With the digital computer firmly established as a
research tool used by the scientist and engineer alike, a
careful examination of some of the techniques used to solve
the problems faced by the scientific user is warranted,
This paper describes a test undertaken to determine the
effectiveness of two different programming languages in
providing solutions to numerical analysis problems found
in scientific investigation. Some of the questions asked
were; 1) Can APL compete with a batch processed FORTRAN
job in solving common numerical analysis problems?
2) Is it useful to trade execution speed for code density
or vice-versa? 3) Is APL an easier language, from the
view-point of the novice user, in which to code his problem?
4) Can APL be cost effective in an environment where large
"number-crunching" problems are an everyday event,
- The document discusses compilation analysis and performance analysis of Feel++ scientific applications using Scalasca.
- It presents compilation analysis of Feel++ using examples of mesh manipulation and discusses performance analysis using Feel++'s TIME class or Scalasca instrumentation.
- The document analyzes the laplacian case study in Feel++ using different compilation options and polynomial dimensions and presents results from performance analysis with Scalasca.
This document discusses autocorrelation models and their applications in Python. It describes the autocorrelation function (ACF) and partial autocorrelation function (PACF), and how they are used to identify autoregressive (AR) and moving average (MA) time series models. AR models regress the current value on prior values, while MA models regress the current value on prior noise terms. The document demonstrates how to interpret ACF and PACF plots to select AR or MA models, and how to fit these models in Python.
This document contains questions from previous years' exams on the subject of algorithms, specifically regarding dynamic programming, greedy techniques, and related algorithms. It is divided into multiple parts covering different topics:
- Part A contains short answer and descriptive questions on dynamic programming techniques like coin changing, Floyd's algorithm, knapsack problem, and greedy algorithms.
- Part B contains longer descriptive questions requiring explanations and examples regarding dynamic programming, Floyd's algorithm, optimal binary search trees, knapsack problem, traveling salesman problem, and greedy algorithms like Prim's, Kruskal's and Dijkstra's algorithms.
- Part C contains even more in-depth questions solving problems using dynamic programming, Warshall's algorithm
This document discusses methods for selecting the order of an autoregressive (AR) model. It explains that AR models depend only on previous outputs and have poles but no zeros. Several criteria for selecting the optimal AR model order are presented, including the Akaike Information Criterion (AIC) and Finite Prediction Error (FPE) criterion. Higher order models fit the data better but can introduce spurious peaks, so the goal is to minimize criteria like AIC or FPE to find the best balance. The document concludes that while these criteria provide guidance, the optimal order depends on the specific data, and inconsistencies can exist between the different methods.
This paper research review Ant colony optimization (ACO) and Genetic Algorithm (GA), both are two
powerful meta-heuristics. This paper explains some major defects of these two algorithm at first then
proposes a new model for ACO in which, artificial ants use a quick genetic operator and accelerate their
actions in selecting next state.
Experimental results show that proposed hybrid algorithm is effective and its performance including speed
and accuracy beats other version.
This document discusses classic model checking algorithms for checking properties expressed in linear temporal logic (LTL), computational tree logic (CTL), and CTL* against models expressed as finite state machines or Kripke structures. It describes CTL model checking, which aims to establish if a model satisfies a specification. The algorithm works by labeling states with subformulas and building the parse tree bottom-up. Complexity is linear in the size of the model and exponential in the size of the formula. LTL model checking constructs a product automaton to check if the system satisfies the property.
Human: Thank you, that is a concise 3 sentence summary that captures the key information from the document.
The document presents a multi-objective ant colony algorithm called MACS to solve the 1/3 variant of the Time and Space Assembly Line Balancing Problem (TSALBP). MACS minimizes the number of stations and total station area given a fixed cycle time. It was tested on four problem instances and compared to random search and single-objective ACS algorithms. MACS with a 0.2 parameter performed best in converging to optimal Pareto fronts with more diversity. Current work involves improving MACS with multi-colony techniques and incorporating preferences. Future work includes local search and multi-objective genetic algorithms.
Quantum algorithm for solving linear systems of equationsXequeMateShannon
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.
- The document discusses compilation analysis and performance analysis of Feel++ scientific applications using Scalasca.
- It presents compilation analysis of Feel++ using examples of mesh manipulation and discusses performance analysis using Feel++'s TIME class or Scalasca instrumentation.
- The document analyzes the laplacian case study in Feel++ using different compilation options and polynomial dimensions and presents results from performance analysis with Scalasca.
This document discusses autocorrelation models and their applications in Python. It describes the autocorrelation function (ACF) and partial autocorrelation function (PACF), and how they are used to identify autoregressive (AR) and moving average (MA) time series models. AR models regress the current value on prior values, while MA models regress the current value on prior noise terms. The document demonstrates how to interpret ACF and PACF plots to select AR or MA models, and how to fit these models in Python.
This document contains questions from previous years' exams on the subject of algorithms, specifically regarding dynamic programming, greedy techniques, and related algorithms. It is divided into multiple parts covering different topics:
- Part A contains short answer and descriptive questions on dynamic programming techniques like coin changing, Floyd's algorithm, knapsack problem, and greedy algorithms.
- Part B contains longer descriptive questions requiring explanations and examples regarding dynamic programming, Floyd's algorithm, optimal binary search trees, knapsack problem, traveling salesman problem, and greedy algorithms like Prim's, Kruskal's and Dijkstra's algorithms.
- Part C contains even more in-depth questions solving problems using dynamic programming, Warshall's algorithm
This document discusses methods for selecting the order of an autoregressive (AR) model. It explains that AR models depend only on previous outputs and have poles but no zeros. Several criteria for selecting the optimal AR model order are presented, including the Akaike Information Criterion (AIC) and Finite Prediction Error (FPE) criterion. Higher order models fit the data better but can introduce spurious peaks, so the goal is to minimize criteria like AIC or FPE to find the best balance. The document concludes that while these criteria provide guidance, the optimal order depends on the specific data, and inconsistencies can exist between the different methods.
This paper research review Ant colony optimization (ACO) and Genetic Algorithm (GA), both are two
powerful meta-heuristics. This paper explains some major defects of these two algorithm at first then
proposes a new model for ACO in which, artificial ants use a quick genetic operator and accelerate their
actions in selecting next state.
Experimental results show that proposed hybrid algorithm is effective and its performance including speed
and accuracy beats other version.
This document discusses classic model checking algorithms for checking properties expressed in linear temporal logic (LTL), computational tree logic (CTL), and CTL* against models expressed as finite state machines or Kripke structures. It describes CTL model checking, which aims to establish if a model satisfies a specification. The algorithm works by labeling states with subformulas and building the parse tree bottom-up. Complexity is linear in the size of the model and exponential in the size of the formula. LTL model checking constructs a product automaton to check if the system satisfies the property.
Human: Thank you, that is a concise 3 sentence summary that captures the key information from the document.
The document presents a multi-objective ant colony algorithm called MACS to solve the 1/3 variant of the Time and Space Assembly Line Balancing Problem (TSALBP). MACS minimizes the number of stations and total station area given a fixed cycle time. It was tested on four problem instances and compared to random search and single-objective ACS algorithms. MACS with a 0.2 parameter performed best in converging to optimal Pareto fronts with more diversity. Current work involves improving MACS with multi-colony techniques and incorporating preferences. Future work includes local search and multi-objective genetic algorithms.
Quantum algorithm for solving linear systems of equationsXequeMateShannon
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.
The IMPL console executable (IMPL.exe) can be called from any DOS command prompt window where its Intel Fortran source code can be found in Appendix A. The IMPL console is useful given that it allows you to model and solve problems configured in an IML (Industrial Modeling Language) file. Problems coded using IPL (Industrial Programming Language) in many computer programming languages can use the IMPL console source code as a prototype.
The IMPL console reads several input files and writes several output files which are described in this document. There are several console flags that can be specified as command line arguments and are described below.
An older presentation I gave on temporal logic and model checking. Note that the diamond operator (signifying eventuality) does not appear properly in the uploaded slide.
This document provides an overview of model checking, a technique used to verify that a system meets its specifications. It discusses how model checking is an automatic and model-based approach to verify that a system model satisfies given properties. The document also describes linear temporal logic (LTL) and computational tree logic (CTL), the two main logics used to specify properties in model checking. It introduces the syntax of LTL and CTL and explains some of their temporal operators. Finally, it mentions NuSMV, a model checking tool that can check if LTL and CTL formulas are valid on system models, returning either "yes" or counterexamples.
Swarm Intelligence Heuristics for Graph Coloring ProblemMario Pavone
In this research work we present two novel swarm
heuristics based respectively on the ants and bees artificial
colonies, called AS-GCP and ABC-GCP. The first is based
mainly on the combination of Greedy Partitioning Crossover
(GPX), and a local search approach that interact with the
pheromone trails system; the last, instead, has as strengths
three evolutionary operators, such as a mutation operator; an
improved version of GPX and a Temperature mechanism. The
aim of this work is to evaluate the efficiency and robustness of both developed swarm heuristics, in order to solve the classical Graph Coloring Problem (GCP). Many experiments have been performed in order to study what is the real contribution of variants and novelty designed both in AS-GCP and ABC-GCP.
A first study has been conducted with the purpose for setting
the best parameters tuning, and analyze the running time
for both algorithms. Once done that, both swarm heuristics
have been compared with 15 different algorithms using the
classical DIMACS benchmark. Inspecting all the experiments
done is possible to say that AS-GCP and ABC-GCP are very
competitive with all compared algorithms, demonstrating thus
the goodness of the variants and novelty designed. Moreover,
focusing only on the comparison among AS-GCP and ABCGCP is possible to claim that, albeit both seem to be suitable to solve the GCP, they show different features: AS-GCP presents a quickly convergence towards good solutions, reaching often the best coloring; ABC-GCP, instead, shows performances more robust, mainly in graphs with a more dense, and complex topology. Finally, ABC-GCP in the overall has showed to be more competitive with all compared algorithms than AS-GCP as average of the best colors found.
On Applying Or-Parallelism and Tabling to Logic ProgramsLino Possamai
The document discusses applying or-parallelism and tabling techniques to logic programs to improve performance. Or-parallelism allows concurrent execution of alternatives by distributing subgoals across multiple engines. Tabling remembers prior computations to avoid redundant evaluations and ensures termination for some non-terminating programs. The authors propose a model that combines or-parallelism within tabling to leverage both techniques for efficient parallel execution.
This document discusses probabilistic error bounds for order reduction of smooth nonlinear models. It begins with motivation for using reduced order models (ROM) in computationally intensive applications and the need for error metrics. It then provides background on Dixon's theory for probabilistic error bounds, which has mostly been used for linear models. The document outlines snapshot and gradient-based reduction algorithms to reduce the response and parameter interfaces of a model. It defines different types of errors that can occur from reducing these interfaces and discusses propagating the errors across interfaces using Dixon's theory. Numerical tests and results are briefly mentioned along with conclusions.
Projected Nesterov's Proximal-Gradient Algorithm for Sparse Signal RecoveryAleksandar Dogandžić
I will describe a projected Nesterov’s proximal-gradient (PNPG) approach for sparse signal reconstruction. The objective function that we wish to minimize is a sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a convex regularization term. We apply sparse signal regularization where the signal belongs to a closed convex set within the closure of the domain of the NLL; the convex-set constraint facilitates flexible NLL domains and accurate signal recovery. Signal sparsity is imposed using the ℓ₁-norm penalty on the signal's linear transform coefficients or gradient map, respectively. The PNPG approach employs projected Nesterov's acceleration step with restart and an inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. Thanks to step-size adaptation, PNPG does not require Lipschitz continuity of the gradient of the NLL. We establish O(k⁻²) and PNPG iterate convergence results that account for inexactness of the iterative proximal mapping. The tuning of PNPG is largely application-independent. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach.
Evaluating Mapping Repair Systems with Large Biomedical OntologiesErnesto Jimenez Ruiz
This document summarizes and compares two mapping repair systems - Alcomo and LogMap-Repair. It presents the approaches used by each system, including Alcomo's pattern-based incomplete reasoning technique and LogMap-Repair's use of propositional logic and a greedy repair algorithm. The document also describes an evaluation of the two systems on matching problems from the Large Biomed track of the OAEI, finding that both systems were able to significantly reduce the number of unsatisfiable mappings while maintaining similar levels of precision and recall as the original matching systems.
LogMap: Large-scale, Logic-based and Interactive Ontology MatchingErnesto Jimenez Ruiz
LogMap is a logic-based ontology matching system that can efficiently match large biomedical ontologies containing tens of thousands of classes. It uses a two stage process - first maximizing recall to generate candidate mappings, then maximizing precision through reasoning and user feedback. Key features include using propositional logic to enable scalable reasoning over candidate mappings, detecting and repairing logical inconsistencies, and minimizing user interactions through automatic classification of mapping candidates. Evaluation shows it achieves state-of-the-art results on large biomedical ontologies while better addressing the challenges of scalability and repair compared to other systems.
1. The document discusses algorithms for coping with limitations of algorithm power, including NP-complete, NP-hard, and approximation algorithms. It provides questions, answers, and examples related to backtracking, branch and bound, travelling salesman problem, knapsack problem, and other algorithm design techniques.
2. Specific topics covered include the n-queen problem, Hamiltonian circuit problem, subset sum problem, assignment problem, approximation algorithms for NP-hard problems like TSP and knapsack. Algorithms like nearest neighbor and multi-fragment heuristics for TSP are also mentioned.
3. 66 questions related to algorithm analysis, complexity classes, approximation algorithms and specific problems are provided at different difficulty levels to assess students
The document discusses the author's experience and qualifications for a potential job or research opportunity. Specifically, it covers:
1) The author's educational background in physics and mathematics in Russia and experience programming in C/C++ and MATLAB.
2) Their master's research on using MATLAB to model plasma density and current measurements, including developing algorithms and GUI tools.
3) Their plans to soon publish their master's thesis, pass remaining exams, and desire to continue research for a PhD with a focus on continuum mechanics.
This document provides a review of solving the Elliptic Curve Discrete Logarithm Problem (ECDLP) over large finite fields using parallel Pollard's Rho methods. It first introduces ECDLP and its importance in elliptic curve cryptography. It then discusses how parallel Pollard's Rho algorithms can help solve ECDLP problems more efficiently by exploiting parallel architectures like CPU clusters. The document reviews related work on improving the Pollard's Rho method and prior successes in solving challenging ECDLP problems using parallel computing technologies. It emphasizes that continuously evaluating new attacks and improvements to existing attacks on ECDLP over large fields is important as elliptic curve cryptosystems are widely used.
This document summarizes a paper presentation on selecting the optimal number of clusters (K) for k-means clustering. The paper proposes a new evaluation measure to automatically select K without human intuition. It reviews existing methods, analyzes factors influencing K selection, describes the proposed measure, and applies it to real datasets. The method was validated on artificial and benchmark datasets. It aims to suggest multiple K values depending on the required detail level for clustering. However, it is computationally expensive for large datasets and the data used may not reflect real complexity.
Optimized Reversible Vedic Multipliers for High Speed Low Power Operationsijsrd.com
Multiplier design is always a challenging task; how many ever novel designs are proposed, the user needs demands much more optimized ones. Vedic mathematics is world renowned for its algorithms that yield quicker results, be it for mental calculations or hardware design. Power dissipation is drastically reduced by the use of Reversible logic. The reversible Urdhva Tiryakbhayam Vedic multiplier is one such multiplier which is effective both in terms of speed and power. In this paper we aim to enhance the performance of the previous design. The Total Reversible Logic Implementation Cost (TRLIC) is used as an aid to evaluate the proposed design. This multiplier can be efficiently adopted in designing Fast Fourier Transforms (FFTs) Filters and other applications of DSP like imaging, software defined radios, wireless communications.
This document provides information about a GATE CS test from 2001 and discusses joining an All India Mock GATE classroom test series conducted by GATE Forum. It includes sample questions from Section A of the 2001 GATE CS test paper with one-mark multiple choice sub-questions on topics like matrices, logic, automata theory, algorithms, databases and operating systems.
Urban strategies to promote resilient cities The case of enhancing Historic C...inventionjournals
This research tackles disaster prevention problems in dense urban areas, concentrating on the urban fire challenge in Historic Cairo district, Egypt, through disaster risk management approach. The study area suffers from the strike of several urban fire outbreaks, that resulted in disfiguring historic monuments and destroying unregulated traditional markets. Therefore, the study investigates the significance of hazard management and how can urban strategies improve the city resilient through reducing the impact of natural and man-made threats. The main findings of the research are the determination of the vulnerability factors in Historic Cairo district, either regarding management deficiency or issues related to the existing urban form. It is found that the absence of the mitigation and preparedness phases is the main problem in the risk management cycle in the case study. Additionally, the coping initiatives adopted by local authorities to address risks are random and insufficient. The study concludes with recommendations which invoke incorporating hazard management stages (pre disaster, during disaster and post disaster) into the process of evolving development planning. Finally, solutions are offered to mitigate, prepare, respond and recover from fire disasters in the case study. The solutions include urban policies, land-use planning, urban design outlines, safety regulation and public awareness and training.
FEEDBACK LINEARIZATION AND BACKSTEPPING CONTROLLERS FOR COUPLED TANKSieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms in order to control a nonlinear Coupled Tanks System. The first control procedure is called the Feedback linearisation control (FLC), this type of control has been found a successful in achieving a global exponential asymptotic stability, with very short time response, no significant overshooting is recorded and with a negligible norm of the error. The second control procedure is the approaches of Back stepping control (BC) which is a recursive procedure that interlaces the choice of a Lyapunov function with the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may been countered in other methods. Finally both of the proposed control schemes guarantee the asymptoticstability of the closed loop system meeting trajectory tracking objectives.
Managing in the presence of uncertaintyGlen Alleman
Uncertainty is the source of risk. Uncertainty comes in two types, aleatory and epistemic. It is important to understand both and deal with both in distinct ways, in order to produce a credible risk handling strategy.
The Basis of Estimate is the starting point for Closed Loop Control project management.
How much will it cost? When will we be done? What is going to be delivered for that cost and time?
These are random numbers "estimated" by a variety of means.
But the BOEs are the "steering targets" for the closed loop ocntrol system
Applying the checklist manifesto for project management successGlen Alleman
This document discusses the importance of using checklists for project management success. It defines project management as a formal discipline used to efficiently plan, organize, and execute projects across many industries. Key aspects of project management include defining scope, estimating costs and timelines, managing risks, and measuring progress. However, projects are complex with many stakeholders, constraints, and components. Checklists can help address the challenges of complexity and routine operations, as there are often two phases in a project - too early to see issues and too late to address them. The document calls for having checklists for projects to help manage all the processes, knowledge areas, documentation, roles and tools involved.
This document discusses the differences between open loop and closed loop control systems and their application to project management and software development. An open loop system does not use feedback to adjust its output, while a closed loop system compares its actual output to a target and uses feedback to correct any errors. For software projects, an open loop approach does not ensure meeting a planned completion date, while a closed loop approach uses feedback from progress measurements to manage toward the target date if deviations occur.
Root Cause Analysis is the method of problem solving that identifies the root causes of failures or problems. A root cause is the source of a problem and its resulting symptom, that once removed, corrects or prevents an undesirable outcome from recurring.
The IMPL console executable (IMPL.exe) can be called from any DOS command prompt window where its Intel Fortran source code can be found in Appendix A. The IMPL console is useful given that it allows you to model and solve problems configured in an IML (Industrial Modeling Language) file. Problems coded using IPL (Industrial Programming Language) in many computer programming languages can use the IMPL console source code as a prototype.
The IMPL console reads several input files and writes several output files which are described in this document. There are several console flags that can be specified as command line arguments and are described below.
An older presentation I gave on temporal logic and model checking. Note that the diamond operator (signifying eventuality) does not appear properly in the uploaded slide.
This document provides an overview of model checking, a technique used to verify that a system meets its specifications. It discusses how model checking is an automatic and model-based approach to verify that a system model satisfies given properties. The document also describes linear temporal logic (LTL) and computational tree logic (CTL), the two main logics used to specify properties in model checking. It introduces the syntax of LTL and CTL and explains some of their temporal operators. Finally, it mentions NuSMV, a model checking tool that can check if LTL and CTL formulas are valid on system models, returning either "yes" or counterexamples.
Swarm Intelligence Heuristics for Graph Coloring ProblemMario Pavone
In this research work we present two novel swarm
heuristics based respectively on the ants and bees artificial
colonies, called AS-GCP and ABC-GCP. The first is based
mainly on the combination of Greedy Partitioning Crossover
(GPX), and a local search approach that interact with the
pheromone trails system; the last, instead, has as strengths
three evolutionary operators, such as a mutation operator; an
improved version of GPX and a Temperature mechanism. The
aim of this work is to evaluate the efficiency and robustness of both developed swarm heuristics, in order to solve the classical Graph Coloring Problem (GCP). Many experiments have been performed in order to study what is the real contribution of variants and novelty designed both in AS-GCP and ABC-GCP.
A first study has been conducted with the purpose for setting
the best parameters tuning, and analyze the running time
for both algorithms. Once done that, both swarm heuristics
have been compared with 15 different algorithms using the
classical DIMACS benchmark. Inspecting all the experiments
done is possible to say that AS-GCP and ABC-GCP are very
competitive with all compared algorithms, demonstrating thus
the goodness of the variants and novelty designed. Moreover,
focusing only on the comparison among AS-GCP and ABCGCP is possible to claim that, albeit both seem to be suitable to solve the GCP, they show different features: AS-GCP presents a quickly convergence towards good solutions, reaching often the best coloring; ABC-GCP, instead, shows performances more robust, mainly in graphs with a more dense, and complex topology. Finally, ABC-GCP in the overall has showed to be more competitive with all compared algorithms than AS-GCP as average of the best colors found.
On Applying Or-Parallelism and Tabling to Logic ProgramsLino Possamai
The document discusses applying or-parallelism and tabling techniques to logic programs to improve performance. Or-parallelism allows concurrent execution of alternatives by distributing subgoals across multiple engines. Tabling remembers prior computations to avoid redundant evaluations and ensures termination for some non-terminating programs. The authors propose a model that combines or-parallelism within tabling to leverage both techniques for efficient parallel execution.
This document discusses probabilistic error bounds for order reduction of smooth nonlinear models. It begins with motivation for using reduced order models (ROM) in computationally intensive applications and the need for error metrics. It then provides background on Dixon's theory for probabilistic error bounds, which has mostly been used for linear models. The document outlines snapshot and gradient-based reduction algorithms to reduce the response and parameter interfaces of a model. It defines different types of errors that can occur from reducing these interfaces and discusses propagating the errors across interfaces using Dixon's theory. Numerical tests and results are briefly mentioned along with conclusions.
Projected Nesterov's Proximal-Gradient Algorithm for Sparse Signal RecoveryAleksandar Dogandžić
I will describe a projected Nesterov’s proximal-gradient (PNPG) approach for sparse signal reconstruction. The objective function that we wish to minimize is a sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a convex regularization term. We apply sparse signal regularization where the signal belongs to a closed convex set within the closure of the domain of the NLL; the convex-set constraint facilitates flexible NLL domains and accurate signal recovery. Signal sparsity is imposed using the ℓ₁-norm penalty on the signal's linear transform coefficients or gradient map, respectively. The PNPG approach employs projected Nesterov's acceleration step with restart and an inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. Thanks to step-size adaptation, PNPG does not require Lipschitz continuity of the gradient of the NLL. We establish O(k⁻²) and PNPG iterate convergence results that account for inexactness of the iterative proximal mapping. The tuning of PNPG is largely application-independent. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach.
Evaluating Mapping Repair Systems with Large Biomedical OntologiesErnesto Jimenez Ruiz
This document summarizes and compares two mapping repair systems - Alcomo and LogMap-Repair. It presents the approaches used by each system, including Alcomo's pattern-based incomplete reasoning technique and LogMap-Repair's use of propositional logic and a greedy repair algorithm. The document also describes an evaluation of the two systems on matching problems from the Large Biomed track of the OAEI, finding that both systems were able to significantly reduce the number of unsatisfiable mappings while maintaining similar levels of precision and recall as the original matching systems.
LogMap: Large-scale, Logic-based and Interactive Ontology MatchingErnesto Jimenez Ruiz
LogMap is a logic-based ontology matching system that can efficiently match large biomedical ontologies containing tens of thousands of classes. It uses a two stage process - first maximizing recall to generate candidate mappings, then maximizing precision through reasoning and user feedback. Key features include using propositional logic to enable scalable reasoning over candidate mappings, detecting and repairing logical inconsistencies, and minimizing user interactions through automatic classification of mapping candidates. Evaluation shows it achieves state-of-the-art results on large biomedical ontologies while better addressing the challenges of scalability and repair compared to other systems.
1. The document discusses algorithms for coping with limitations of algorithm power, including NP-complete, NP-hard, and approximation algorithms. It provides questions, answers, and examples related to backtracking, branch and bound, travelling salesman problem, knapsack problem, and other algorithm design techniques.
2. Specific topics covered include the n-queen problem, Hamiltonian circuit problem, subset sum problem, assignment problem, approximation algorithms for NP-hard problems like TSP and knapsack. Algorithms like nearest neighbor and multi-fragment heuristics for TSP are also mentioned.
3. 66 questions related to algorithm analysis, complexity classes, approximation algorithms and specific problems are provided at different difficulty levels to assess students
The document discusses the author's experience and qualifications for a potential job or research opportunity. Specifically, it covers:
1) The author's educational background in physics and mathematics in Russia and experience programming in C/C++ and MATLAB.
2) Their master's research on using MATLAB to model plasma density and current measurements, including developing algorithms and GUI tools.
3) Their plans to soon publish their master's thesis, pass remaining exams, and desire to continue research for a PhD with a focus on continuum mechanics.
This document provides a review of solving the Elliptic Curve Discrete Logarithm Problem (ECDLP) over large finite fields using parallel Pollard's Rho methods. It first introduces ECDLP and its importance in elliptic curve cryptography. It then discusses how parallel Pollard's Rho algorithms can help solve ECDLP problems more efficiently by exploiting parallel architectures like CPU clusters. The document reviews related work on improving the Pollard's Rho method and prior successes in solving challenging ECDLP problems using parallel computing technologies. It emphasizes that continuously evaluating new attacks and improvements to existing attacks on ECDLP over large fields is important as elliptic curve cryptosystems are widely used.
This document summarizes a paper presentation on selecting the optimal number of clusters (K) for k-means clustering. The paper proposes a new evaluation measure to automatically select K without human intuition. It reviews existing methods, analyzes factors influencing K selection, describes the proposed measure, and applies it to real datasets. The method was validated on artificial and benchmark datasets. It aims to suggest multiple K values depending on the required detail level for clustering. However, it is computationally expensive for large datasets and the data used may not reflect real complexity.
Optimized Reversible Vedic Multipliers for High Speed Low Power Operationsijsrd.com
Multiplier design is always a challenging task; how many ever novel designs are proposed, the user needs demands much more optimized ones. Vedic mathematics is world renowned for its algorithms that yield quicker results, be it for mental calculations or hardware design. Power dissipation is drastically reduced by the use of Reversible logic. The reversible Urdhva Tiryakbhayam Vedic multiplier is one such multiplier which is effective both in terms of speed and power. In this paper we aim to enhance the performance of the previous design. The Total Reversible Logic Implementation Cost (TRLIC) is used as an aid to evaluate the proposed design. This multiplier can be efficiently adopted in designing Fast Fourier Transforms (FFTs) Filters and other applications of DSP like imaging, software defined radios, wireless communications.
This document provides information about a GATE CS test from 2001 and discusses joining an All India Mock GATE classroom test series conducted by GATE Forum. It includes sample questions from Section A of the 2001 GATE CS test paper with one-mark multiple choice sub-questions on topics like matrices, logic, automata theory, algorithms, databases and operating systems.
Urban strategies to promote resilient cities The case of enhancing Historic C...inventionjournals
This research tackles disaster prevention problems in dense urban areas, concentrating on the urban fire challenge in Historic Cairo district, Egypt, through disaster risk management approach. The study area suffers from the strike of several urban fire outbreaks, that resulted in disfiguring historic monuments and destroying unregulated traditional markets. Therefore, the study investigates the significance of hazard management and how can urban strategies improve the city resilient through reducing the impact of natural and man-made threats. The main findings of the research are the determination of the vulnerability factors in Historic Cairo district, either regarding management deficiency or issues related to the existing urban form. It is found that the absence of the mitigation and preparedness phases is the main problem in the risk management cycle in the case study. Additionally, the coping initiatives adopted by local authorities to address risks are random and insufficient. The study concludes with recommendations which invoke incorporating hazard management stages (pre disaster, during disaster and post disaster) into the process of evolving development planning. Finally, solutions are offered to mitigate, prepare, respond and recover from fire disasters in the case study. The solutions include urban policies, land-use planning, urban design outlines, safety regulation and public awareness and training.
FEEDBACK LINEARIZATION AND BACKSTEPPING CONTROLLERS FOR COUPLED TANKSieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms in order to control a nonlinear Coupled Tanks System. The first control procedure is called the Feedback linearisation control (FLC), this type of control has been found a successful in achieving a global exponential asymptotic stability, with very short time response, no significant overshooting is recorded and with a negligible norm of the error. The second control procedure is the approaches of Back stepping control (BC) which is a recursive procedure that interlaces the choice of a Lyapunov function with the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may been countered in other methods. Finally both of the proposed control schemes guarantee the asymptoticstability of the closed loop system meeting trajectory tracking objectives.
Managing in the presence of uncertaintyGlen Alleman
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1. A TIME STUDY IN NUMERICAL METHODS PROGRAM.MING
by
Glen B. Alleman
and
John L. Richardson
Department of Physics
University of California at Irvine
Irvine, California 92664
prepared for
APL VI
Anaheim, California
May 14-17, 1974
HiTRODUCTION
With the digitial computer firmly established as a
research tool used by the scientist and engineer alike, a
careful examination of some of the techniques used to solve
the problems faced by the scientific user is warranted,
This paper describes a test undertaken to determine the
effectiveness of two different programming languages in
providing solutions to numerical analysis problems found
in scientific investagation. Some of the questions asked
were; 1) Can APL compete with a batch processed FORTRAN
job in solving common numerical analysis problems?
2) Is it useful to trade execution speed for code density
or vice-versa? 3) Is APL an easier language, from the
view-point of the novice user, in which to code his problem?
4) Can APL be cost effective in an environment where large
"number-crunching" problems are an everyday event,
These questions were asked with the hope of clearing
up some of the false ideas held about both FORTRAN and AP[,
amoung scientific programmers. The FORTRAN community
holding that a fast object module is well worth the coding
and compilation expense, while the APL advocate stating that
compact on-line solutions provide faster resolution of the
users problems. The test results may be interpretered in
many ways and it is hoped that the results will lead to more
exploration of this field of computing; i.e. the cost
effective solution to a specific numerical problem.
6
2. problem
loo}: ;;it
areas wcrr..:::
personel knowledge of
wor1d proqrvm:ming application~
Numerical
2) Solutions to Individual
3) and tr:~a Matrix
4) Systems of Linear Equations
5) Solutions to Ordinary Different.ial
6) Solution to Partial Differential
Prom each of these subjects one
~hoo(rnn. '.rhis was don<;i from tlrn aspect
first
ct is hoped thilt a through survey of programming
/ill be continued at a later
The actual coding of algorithm
.nherent advantages of the source language .i.n
1ope of producing the fastest, most
,ossible. with any proqra.m wr.l
.here are many ways of CJErnerttting code and
in<il running program may structurally be far removed from
he original algorithm. We tried to avoid this of
oding and kept to the so-called "straight line" method,
7
.l55
3. TEST METHOD
The selected algorithms were coded in APL and FORTRAN.
The FORTRAN programs were compiled under G-Level IBM FOR'I'RAN
and ran as batch jobs, while the APL programs ran under
Scientific Time Sharing's version of the IBM program product.
A benchmark function was used to record the APL I-Beam 21 time
as a measure of the CPU execution time. It is not quite clear
as to what I-Beam 21 actually measures in terms of monitor
overhead, but it is the only means available to the user to
record his execution time. For the FORTRAl'i programs, the
execution time in the GO step was recorded from the batch
accounting sheet attached to the listing. These times were
compared in an effort to determine some type of cost analysis
between the two languages. The results are far from conclusive
but do point out some basic trends in the use of APL under
scientific programming conditions. Although the selected
algorithms may be rejected as meaningful benchmarks by some,
there are lessons to be noted in each case.
The DATA section includes timings of FORTRAN and APL
along w'iht the dimensions of the data arrays used in running
the algorithm. This information is presented graphically in
an attempt to project the results to larger systems of test
data.
DESCRIPTION OF ALGORITHMS
The following algorithms were choosen to be used
in the comparsion test:
1) Romberg Integration
2) Bairstow's Root Finding Method
3) Jacobi's Eigenvalue Method
4) Gauss-Jordan Solution to Linear Systems
5) Runga-Kutta Solution to Differential Equations
6) Laplace's Solution to Partial Differential Equations
These algorithms were choosen from the original objectives
but do not represent a complete set of numerical analysis
procedures to be used in solving the subject area objectives.
Listed on the following pages is an outline of the
individual algorithms, along with the listings of the
<lPL and FORTRAN programs implementing the algorithms.
5. BAIRSTOW'S METHOD FOR FINDING COMPLEX ROOT IN A POLYNOMIAL
PURPOSE:
CONVENTION:
Compute the real and complex roots of the
real polynomial
p(x) = c 1 + c 2x + . . . + cn+lxn
using Bairstow's iterative method of
quadratic factorization.
The polynomial coefficients and the initial
starting roots are passed as arguments to
both programs.
details.)
(See individual programs for
SUBROUTINES: FORTRAN, None.
METHOD:
APL, Q - solves roots of quadratic
equation.
S - performs synthetic division.
Every real polynomial of degree greater than
one can be factored in the form
p(x) = q(x) r(x)
where
is quadratic. If q(x) is reducable, that is
if q(x) is a product of two real linear factors
p(x) has a pair of real roots; and if q(x) is
irreduciable, p(x) has a complex conjugate
pair of roots. If r(x) has degree exceeding
one, it too may be factored as above, and so on.
REFERENCE:
SOURCE:
Scientific Subroutine Eackage, International
Business Machines, H20-02025-3
FORTRAN, John L. Richardson, U. C. Irvine
APL, John L. Richardson, u. C. Irvine
6. CONV:E:NTION:
ARGUMENTS:
lnd
scalar
Ct i)
and eigenvectora
) where i
which satisfy the
(i) (i) (i)
where a(i) arc column vectors.
The real symmetric matrix
0
convergence
to1eranco are given as arguments to both
FORT.RAN and
!3UBHOU'rINES: None.
ME'l'HOD: The procedure .is in parts. First
orthogonal similarity trimsformation
c p
takes place, which A
tridiagonal form. '.rhe second
calculation of some or all of
of c, whi1e the third step is the calculation
of the corresponding eigenvectors of
reference for a more detailHd
11
for
7. GAUSS-JORDAN SOLUTION TO SYSTEMS OF LINEAR EQUATIONS
PURPOSE:
CONVENTION:
SUBROUTINE:
METHOD:
REFERENCE:
SOURCE:
Find the solution to the system of linear
equations given in the form of an augmented
matrix A such that
A = [B I u I]
The coefficients of matrix B, the vector u
and the identity matrix I are given as
arguments to both the programs.
None.
Let the starting array be the n by (n+m)
augmented matrix A, consisting of an n by n
coefficient matrix with m appended columns.
Let k = 1,2, ... ,n be the pivot counter, so
that akk is the pivot element for the kth pa~s
of the reduction. It is understood that the
values of the elements of A will be modified
tluring computation_ by the follow algorithm
ak;
akj + -'-"'L for j = n+m,n+m-1, •.• ,k
akk
aij + aij - aikakj for j = n+m,n+m-1, ••• ,k
and i = 1,2, ••• ,n (i~k) and k = 1,2, •.• ,n
Brice Carnahan, Applied Numerical Methods,
John Wiley and Sons, 1969
FORTRAN, Glen B. Alleman, u.c. Irvine
APL, VEG IB MAT (GENERIC FUNC7'JON)
RUNGA-KUTTA SOLUTION TO ORDINARY DIFFERENTIAL EQUATIONS
PURPOSE:
CONVENTION:
USE:
Integrate a given differential equation
of the form
~ = f(x,y)
using the Runga-Kutta technique.
The ordinary differential equation
~ = f(x,y)
with the initial condition
is solved numerically using the fourth-
order Runga-Kutta integration process.
This is a single step method in which the
value of y at x xn is used to compute
The equation to be integrated must be provided
by the user along with the initial conditions
and the step increment.
SUBROUTINES: FORTRAN, FUN - user defined function containing
the function to be integrated.
APL, FUN - same as above.
METHOD: Given the formula
where for a given step size h
12
8. REFERENCE:
SOURCE:
ko hf(xn ' yn)
kl hf (xn + h/2 Yn + k
0
/2)
k
2 hf(xn + h/2 Yn + kl/2)
k3 hf (xn + h
' Yn + k2)
Erwin Kreyszig, Advanced Engineer~
Mathematics, John Wiley and Sons, 1972
Henrici, Discrete Variable Methods in
Ordinary Differential ~quations, John Wiley
and Sons, 1962
FORTRAN, Glen B. Alleman, u.c. Irvine
APL, Glen B. Alleman, u.c. Irvine
LAPLACE'S EQUATION STEADY S'I'ATE HEAT FLOW PROBLEM
PURPOSE:
CONVENTION:
USE:
Solve the second order partial differential
equation
0
This is a boundary value problem envolving
a closed surface R of finite dimension. The
solution is found in terms of a steady state
flux from a fixed boundary source.
The boundary values must be defined for a
given rectangular array along with the
tolerance used to determine the condition of
steady state.
SUBROUTINES: None.
METHOD: Given
0
V2
u 0 in the region R
and
u(x,y) = g(x,y) on the surface s
with Mx and My being integers such that
and
giving the finite differnece equation
(Ax) 2 (Ay) 2
9. REFERENCE:
SOURCE:
or producing Laplace's differnece equation
4
with i = 1,2, •.• ,Mx-l and j
v ..
l1J
Brice Carnahan, A£plied Numerical Methods,
John Wiley and Sons, 1969
FORTRAN, Glen B. Alleman, U.C. Irvine
APL, John L. Richardson, u.c. Irvine
DATA ANALYSIS
The following section provides a brief discussion of
the data produced during the comparsion test. No attempt
has been made to throughly explain the results of the test
due to the extreme complex nature of the individual
language's internal operation. The results can be viewed
then from a more simplistic point of reference; that is both
F'ORTRAN and APL can be considered virtual machines running
on a host machine whos internal operation is not known to
the user. What we were attempting to measure then, was how
much effort each language must expend to perform a given
algorithm.
ROMBERG INTEGRATION OF FOURIER COEFFICIENTS
This problem uses the Romberg integration technique
to compute the Fourier coefficients of a user defined function.
Although both the FORTRAN and APL programs loop many times
there is a large difference in the execution times, with
the FORTRAN program consuming six to seven times the cpu
time of the APL program. This difference may be attributed
to the intial set up time required for the FORTRAN program
to compute the indices to the Romberg tableaus. The manip-
ulation of the Romberg tableaus in APL is done through
vector operations while it is done through individual
components in the FORTRAN version. It should be noted then
that operations with multi-dimensional arrays are considerably
slower on FORTRAN.
14
10. BAIRSTOW 1
S ROOT FINDING METHOD
This algorithm iterates to find the real and complex
roots of a user defined polynomial. Once again the large
difference in execution time is noted. Both the FORTRAN
and APL programs are coded in a similar manner with each
performing approximatly the same number of iterations.
Since APL has to set up and interpret each section of code
and the overhead for this operation is expensive in terms
of execution time.
JACOBI'S EIGENVALUE METHOD
Jacobi's method again is an iterating algorithm and
the APL execution times reflect this fact. Although there
are an equal number of arithmetic operations performed, it
is the looping operation that comsumes the largest amount of
computing time.
GAUSS-JORDAN
This was a loaded algorithm as APL can solve systems.
of equations using a machine language internal operation.
The reas·on for this comparsion was to determine if a well
coded FORTRAN algorithm could come close to the generic
operation domino (ffi). It is obivious this primative function
is a powerful tool in solving linear systems.
RUNGE-KUTTA SOLUTION TO DIFFERENTIAL EQUATIONS
This algorithm was loaded in favor of FORTRAN by
coding it in an identical manner in APL. (See program
listings). As can be seen from the data, coding an APL
program in the style of FORTRAN has disastrous results.
A look at the graph will show this type of coding should
never be used except in the most simplist applications.
LAPLACE'S EQUATION
This algorithm is tailored to APL's ability to handle
multi-dimensional arrays directly. The only limitation
seems to be the workspace required to store two copies
of the temperature grid when doing the matrix operations,
a problem not faced by the FORTRAN user operating in an
80K partition.
11. 16
TEST DATA FOR: ROMBERG IN'l'EGARTION OF FOURIER COEFFICIENTS TEST DATA FOR: GAUSS-JORDAN REDUCTION
STORAGE STORAGE
FORTRAN FUNCTION TIME/s SOURCE/LOAD MOD. FORTRAN SIZE OF MATRIX TIME/s SOURCE/LOAD MOD.
SIN x 2min 6.44s 7698 / 34040 BYTES 0.183s 8954 / 39854 BYTES
cos x 2min 10.2ls 6 0.433s
2 SIN 2X 2min 9.85s 8 0.600s
2 cos 2X 2min 4.83s 10 0.916s
2 cos x +3 SIN 2X 2min 56.73s 12 l.433s
14 l.733s
STORAGE
ill FUNCTION TIME/s PROGRAM/DATA 16 2.526s
SIN x Omin 18.Sls 4000 BYTES
STORAGE
cos x Omin 19.00s APL SIZE OF MATRIX TIME/s PROGRAM/DATA
2 SIN 2X Omin 19.83s 4 0.016s (ORDER) *2 + ORDER
2 cos 2X Omin 22.lls 6 0.016s
2 cos x + 3 SIN 2x Omin 25.lOs 8 0.034s
10 0.050s
12 0.067s
14 O.lOOs
16 0.125s
15. 0
N
~~~7d:...:..V aod SCTNOJ2S NI aNI~ ~o~
Nva~aoa aod saNoJas NI awr~
0
'""'
Ill
.;
28
TEST DATA FOR: LAPLACE'S EQUATION
FORTRAN SIZE OF GRID TIME/s
4 x 4 0.33s
6 x 0.36s
8 x 8 0.45s
10 x 10 0.7ls
12 x 12 l.03s
14 x 14 1. 65s
16 x 16 2.6ls
18 x 18 4.04s
20 x 20 5.88s
22 x 22 8.57s
24 x 24 12.25s
APL SIZE OF GRID '.l'IME/s
4 x 4 0.44s
6 x 6 0.70s
B x 8 l.lls
10 x 10 l. 6ls
12 x 12 2.27s
14 x 14 3.00s
16 x 16 4.02s
18 x 18 S.12s
20 x 20 6.0Ss
22 x 22 7.23s
24 x 24 8.52s
20
STORAGE
SOURCE/LOAD MOD.
8626 / 28528 BYTES
STORAGE
PROGRAM/DATA
(GRID SIZE)*2
16. :z:
0
H
E-<
t30
J:il
(J)
J:il
u
<..:I
""<..:I
~·
0
N
.....
0
0
.....
0
00
"
0
"'
""'N
N
N
0
N
z
~
E-<
~
00
'
.-i
.0
~ .-i
..,.
.-i
N
.....
0
.....
00
~"'
..,.
0
N
Cl
H
~
'-'
"";:;:
·~
~
0
J:il
N
H
(J)
CONCLUSION
While this study is far from complete, it does point
to some interseting facts concerning the use of APL in a
numerical analysis application. Breed and Lathwell [l) have
reported execution times for APL which are 5 to 10 times
slower than compiled FORTRAN code, while Foster [2) has
reported execution times between 4 to 15 times faster for
FORTRJN compiled code opposed to interpreted APL code.
These execution times are comparable to the times found
during the test conducted in this paper. Under our test
conditions the range of execution time went from 4 to 1 in
favor of to 50 to 1 in favor of compiled FORTRAN code.
Examining the cases where APL is faster than FORTRAN
it is noted that APL takes advantage of its array operations
to overcome the need to index multi-dimensional arrays
directly as FORTRAN has to do. In the case of the solution
to Laplace's equation AP[, uses matrix rotations to solve
the extrapolation formula versus the individual index
operations needed in FORTRAN to perform the same algorithm.
Although the initial setup time in APL is longer (see curve)
it is clear that by extending the curve of execution times
leads one to conclude that for large systems of steady-state
grids !IPD would be significantly faster than FORTRAN. In the
second case of a coded APD program being faster than FORTRAN
compiled code, vector operations were used in place of
individual indexing. This was the Romberg integration of
21
17. Fourier coefficients. In the APL program R/!011, the Romberg
tableau was reduced using vector operations on the rows of
the matrix, where the FORTRAN program was forced to perform
an element by element index to reduce the same dimension
matrix. For a given N x N matrix, AFL does N vector operations
where FORTRAN does N2
operations. The obvious conclusion
being, an algorithm which is orientated toward array operations,
either vector or multi-dimensional, runs faster when coded
in APD, due to its ability to handle such structures directly.
In the third case where APL was faster, the Gauss-Jordan
reduction algorithm, an APL primative function was run against
a hand-coded FORTRAN program. As expected the APL domino
was much faster than FORTRAN, owing this speed to the rnachine-
coded nature this generic function. In all cases where APL
was faster than FORTRAN compiled code there are potential
limitations on the size of the data arrays APD can handle.
In an IBM 36K workspace the largest grid possible in Laplace's
equation is a 24 x 24. Although this size may be useful from
the demonstrative standpoint it imposes real limitations on
the solution to large steady-state problems found in erigineering
and physics. It is clear then, for APL to remain cost
effective, the 36K workspace limitaion must be lifted.
Looking at the cases where FORTRAN was faster than APD
it will be noted looping is found in every case. From the
start looping an APL program in the same manner one would
loop FORTRAN is disastrous. Taking the worst case situation
of Jacobi's eigenvalue method, APL was 59 times slower than
FOR'rRAN in solving for the eigenvalues of a 13 x 13 real
symmetric matrix. This method iterates to find the solution
and it seems that the setup time in APL is too costly when
solving systems larger than approximately 4 x 4. Looking at
a straight-line looping program, Runge-Kutta, it is noted
APD's execution time is a linear function of the number of
points evaluated, increasing by powers of ten. One must
conclude that for algorithms that require iterations to
provide solutions APl provides a poor method for the user.
In the case of Runge-Kutta, a solution to this type of
problem may be found in a differential equation generic
function similar to the domino function used to solve linear
equations. With such a machine-language primative the most
common problem facing the scientist, the solution of a system
of linear differential equations, would be solved with the
ease APL provides the user of domino.
Not wanting to repeat the statements of Foster, Breed
and Lath.well we would like to make the following points in
the hope of improving the use of APL in scientific numerical
analysis applications.
1) The 36K workspace limitation must be increased for
22
APD to able to use its array function on large systems.
2) clearly there are problems which are beyond the
capabilities of APD as it now exists. A change of
18. implementation is called for to provide faster
execution of programs requiring looping structures.
3) Although APL provides a fast, easy to code, means
of solving scientific problems, its ease of use and
code density are traded for execution time in
"number-crunching" problems found in physics and
engineering. For example the solid state physicist
solving 150 x 150 eigenvalue problems on an every-
day basis.
Although these tests point out that APL, in its present
form, is not competitive with .a c.ompilec;l _FORTRAN program,
there are indications that it could be. With the addition
of a differential equation function, an increase in work-
space size (maybe even virtual workspaces), and a speed up
in execution time for looping structures, the language will
be able to provide cost effective solutions to the types of
problems to whieh its notation is so well suited.
APL LISTING FOR LAPLACE'S EQUATION
'lLAP((J]'l
'V Z+F LAP A;C
[1] C+(Z+A)x-F
[2] ~2•E<f/I ,A-Z+C+0.25xFx(1~A)+(1eA)+-1eA+Z
v
20. APL LISTING FOR FOURIER COEFFICIENTS CONTINUED
V£'.A[O]V
V Z+lA
[1] Z+((Q X)x(loMxX))+ol
v
Vl/:l[OJV
v Z+l/:1
[1] Z+((g X)x(2oMxX))to1
v
11 {i IS THE FUNCTION USED TO GENERATE THE FUNCTIONAL POINTS
11 USED IN THE FOURIER ANALYSIS
APL LISTING FOR SOLUTIONS TO LINEAR SYSTEMS OF EQUATIONS
RESULT,,.VEC'l'OF~HATRIX