down y error e propagated upwards y At each node, check e p y Check passes automatically at leaves y Iterate if check fails; use root bounds to terminate

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Oct 18, 2001Talk @ MIT44 Example n Line intersection (2-D): y generate 2500 pairs of lines y compute their intersections y check if intersection lies on one line y 40% failure rate at Level I n In 3-D: y classify pairs of lines as skew, parallel, intersecting, or identical. y At Level I, some pairs are parallel and intersecting, etc.

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Oct 18, 2001Talk @ MIT45 Example: Theorem Proving Application n Kahan’s example (4/26/00) y “To show that you need theorem proving, or why significance arithmetic is doomed” y F(z): if (z=0) return 1; else (exp(z)-1)/z; y Q(y): |y-sqrt(y**2 +1)| - 1/(y-sqrt(y**2+1)); y G(x): F(Q(x)**2). y Compute G(x) for x=15 to 9999. n Theorem proving with Core Library [Tulone-Yap-Li’00] y Generalized Schwartz Lemma for radical expressions

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Oct 18, 2001Talk @ MIT46 Constructive Root Bounds

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Oct 18, 2001Talk @ MIT47 Problem of Constructive Root Bounds n Classical root bounds (e.g. Cauchy’s) are not constructive n Wanted: recursive rules for a family of expressions to maintain parameters p 1, p 2, etc, for any expression E so that B(p 1, p 2,...) is a root bound for E. n We will survey various bounds

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Oct 18, 2001Talk @ MIT48 Illustration n is root of A(X) = i=0 a i X i of degree n n Height of A(X) is A n Degree-height [Yap-Dube 95] y Cauchy’s bound: y maintain bound on heights y but this requires the maintenance of bounds on degrees n Degree-length [Yap-Dube 95] y Landau’s bound:

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Oct 18, 2001Talk @ MIT49 Degree-Measure Bounds n [Mignotte 82, Burnikel et al 00] where m( is the measure of . n It turns out that we also need to maintain the degree bound

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Oct 18, 2001Talk @ MIT50 BFMS Bound n [Burnikel-Fleischer-Mehlhorn-Schirra’99] y For radical expressions y Tight for division-free expressions y For those with divisions: where E is transformed to U(E)/L(E) y Improvement [‘01]

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Oct 18, 2001Talk @ MIT51 New Constructive Bound n Applies to general - expressions. n The bounding function is y : an upper bound on the absolute value of conjugates of y : an upper bound on the leading coefficient of y : an upper bound on the degree of

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Oct 18, 2001Talk @ MIT52 Inductive Rules

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Oct 18, 2001Talk @ MIT53 Comparative Study n Major open question: is there a root bit- bound that depends linearly on degree? n No single constructive root bound is always better than the others. y BFMS, degree-measure and our bound y Compare their behavior on interesting classes of expressions: F sum of square roots, F continued fraction, etc. y In Core Library, we compute all three bounds and choose the best one.

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Oct 18, 2001Talk @ MIT54 Experimental Results n A critical predicate in Fortune’s sweepline algorithm for Voronoi diagrams is. n Input coordinates are L-bit long, a’s, b’s and d’s are 3L-, 6L- and 2L-bit integers, respectively. y The BFMS bound: (79 L + 30) bits y The degree-measure bound: (64 L + 12) bits y Our new bound: (19 L + 9) bits y Best possible [Sellen-Yap]: (15 L + O(1)) bits

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Oct 18, 2001Talk @ MIT55 Timing on Synthetic Input Timings for Fortune’s predicate on degenerate inputs (in seconds) Tested on a Sun UltraSparc (440MHz, 512 MB) 84.5410.991.620.220.03D-M 79.4311.691.630.240.03BFMS 3.900.690.120.030.01NEW 200100502010L

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Oct 18, 2001Talk @ MIT56 Timing for Degenerate input Timings on Fortune’s algorithm on degenerate inputs on a uniform (32 x 32) grid with L-bit coordinates (in seconds) > 2 hrs1874.41681.6418.5D-M 5892.21218.11014.186.1BFMS 112.347.541.735.2NEW 50302010L

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Oct 18, 2001Talk @ MIT57 Moore’s Law and Non-robustness Trends

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Oct 18, 2001Talk @ MIT58 Part III: OVERVIEW n New Directions n Robustness as Resource F (or, how to exploit Moore’s Law) n Certification Paradigm F (or, how to use incorrect algorithms)

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Oct 18, 2001Talk @ MIT59 Moore’s Law and Robustness n Computers are becoming faster (Moore’s Law, 1965) n Software are becoming less robust y crashes more often y must solves larger/harder problems (e.g. Meshes, CAD) y expectation is increasing n Inverse correlation y is this inevitable?

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Oct 18, 2001Talk @ MIT60 Reversing the Trend n Robustness: all-or-nothing ? n Computational Paradigm y robustness as computational resource y exchange some speed for robustness n Goal: a positive correlation between Moore’s Law and robustness

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Oct 18, 2001Talk @ MIT61 Robustness Trade-off Curves n Each program P, for given input, defines a family of speed- robustness trade-off curves, one curve for each CPU speed What is 100% robustness?

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Oct 18, 2001Talk @ MIT62 Computing on the Curve n P only needs to be recompiled to operate at any chosen point on a curve. n With each new hardware upgrade, we can automatically recompile program to achieve the “sweetspot” on the new curve

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Oct 18, 2001Talk @ MIT63 Architecture n Automatically generated Makefiles y a registry of programs that wants to join y sample inputs (various sizes) for each program y robustness targets/parameters y optimization function (e.g., min acceptable speed)

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Oct 18, 2001Talk @ MIT64 Hardware Change/Upgrade n For given hardware configuration, for each program: y compile different robust versions of the program y run each on the test inputs to determine trade-off curve y optimize the parameters n Layer above the usual compiler n Application: libraries, CPU upgrades n Can be automatic and transparent (like the effects of Moore’s law)

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Oct 18, 2001Talk @ MIT65 Certification Paradigm

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Oct 18, 2001Talk @ MIT66 Another Paradigm Shift n Motivation: programs based on machine arithmetic are fast, but not always correct. n Floating point filter phenomenon n Need new framework for working with ``useful but incorrect algorithms’’

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Oct 18, 2001Talk @ MIT67 Verification vs. Checking n Program verification movement (1970s) n Another reason why it failed: y Algorithm: “always correct” y H-algorithm: “always fast, often correct” n Checking paradigm [Blum,Kannan] y use H-algorithms y only check specific input/output pairs, not programs.

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Oct 18, 2001Talk @ MIT68 Checking vs. Certifying n Checker: given an input-output pair, always accept or reject correctly n Certifier (or Filter): given an input- output pair, either accept or “unsure” n Why certify? y Certifiers are easier to design y Nontrivial checkers may not be known (e.g. sign of determinant)

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Oct 18, 2001Talk @ MIT69 Filtered Algorithms n How to use Certifiers? Ingredients: F Algorithm A F H-algorithm H F Filter or Certifier F n Basic Architecture y run H, filter with F, run A if necessary

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Oct 18, 2001Talk @ MIT70 Some Floating Point Filters n FvW Filter, BFS Filter n Static, Semi-static and dynamic filters n Lemma: y given expression E with k operations and floating point inputs, can compute an upper bound on E with 3k flops n Static case [FwW]: about 10 flops per arithmetic operations

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Oct 18, 2001Talk @ MIT71 Model for Filtering n Case: 2D Delaunay Triangulation y Base line y Top line y sigma = top line/base line = 60 y phi = filtering performance = 20 n efficacy = sigma/phi = 3 n Synthetic vs. Realistic algorithms n beta factor y = filterable fraction of work at top line n estimating beta [Mehlhorn et al]

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Oct 18, 2001Talk @ MIT72 Extensions n Different levels of granularity y Checking Geometric structures [Mehlhorn, et al] y Determinant filter [Pan] n Filter compiler [Schirra] n Cascaded bank of filter [Funke et al] y skip A if necessary

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Oct 18, 2001Talk @ MIT73 Conclusion

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Oct 18, 2001Talk @ MIT74 REVIEW n Part I: NonRobustness Survey n Part II: Exact Geometric Computation F Core Library F Constructive Root Bounds n Part III: New Directions F NonRobustness as Resource (or, how to exploit Moore’s Law) F Certification Paradigm (or, how to use incorrect algorithms)

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Oct 18, 2001Talk @ MIT75 Summary n Non-robustness has major economic/productivity impact n Scientifically sound and systematic solution based on EGC is feasible n Main open problem: y Is EGC for possible for non-algebraic problems? (CSP) n Program filtering goes beyond program checking n Robustness as resource paradigm can be exploited (with Moore’s Law)

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Oct 18, 2001Talk @ MIT76 Download Software n Core Library Version 1.4 n small and easy-to-use n Project Homepage: http://cs.nyu.edu/exact/

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Oct 18, 2001Talk @ MIT77 Last Slide

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Oct 18, 2001Talk @ MIT78 Extras

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Oct 18, 2001Talk @ MIT79 Algebraic Degree Bound n The expression where is the radical or root node in the expression. n The degree bound, where is either the index of radical nodes or the degree in polynomial root nodes.

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Oct 18, 2001Talk @ MIT80 Leading Coefficients and Conjugates Bound n Basic tool: resultant calculus n Case: given two algebraic numbers and with minimal polynomials and and a defining polynomial for is y Represented in the Sylvester matrix form y Leading coefficient: y Constant term: y Degree:

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Oct 18, 2001Talk @ MIT81 (cont.) n Due to the admission of divisions, we also need to bound: y Tail coefficients y Lower bounds on conjugates n Measures are involved in bounding tail coefficients.

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Oct 18, 2001Talk @ MIT82 End of Extra Slides

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