Front-end. Organization of a Modern Compiler. Middle1. Middle2. Back-end. converted to control flow) Representation
|
|
- Sydney James
- 5 years ago
- Views:
Transcription
1
2 register allocation instruction selection Code Low-level intermediate Representation Back-end Assembly array references converted into low level operations, loops converted to control flow Middle2 Low-level Intermediate Representation conventional optimizations loops,array references are preserved Middle1 loop-level transformations High-level Intermediate Representation Source Program Front-end syntax analysis type-checking symbol table Organization of a Modern Compiler
3 "! $ &,.- / / =< > A 2 CB D@EFG IHE J KML N IOE PE J Q Q DSRT Q Q D N IOE NU VW U H W O QC XE OO U CN E N I IXIO E C E N I O Q ZHH F V U I W NU O QC Z[E I N Q I X JJ J Q D OQ Q G J Q Q D N IOE / 01 \ 3 4 ] \ \ 3 ^ _ 1 ` ] \ \ / 01 a 3 \< 4 ] a \ 3 ] a \ b ] \ \ / 01 a 3 \< 4 / 01 c 3 \< 4 a ] a c ed 3 ] a \? ] c \ f
4 GF Q ZH OG Q F J IOEN PE h Z g N QVR D@EFG IHE K L J Q D OQ
5 i j k l & m "! - j n & & - j & & m! o p, rq vuts s r xw y z v t{ p q v { x } t ~ w v y r y
6 Š ŒŒŒ o &, l & $ ˆ Š ˆ. Š ˆ. Š ˆ Š ˆ. Š Œ ŒŒ Ž ˆ. oƒ 1 N I S 1 DO I = 1, N DO J = 1,M M J!! - ƒ!
7 $ 1 DO K = 1, M DO L = 1, N SK,L N L DO I = 1, N DO J = 1,M SI,J 1 M J 1 M I J 1 N I = K K L
8 $ $,,, -
9 , Question How do we determine when loop permutation is legal? Transformed loop will produce different values A[3,1] for example => permutation is illegal for this loop. A[I,J] = A[I-1,J1] 1 DO J = 1, M DO I = 2, N After loop permutation Assume that array has 1 s stored everywhere before loop begins. DO I = 2, N DO J = 1, M A[I,J] = A[I-1,J1] 1 N 1 I 2 N J,
10 ,,! l oƒ, l m š œ. ž Ÿ š ž Ÿ œ ž, o & &, -
11
12 Question How do we generate loop bounds for transformed loop nest? 1 K 1 N L DO K = 1, N DO L = I, N N SK,L I J = K L 1 N I DO I = 1, N DO J = 1, I SI,J 1 N J
13 i p & p &,, l p &, - ª, l l f
14 «h
15 -, - $ $, i
16 Most problems regarding correctness of transformations and code generation can be reduced to these problems. ii Enumerate all integer solutions. i Are there integer solutions? x is an n vector of unknowns, b is an m vector of integers, where A is a m X n matrix of integers, Given a system of linear inequalities A x < b Two problems
17 Region described by inequality is convex if two points are in region, all points in between them are in region 3x 4y <= 12 x 3x4y = 12 y Inequality half-plane 2D, half-space>2d Equality line 2D, plane 3D, hyperplane > 3D Intuition about systems of linear inequalities
18 Region described by inequalities is a convex polyhedron if two points are in region, all points in between them are in region 3x - 3y <= 9 x x >= - 3x4y <= 12 y <= 4 y Conjunction of inequalties = intersection of half-spaces => some convex region Intuition about systems of linear inequalities
19 $ $, - - n &, - k $ m -
20 - i S1 executes before S2 ii S1 and S2 both read from the same location Input dependence S1 -> S2 i S1 executes before S2 ii S1 and S2 write to the same location Output dependence S1 -> S2 i S1 executes before S2 ii S1 reads from a location that is overwritten later by S2 output Anti-dependence S1 -> S2 x = 2 y = x 1 x = 3 y = output anti flow i S1 executes before S2 in program order ii S1 writes into a location that is read by S2 Flow dependence S1 -> S2 control Dependences data flow anti output
21 Key notion Aliasing two program names may refer to the same location like Xi and Xj May-dependence vs must-dependence More precise analysis may eliminate may-dependences => Unless we know from interprocedural analysis that the parameters i and j are always distinct, we must play it safe and insert the dependence. Answer If i = j, there is a dependence otherwise, not. Question Is there an output dependence from the first assignment to the second? procedure f X,i,j begin Xi = 10 Xj = end Example When you are not sure whether a dependence exists, you must assume it does. - Real programs imprecise information => need for safe approximation Conservative Approximation
22 How do we compute dependences between iterations of a loop nest? Iteration I1,J1 is said to be dependent on iteration I2,J2 if a dynamic instance I1,J1 of a statement in loop body is dependent on a dynamic instance I2,J2 of a statement in the loop body. Dependence between iterations Dynamic instance of a statement Execution of a statement for given loop index values each I,J value of loop indices corresponds to one point in picture I DO I = 1, 100 DO J = 1, 100 S J Loop level Analysis granularity is a loop iteration
23 ± ² ³ ² µ ² ± µ š ž ¹ œ ž Ÿ º»¼»œ ½ ½ ž ¹ º»»œ ½ ½ p À & Á  Á à p Ä Á Â Ä Á Ã Ä Ž ÅÇÆur ts t ~ uw ut vt È É p Ê Á Â Ë Á à Šy v Ì vt vuu ÍÎ w { v s r xw É p! Á Ï Á Ð p Ä Á ÏÒÑ Á Ð Ä Ž Å u vt È t ~ uw t ruæ ts É p Ê Á Ð Ë Á Ï Å y v Ì vt vuu ÍÎ w { v s r xw É p Á  Á  p Ä Á Â Ñ Á Â Ä Ž ÅÇÆur rts x z u } uwv Ì uw utè É p Ê Á Â Ê Á  Šy v Ì vt vuu Í Î w { v s r xw É f
24 í ä ä î è æè é ä ä ì è æè ä ä è æè Ú Û Ú Û Û Ú Û Ú Û Û Š Á  Á à à ½ ½ ½ ž Ÿ ± ³ ðï ã Ü å ß æëê ã Ü å ß ã Ü å ß æëê ã Ü å ß ã Ü å ß æ ç ã Ü å ß ß â Û à Þ Þ Ü â Ú Þ Þ ß Ý Ûáà Þ Þ Ü Ý Ú Þ Þ Ù o - À & ˆØ  ˆ ž ¹ º»Õ»œ Ÿ ½ Ÿ Ö»œ Ÿ º»¼»œ Ÿ Ÿ»œ Ÿ ½ ½ š ž ¹ ž Ÿ Ô ¹ ¹ š ž ¹ œ ž ¹ ¹ ² ³ ² µ ² ² Óµ ± µ
25 ø ö ù óø ï òñ óô õò ó úö ù úò öúò üúû ûõò úý öúþ ò ú ö õò õÿ ø ü ÿ
26 ï
27 ! " $ &, - &...., / < ê < = >? û? ô ûõò úý >@ ô óóúò õ ü BA < < = < = < ï C
28 0D BA < < = < = < ûõò úý úe üüú ú ö òô úó? G H I KJ õ ü F ÿÿ ü> L L L L L L L A A < A < A < < M M M M M M M N O O O O O O O P A A A Q R R R R R R R S ï T
29 ! U " $ & " $ V & V - &.. XW V/ Z [ \ ] Z [ \ZB^_ a`i b c d egf a` b _h \ A i = i j ò kú@ óÿ úe ô õ ù? ô öú@ ] òô ó òô úó ú@ ú l ø õÿ ô üúóô m ôò ú l ø õÿ ô üúóô ï n
30 oqp r sut r vxw o p y s t y v z p r { p y } oop r p y vq~ o t r { t y vv z 9 BA i = i ƒ i 9 BA i = i 0 0 ˆ
31 0 4 " $ & " $ V & V - &.. XW V/ -... Z [ \ ] Z [ \ZB^_ a`i b c d egf a` b _h \ BA i = i ˆŠ
32 0 Œ Ž Ž š š œ š ž šÿ š Ÿœ š š š Z \ < Z \ Z \ < Z \ j ª k «[ < ª G ª ± ² F ² ª ³ ² ˆµ
33 ˆ ˆ For a given I, the J co-ordinate of a point in the iteration space of the loop nest satisfies maxl1i,l2i <= J <= minu1i,u2i I U1 L1 L2 U2 J general polyhedral iteration spaces! Min s and max s in loop bounds mayseem weird, but actually they describe
34 ¹ º» ¼ ½» ¹ ¼ º ½ ½»¹ ¹ ¹ ¼ À º º Á 2 À¼ Ž àŽ ÃÂ Ä Å ª ³ ² Æ«@ ª Ç ª? È? ³ ª > ² ɲ ² > ² ²³ Ê ª G ÌË ÌË A Æ? ³ ³ l ³ k Å ² ª ª ²³ È F³ ³ Å ª ³ ² Ê k Å F Ç Ç ª «F³ >? k G ª ³ ² ª ² È F Ç Í > ² ų Ê Î Î ³ ª Î Î ³ ª ³ ² Å ²³ l Å ³ ª ³ ˆÏ
35 Ð À Ñ» ¹ ¹¼ ¼ ¹ º ¼ ½¼ ¹» 2» ¼ 2» ¹»»¹ ¼ º ½» Ñ ¼» º ½ ¼ º» 0» 0 ¼ ¹ ¼» À»¹ ¼» Ñ» ~ Ò ÓKÔ 4 Õ 2Ö 2Ö Ö» 0 ¹ ½ ¹ ¹ Ö¼ º Ö 4 ؈
36 » 0 ¼ ¹ ¼» Ö À»¹ Ö¼» Ö Ñ» ~ Ò ÓKÔ 4» Ö À»¹ Ö¼» ½0 ¹¼ z À! Ö À ¹ ÚÙ Ö»ÛÜ ¹ ¼ ¹ ¹ ¼»¹ Ö¼ ¼»À¹»¹ Ö¼ ŒÝ Þ À ¹ ¼ ¹ ¹ ¼»¹ Ö¼ Ö ß ¹ ¼ z À»¹ Ý Ö Ö ¼» ½0 ¹¼ z À ¹» à À» Ü ¼ 2Ö ¼ Ö À»¹ Ö¼ z À¹»¹ º Ö¼ ¼»¹ ¹ ¼» 0 ¼ À Ö ¹ß ¼ z À ¹»¹ ¼á Ñ» 0 ¹ ¼ Ñ Ö ½À ¼ Ö» Ö ½À» 0 ¼ À Ö ¹ß ¼ z À ¹»¹ ¹ Ö Ö â z À ¹»¹ 0 Ö À ¼ Ö» ½ Ö¼» ¹ ¼ Ñ ¹ ¼» Ö¹ ¼ z À ¹»¹ À» 0 ¼ º» Ñ ˆ
37 ˆ C 2x 3 y = x <= y <= -9 - equations & inequalities 2x 3 y z = 3x 4 y = 3 - several equations, several variables 2 x 3 y = - one equation, several variables Presentation sequence Diophatine equations use integer Gaussian elimination Solve equalities first then use Fourier-Motzkin elimination
38 ˆ T Intuition Think of underdetermined systems of eqns over reals. Caution Integer constraint => Diophantine system may have no solns Solutions z = t => x = 3t - 3 y = 3-2t y = 3-2t Rewrite equation as 2z y = 3 GCD2,3 = 1 which divides 3. Let z = x floor3/2 y = x y 4 2x 3y = 3 GCD2,1 = 1 which divides 3. Solutions x = t, y = 3-2t 3 2x y = 3 2 2x = One solution x = 3 1 2x = 3 No solutions Examples One equation, many variables Thm The linear Diophatine equation a1 x1 a2 x2... an xn = c has integer solutions iff gcda1,a2,...,an divides c.
39 ˆ n If a1 is the smallest co-efficient in 1, we are left with 1 variable base case. Otherwise, the size of the smallest co-efficient has decreased, so we have made progress in the induction. Observe that 1 has integer solutions iff original equation does too. Now gcda,b = gcda mod b, b => gcda1,a2,...,an = gcda1, a2 mod a1,..,an mod a1 => gcda1, a2 mod a1,..,an mod a1 divides c. where we assume that all terms with zero coefficient have been deleted. a1 t a2 mod a1 x2... an mod a1 xn = c 1 Suppose smallest coefficient is a1, and let t = x1 floora2/a1 x2... flooran/a1 xn In terms of this variable, the equation can be rewritten as If of variables = 1, then equation is a1 x1 = c which has integer solutions if a1 divides c. If smallest coefficient = 1, then gcda1,a2,...,an = 1 which divides c. Wlog, assume that a1 = 1, and observe that the equation has solutions of the form c - a2 t2 - a3 t an tn, t2, t3,...tn. Inductive case Base case Proof WLOG, assume that all coefficients a1,a2,...an are positive. We prove only the IF case by induction, the proof in the other direction is trivial. Induction is on minsmallest coefficient, number of variables. Thm The linear Diophatine equation a1 x1 a2 x2... an xn = c has integer solutions iff gcda1,a2,...,an divides c.
40 Ï = Does gcda1,a2,...,an divide c? - Does this have integer solutions? Eqn a1 x1 a2 x2... an xn = c Summary
41 Ï Š For a matrix with a single row, column echelon form is x Key concept column echelon form - "lower triangular form for underdetermined systems" looks lower triangular, right? Solution is a = 4, b = t 2 0a b T = It is hard to read off solution from this, but for special matrices, it is easy. T 3 x y z = We can write single equation as It is useful to consider solution process in matrix-theoretic terms.
42 3x y z = Substitution t = x y 2z New equation 3t 2y 2z = Substitution u = yzt New equation 2u t = Solution u = p1 t = -2p1 Backsubstitution y = p2 t = -2p1 z = 3p1-p2- Backsubstitution x = 1-p1p2 y = p2 z = 3p1-p2- Ï µ Solution to original system 12-a-b U1U2U3 a b T -3a-b b Product of matrices = = T Solution a b U1U2U3 U = U2 = U1 3
43 Ï ˆ INTEGER GAUSSIAN ELIMINATION Question Can we convert general integer matrix into equivalent lower triangular system? z z = arbitrary integer y = => y = 3 x x = Easy special case lower triangular matrix It is not easy to determine if this Diophatine system has solutions. z Example 2x 3y 4z = => x - y 2z = y x = Key idea use integer Gaussian elimination Systems of Diophatine Equations
44 Ï Ï AU1U2...Ukx = b => AU1U2...Ukx = b => x = U1U2...Ukx Proof AU1U2...Uk is lower triangular say L Solve Lx = b easy Compute x = U1U2...Ukx Overall strategy Given Ax = b Find matrices U1, U2,...Uk such that - Use row/column operations to get matrix into triangular form - For us, column operations are more important because we usually have more unknowns than equations Integer gaussian Elimination
45 Ï Ø Question Can we stay purely in the integer domain? One solution Use only unimodular column operations using rationals. of original system requires solving lower triangular system Intuition With some column operations, recovering solution -4 y 2 0 x = which has no integer solutions! x y = Solution x = -, y = 1 Caution Not all column operations preserve integer solutions.
46 Ï ã n = x = x n y y = y Check c Add an integer multiple of one column to another b Negate a column Check x = x, y = - y x = y, y = x Let x,y satisfy second eqn. Let x,y satisfy first eqn. a Interchange two columns Check Unimodular Column Operations
47 Ï ä x x = = => => y y = 3 z z = t z t y = x t = -1 t => => => => z y x = Example
48 Ï å 4. The product of two unimodular matrices is also unimodular. of 1 or A unimodular matrix has integer entries and a determinant 2. Unimodular column operations can be used to reduce a matrix A into lower triangular form. on the matrix A of the system A x = b preserve integer solutions, as do sequences of these operations. - interchanging two columns - negating a column - adding an integer multiple of one column to another 1. The three unimodular column operations Facts
49 Ï æ
50 çø Note Even in regular Gaussian elimination, we want column echelon form rather than lower triangular form when we have under-determined systems. x x x x 0 0 x 0 0 is lower triangular but not column echelon. Point writing down the solution for this system requires additional work with the last equation 1 equation, 2 variables. This work is precisely what is required to produce the column echelon form. ii column j1 is zero if column j is. Column echelon form Let rj be the row containing the first non-zero in column j. i rj1 > rj if column j is not entirely zero. Detail Instead of lower triangular matrix, you should to compute column echelon form of matrix. x = U x where x is the solution of the system Lx = b 3. If explicit form of solutions is desired, let U be the product of unimodular matrices corresponding to the column operations. 2. If Lx = b has integer solutions, so does the original system. 1. Use unimodular column operations to reduce matrix A to lower triangular form L. Algorithm Given a system of Diophantine equations Ax = b
Loop parallelization using compiler analysis
Loop parallelization using compiler analysis Which of these loops is parallel? How can we determine this automatically using compiler analysis? Organization of a Modern Compiler Source Program Front-end
More informationOrganization of a Modern Compiler. Middle1
Organization of a Modern Compiler Source Program Front-end syntax analysis + type-checking + symbol table High-level Intermediate Representation (loops,array references are preserved) Middle1 loop-level
More informationOrganization of a Modern Compiler. Front-end. Middle1. Back-end DO I = 1, N DO J = 1,M S 1 1 N. Source Program
Organization of a Modern Compiler ource Program Front-end snta analsis + tpe-checking + smbol table High-level ntermediate Representation (loops,arra references are preserved) Middle loop-level transformations
More information' $ Dependence Analysis & % 1
Dependence Analysis 1 Goals - determine what operations can be done in parallel - determine whether the order of execution of operations can be altered Basic idea - determine a partial order on operations
More information! " # $! % & '! , ) ( + - (. ) ( ) * + / 0 1 2 3 0 / 4 5 / 6 0 ; 8 7 < = 7 > 8 7 8 9 : Œ Š ž P P h ˆ Š ˆ Œ ˆ Š ˆ Ž Ž Ý Ü Ý Ü Ý Ž Ý ê ç è ± ¹ ¼ ¹ ä ± ¹ w ç ¹ è ¼ è Œ ¹ ± ¹ è ¹ è ä ç w ¹ ã ¼ ¹ ä ¹ ¼ ¹ ±
More informationHomework #2: assignments/ps2/ps2.htm Due Thursday, March 7th.
Homework #2: http://www.cs.cornell.edu/courses/cs612/2002sp/ assignments/ps2/ps2.htm Due Thursday, March 7th. 1 Transformations and Dependences 2 Recall: Polyhedral algebra tools for determining emptiness
More informationLA PRISE DE CALAIS. çoys, çoys, har - dis. çoys, dis. tons, mantz, tons, Gas. c est. à ce. C est à ce. coup, c est à ce
> ƒ? @ Z [ \ _ ' µ `. l 1 2 3 z Æ Ñ 6 = Ð l sl (~131 1606) rn % & +, l r s s, r 7 nr ss r r s s s, r s, r! " # $ s s ( ) r * s, / 0 s, r 4 r r 9;: < 10 r mnz, rz, r ns, 1 s ; j;k ns, q r s { } ~ l r mnz,
More informationFramework for functional tree simulation applied to 'golden delicious' apple trees
Purdue University Purdue e-pubs Open Access Theses Theses and Dissertations Spring 2015 Framework for functional tree simulation applied to 'golden delicious' apple trees Marek Fiser Purdue University
More informationAn Example file... log.txt
# ' ' Start of fie & %$ " 1 - : 5? ;., B - ( * * B - ( * * F I / 0. )- +, * ( ) 8 8 7 /. 6 )- +, 5 5 3 2( 7 7 +, 6 6 9( 3 5( ) 7-0 +, => - +< ( ) )- +, 7 / +, 5 9 (. 6 )- 0 * D>. C )- +, (A :, C 0 )- +,
More information" #$ P UTS W U X [ZY \ Z _ `a \ dfe ih j mlk n p q sr t u s q e ps s t x q s y i_z { U U z W } y ~ y x t i e l US T { d ƒ ƒ ƒ j s q e uˆ ps i ˆ p q y
" #$ +. 0. + 4 6 4 : + 4 ; 6 4 < = =@ = = =@ = =@ " #$ P UTS W U X [ZY \ Z _ `a \ dfe ih j mlk n p q sr t u s q e ps s t x q s y i_z { U U z W } y ~ y x t i e l US T { d ƒ ƒ ƒ j s q e uˆ ps i ˆ p q y h
More informationRedoing the Foundations of Decision Theory
Redoing the Foundations of Decision Theory Joe Halpern Cornell University Joint work with Larry Blume and David Easley Economics Cornell Redoing the Foundations of Decision Theory p. 1/21 Decision Making:
More informationGeneral Neoclassical Closure Theory: Diagonalizing the Drift Kinetic Operator
General Neoclassical Closure Theory: Diagonalizing the Drift Kinetic Operator E. D. Held eheld@cc.usu.edu Utah State University General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator
More informationOptimal Control of PDEs
Optimal Control of PDEs Suzanne Lenhart University of Tennessee, Knoville Department of Mathematics Lecture1 p.1/36 Outline 1. Idea of diffusion PDE 2. Motivating Eample 3. Big picture of optimal control
More informationMatrices and Determinants
Matrices and Determinants Teaching-Learning Points A matri is an ordered rectanguar arra (arrangement) of numbers and encosed b capita bracket [ ]. These numbers are caed eements of the matri. Matri is
More informationAn Introduction to Optimal Control Applied to Disease Models
An Introduction to Optimal Control Applied to Disease Models Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Lecture1 p.1/37 Example Number of cancer cells at time (exponential
More informationB œ c " " ã B œ c 8 8. such that substituting these values for the B 3 's will make all the equations true
System of Linear Equations variables Ð unknowns Ñ B" ß B# ß ÞÞÞ ß B8 Æ Æ Æ + B + B ÞÞÞ + B œ, "" " "# # "8 8 " + B + B ÞÞÞ + B œ, #" " ## # #8 8 # ã + B + B ÞÞÞ + B œ, 3" " 3# # 38 8 3 ã + 7" B" + 7# B#
More informationData Dependences and Parallelization. Stanford University CS243 Winter 2006 Wei Li 1
Data Dependences and Parallelization Wei Li 1 Agenda Introduction Single Loop Nested Loops Data Dependence Analysis 2 Motivation DOALL loops: loops whose iterations can execute in parallel for i = 11,
More informationExamination paper for TFY4240 Electromagnetic theory
Department of Physics Examination paper for TFY4240 Electromagnetic theory Academic contact during examination: Associate Professor John Ove Fjærestad Phone: 97 94 00 36 Examination date: 16 December 2015
More informationComplex Analysis. PH 503 Course TM. Charudatt Kadolkar Indian Institute of Technology, Guwahati
Complex Analysis PH 503 Course TM Charudatt Kadolkar Indian Institute of Technology, Guwahati ii Copyright 2000 by Charudatt Kadolkar Preface Preface Head These notes were prepared during the lectures
More informationPlanning for Reactive Behaviors in Hide and Seek
University of Pennsylvania ScholarlyCommons Center for Human Modeling and Simulation Department of Computer & Information Science May 1995 Planning for Reactive Behaviors in Hide and Seek Michael B. Moore
More informationOC330C. Wiring Diagram. Recommended PKH- P35 / P50 GALH PKA- RP35 / RP50. Remarks (Drawing No.) No. Parts No. Parts Name Specifications
G G " # $ % & " ' ( ) $ * " # $ % & " ( + ) $ * " # C % " ' ( ) $ * C " # C % " ( + ) $ * C D ; E @ F @ 9 = H I J ; @ = : @ A > B ; : K 9 L 9 M N O D K P D N O Q P D R S > T ; U V > = : W X Y J > E ; Z
More informationConstructive Decision Theory
Constructive Decision Theory Joe Halpern Cornell University Joint work with Larry Blume and David Easley Economics Cornell Constructive Decision Theory p. 1/2 Savage s Approach Savage s approach to decision
More informationQUESTIONS ON QUARKONIUM PRODUCTION IN NUCLEAR COLLISIONS
International Workshop Quarkonium Working Group QUESTIONS ON QUARKONIUM PRODUCTION IN NUCLEAR COLLISIONS ALBERTO POLLERI TU München and ECT* Trento CERN - November 2002 Outline What do we know for sure?
More informationPrincipal Secretary to Government Haryana, Town & Country Planning Department, Haryana, Chandigarh.
1 From To Principal Secretary to Government Haryana, Town & Country Planning Department, Haryana, Chandigarh. The Director General, Town & Country Planning Department, Haryana, Chandigarh. Memo No. Misc-2339
More informationAN IDENTIFICATION ALGORITHM FOR ARMAX SYSTEMS
AN IDENTIFICATION ALGORITHM FOR ARMAX SYSTEMS First the X, then the AR, finally the MA Jan C. Willems, K.U. Leuven Workshop on Observation and Estimation Ben Gurion University, July 3, 2004 p./2 Joint
More informationMax. Input Power (W) Input Current (Arms) Dimming. Enclosure
Product Overview XI025100V036NM1M Input Voltage (Vac) Output Power (W) Output Voltage Range (V) Output urrent (A) Efficiency@ Max Load and 70 ase Max ase Temp. ( ) Input urrent (Arms) Max. Input Power
More informationVectors. Teaching Learning Point. Ç, where OP. l m n
Vectors 9 Teaching Learning Point l A quantity that has magnitude as well as direction is called is called a vector. l A directed line segment represents a vector and is denoted y AB Å or a Æ. l Position
More informationALTER TABLE Employee ADD ( Mname VARCHAR2(20), Birthday DATE );
!! "# $ % & '( ) # * +, - # $ "# $ % & '( ) # *.! / 0 "# "1 "& # 2 3 & 4 4 "# $ % & '( ) # *!! "# $ % & # * 1 3 - "# 1 * #! ) & 3 / 5 6 7 8 9 ALTER ; ? @ A B C D E F G H I A = @ A J > K L ; ?
More informationDM559 Linear and Integer Programming. Lecture 2 Systems of Linear Equations. Marco Chiarandini
DM559 Linear and Integer Programming Lecture Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. Outline 1. 3 A Motivating Example You are organizing
More informationVector analysis. 1 Scalars and vectors. Fields. Coordinate systems 1. 2 The operator The gradient, divergence, curl, and Laplacian...
Vector analysis Abstract These notes present some background material on vector analysis. Except for the material related to proving vector identities (including Einstein s summation convention and the
More information3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 REDCLIFF MUNICIPAL PLANNING COMMISSION FOR COMMENT/DISCUSSION DATE: TOPIC: April 27 th, 2018 Bylaw 1860/2018, proposed amendments to the Land Use Bylaw regarding cannabis
More informationAdvanced Restructuring Compilers. Advanced Topics Spring 2009 Prof. Robert van Engelen
Advanced Restructuring Compilers Advanced Topics Spring 2009 Prof. Robert van Engelen Overview Data and control dependences The theory and practice of data dependence analysis K-level loop-carried dependences
More informationThe University of Bath School of Management is one of the oldest established management schools in Britain. It enjoys an international reputation for
The University of Bath School of Management is one of the oldest established management schools in Britain. It enjoys an international reputation for the quality of its teaching and research. Its mission
More informationNew method for solving nonlinear sum of ratios problem based on simplicial bisection
V Ù â ð f 33 3 Vol33, No3 2013 3 Systems Engineering Theory & Practice Mar, 2013 : 1000-6788(2013)03-0742-06 : O2112!"#$%&')(*)+),-))/0)1)23)45 : A 687:9 1, ;:= 2 (1?@ACBEDCFHCFEIJKLCFFM, NCO 453007;
More informationThis document has been prepared by Sunder Kidambi with the blessings of
Ö À Ö Ñ Ø Ò Ñ ÒØ Ñ Ý Ò Ñ À Ö Ñ Ò Ú º Ò Ì ÝÊ À Å Ú Ø Å Ê ý Ú ÒØ º ÝÊ Ú Ý Ê Ñ º Å º ² ºÅ ý ý ý ý Ö Ð º Ñ ÒÜ Æ Å Ò Ñ Ú «Ä À ý ý This document has been prepared by Sunder Kidambi with the blessings of Ö º
More informationA Functional Quantum Programming Language
A Functional Quantum Programming Language Thorsten Altenkirch University of Nottingham based on joint work with Jonathan Grattage and discussions with V.P. Belavkin supported by EPSRC grant GR/S30818/01
More informationMatrices and Matrix Algebra.
Matrices and Matrix Algebra 3.1. Operations on Matrices Matrix Notation and Terminology Matrix: a rectangular array of numbers, called entries. A matrix with m rows and n columns m n A n n matrix : a square
More information$%! & (, -3 / 0 4, 5 6/ 6 +7, 6 8 9/ 5 :/ 5 A BDC EF G H I EJ KL N G H I. ] ^ _ ` _ ^ a b=c o e f p a q i h f i a j k e i l _ ^ m=c n ^
! #" $%! & ' ( ) ) (, -. / ( 0 1#2 ' ( ) ) (, -3 / 0 4, 5 6/ 6 7, 6 8 9/ 5 :/ 5 ;=? @ A BDC EF G H I EJ KL M @C N G H I OPQ ;=R F L EI E G H A S T U S V@C N G H IDW G Q G XYU Z A [ H R C \ G ] ^ _ `
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationSolution of Linear Equations
Solution of Linear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 7, 07 We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations A subclass
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationApplications of Discrete Mathematics to the Analysis of Algorithms
Applications of Discrete Mathematics to the Analysis of Algorithms Conrado Martínez Univ. Politècnica de Catalunya, Spain May 2007 Goal Given some algorithm taking inputs from some set Á, we would like
More informationETIKA V PROFESII PSYCHOLÓGA
P r a ž s k á v y s o k á š k o l a p s y c h o s o c i á l n í c h s t u d i í ETIKA V PROFESII PSYCHOLÓGA N a t á l i a S l o b o d n í k o v á v e d ú c i p r á c e : P h D r. M a r t i n S t r o u
More informationBlock-tridiagonal matrices
Block-tridiagonal matrices. p.1/31 Block-tridiagonal matrices - where do these arise? - as a result of a particular mesh-point ordering - as a part of a factorization procedure, for example when we compute
More informationPharmacological and genomic profiling identifies NF-κB targeted treatment strategies for mantle cell lymphoma
CORRECTION NOTICE Nat. Med. 0, 87 9 (014) Pharmacoogica and genomic profiing identifies NF-κB targeted treatment strategies for mante ce ymphoma Rami Raha, Mareie Fric, Rodrigo Romero, Joshua M Korn, Robert
More informationLoop Parallelization Techniques and dependence analysis
Loop Parallelization Techniques and dependence analysis Data-Dependence Analysis Dependence-Removing Techniques Parallelizing Transformations Performance-enchancing Techniques 1 When can we run code in
More informationSubrings and Ideals 2.1 INTRODUCTION 2.2 SUBRING
Subrings and Ideals Chapter 2 2.1 INTRODUCTION In this chapter, we discuss, subrings, sub fields. Ideals and quotient ring. We begin our study by defining a subring. If (R, +, ) is a ring and S is a non-empty
More informationLectures on Linear Algebra for IT
Lectures on Linear Algebra for IT by Mgr Tereza Kovářová, PhD following content of lectures by Ing Petr Beremlijski, PhD Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 3 Inverse Matrix
More information1. Allstate Group Critical Illness Claim Form: When filing Critical Illness claim, please be sure to include the following:
Dear Policyholder, Per your recent request, enclosed you will find the following forms: 1. Allstate Group Critical Illness Claim Form: When filing Critical Illness claim, please be sure to include the
More informationConnection equations with stream variables are generated in a model when using the # $ % () operator or the & ' %
7 9 9 7 The two basic variable types in a connector potential (or across) variable and flow (or through) variable are not sufficient to describe in a numerically sound way the bi-directional flow of matter
More informationBooks. Book Collection Editor. Editor. Name Name Company. Title "SA" A tree pattern. A database instance
"! # #%$ $ $ & & & # ( # ) $ + $, -. 0 1 1 1 2430 5 6 78 9 =?
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More information1 - Systems of Linear Equations
1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are
More informationAMS 209, Fall 2015 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems
AMS 209, Fall 205 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems. Overview We are interested in solving a well-defined linear system given
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationA Study on the Analysis of Measurement Errors of Specific Gravity Meter
HWAHAK KONGHAK Vol. 40, No. 6, December, 2002, pp. 676-680 (2001 7 2, 2002 8 5 ) A Study on the Analysis of Measurement Errors of Specific Gravity Meter Kang-Jin Lee, Jae-Young Her, Young-Cheol Ha, Seung-Hee
More informationMatrices. Chapter Definitions and Notations
Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which
More informationSOLAR MORTEC INDUSTRIES.
SOLAR MORTEC INDUSTRIES www.mortecindustries.com.au Wind Regions Region A Callytharra Springs Gascoyne Junction Green Head Kununurra Lord Howe Island Morawa Toowoomba Wittanoom Bourke Region B Adelaide
More information: œ Ö: =? À =ß> real numbers. œ the previous plane with each point translated by : Ðfor example,! is translated to :)
â SpanÖ?ß@ œ Ö =? > @ À =ß> real numbers : SpanÖ?ß@ œ Ö: =? > @ À =ß> real numbers œ the previous plane with each point translated by : Ðfor example, is translated to :) á In general: Adding a vector :
More informationGAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)
GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.
More informationSystems of Equations 1. Systems of Linear Equations
Lecture 1 Systems of Equations 1. Systems of Linear Equations [We will see examples of how linear equations arise here, and how they are solved:] Example 1: In a lab experiment, a researcher wants to provide
More informationSymbols and dingbats. A 41 Α a 61 α À K cb ➋ à esc. Á g e7 á esc. Â e e5 â. Ã L cc ➌ ã esc ~ Ä esc : ä esc : Å esc * å esc *
Note: Although every effort ws tken to get complete nd ccurte tble, the uhtor cn not be held responsible for ny errors. Vrious sources hd to be consulted nd MIF hd to be exmined to get s much informtion
More informationSolving Linear Systems Using Gaussian Elimination
Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n
More informationSaturday, April 23, Dependence Analysis
Dependence Analysis Motivating question Can the loops on the right be run in parallel? i.e., can different processors run different iterations in parallel? What needs to be true for a loop to be parallelizable?
More informationFast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation
Fast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation Christina C. Christara and Kit Sun Ng Department of Computer Science University of Toronto Toronto, Ontario M5S 3G4,
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationn P! " # $ % & % & ' ( ) * + % $, $ -. $ -..! " # ) & % $/ $ - 0 1 2 3 4 5 6 7 3 8 7 9 9 : 3 ; 1 7 < 9 : 5 = > : 9? @ A B C D E F G HI J G H K D L M N O P Q R S T U V M Q P W X X Y W X X Z [ Y \ ] M
More informationHarlean. User s Guide
Harlean User s Guide font faq HOW TO INSTALL YOUR FONT You will receive your files as a zipped folder. For instructions on how to unzip your folder, visit LauraWorthingtonType.com/faqs/. Your font is available
More informationObtain from theory/experiment a value for the absorbtion cross-section. Calculate the number of atoms/ions/molecules giving rise to the line.
Line Emission Observations of spectral lines offer the possibility of determining a wealth of information about an astronomical source. A usual plan of attack is :- Identify the line. Obtain from theory/experiment
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More informationQueues, Stack Modules, and Abstract Data Types. CS2023 Winter 2004
Queues Stack Modules and Abstact Data Types CS2023 Wnte 2004 Outcomes: Queues Stack Modules and Abstact Data Types C fo Java Pogammes Chapte 11 (11.5) and C Pogammng - a Moden Appoach Chapte 19 Afte the
More informationMathematical Optimisation, Chpt 2: Linear Equations and inequalities
Mathematical Optimisation, Chpt 2: Linear Equations and inequalities Peter J.C. Dickinson p.j.c.dickinson@utwente.nl http://dickinson.website version: 12/02/18 Monday 5th February 2018 Peter J.C. Dickinson
More informationBlock vs. Stream cipher
Block vs. Stream cipher Idea of a block cipher: partition the text into relatively large (e.g. 128 bits) blocks and encode each block separately. The encoding of each block generally depends on at most
More informationTesting SUSY Dark Matter
Testing SUSY Dark Matter Wi de Boer, Markus Horn, Christian Sander Institut für Experientelle Kernphysik Universität Karlsruhe Wi.de.Boer@cern.ch http://hoe.cern.ch/ deboerw SPACE Part Elba, May 7, CMSSM
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationA PRACTICAL ALGORITHM FOR EXACT ARRAY DEPENDENCE ANALYSIS
A PRACTICAL ALGORITHM FOR EXACT ARRAY DEPENDENCE ANALYSIS 10/11/05 Slide 1 Introduction We describe the Omega Test which is a new method for dependence analysis. It combines new methods from integer programming
More informationTowards a High Level Quantum Programming Language
MGS Xmas 04 p.1/?? Towards a High Level Quantum Programming Language Thorsten Altenkirch University of Nottingham based on joint work with Jonathan Grattage and discussions with V.P. Belavkin supported
More informationDamping Ring Requirements for 3 TeV CLIC
Damping Ring Requirements for 3 TeV CLIC Quantity Symbol Value Bunch population N b 9 4.1 10 No. of bunches/train k bt 154 Repetition frequency f r 100 Hz Horizontal emittance γε x 7 4.5 10 m Vertical
More informationSample Exam 1: Chapters 1, 2, and 3
L L 1 ' ] ^, % ' ) 3 Sample Exam 1: Chapters 1, 2, and 3 #1) Consider the lineartime invariant system represented by Find the system response and its zerostate and zeroinput components What are the response
More informationBuilding Products Ltd. G Jf. CIN No. L26922KA199SPLC018990
Building Products Ltd.! #$%&'( )*$+.+/'12 3454*67 89 :;= >?? @*;;* ABCDEFGHH @!EJ KLM OGGO RS*4$ST U$+V.+W'X6 $Y Z74' [4%\*67 ]^_ `?a bc>de fg>h> iddjkl mno ApqCDEFrHHHO @ce @sbt?u @vwx yz PD @ Q { }~
More informationLecture 16: Modern Classification (I) - Separating Hyperplanes
Lecture 16: Modern Classification (I) - Separating Hyperplanes Outline 1 2 Separating Hyperplane Binary SVM for Separable Case Bayes Rule for Binary Problems Consider the simplest case: two classes are
More informationA Language for Task Orchestration and its Semantic Properties
DEPARTMENT OF COMPUTER SCIENCES A Language for Task Orchestration and its Semantic Properties David Kitchin, William Cook and Jayadev Misra Department of Computer Science University of Texas at Austin
More informationLinear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway
Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row
More informationFinding small factors of integers. Speed of the number-field sieve. D. J. Bernstein University of Illinois at Chicago
The number-field sieve Finding small factors of integers Speed of the number-field sieve D. J. Bernstein University of Illinois at Chicago Prelude: finding denominators 87366 22322444 in R. Easily compute
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationSynchronizing Automata Preserving a Chain of Partial Orders
Synchronizing Automata Preserving a Chain of Partial Orders M. V. Volkov Ural State University, Ekaterinburg, Russia CIAA 2007, Prague, Finland, 16.07.07 p.1/25 Synchronizing automata We consider DFA:.
More informationF O R SOCI AL WORK RESE ARCH
7 TH EUROPE AN CONFERENCE F O R SOCI AL WORK RESE ARCH C h a l l e n g e s i n s o c i a l w o r k r e s e a r c h c o n f l i c t s, b a r r i e r s a n d p o s s i b i l i t i e s i n r e l a t i o n
More informationUNIQUE FJORDS AND THE ROYAL CAPITALS UNIQUE FJORDS & THE NORTH CAPE & UNIQUE NORTHERN CAPITALS
Q J j,. Y j, q.. Q J & j,. & x x. Q x q. ø. 2019 :. q - j Q J & 11 Y j,.. j,, q j q. : 10 x. 3 x - 1..,,. 1-10 ( ). / 2-10. : 02-06.19-12.06.19 23.06.19-03.07.19 30.06.19-10.07.19 07.07.19-17.07.19 14.07.19-24.07.19
More information7.2 Linear equation systems. 7.3 Linear least square fit
72 Linear equation systems In the following sections, we will spend some time to solve linear systems of equations This is a tool that will come in handy in many di erent places during this course For
More informationMATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra
MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationComputer Systems Organization. Plan for Today
Computer Systems Organization "!#%$&(' *),+-/. 021 345$&7634 098 :;!/.:?$&4@344A$&76 Plan for Today 1 ! ILP Limit Study #"%$'&)('*+-,.&/ 0 1+-*2 435&)$'768&9,:+8&.* &:>@7ABAC&ED(
More informationNUMERICAL MATHEMATICS & COMPUTING 7th Edition
NUMERICAL MATHEMATICS & COMPUTING 7th Edition Ward Cheney/David Kincaid c UT Austin Engage Learning: Thomson-Brooks/Cole wwwengagecom wwwmautexasedu/cna/nmc6 October 16, 2011 Ward Cheney/David Kincaid
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 17, pp. 76-92, 2004. Copyright 2004,. ISSN 1068-9613. ETNA STRONG RANK REVEALING CHOLESKY FACTORIZATION M. GU AND L. MIRANIAN Ý Abstract. For any symmetric
More informationÇÙÐ Ò ½º ÅÙÐ ÔÐ ÔÓÐÝÐÓ Ö Ñ Ò Ú Ö Ð Ú Ö Ð ¾º Ä Ò Ö Ö Ù Ð Ý Ó ËÝÑ ÒÞ ÔÓÐÝÒÓÑ Ð º Ì ÛÓ¹ÐÓÓÔ ÙÒÖ Ö Ô Û Ö Ö ÖÝ Ñ ¹ ÝÓÒ ÑÙÐ ÔÐ ÔÓÐÝÐÓ Ö Ñ
ÅÙÐ ÔÐ ÔÓÐÝÐÓ Ö Ñ Ò ÝÒÑ Ò Ò Ö Ð Ö Ò Ó Ò Ö ÀÍ ÖÐ Òµ Ó Ò ÛÓÖ Û Ö Ò ÖÓÛÒ Ö Ú ½ ¼¾º ¾½ Û Åº Ä Ö Ö Ú ½ ¼¾º ¼¼ Û Äº Ñ Ò Ëº Ï ÒÞ ÖÐ Å ÒÞ ½ º¼ º¾¼½ ÇÙÐ Ò ½º ÅÙÐ ÔÐ ÔÓÐÝÐÓ Ö Ñ Ò Ú Ö Ð Ú Ö Ð ¾º Ä Ò Ö Ö Ù Ð Ý Ó ËÝÑ
More informationA Theory of Universal AI
A Theory of Universa AI Literature Marcus Hutter Kircherr, Li, and Vitanyi Presenter Prashant J. Doshi CS594: Optima Decision Maing A Theory of Universa Artificia Inteigence p.1/18 Roadmap Caim Bacground
More informationMath 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
More information