The document discusses techniques for improving code through transformations, including eliminating common subexpressions, copy propagation, detecting loop-invariant computations, performing code motion, and eliminating induction variables. Code transformations aim to reduce code size and running time by taking advantage of properties like invariant values to optimize programs without changing their meaning or output. Algorithms described include analyzing flow graphs, data flow, and control flow to identify optimization opportunities and apply transformations correctly and safely.
Modern Control - Lec 04 - Analysis and Design of Control Systems using Root L...Amr E. Mohamed
The document provides an overview of root locus analysis and design of control systems. It begins with an introduction to root locus including motivation, definition, and the basic feedback control system model. It then covers the key rules and steps for constructing and interpreting root loci, including determining asymptotes, breakaway/break-in points, and imaginary axis crossings. Three examples are worked through step-by-step to demonstrate how to apply the rules and steps to sketch root loci for different open-loop transfer functions. The document explains how root locus can be used to choose controller parameters to satisfy transient performance requirements.
The document discusses multiple topics related to analog control systems, including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they relate to system response.
4. Defining characteristics of second order systems and analyzing steady state error.
5. Discussing stability analysis in the complex s-plane and conditions for stable, unstable and marginally stable systems.
This document discusses discrete state space models. It begins with an introduction to state variable models and their generic structure. It then discusses various canonical forms for state space models including controllable, observable, and Jordan canonical forms. It also covers computing the characteristic equation, eigenvalues, state transition matrices using different techniques like inverse Laplace transform, similarity transformations, and Cayley-Hamilton theorem. Examples are provided to illustrate finding state space models from transfer functions and computing the state transition matrix.
The document discusses the Nyquist stability criterion for analyzing the stability of sampled-data control systems. It begins by defining the Nyquist criterion and contour, and how they can be used to determine the number of closed-loop poles outside the unit circle (Z). It then provides an example showing how to apply the Nyquist criterion by plotting the loop gain and counting encirclements of the critical point. The document also discusses modifications to the Nyquist contour when open-loop poles are on the unit circle and defines the Nyquist criterion theorem.
Modern Control - Lec 04 - Analysis and Design of Control Systems using Root L...Amr E. Mohamed
The document provides an overview of root locus analysis and design of control systems. It begins with an introduction to root locus including motivation, definition, and the basic feedback control system model. It then covers the key rules and steps for constructing and interpreting root loci, including determining asymptotes, breakaway/break-in points, and imaginary axis crossings. Three examples are worked through step-by-step to demonstrate how to apply the rules and steps to sketch root loci for different open-loop transfer functions. The document emphasizes that root locus allows choosing controller parameters to place closed-loop poles in desired performance regions.
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
This presentation discusses transfer functions. It defines a transfer function as the ratio of the Laplace transform of the output to the Laplace transform of the input with all initial conditions equal to zero. It classifies transfer functions as proper, strictly proper, or improper based on the order of the numerator and denominator polynomials. Poles and zeros are also discussed as values of s that make the transfer function tend to infinity or zero. The presentation provides examples of finding the transfer function of electrical and mechanical systems using differential equations. Advantages and disadvantages of using transfer functions to model systems are also summarized.
Modern Control - Lec 04 - Analysis and Design of Control Systems using Root L...Amr E. Mohamed
The document provides an overview of root locus analysis and design of control systems. It begins with an introduction to root locus including motivation, definition, and the basic feedback control system model. It then covers the key rules and steps for constructing and interpreting root loci, including determining asymptotes, breakaway/break-in points, and imaginary axis crossings. Three examples are worked through step-by-step to demonstrate how to apply the rules and steps to sketch root loci for different open-loop transfer functions. The document explains how root locus can be used to choose controller parameters to satisfy transient performance requirements.
The document discusses multiple topics related to analog control systems, including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they relate to system response.
4. Defining characteristics of second order systems and analyzing steady state error.
5. Discussing stability analysis in the complex s-plane and conditions for stable, unstable and marginally stable systems.
This document discusses discrete state space models. It begins with an introduction to state variable models and their generic structure. It then discusses various canonical forms for state space models including controllable, observable, and Jordan canonical forms. It also covers computing the characteristic equation, eigenvalues, state transition matrices using different techniques like inverse Laplace transform, similarity transformations, and Cayley-Hamilton theorem. Examples are provided to illustrate finding state space models from transfer functions and computing the state transition matrix.
The document discusses the Nyquist stability criterion for analyzing the stability of sampled-data control systems. It begins by defining the Nyquist criterion and contour, and how they can be used to determine the number of closed-loop poles outside the unit circle (Z). It then provides an example showing how to apply the Nyquist criterion by plotting the loop gain and counting encirclements of the critical point. The document also discusses modifications to the Nyquist contour when open-loop poles are on the unit circle and defines the Nyquist criterion theorem.
Modern Control - Lec 04 - Analysis and Design of Control Systems using Root L...Amr E. Mohamed
The document provides an overview of root locus analysis and design of control systems. It begins with an introduction to root locus including motivation, definition, and the basic feedback control system model. It then covers the key rules and steps for constructing and interpreting root loci, including determining asymptotes, breakaway/break-in points, and imaginary axis crossings. Three examples are worked through step-by-step to demonstrate how to apply the rules and steps to sketch root loci for different open-loop transfer functions. The document emphasizes that root locus allows choosing controller parameters to place closed-loop poles in desired performance regions.
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
This presentation discusses transfer functions. It defines a transfer function as the ratio of the Laplace transform of the output to the Laplace transform of the input with all initial conditions equal to zero. It classifies transfer functions as proper, strictly proper, or improper based on the order of the numerator and denominator polynomials. Poles and zeros are also discussed as values of s that make the transfer function tend to infinity or zero. The presentation provides examples of finding the transfer function of electrical and mechanical systems using differential equations. Advantages and disadvantages of using transfer functions to model systems are also summarized.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
The document discusses open loop transfer functions and stability analysis using Nyquist plots. It begins with an outline of topics including partial fraction expansion, open loop systems, Nyquist plots, and stability criteria. It then provides examples of using partial fraction expansion to decompose transfer functions with real distinct roots, complex conjugate roots, and repeated roots. The document explains open loop and closed loop system nomenclature. It introduces the Nyquist stability criterion, which involves plotting the open loop transfer function on the Nyquist plot and checking if it encircles the critical point at -1.
This document discusses deadbeat response design for digital control systems. It covers:
1. Designing controllers to achieve a deadbeat response when plant poles and zeros are inside the unit circle. The controller must cancel plant poles to achieve zero steady state error within a finite number of samples.
2. Examples where the controller achieves a deadbeat response of 1 sample for a step input and 2 samples for a ramp input.
3. Considerations for designing deadbeat responses when some plant poles and zeros are on or outside the unit circle, where imperfect cancellation could lead to instability. The controller must not cancel these poles and zeros.
4. Achieving a deadbeat response in sampled data control systems without
The document discusses state space modeling of dynamic systems. It introduces state space analysis as a modern technique for modeling nonlinear, time-varying, and multi-input multi-output systems. The key concepts of state variables, state vector, and state space representations using state equations and output equations are explained. Examples of modeling RL, RC, and RLC circuits using state space approaches are provided to illustrate the technique.
This document discusses different models for representing digital control systems and filters in MATLAB, including transfer functions, zero-pole-gain, state space, partial fraction expansion, second-order sections, lattice structure, and convolution matrix representations. It provides examples of converting between these different representations, such as using the poly and roots functions to convert between polynomial and zero-pole-gain forms. The toolbox functions residuez, tf2latc, and convmtx are also demonstrated for partial fraction expansion, converting to lattice structure, and generating a convolution matrix respectively.
Here are the steps to solve for the transfer function G(s) = X2(s)/F(s) for the given system:
1. Draw the free body diagrams for both masses M1 and M2 showing all the forces acting on each mass.
2. Write the Newton's second law equation for M1:
(M1s2 + f1vs + k1)X1(s) - k2(X1(s) - X2(s)) = 0
3. Write the Newton's second law equation for M2:
-k2(X1(s) - X2(s)) + (M2s2 + f2vs + k
This document discusses properties of the Laplace transform. It begins by defining the Laplace transform and common notation used. It then provides examples of Laplace transforms for some common functions like the impulse function, unit step function, ramp function, sine and cosine functions, and exponential functions. It also discusses rules like linearity and time shifting. The document concludes by summarizing some of the key Laplace transforms and properties discussed.
Introduction, Types of Stable System, Routh-Hurwitz Stability Criterion, Disadvantages of Hurwitz Criterion, Techniques of Routh-Hurwitz criterion, Examples, Special Cases of Routh Array, Advantages and Disadvantages of Routh-Hurwitz Stability Criterion, and examples.
This document discusses the design of digital controllers using root locus analysis. It provides examples of designing proportional controllers for first and second order systems to meet specifications on damping ratio, natural frequency, and settling time. The procedures involve constructing root loci, determining breakaway points and critical gains, and using the MATLAB root locus tool to plot contours and obtain design values for proportional gain.
The document discusses root locus analysis and controller design techniques.
1) Root locus analysis involves plotting the trajectories of a system's closed-loop poles as a parameter (such as a controller gain) is varied. Properties of root loci like pole departure angles and asymptotes are examined.
2) Design specifications like overshoot and settling time can be translated to desired pole regions. A simple proportional controller can place poles near this region.
3) Additional controller types like integral and derivative are introduced to modify the root locus for improved steady-state response or damping. Combining controller types benefits stability.
This document discusses the differences between non-deterministic finite automata (NFA) and deterministic finite automata (DFA). It states that NFAs can recognize the union, concatenation, and Kleene closure of regular languages. It also notes that any NFA can be converted to an equivalent DFA by constructing transition sets representing the lambda-closures of NFA states. The conversion procedure involves identifying the lambda-closure of each NFA state and making those closures the states of the DFA. Transitions between DFA states are determined by the possible transitions between NFA lambda-closures on given input symbols. This ensures the resulting DFA recognizes the same language as the original NFA.
This chapter discusses transient response in control systems. It describes how to determine the time response from a transfer function using poles and zeros. For a first order system, the chapter defines the time constant, rise time and settling time. For a second order system, it defines damping ratio, percent overshoot, settling time and peak time. The chapter also discusses higher order systems and how to approximate them as second order systems. Exercises are provided to analyze systems and design feedback control systems based on desired transient response specifications.
The document discusses root locus analysis, a technique for analyzing the stability and transient response of control systems. It provides rules for sketching root loci, including that branches represent closed-loop poles and the locus is symmetric about the real axis. The document also describes refining the root locus sketch by finding the imaginary axis crossing, angles of departure and arrival, and approximating higher-order systems as second-order. An example problem is given to apply these techniques.
1) A Petri net is a graphical modeling tool used to represent systems with concurrent, asynchronous components. It consists of places, transitions, and arcs between them.
2) The document discusses using Petri nets to model resource sharing between two processors to avoid deadlocks. It defines deadlocks and describes the banker's algorithm for deadlock avoidance.
3) The banker's algorithm works by ensuring the system is always in a safe state where all processes can complete even if maximum resources are requested. It maintains data structures to track available, allocated, and needed resources.
This document discusses closed loop control systems. It begins by recapping the components of a closed loop system including a plant, reference input, sensor, controller, and actuator. The plant being considered is a linear time invariant system represented by differential equations. The document then discusses representing the closed loop system using transfer functions for each component in a block diagram. It explains how to calculate the closed loop transfer function and closed loop poles, which must lie in the left half of the complex plane for stability. Finally, it notes the process is similar for multi-input multi-output systems using transfer function matrices.
Invited talk held by Karsten Wolf on June 26, 2007 on the 28th International Conference on Application and Theory of Petri Nets and Other Models of Concurrency (PETRI NETS 2007) in Siedlce, Poland.
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
Digital control systems (dcs) lecture 18-19-20Ali Rind
This document discusses digital control systems and related topics such as difference equations, z-transforms, and mapping between the s-plane and z-plane. It begins with an outline of topics to be covered including difference equations, z-transforms, inverse z-transforms, and the relationship between the s-plane and z-plane. Examples are provided to illustrate difference equations, z-transforms, and mapping poles between the two planes. Standard z-transforms of discrete-time signals like the unit impulse and sampled step are also defined.
This document discusses responsible citizenship and outlines some of the key characteristics and responsibilities of good citizens. It notes that citizenship involves being a member of a community and working to make that community a better place. Good citizens are aware of their rights and responsibilities, informed about social and political issues, and concerned about others. The document also lists some skills that good citizens possess, such as critical thinking, analyzing information, and participating in discussions. Good citizens are described as open-minded, tolerant, and respectful of justice and democracy. The document concludes by outlining an assignment for students to undertake a 10-hour community service activity and write a reflection on their experience.
To prepare a clean, quality weld seam, it’s important to have a superior edge, and the correct preparation sequence prevents costly rework. The NorBevel-6 and NorBevel-12 range of machines and accessories are new additions to Norton’s metal fabrication solutions.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
The document discusses open loop transfer functions and stability analysis using Nyquist plots. It begins with an outline of topics including partial fraction expansion, open loop systems, Nyquist plots, and stability criteria. It then provides examples of using partial fraction expansion to decompose transfer functions with real distinct roots, complex conjugate roots, and repeated roots. The document explains open loop and closed loop system nomenclature. It introduces the Nyquist stability criterion, which involves plotting the open loop transfer function on the Nyquist plot and checking if it encircles the critical point at -1.
This document discusses deadbeat response design for digital control systems. It covers:
1. Designing controllers to achieve a deadbeat response when plant poles and zeros are inside the unit circle. The controller must cancel plant poles to achieve zero steady state error within a finite number of samples.
2. Examples where the controller achieves a deadbeat response of 1 sample for a step input and 2 samples for a ramp input.
3. Considerations for designing deadbeat responses when some plant poles and zeros are on or outside the unit circle, where imperfect cancellation could lead to instability. The controller must not cancel these poles and zeros.
4. Achieving a deadbeat response in sampled data control systems without
The document discusses state space modeling of dynamic systems. It introduces state space analysis as a modern technique for modeling nonlinear, time-varying, and multi-input multi-output systems. The key concepts of state variables, state vector, and state space representations using state equations and output equations are explained. Examples of modeling RL, RC, and RLC circuits using state space approaches are provided to illustrate the technique.
This document discusses different models for representing digital control systems and filters in MATLAB, including transfer functions, zero-pole-gain, state space, partial fraction expansion, second-order sections, lattice structure, and convolution matrix representations. It provides examples of converting between these different representations, such as using the poly and roots functions to convert between polynomial and zero-pole-gain forms. The toolbox functions residuez, tf2latc, and convmtx are also demonstrated for partial fraction expansion, converting to lattice structure, and generating a convolution matrix respectively.
Here are the steps to solve for the transfer function G(s) = X2(s)/F(s) for the given system:
1. Draw the free body diagrams for both masses M1 and M2 showing all the forces acting on each mass.
2. Write the Newton's second law equation for M1:
(M1s2 + f1vs + k1)X1(s) - k2(X1(s) - X2(s)) = 0
3. Write the Newton's second law equation for M2:
-k2(X1(s) - X2(s)) + (M2s2 + f2vs + k
This document discusses properties of the Laplace transform. It begins by defining the Laplace transform and common notation used. It then provides examples of Laplace transforms for some common functions like the impulse function, unit step function, ramp function, sine and cosine functions, and exponential functions. It also discusses rules like linearity and time shifting. The document concludes by summarizing some of the key Laplace transforms and properties discussed.
Introduction, Types of Stable System, Routh-Hurwitz Stability Criterion, Disadvantages of Hurwitz Criterion, Techniques of Routh-Hurwitz criterion, Examples, Special Cases of Routh Array, Advantages and Disadvantages of Routh-Hurwitz Stability Criterion, and examples.
This document discusses the design of digital controllers using root locus analysis. It provides examples of designing proportional controllers for first and second order systems to meet specifications on damping ratio, natural frequency, and settling time. The procedures involve constructing root loci, determining breakaway points and critical gains, and using the MATLAB root locus tool to plot contours and obtain design values for proportional gain.
The document discusses root locus analysis and controller design techniques.
1) Root locus analysis involves plotting the trajectories of a system's closed-loop poles as a parameter (such as a controller gain) is varied. Properties of root loci like pole departure angles and asymptotes are examined.
2) Design specifications like overshoot and settling time can be translated to desired pole regions. A simple proportional controller can place poles near this region.
3) Additional controller types like integral and derivative are introduced to modify the root locus for improved steady-state response or damping. Combining controller types benefits stability.
This document discusses the differences between non-deterministic finite automata (NFA) and deterministic finite automata (DFA). It states that NFAs can recognize the union, concatenation, and Kleene closure of regular languages. It also notes that any NFA can be converted to an equivalent DFA by constructing transition sets representing the lambda-closures of NFA states. The conversion procedure involves identifying the lambda-closure of each NFA state and making those closures the states of the DFA. Transitions between DFA states are determined by the possible transitions between NFA lambda-closures on given input symbols. This ensures the resulting DFA recognizes the same language as the original NFA.
This chapter discusses transient response in control systems. It describes how to determine the time response from a transfer function using poles and zeros. For a first order system, the chapter defines the time constant, rise time and settling time. For a second order system, it defines damping ratio, percent overshoot, settling time and peak time. The chapter also discusses higher order systems and how to approximate them as second order systems. Exercises are provided to analyze systems and design feedback control systems based on desired transient response specifications.
The document discusses root locus analysis, a technique for analyzing the stability and transient response of control systems. It provides rules for sketching root loci, including that branches represent closed-loop poles and the locus is symmetric about the real axis. The document also describes refining the root locus sketch by finding the imaginary axis crossing, angles of departure and arrival, and approximating higher-order systems as second-order. An example problem is given to apply these techniques.
1) A Petri net is a graphical modeling tool used to represent systems with concurrent, asynchronous components. It consists of places, transitions, and arcs between them.
2) The document discusses using Petri nets to model resource sharing between two processors to avoid deadlocks. It defines deadlocks and describes the banker's algorithm for deadlock avoidance.
3) The banker's algorithm works by ensuring the system is always in a safe state where all processes can complete even if maximum resources are requested. It maintains data structures to track available, allocated, and needed resources.
This document discusses closed loop control systems. It begins by recapping the components of a closed loop system including a plant, reference input, sensor, controller, and actuator. The plant being considered is a linear time invariant system represented by differential equations. The document then discusses representing the closed loop system using transfer functions for each component in a block diagram. It explains how to calculate the closed loop transfer function and closed loop poles, which must lie in the left half of the complex plane for stability. Finally, it notes the process is similar for multi-input multi-output systems using transfer function matrices.
Invited talk held by Karsten Wolf on June 26, 2007 on the 28th International Conference on Application and Theory of Petri Nets and Other Models of Concurrency (PETRI NETS 2007) in Siedlce, Poland.
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
Digital control systems (dcs) lecture 18-19-20Ali Rind
This document discusses digital control systems and related topics such as difference equations, z-transforms, and mapping between the s-plane and z-plane. It begins with an outline of topics to be covered including difference equations, z-transforms, inverse z-transforms, and the relationship between the s-plane and z-plane. Examples are provided to illustrate difference equations, z-transforms, and mapping poles between the two planes. Standard z-transforms of discrete-time signals like the unit impulse and sampled step are also defined.
This document discusses responsible citizenship and outlines some of the key characteristics and responsibilities of good citizens. It notes that citizenship involves being a member of a community and working to make that community a better place. Good citizens are aware of their rights and responsibilities, informed about social and political issues, and concerned about others. The document also lists some skills that good citizens possess, such as critical thinking, analyzing information, and participating in discussions. Good citizens are described as open-minded, tolerant, and respectful of justice and democracy. The document concludes by outlining an assignment for students to undertake a 10-hour community service activity and write a reflection on their experience.
To prepare a clean, quality weld seam, it’s important to have a superior edge, and the correct preparation sequence prevents costly rework. The NorBevel-6 and NorBevel-12 range of machines and accessories are new additions to Norton’s metal fabrication solutions.
Norton has introduced two new bevelling machine ranges, the NorBevel-6 and NorBevel-12, which allow for faster, easier, and more accurate bevelling and rounding of metal compared to traditional methods. The electric and pneumatic NorBevel machines are compact and ergonomic, can bevel metal up to depths of 6mm or 12mm respectively, and include interchangeable heads for different radii and angles. The document provides details on the specifications and features of the various NorBevel machine models.
L’après-midi, visite de New Delhi – Sur le Rajpath, vous passerez devant les bâtiments du gouvernement de l’ère britannique, dessinés par Lutyens, y compris l’India Gate (Porte de l’Inde) et le Palais présidentiel.
This document provides information about purchasing a 3Com 3C12091 LINKBUILDER ECS REMOTE MODULE from Launch 3 Telecom. It describes how to purchase the product via phone, email, or by sending a request for quote online. It also provides details about payment methods, same-day shipping and order tracking, warranty, and additional services offered by Launch 3 Telecom such as repairs, maintenance contracts, de-installation, and recycling.
This document provides information about purchasing a 3Com 3C125A SuperStack II Hub 100 Management Unit from Launch 3 Telecom. It describes how to purchase the product via phone, email, or by submitting a request for quote online. It also details Launch 3 Telecom's payment options, same-day shipping and tracking, warranty, and additional services offered like repairs.
This document provides information about purchasing a 3Com 3C39036 36-port switch, including:
- Contact and payment details for purchasing the switch
- Same-day shipping and order tracking procedures
- Warranty and customer service information for the switch
This document discusses convolutional codes. It defines basic concepts like constraint length and generator polynomials that define convolutional codes. It describes representations of convolutional codes using state diagrams and trellis diagrams. It also discusses decoding convolutional codes using the Viterbi algorithm, which finds the most likely path through the trellis. The document concludes by discussing properties of convolutional codes like free distance, which is the minimum Hamming distance between codewords.
The document discusses iteration spaces in compiler design. It describes how to construct iteration spaces from loop nests by representing them mathematically using integer matrices and vectors. It also covers controlling the execution order of iterations by changing the ordering of loop indices, projecting iteration spaces to lower dimensions to generate loop bounds, and changing the axes used to sweep through an iteration space.
Analysis and Design of Control System using Root LocusSiyum Tsega Balcha
Root locus analysis is a powerful tool in control systems engineering used to analyze the behavior of a system's closed-loop poles as a function of a parameter, typically a controller gain. It provides engineers with valuable insights into how changing system parameters affect stability and performance, helping them design robust and stable control systems. Let's explore the key concepts, techniques, and practical implications of root locus analysis. At its core, root locus analysis focuses on the movement of the closed-loop poles in the complex plane as a control parameter varies. These poles represent the characteristic equation's roots, which determine the system's stability and transient response. By examining the pole locations as the parameter changes, engineers can gain a deeper understanding of the system's behavior and make informed design decisions.
The document discusses optimization problems and graph search algorithms. It begins by defining an optimization problem and providing examples like the longest common subsequence problem and course scheduling problem. It then defines graph search algorithms and discusses types of graphs. It explains the generic search algorithm and ingredients like the loop invariant. It provides details on specific graph search algorithms like breadth-first search, Dijkstra's shortest path algorithm, and depth-first search. It includes pseudocode for the algorithms.
This document provides lecture notes on dataflow analysis techniques. It introduces liveness analysis, neededness analysis, and reaching definitions analysis. Liveness analysis is extended to handle memory references. Neededness analysis is introduced to identify dead code by determining which variables are needed, rather than just live. Reaching definitions analysis is a forward dataflow analysis used for optimizations like constant propagation. Examples are provided and the analysis rules for each technique are specified.
Support Vector Machine (SVM) is a supervised machine learning algorithm that can be used for both classification and regression tasks. It works by finding a hyperplane in an N-dimensional space that distinctly classifies the data points. SVM finds this optimal separation by maximizing the margin between the two classes. The algorithm transforms the data into a higher dimension using kernels to handle non-linear classification. Popular kernels include polynomial kernels and Gaussian radial basis function kernels. SVMs are effective because they use a convex optimization which guarantees a global optimal solution and avoids local optima issues.
Bounded model checking encodes executions of a system up to a bounded length k as a propositional formula along with a property violation. If the formula is satisfiable, a counterexample is found; if unsatisfiable, no counterexample up to length k exists. The technique leverages efficient SAT solvers but is incomplete. Modern SAT solvers use conflict-driven learning and backtracking based on the Davis-Putnam-Logemann-Loveland algorithm to efficiently solve the propositional formulas generated by bounded model checking.
The document discusses abstraction in programming. It defines abstraction as focusing on essential characteristics and filtering out unnecessary details to simplify concepts. Abstraction allows for generalization by eliminating details to create generic representations that can apply to many specific situations. While abstraction simplifies concepts, it also introduces challenges in accurately representing real-world concepts and ensuring different abstraction levels are consistent.
An Efficient And Safe Framework For Solving Optimization ProblemsLisa Muthukumar
This document describes a new optimization framework called QuadOpt that combines interval analysis techniques with safe linear relaxations to provide rigorous and efficient global optimization. QuadOpt uses consistency techniques from QuadSolver to reduce variable domains and computes a safe lower bound on a linear relaxation of the problem. It performs branch and bound search to rigorously bound the global optimum. Experimental results on test problems show that QuadOpt provides certified solutions with fewer splits than other rigorous methods while being faster than nonsafe solvers.
This document provides information on combinational logic circuits and summarizes steps for analyzing combinational logic problems using truth tables and Karnaugh maps. It begins by defining combinational circuits as those whose outputs solely depend on current inputs, as opposed to sequential circuits which use memory elements. It then provides examples of writing truth tables and deriving Boolean expressions from problem statements. The document also covers standard forms of sum of products and product of sums, and methods for simplifying expressions using Karnaugh maps including grouping cells and rules for grouping.
This document provides information on combinational logic circuits and techniques for analyzing them, including:
1. Combinational circuits have outputs that solely depend on current inputs, unlike sequential circuits which use memory elements.
2. Truth tables are used to represent the relationships between inputs and outputs, and techniques like Karnaugh maps can simplify Boolean expressions.
3. Karnaugh maps arrange minterms or maxterms in a grid, allowing groups of redundant variables to be identified and simplified. Standard forms like sum of products can be plotted and minimized on the map.
White Box testing by Pankaj Thakur, NITTTR ChandigarhPankaj Thakur
The document provides an overview of white box testing techniques. It begins with an introduction to white box testing and its various forms. It then compares white box and black box testing and discusses cyclomatic complexity. The document outlines several white box testing techniques including control flow testing, data flow testing, slice based testing, and mutation testing. It discusses the advantages and disadvantages of white box testing and provides examples to illustrate control flow testing, branch coverage, condition coverage, and path coverage. The document also briefly covers loop testing and data flow testing.
Verilog Modeling of Basic Combinational Logic. This class notes present main combinational building blocks including multiplexers, decoders, adders, and comparators. Basic if-then-else, case, and loop constructs are also described.
BCA DATA STRUCTURES ALGORITHMS AND PRELIMINARIES MRS SOWMYA JYOTHISowmya Jyothi
This document discusses algorithms and their notations. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. Algorithms use three main types of logic: sequential flow, conditional flow, and repetitive flow. Conditional flow uses if structures like single alternative (if/then), double alternative (if/then else), and multiple alternative (if/else if/else). Repetitive flow uses repeat structures like repeat-for (with an index from start to end) and repeat-while (with a condition). Proper comments, variable names, assignments, inputs/outputs, and procedures are also discussed for writing clear algorithms.
This document discusses test generation for digital circuits. It covers fault detection and location in digital systems, as well as various test generation methods for combinational and sequential logic circuits. For combinational circuits, it describes path sensitization and Boolean difference methods. For sequential circuits, it discusses converting the circuit to combinational form and verifying the state table. The document also discusses design for testability, including testability measures and techniques like LSSD. Reed-Muller expansion is presented as a method to derive and implement logic functions.
This document provides an overview of control structures and flow altering statements in Ruby. It discusses conditionals like if/else, unless, case; loops like while, until, for/in; iterators and enumerable objects; and flow altering statements like return, break, next, redo. Conditionals and loops allow altering the sequential execution of code. Iterators and enumerables provide concise ways to iterate over objects. Statements like return and break change the normal flow of control.
IntroductionTopological sorting is a common operation performed .docxmariuse18nolet
Introduction
Topological sorting is a common operation performed on directed acyclic graphs (dags). It arises in numerous applications including task scheduling, serialization and dependency determinations, and graph drawing. The goal of this project is to manipulate graph representations, perform topological sorting on a dag, and use the topological numbers in a longest path computation as well as in a heuristic solving a scheduling problem.
Degree-based Topological Sort
Given is a dag G=(V,E), V={0, 1, … ,n-1}, having n vertices and m edges. The graph is represented in the form of adjacency lists. For each vertex u, Aout[u] points to a list containing all vertices v such that <u,v> is an edge from u to v and Ain[u] points to a list containing all vertices v such that <v,u> is an edge from v to u. The length of list Aout[u] is the out-degree of node u and the length of list Ain[u] is the in-degree of node u.
In any dag there exists at least one vertex having in-degree 0 (called the sources) and at least one vertex with out-degree 0 (called the sinks). The dags used in the project can have an arbitrary number of sources, but will always have one sink. The dag shown in Figure 1 has two sources (vertices A and G) and one sink (vertex F).
Figure 1: A 7-vertex dag with two sources and one sink.
Topological sorting assigns to each vertex an integer (also called its topological number). See Chapter 4.2 in the text for details. The topological numbers represent a linear ordering of the vertices such that for every edge <u,v> vertex u appears before v in the ordering. For the graph in Figure 1, the ordering A G B D C E F is a topological ordering and the topological numbers are A=0, B=2, C=4, D=3, E=5, F=6, G=1. Note that topological orderings are not unique.
We note that the textbook presents topological sorting as an application of DFS (Algorithm 4.5 in Chapter 4.2). Another approach is a non-recursive implementation based on a degree-driven graph exploration. In this project you need to implement the degree-based topological sorting approach described below.
All source vertices are ready to be placed. For Figure 1, vertices A and G can be placed right away. Once a vertex u has all the vertices incident to its incoming edges placed, u can be placed. This observation forms the basis of degree-based topological sorting algorithm.
The algorithm uses a queue TSQ. Whether one uses a FIFO or a LIFO (i.e., stack) queue does not matter for determining a topological ordering (different queues will produce different orderings). The topological numbers are maintained in an array, say array T.
The high-level description of the degree-driven algorithm is as follows:
1 Make all initializations; includes a variable count set to 0 and placing all sources in queue TSQ.
2 u = dequeue() and we set T[u]=count
3 All of u’s out-going edges are examined. For each edge <u,v>, reduce the number of edges into v by 1. Once all incoming edges of vertex v have received topolo.
An array is a fixed size sequence of elements of the same type. It allows direct access to elements by their index position. Arrays can store multiple values in an organized way and are a fundamental abstract data type. The document then provides algorithms for matrix addition, subtraction, and multiplication using two-dimensional arrays. It also discusses sparse matrices and different types of control statements like if/else, loops (while, do-while, for), and nested structures.
This document provides an overview of discrete-time systems and models. It begins by explaining that digital control involves systems that are updated at discrete time instants, which can be described by discrete-time models. The chapter objectives are then listed, which include explaining difference equations, obtaining z-transforms, solving difference equations, and more. An example is provided to illustrate how a discrete-time model arises from an analog system with piecewise constant control. Key concepts such as difference equations, the z-transform, and its properties are then defined and discussed.
Lect 7: Verilog Behavioral model for Absolute BeginnersDr.YNM
The document discusses Verilog behavioral modeling, which provides the highest level of abstraction for designing complex systems through procedural statements. It describes two main procedural blocks - the always block and initial block. The always block uses procedural assignments to continuously update variables, while the initial block executes stimulus code once at startup for simulation testing. Several procedural statements like if/else, case, and forever are also covered.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
2. Contents
Introduction
Elimination of Global Common Sub-expressions
Copy Propagation
Detection of Loop-Invariant Computations
Performing Code Motion
Alternative Code Motion Strategies
Maintaining Data-Flow Information After Code
Motion
Elimination of Induction Variables
3. What is code optimization?
• The code produced by the straight forward compiling
algorithms can often be made to run faster or take less
space or both.
• This improvement is achieved by program transformations
that are traditionally called optimizations.
• Machine independent optimizations are program
transformations that improve the target code without
taking into consideration any properties of the target
machine.
• Machine dependent optimizations are based on register
allocation and utilization of special machine-instruction
sequences.
4. Why Code-improving Transformation?
Code transformations are performed for a
variety of reasons among which are:
• To reduce the size of the code
• To reduce the running time of the
program
• To take advantage of machine idioms
5. Criteria for code improvement
transformations:
Simply stated , the best program transformations are
those that yield the most benefit for the least effort.
• The transformation must preserve the meaning of
programs. That is, the optimization must not change
the output produced by a program for a given input, or
cause an error.
• A transformation must, on the average, speed up
programs by a measurable amount.
• The transformation must be worth the effort.
6. Global Common Sub expression Elimination
• Performed on flow graph
• Requires available expression information
• In addition to finding what expressions are available
at the endpoints of basic blocks, we need to know
where each of those expressions was most recently
evaluated (which block and which position within that
block).
7. ALGORITHM: Global common sub- expression elimination.
INPUT: A flow graph with available expression information.
OUTPUT: A revised flow graph.
METHOD: For every statement s of the form x := y+z^6 such that y+z is
available at the beginning of block and neither y nor r z is defined prior to
statement s in that block, do the following:
• To discover the evaluations of y+z that reach s’s block, we follow flow graph
edges, searching backward from s’s block. However, we do not go through any
block that evaluates y+z. The last evaluation of y+z in each block encountered
is an evaluation of y+z that reaches s.
• Create new variable u.
• Replace each statement w: =y+z found in (1) by
u : = y + z
w : = u
• Replace statement s by x:=u.
9. Copy Propagation
• Performed on flow graph.
• Given copy statement x=y and use w=x, we can replace w=x
with w=y only if the following conditions are met:
1. x=y must be the only definition of x reaching w=x
-This can be determined through ud-chains
2. There may be no definitions of y on any path from
x=y to w=x.
-Use iterative data flow analysis to solve this.
-Even, better, use iterative data flow analysis to solve both
problems at the same time.
10. ALGORITHM: Copy propagation.
INPUT: a flow graph G, with ud-chains giving the definitions reaching
block B, and with c in[B] representing the solution to equations that is the
set of copies x:=y that reach block B along every path, with no assignment
to x or y following the last occurrence of x:=y on the path. We also need
ud-chains giving the uses of each definition.
OUTPUT: A revised flow graph.
METHOD: For each copy s : x:=y do the following:
• Determine those uses of x that are reached by this definition of
namely, s: x: =y.
• Determine whether for every use of x found in (1) , s is in c in[B],
where B is the block of this particular use, and moreover, no
definitions of x or y occur prior to this use of x within B. Recall that if
s is in c in[B]then s is the only definition of x that reaches B.
• If s meets the conditions of (2), then remove s and replace all uses of
x found in (1) by y.
11.
12. Detection of loop-invariant computations
• Ud-chains can be used to detect those computations in a loop that are
loop-invariant, that is, whose value does not change as long as control
stays within the loop.
• Loop is a region consisting of set of blocks with a header that dominates
all the other blocks, so the only way to enter the loop is through the
header.
• If an assignment x := y+z is at a position in the loop where all possible
definitions of y and z are outside the loop, then y+z is loop-invariant
because its value will be the same each time x:=y+z is encountered.
• Having recognized that value of x will not change, consider v := x+w,
where w could only have been defined outside the loop, then x+w is also
loop-invariant.
13. ALGORITHM: Detection of loop-invariant computations.
INPUT: A loop L consisting of a set of basic blocks, each block containing
sequence of three-address statements. We assume ud-chains are available
for the Individual statements.
OUTPUT: the set of three-address statements that compute the same
value each time executed, from the time control enters the loop L until
control next leaves L.
METHOD: we shall give a rather informal specification of the algorithm,
trusting that the principles will be clear.
• Mark “invariant” those statements whose operands are all either
constant or have all their reaching definitions outside L.
• Repeat step (3) until at some repetition no new statements are marked
“invariant”.
• Mark “invariant” all those statements not previously so marked all of
whose operands either are constant, have all their reaching definitions
outside L, or have exactly one reaching definition, and that definition is a
statement in L marked invariant.
14. Performing Code Motion
Having found the invariant statements within a loop, we can apply to
some of them an optimization known as code motion, in which the
statements are moved to pre-header of the loop.
The following three conditions ensure that code motion does not
change what the program computes. Consider s: x: =y+z.
• The block containing s dominates all exit nodes of the loop, where an
exit of a loop is a node with a successor not in the loop.
• There is no other statement in the loop that assigns to x. Again, if x
is a temporary assigned only once, this condition is surely satisfied
and need not be changed.
• No use of x in the loop is reached by any definition of x other than
s. This condition too will be satisfied, normally, if x is temporary.
15.
16. ALGORITHM: Code motion.
INPUT: A loop L with ud-chaining information and dominator information.
OUTPUT: A revised version of the loop with a pre-header and some statements
moved to the pre-header.
METHOD:
• Use loop-invariant computation algorithm to find loop-invariant statements.
• For each statement s defining x found in step(1), check:
i) That it is in a block that dominates all exits of L,
ii) That x is not defined elsewhere in L, and
iii) That all uses in L of x can only be reached by the definition of x in
statement s.
• Move, in the order found by loop-invariant algorithm, each statement s found
in (1) and meeting conditions (2i), (2ii), (2iii) , to a newly created pre-header,
provided any operands of s that are defined in loop L have previously had
their definition statements moved to the pre-header.
17. Alternative Code Motion Strategies
The condition (1) can be relaxed if we are willing to take the risk that we
may actually increase the running time of the program a bit; of course, we
never change what the program computes. The relaxed version of code
motion condition (1) is that we may move a statement s assigning x only if:
1’. The block containing s either dominates all exists of the loop, or x is not
used outside the loop. For example, if x is a temporary variable, we can be
sure that the value will be used only in its own block.
• If code motion algorithm is modified to use condition (1’), occasionally
the running time will increase, but we can expect to do reasonably well
on the average.
• The modified algorithm may move to pre-header certain computations
that may not be executed in the loop. Not only does this risk slowing
down the program significantly, it may also cause an error in certain
circumstances.
18. • Even if none of the conditions of (2i), (2ii), (2iii) of code motion
algorithm are met by an assignment x: =y+z, we can still take the
computation y+z outside a loop. Create a new temporary t, and set t: =y+z
in the pre-header. Then replace x: =y+z by x: =t in the loop.
• In many cases we can propagate out the copy statement x: = t.
Maintaining data-flow information after code
motion
• The transformations of code motion algorithm do not change ud-chaining
information, since by condition (2i), (2ii), and (2iii), all uses of the
variable assigned by a moved statement s that were reached by s are
still reached by s from its new position.
• Definitions of variables used by s are either outside L, in which case they
reach the preheader, or they are inside L, in which case by step (3) they
were moved to pre-header ahead of s.
19. • If the ud-chains are represented by lists of pointers to pointers
to statements, we can maintain ud-chains when we move
statement s by simply changing the pointer to s when we move it.
That is, we create for each statement s pointer ps, which always
points to s.
• We put the pointer on each ud-chain containing s. Then, no matter
where we move s, we have only to change ps , regardless of how
many ud-chains s is on.
• The dominator information is changed slightly by code motion.
The pre-header is now the immediate dominator of the header,
and the immediate dominator of the pre-header is the node that
formerly was the immediate dominator of the header. That is, the
pre-header is inserted into the dominator tree as the parent of
the header.
20. Elimination of induction variable
A variable x is called an induction variable of a loop L if every time the
variable x changes values, it is incremented or decremented by some
constant. Often, an induction variable is incremented by the same
constant each time around the loop, as in a loop headed by for i := 1 to 10.
• However, our methods deal with variables that are incremented or
decremented zero, one, two, or more times as we go around a loop. The
number of changes to an induction variable may even differ at
different iterations.
• A common situation is one in which an induction variable, say i, indexes
an array, and some other induction variable, say t, whose value is a
linear function of i, is the actual offset used to access the array.
• Often, the only use made of i is in the test for loop termination. We
can then get rid of i by replacing its test by one on t.
• We shall look for basic induction variables, which are those variables i
whose only assignments within loop L are of the form i := i+c or i-c,
where c is a constant.
21. ALGORITHM: Elimination of induction variables.
INPUT: A loop L with reaching definition information, loop-invariant
computation information and live variable information.
OUTPUT: A revised loop.
METHOD:
• Consider each basic induction variable i whose only uses are to
compute other induction variables in its family and in conditional
branches. Take some j in i’s family, preferably one such that c and
d in its triple are as simple as possible and modify each test that i
appears in to use j instead. We assume in the following that c is
positive. A test of the form ‘if i relop x goto B’, where x is not an
induction variable, is replaced by
r := c*x /* r := x if c is 1. */
r := r+d /* omit if d is 0 */
22. • if j relop r goto B where, r is a new temporary. The case ‘if x relop i goto
B’ is handled analogously. If there are two induction variables i1 and i2 in
the test if i1 relop i2 goto B, then we check if both i1 and i2 can be
replaced. The easy case is when we have j1 with triple and j2 with triple,
and c1=c2 and d1=d2. Then, i1 relop i2 is equivalent to j1 relop j2.
• Now, consider each induction variable j for which a statement j: =s was
• introduced. First check that there can be no assignment to s between
the
• introduced statement j :=s and any use of j.
• In the usual situation, j is used in the block in which it is defined,
simplifying this check; otherwise, reaching definitions information, plus
some graph analysis is needed to implement the check. Then replace all
uses of j by uses of s and delete statement j: =s