This document provides an overview of TensorFlow and how to implement machine learning models using TensorFlow. It discusses:
1) How to install TensorFlow either directly or within a virtual environment.
2) The key concepts of TensorFlow including computational graphs, sessions, placeholders, variables and how they are used to define and run computations.
3) An example one-layer perceptron model for MNIST image classification to demonstrate these concepts in action.
This document provides an overview of the Go programming language. It discusses that Go was initially developed at Google in 2007 and is now an open source language used by many companies. The document then covers Why Go is useful, including its memory management, concurrency support, and cross-platform capabilities. It also summarizes some of Go's basic syntax like packages, functions, variables, types, and control structures. Finally, it discusses some key Go concepts like methods, interfaces, channels, and the net/http package.
(1) The document provides instructions for installing the CounterClockwise plugin for Eclipse to get an IDE for Clojure development. (2) It describes how to create and load Clojure files and launch a REPL for evaluation. (3) The document includes exercises on Clojure basics like functions, macros, and functional programming techniques as well as examples for implementing macros.
This document is the preface to a textbook on algorithmic mathematics. It introduces algorithms as unambiguous instructions to solve mathematical problems. The textbook aims to teach algorithm design and analysis, and provide constructive tools for abstract algebra. Algorithms are written in a simple language using standard notation, with if-statements and while-loops. Exercises similar to exam questions are included to aid learning.
Functions allow programmers to organize and reuse code. There are three types of functions: built-in functions, modules, and user-defined functions. User-defined functions are created using the def keyword and can take parameters and arguments. Functions can return values and have different scopes depending on if a variable is local or global. Recursion is when a function calls itself, and is useful for breaking down complex problems into simpler sub-problems. Common recursive functions calculate factorials, Fibonacci numbers, and generate the Pascal's triangle.
The document is a report on the topic of computer programming and utilization prepared by group C. It discusses functions, including the definition of a function, function examples, benefits of functions, function prototypes, function arguments, and recursion. It provides examples of math library functions, global and local variables, and external variables. It also includes examples of recursive functions to calculate factorials and the Fibonacci series recursively.
This document is the student solutions manual for the 4th edition of the textbook "Essential Mathematics for Economic Analysis" by Knut Sydsæter, Peter Hammond, and Arne Strøm. The manual provides detailed solutions to problems marked in the textbook. The preface states that the solutions should be used along with any abbreviated answers in the textbook. The authors welcome feedback to improve accuracy. The manual is organized by chapter and covers topics like algebra, equations, functions, differentiation, integration, optimization, and matrices.
Computational language have been used in physics research
for many years and there is a plethora of programs and packages on the Web which can be used to solve dierent problems. In this report I trying to use as many of these available solutions as possible and not reinvent the wheel. Some of these packages have been written in C program. As I stated above, physics relies heavily on graphical representations. Usually,the scientist would save the results
from some calculations into a file, which then can be read and used for display by a graphics package like Gnuplot.
This document provides an overview of TensorFlow and how to implement machine learning models using TensorFlow. It discusses:
1) How to install TensorFlow either directly or within a virtual environment.
2) The key concepts of TensorFlow including computational graphs, sessions, placeholders, variables and how they are used to define and run computations.
3) An example one-layer perceptron model for MNIST image classification to demonstrate these concepts in action.
This document provides an overview of the Go programming language. It discusses that Go was initially developed at Google in 2007 and is now an open source language used by many companies. The document then covers Why Go is useful, including its memory management, concurrency support, and cross-platform capabilities. It also summarizes some of Go's basic syntax like packages, functions, variables, types, and control structures. Finally, it discusses some key Go concepts like methods, interfaces, channels, and the net/http package.
(1) The document provides instructions for installing the CounterClockwise plugin for Eclipse to get an IDE for Clojure development. (2) It describes how to create and load Clojure files and launch a REPL for evaluation. (3) The document includes exercises on Clojure basics like functions, macros, and functional programming techniques as well as examples for implementing macros.
This document is the preface to a textbook on algorithmic mathematics. It introduces algorithms as unambiguous instructions to solve mathematical problems. The textbook aims to teach algorithm design and analysis, and provide constructive tools for abstract algebra. Algorithms are written in a simple language using standard notation, with if-statements and while-loops. Exercises similar to exam questions are included to aid learning.
Functions allow programmers to organize and reuse code. There are three types of functions: built-in functions, modules, and user-defined functions. User-defined functions are created using the def keyword and can take parameters and arguments. Functions can return values and have different scopes depending on if a variable is local or global. Recursion is when a function calls itself, and is useful for breaking down complex problems into simpler sub-problems. Common recursive functions calculate factorials, Fibonacci numbers, and generate the Pascal's triangle.
The document is a report on the topic of computer programming and utilization prepared by group C. It discusses functions, including the definition of a function, function examples, benefits of functions, function prototypes, function arguments, and recursion. It provides examples of math library functions, global and local variables, and external variables. It also includes examples of recursive functions to calculate factorials and the Fibonacci series recursively.
This document is the student solutions manual for the 4th edition of the textbook "Essential Mathematics for Economic Analysis" by Knut Sydsæter, Peter Hammond, and Arne Strøm. The manual provides detailed solutions to problems marked in the textbook. The preface states that the solutions should be used along with any abbreviated answers in the textbook. The authors welcome feedback to improve accuracy. The manual is organized by chapter and covers topics like algebra, equations, functions, differentiation, integration, optimization, and matrices.
Computational language have been used in physics research
for many years and there is a plethora of programs and packages on the Web which can be used to solve dierent problems. In this report I trying to use as many of these available solutions as possible and not reinvent the wheel. Some of these packages have been written in C program. As I stated above, physics relies heavily on graphical representations. Usually,the scientist would save the results
from some calculations into a file, which then can be read and used for display by a graphics package like Gnuplot.
This document provides an overview of key concepts in programming and Python. It defines terms like code, syntax, output, console, compiling, interpreting, and variables. It explains Python as an interpreted language and shows examples of printing output, taking user input, performing calculations with numbers and math commands, using variables, and basic control structures like if/else and loops. It also covers data types like integers, floats, strings, lists, and how to modify and format them.
The Trailblazer Ride from the If Jungle into a Civilised Railway Station - Or...Ruby Meditation
Speech of Orban Botond, Ruby Developer at Toptal, at Ruby Meditation 27, Dnipro, 19.05.2019
Slideshare -
Next conference - http://www.rubymeditation.com/
Software development is a domain where everybody can make a beautiful sculpture or can quickly build an unsolvable maze. According to my observations the later happens more often unfortunately.
In the title for my presentation both the If Jungle and the Civilised Railway Station are methaphors representing the opposite ends of the scale of quality code.
In my presentation I am going to present my personal experiences on how to get out from the trap of the if jungle by making the code to adhere to the SRP and DRY principles. I am also going to show the advantages of the Railway Oriented Programing using the 3 different libraries.
The examples are going to be stereotypical errors, fun and easy to follow.
Announcements and conference materials https://www.fb.me/RubyMeditation
News https://twitter.com/RubyMeditation
Photos https://www.instagram.com/RubyMeditation
The stream of Ruby conferences (not just ours) https://t.me/RubyMeditation
Software development is a domain where everybody can make a beautiful sculpture or can quickly build an unsolvable maze. According to my observations the later happens more often unfortunately.
In the title for my presentation both the If Jungle and the Civilised Railway Station are methaphors representing the opposite ends of the scale of quality code.
In this presentation I am going to present my personal experiences on how to get out from the trap of the if jungle by making the code to adhere to the SRP and DRY principles. I am also going to show the advantages of the Railway Oriented Programing using the 3 different libraries.
The examples are going to be stereotypical errors, fun and easy to follow.
This document provides examples of built-in functions and decorators in Python like map, filter, all, any, getattr, hasattr, setattr, callable, isinstance, issubclass, closures, and memoization decorators. It demonstrates how to use these functions and decorators through examples. Built-in functions like map, filter and decorators allow extending functionality of functions. Closures enable functions to remember values in enclosing scopes. The @decorator syntax is demonstrated to be equivalent to applying a function to another function.
This document provides a summary of key concepts in artificial intelligence, organized into the following sections:
1. Reflex-based models such as linear predictors for classification and regression using techniques like loss minimization and regularization.
2. States-based models including search optimization techniques like tree search, graph search, A* search and Markov decision processes.
3. Variables-based models covering constraint satisfaction problems, Bayesian networks and inference.
4. Logic-based models discussing knowledge bases, propositional logic and first-order logic.
The document defines important terms and algorithms at a high level across different areas of AI like supervised and unsupervised learning, neural networks, optimization, probabilistic modeling and logic.
Finding root of equation (numarical method)Rajan Thakkar
▪ The numerical methods provide a real solution to problems where f (x) = 0.
▪ Here we are not only programing these methods but also to analyze which method converges who faster than the other method.
▪ We measure the average time to solve each method (how much time conduct for solving by each method) and the iteration (In how many iterations the method converges.)
▪ Use dynamic input so that user can input its own choice of function
▪ We use the graph for new point so that if f(x)=0 than what is the value x we can easily find out by viewing the graph
▪ In That we also compare two different language which is faster c++ or python.
▪ We conclude that among all the Newton Raphson is efficient method.
▪ Note: calculation time is low in bisection method and give answer in too many iterations but depending upon system newton Raphson is take little bit more time but find the root in few iterations.
Столпы функционального программирования для адептов ООП, Николай МозговойSigma Software
This document provides an overview of functional programming concepts for object-oriented programmers. It discusses the fundamentals of FP including immutability, purity, first-class and higher-order functions, closures, and recursion. It provides examples of these concepts in languages like Lisp, F#, C#, and JavaScript. The document also compares OO and FP concepts and discusses derived FP concepts like partial application, lazy evaluation, and pattern matching.
This document discusses recursion and recursive algorithms. It begins with definitions of recursion and recursive thinking. Examples of recursively defined concepts and recursively structured algorithms are provided, such as defining a list recursively and recursively searching all subfolders of a folder. The document then covers implementing recursion in programming through recursive methods and functions. Examples of recursively implemented algorithms like factorial and power functions are analyzed. Tracing the execution of recursive methods is discussed through activation records and recursion stacks. Finally, searching algorithms like binary search are described, which can be implemented recursively.
Principle of Function Analysis - by Arun Umraossuserd6b1fd
This note explains about functions, type of function, their behaviour, conversion and derivation. This note is best for those who are going to study calculus or phys
Functions
Function
Type of Function
Algebraic Function
Trigonometric Function
Logarithmic Function
Integral Function
Rational Fraction
Rational Function
Explicie & Implicit Function
Unique Values of Function
Odd & Even Function
Properties of Odd-Even Functions
Homogeneous Function
Linear Function
Inverse Function
Sampling of Function
Piece-wise Function
Sketch the Function
Straight Line
Domain & Range
Ordered Pairs
Ordered Pairs from Function
Function in Different Domains
Increasing-Decreasing Function
Modulo Function
Sub-equations of Modulo Function
Inequality
Inequalities
Properties of Inequalities
Transitivity
Addition and subtraction
Multiplication and Division
Additive inverse
Multiplicative Inverse
Interval Notion In Inequality
Answer Set
For Real Numbers
For Integers
Compound Statements
Linear Inequality
Quadratic Inequality
Quotients and Absolute Inequalities
This document discusses various techniques for optimizing Python code to improve performance. It begins by explaining that Python is an interpreted language and is generally slower than compiled languages like C/C++. Several methods for speeding up Python code are then presented: using local variables instead of global variables, leveraging built-in functions, list comprehensions, generator expressions, NumPy for numeric computing, Numba for just-in-time compilation, and algorithm/data structure optimization. Specific code examples are provided to demonstrate how these techniques can significantly reduce runtime. The key message is that with the right optimizations, Python code can achieve speeds comparable to lower-level languages while retaining the benefits of a high-level, interpreted language.
This document introduces vectors and vector calculus. It defines a vector as a directed line segment with a magnitude and direction. Vectors are used to represent physical quantities that have both magnitude and direction, such as velocity, acceleration, and force. The document discusses Euclidean spaces R^2 and R^3, which are used to graph functions of two and three variables. It introduces right-handed coordinate systems for labeling points and directions in three-dimensional space.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where pointing thumb, index, and middle fingers represent the positive x, y, z axes. Vector operations like dot and cross products are introduced.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where the thumb points in the positive z direction and the fingers curl from x to y. Vector operations like the dot and cross products are introduced.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where pointing thumb, index, and middle fingers represent the positive x, y, z axes. Vector operations like dot and cross products are introduced.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where the thumb points in the positive z direction and the fingers curl from x to y. Vector operations like the dot and cross products are introduced.
This document introduces vectors and vector calculus. It defines a vector as a directed line segment with a magnitude and direction. Vectors are used to represent phenomena with both magnitude and direction, such as velocity and force. The document discusses Euclidean spaces R^2 and R^3, which are 2-dimensional and 3-dimensional spaces. It introduces right-handed coordinate systems for labeling points and representing graphs of functions of two or three variables in these spaces.
This document provides an overview of using R for financial modeling. It covers basic R commands for calculations, vectors, matrices, lists, data frames, and importing/exporting data. Graphical functions like plots, bar plots, pie charts, and boxplots are demonstrated. Advanced topics discussed include distributions, parameter estimation, correlations, linear and nonlinear regression, technical analysis packages, and practical exercises involving financial data analysis and modeling.
This document provides an introduction and overview of the contents of a textbook on numerical analysis. The textbook covers topics like differential and difference equations, numerical solutions to differential equations, and numerical linear algebra. Chapter 1 introduces differential equations and discusses methods for solving certain types analytically. It also introduces difference equations. Subsequent chapters will discuss numerical methods for solving differential equations, and numerical techniques for linear algebra problems like solving systems of linear equations.
This document describes a maximum entropy modeling toolkit that provides conditional maximum entropy models, L-BFGS and GIS parameter estimation algorithms, Gaussian prior smoothing, C++ and Python APIs. It discusses building, installing and using the toolkit on various operating systems. It also provides tutorials on maximum entropy modeling concepts and using the toolkit's features and APIs.
This document provides an overview of key concepts in programming and Python. It defines terms like code, syntax, output, console, compiling, interpreting, and variables. It explains Python as an interpreted language and shows examples of printing output, taking user input, performing calculations with numbers and math commands, using variables, and basic control structures like if/else and loops. It also covers data types like integers, floats, strings, lists, and how to modify and format them.
The Trailblazer Ride from the If Jungle into a Civilised Railway Station - Or...Ruby Meditation
Speech of Orban Botond, Ruby Developer at Toptal, at Ruby Meditation 27, Dnipro, 19.05.2019
Slideshare -
Next conference - http://www.rubymeditation.com/
Software development is a domain where everybody can make a beautiful sculpture or can quickly build an unsolvable maze. According to my observations the later happens more often unfortunately.
In the title for my presentation both the If Jungle and the Civilised Railway Station are methaphors representing the opposite ends of the scale of quality code.
In my presentation I am going to present my personal experiences on how to get out from the trap of the if jungle by making the code to adhere to the SRP and DRY principles. I am also going to show the advantages of the Railway Oriented Programing using the 3 different libraries.
The examples are going to be stereotypical errors, fun and easy to follow.
Announcements and conference materials https://www.fb.me/RubyMeditation
News https://twitter.com/RubyMeditation
Photos https://www.instagram.com/RubyMeditation
The stream of Ruby conferences (not just ours) https://t.me/RubyMeditation
Software development is a domain where everybody can make a beautiful sculpture or can quickly build an unsolvable maze. According to my observations the later happens more often unfortunately.
In the title for my presentation both the If Jungle and the Civilised Railway Station are methaphors representing the opposite ends of the scale of quality code.
In this presentation I am going to present my personal experiences on how to get out from the trap of the if jungle by making the code to adhere to the SRP and DRY principles. I am also going to show the advantages of the Railway Oriented Programing using the 3 different libraries.
The examples are going to be stereotypical errors, fun and easy to follow.
This document provides examples of built-in functions and decorators in Python like map, filter, all, any, getattr, hasattr, setattr, callable, isinstance, issubclass, closures, and memoization decorators. It demonstrates how to use these functions and decorators through examples. Built-in functions like map, filter and decorators allow extending functionality of functions. Closures enable functions to remember values in enclosing scopes. The @decorator syntax is demonstrated to be equivalent to applying a function to another function.
This document provides a summary of key concepts in artificial intelligence, organized into the following sections:
1. Reflex-based models such as linear predictors for classification and regression using techniques like loss minimization and regularization.
2. States-based models including search optimization techniques like tree search, graph search, A* search and Markov decision processes.
3. Variables-based models covering constraint satisfaction problems, Bayesian networks and inference.
4. Logic-based models discussing knowledge bases, propositional logic and first-order logic.
The document defines important terms and algorithms at a high level across different areas of AI like supervised and unsupervised learning, neural networks, optimization, probabilistic modeling and logic.
Finding root of equation (numarical method)Rajan Thakkar
▪ The numerical methods provide a real solution to problems where f (x) = 0.
▪ Here we are not only programing these methods but also to analyze which method converges who faster than the other method.
▪ We measure the average time to solve each method (how much time conduct for solving by each method) and the iteration (In how many iterations the method converges.)
▪ Use dynamic input so that user can input its own choice of function
▪ We use the graph for new point so that if f(x)=0 than what is the value x we can easily find out by viewing the graph
▪ In That we also compare two different language which is faster c++ or python.
▪ We conclude that among all the Newton Raphson is efficient method.
▪ Note: calculation time is low in bisection method and give answer in too many iterations but depending upon system newton Raphson is take little bit more time but find the root in few iterations.
Столпы функционального программирования для адептов ООП, Николай МозговойSigma Software
This document provides an overview of functional programming concepts for object-oriented programmers. It discusses the fundamentals of FP including immutability, purity, first-class and higher-order functions, closures, and recursion. It provides examples of these concepts in languages like Lisp, F#, C#, and JavaScript. The document also compares OO and FP concepts and discusses derived FP concepts like partial application, lazy evaluation, and pattern matching.
This document discusses recursion and recursive algorithms. It begins with definitions of recursion and recursive thinking. Examples of recursively defined concepts and recursively structured algorithms are provided, such as defining a list recursively and recursively searching all subfolders of a folder. The document then covers implementing recursion in programming through recursive methods and functions. Examples of recursively implemented algorithms like factorial and power functions are analyzed. Tracing the execution of recursive methods is discussed through activation records and recursion stacks. Finally, searching algorithms like binary search are described, which can be implemented recursively.
Principle of Function Analysis - by Arun Umraossuserd6b1fd
This note explains about functions, type of function, their behaviour, conversion and derivation. This note is best for those who are going to study calculus or phys
Functions
Function
Type of Function
Algebraic Function
Trigonometric Function
Logarithmic Function
Integral Function
Rational Fraction
Rational Function
Explicie & Implicit Function
Unique Values of Function
Odd & Even Function
Properties of Odd-Even Functions
Homogeneous Function
Linear Function
Inverse Function
Sampling of Function
Piece-wise Function
Sketch the Function
Straight Line
Domain & Range
Ordered Pairs
Ordered Pairs from Function
Function in Different Domains
Increasing-Decreasing Function
Modulo Function
Sub-equations of Modulo Function
Inequality
Inequalities
Properties of Inequalities
Transitivity
Addition and subtraction
Multiplication and Division
Additive inverse
Multiplicative Inverse
Interval Notion In Inequality
Answer Set
For Real Numbers
For Integers
Compound Statements
Linear Inequality
Quadratic Inequality
Quotients and Absolute Inequalities
This document discusses various techniques for optimizing Python code to improve performance. It begins by explaining that Python is an interpreted language and is generally slower than compiled languages like C/C++. Several methods for speeding up Python code are then presented: using local variables instead of global variables, leveraging built-in functions, list comprehensions, generator expressions, NumPy for numeric computing, Numba for just-in-time compilation, and algorithm/data structure optimization. Specific code examples are provided to demonstrate how these techniques can significantly reduce runtime. The key message is that with the right optimizations, Python code can achieve speeds comparable to lower-level languages while retaining the benefits of a high-level, interpreted language.
This document introduces vectors and vector calculus. It defines a vector as a directed line segment with a magnitude and direction. Vectors are used to represent physical quantities that have both magnitude and direction, such as velocity, acceleration, and force. The document discusses Euclidean spaces R^2 and R^3, which are used to graph functions of two and three variables. It introduces right-handed coordinate systems for labeling points and directions in three-dimensional space.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where pointing thumb, index, and middle fingers represent the positive x, y, z axes. Vector operations like dot and cross products are introduced.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where the thumb points in the positive z direction and the fingers curl from x to y. Vector operations like the dot and cross products are introduced.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where pointing thumb, index, and middle fingers represent the positive x, y, z axes. Vector operations like dot and cross products are introduced.
This document introduces vectors and vector operations in Euclidean space. It defines Euclidean space as R3, with three perpendicular coordinate axes (x, y, z). The graph of a function f(x,y) lies in R3, consisting of points (x,y,f(x,y)). It describes the right-handed coordinate system, where the thumb points in the positive z direction and the fingers curl from x to y. Vector operations like the dot and cross products are introduced.
This document introduces vectors and vector calculus. It defines a vector as a directed line segment with a magnitude and direction. Vectors are used to represent phenomena with both magnitude and direction, such as velocity and force. The document discusses Euclidean spaces R^2 and R^3, which are 2-dimensional and 3-dimensional spaces. It introduces right-handed coordinate systems for labeling points and representing graphs of functions of two or three variables in these spaces.
This document provides an overview of using R for financial modeling. It covers basic R commands for calculations, vectors, matrices, lists, data frames, and importing/exporting data. Graphical functions like plots, bar plots, pie charts, and boxplots are demonstrated. Advanced topics discussed include distributions, parameter estimation, correlations, linear and nonlinear regression, technical analysis packages, and practical exercises involving financial data analysis and modeling.
This document provides an introduction and overview of the contents of a textbook on numerical analysis. The textbook covers topics like differential and difference equations, numerical solutions to differential equations, and numerical linear algebra. Chapter 1 introduces differential equations and discusses methods for solving certain types analytically. It also introduces difference equations. Subsequent chapters will discuss numerical methods for solving differential equations, and numerical techniques for linear algebra problems like solving systems of linear equations.
This document describes a maximum entropy modeling toolkit that provides conditional maximum entropy models, L-BFGS and GIS parameter estimation algorithms, Gaussian prior smoothing, C++ and Python APIs. It discusses building, installing and using the toolkit on various operating systems. It also provides tutorials on maximum entropy modeling concepts and using the toolkit's features and APIs.
Similar to Vina Score and Vin Min for almost all the models 2024 (20)
Welcome to ASP Cranes, your trusted partner for crane solutions in Raipur, Chhattisgarh! With years of experience and a commitment to excellence, we offer a comprehensive range of crane services tailored to meet your lifting and material handling needs.
At ASP Cranes, we understand the importance of reliable and efficient crane operations in various industries, from construction and manufacturing to logistics and infrastructure development. That's why we strive to deliver top-notch solutions that enhance productivity, safety, and cost-effectiveness for our clients.
Our services include:
Crane Rental: Whether you need a crawler crane for heavy lifting or a hydraulic crane for versatile operations, we have a diverse fleet of well-maintained cranes available for rent. Our rental options are flexible and can be customized to suit your project requirements.
Crane Sales: Looking to invest in a crane for your business? We offer a wide selection of new and used cranes from leading manufacturers, ensuring you find the perfect equipment to match your needs and budget.
Crane Maintenance and Repair: To ensure optimal performance and safety, regular maintenance and timely repairs are essential for cranes. Our team of skilled technicians provides comprehensive maintenance and repair services to keep your equipment running smoothly and minimize downtime.
Crane Operator Training: Proper training is crucial for safe and efficient crane operation. We offer specialized training programs conducted by certified instructors to equip operators with the skills and knowledge they need to handle cranes effectively.
Custom Solutions: We understand that every project is unique, which is why we offer custom crane solutions tailored to your specific requirements. Whether you need modifications, attachments, or specialized equipment, we can design and implement solutions that meet your needs.
At ASP Cranes, customer satisfaction is our top priority. We are dedicated to delivering reliable, cost-effective, and innovative crane solutions that exceed expectations. Contact us today to learn more about our services and how we can support your project in Raipur, Chhattisgarh, and beyond. Let ASP Cranes be your trusted partner for all your crane needs!
What Could Be Behind Your Mercedes Sprinter's Power Loss on Uphill RoadsSprinter Gurus
Unlock the secrets behind your Mercedes Sprinter's uphill power loss with our comprehensive presentation. From fuel filter blockages to turbocharger troubles, we uncover the culprits and empower you to reclaim your vehicle's peak performance. Conquer every ascent with confidence and ensure a thrilling journey every time.
Understanding Catalytic Converter Theft:
What is a Catalytic Converter?: Learn about the function of catalytic converters in vehicles and why they are targeted by thieves.
Why are They Stolen?: Discover the valuable metals inside catalytic converters (such as platinum, palladium, and rhodium) that make them attractive to criminals.
Steps to Prevent Catalytic Converter Theft:
Parking Strategies: Tips on where and how to park your vehicle to reduce the risk of theft, such as parking in well-lit areas or secure garages.
Protective Devices: Overview of various anti-theft devices available, including catalytic converter locks, shields, and alarms.
Etching and Marking: The benefits of etching your vehicle’s VIN on the catalytic converter or using a catalytic converter marking kit to make it traceable and less appealing to thieves.
Surveillance and Monitoring: Recommendations for using security cameras and motion-sensor lights to deter thieves.
Statistics and Insights:
Theft Rates by Borough: Analysis of data to determine which borough in NYC experiences the highest rate of catalytic converter thefts.
Recent Trends: Current trends and patterns in catalytic converter thefts to help you stay aware of emerging hotspots and tactics used by thieves.
Benefits of This Presentation:
Awareness: Increase your awareness about catalytic converter theft and its impact on vehicle owners.
Practical Tips: Gain actionable insights and tips to effectively prevent catalytic converter theft.
Local Insights: Understand the specific risks in different NYC boroughs, helping you take targeted preventive measures.
This presentation aims to equip you with the knowledge and tools needed to protect your vehicle from catalytic converter theft, ensuring you are prepared and proactive in safeguarding your property.
Ever been troubled by the blinking sign and didn’t know what to do?
Here’s a handy guide to dashboard symbols so that you’ll never be confused again!
Save them for later and save the trouble!
Implementing ELDs or Electronic Logging Devices is slowly but surely becoming the norm in fleet management. Why? Well, integrating ELDs and associated connected vehicle solutions like fleet tracking devices lets businesses and their in-house fleet managers reap several benefits. Check out the post below to learn more.
EV Charging at MFH Properties by Whitaker JamiesonForth
Whitaker Jamieson, Senior Specialist at Forth, gave this presentation at the Forth Addressing The Challenges of Charging at Multi-Family Housing webinar on June 11, 2024.
Expanding Access to Affordable At-Home EV Charging by Vanessa WarheitForth
Vanessa Warheit, Co-Founder of EV Charging for All, gave this presentation at the Forth Addressing The Challenges of Charging at Multi-Family Housing webinar on June 11, 2024.
2. 1 Description
Autodiff is a header-only C++ library that facilitates the automatic differentiation (forward mode) of mathematical
functions of single and multiple variables.
This implementation is based upon the Taylor series expansion of an analytic function f at the point x0:
f(x0 + ε) = f(x0) + f0
(x0)ε +
f00
(x0)
2!
ε2
+
f000
(x0)
3!
ε3
+ · · ·
=
N
X
n=0
f(n)
(x0)
n!
εn
+ O εN+1
.
The essential idea of autodiff is the substitution of numbers with polynomials in the evaluation of f(x0). By
substituting the number x0 with the first-order polynomial x0 +ε, and using the same algorithm to compute f(x0 +ε),
the resulting polynomial in ε contains the function’s derivatives f0
(x0), f00
(x0), f000
(x0), ... within the coefficients.
Each coefficient is equal to the derivative of its respective order, divided by the factorial of the order.
In greater detail, assume one is interested in calculating the first N derivatives of f at x0. Without loss of
precision to the calculation of the derivatives, all terms O εN+1
that include powers of ε greater than N can be
discarded. (This is due to the fact that each term in a polynomial depends only upon equal and lower-order terms
under arithmetic operations.) Under these truncation rules, f provides a polynomial-to-polynomial transformation:
f : x0 + ε 7→
N
X
n=0
ynεn
=
N
X
n=0
f(n)
(x0)
n!
εn
.
C++’s ability to overload operators and functions allows for the creation of a class fvar (forward-mode autodiff
variable) that represents polynomials in ε. Thus the same algorithm f that calculates the numeric value of y0 = f(x0),
when written to accept and return variables of a generic (template) type, is also used to calculate the polynomial
PN
n=0 ynεn
= f(x0 + ε). The derivatives f(n)
(x0) are then found from the product of the respective factorial n! and
coefficient yn:
dn
f
dxn
(x0) = n!yn.
2 Examples
2.1 Example 1: Single-variable derivatives
2.1.1 Calculate derivatives of f(x) = x4
at x = 2.
In this example, make fvardouble, Order(2.0) instantiates the polynomial 2 + ε. The Order=5 means that
enough space is allocated (on the stack) to hold a polynomial of up to degree 5 during the proceeding computation.
Internally, this is modeled by a std::arraydouble,6 whose elements {2, 1, 0, 0, 0, 0} correspond to the
6 coefficients of the polynomial upon initialization. Its fourth power, at the end of the computation, is a polynomial
with coefficients y = {16, 32, 24, 8, 1, 0}. The derivatives are obtained using the formula f(n)
(2) = n! ∗ y[n].
#include boost/math/differentiation/autodiff.hpp
#include iostream
template typename T
T fourth_power(T const x) {
T x4 = x * x; // retval in operator*() uses x4’s memory via NRVO.
x4 *= x4; // No copies of x4 are made within operator*=() even when squaring.
return x4; // x4 uses y’s memory in main() via NRVO.
}
int main() {
using namespace boost::math::differentiation;
2
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
3. constexpr unsigned Order = 5; // Highest order derivative to be calculated.
auto const x = make_fvardouble, Order(2.0); // Find derivatives at x=2.
auto const y = fourth_power(x);
for (unsigned i = 0; i = Order; ++i)
std::cout y.derivative( i ) = y.derivative(i) std::endl;
return 0;
}
/*
Output:
y.derivative(0) = 16
y.derivative(1) = 32
y.derivative(2) = 48
y.derivative(3) = 48
y.derivative(4) = 24
y.derivative(5) = 0
*/
The above calculates
y.derivative(0) = f(2) = x4
x=2
= 16
y.derivative(1) = f0
(2) = 4 · x3
x=2
= 32
y.derivative(2) = f00
(2) = 4 · 3 · x2
x=2
= 48
y.derivative(3) = f000
(2) =4 · 3 · 2 · x|x=2 = 48
y.derivative(4) =f(4)
(2) = 4 · 3 · 2 · 1 = 24
y.derivative(5) =f(5)
(2) = 0
2.2 Example 2: Multi-variable mixed partial derivatives with multi-precision data
type
2.2.1 Calculate ∂12
f
∂w3∂x2∂y4∂z3 (11, 12, 13, 14) with a precision of about 50 decimal digits,
where f(w, x, y, z) = exp
w sin
x log(y)
z
+
q
wz
xy
+ w2
tan(z) .
In this example, make ftuplefloat50, Nw, Nx, Ny, Nz(11, 12, 13, 14) returns a std::tuple of 4 indepen-
dent fvar variables, with values of 11, 12, 13, and 14, for which the maximum order derivative to be calculated for
each are 3, 2, 4, 3, respectively. The order of the variables is important, as it is the same order used when calling
v.derivative(Nw, Nx, Ny, Nz) in the example below.
#include boost/math/differentiation/autodiff.hpp
#include boost/multiprecision/cpp_bin_float.hpp
#include iostream
using namespace boost::math::differentiation;
template typename W, typename X, typename Y, typename Z
promoteW, X, Y, Z f(const W w, const X x, const Y y, const Z z) {
using namespace std;
return exp(w * sin(x * log(y) / z) + sqrt(w * z / (x * y))) + w * w / tan(z);
}
int main() {
using float50 = boost::multiprecision::cpp_bin_float_50;
constexpr unsigned Nw = 3; // Max order of derivative to calculate for w
constexpr unsigned Nx = 2; // Max order of derivative to calculate for x
3
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
4. constexpr unsigned Ny = 4; // Max order of derivative to calculate for y
constexpr unsigned Nz = 3; // Max order of derivative to calculate for z
// Declare 4 independent variables together into a std::tuple.
auto const variables = make_ftuplefloat50, Nw, Nx, Ny, Nz(11, 12, 13, 14);
auto const w = std::get0(variables); // Up to Nw derivatives at w=11
auto const x = std::get1(variables); // Up to Nx derivatives at x=12
auto const y = std::get2(variables); // Up to Ny derivatives at y=13
auto const z = std::get3(variables); // Up to Nz derivatives at z=14
auto const v = f(w, x, y, z);
// Calculated from Mathematica symbolic differentiation.
float50 const answer(1976.319600747797717779881875290418720908121189218755);
std::cout std::setprecision(std::numeric_limitsfloat50::digits10)
mathematica : answer ’n’
autodiff : v.derivative(Nw, Nx, Ny, Nz) ’n’
std::setprecision(3)
relative error: (v.derivative(Nw, Nx, Ny, Nz) / answer - 1) ’n’;
return 0;
}
/*
Output:
mathematica : 1976.3196007477977177798818752904187209081211892188
autodiff : 1976.3196007477977177798818752904187209081211892188
relative error: 2.67e-50
*/
2.3 Example 3: Black-Scholes Option Pricing with Greeks Automatically Calculated
2.3.1 Calculate greeks directly from the Black-Scholes pricing function.
Below is the standard Black-Scholes pricing function written as a function template, where the price, volatility
(sigma), time to expiration (tau) and interest rate are template parameters. This means that any Greek based on
these 4 variables can be calculated using autodiff. The below example calculates delta and gamma where the variable
of differentiation is only the price. For examples of more exotic greeks, see example/black scholes.cpp.
#include boost/math/differentiation/autodiff.hpp
#include iostream
using namespace boost::math::constants;
using namespace boost::math::differentiation;
// Equations and function/variable names are from
// https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks
// Standard normal cumulative distribution function
template typename X
X Phi(X const x) {
return 0.5 * erfc(-one_div_root_twoX() * x);
}
enum class CP { call, put };
// Assume zero annual dividend yield (q=0).
template typename Price, typename Sigma, typename Tau, typename Rate
promotePrice, Sigma, Tau, Rate black_scholes_option_price(CP cp,
double K,
Price const S,
Sigma const sigma,
Tau const tau,
4
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
5. Rate const r) {
using namespace std;
auto const d1 = (log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
auto const d2 = (log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
switch (cp) {
case CP::call:
return S * Phi(d1) - exp(-r * tau) * K * Phi(d2);
case CP::put:
return exp(-r * tau) * K * Phi(-d2) - S * Phi(-d1);
}
}
int main() {
double const K = 100.0; // Strike price.
auto const S = make_fvardouble, 2(105); // Stock price.
double const sigma = 5; // Volatility.
double const tau = 30.0 / 365; // Time to expiration in years. (30 days).
double const r = 1.25 / 100; // Interest rate.
auto const call_price = black_scholes_option_price(CP::call, K, S, sigma, tau, r);
auto const put_price = black_scholes_option_price(CP::put, K, S, sigma, tau, r);
std::cout black-scholes call price = call_price.derivative(0) ’n’
black-scholes put price = put_price.derivative(0) ’n’
call delta = call_price.derivative(1) ’n’
put delta = put_price.derivative(1) ’n’
call gamma = call_price.derivative(2) ’n’
put gamma = put_price.derivative(2) ’n’;
return 0;
}
/*
Output:
black-scholes call price = 56.5136
black-scholes put price = 51.4109
call delta = 0.773818
put delta = -0.226182
call gamma = 0.00199852
put gamma = 0.00199852
*/
3 Advantages of Automatic Differentiation
The above examples illustrate some of the advantages of using autodiff:
• Elimination of code redundancy. The existence of N separate functions to calculate derivatives is a form of
code redundancy, with all the liabilities that come with it:
– Changes to one function require N additional changes to other functions. In the 3rd
example above,
consider how much larger and inter-dependent the above code base would be if a separate function were
written for each Greek value.
– Dependencies upon a derivative function for a different purpose will break when changes are made to the
original function. What doesn’t need to exist cannot break.
– Code bloat, reducing conceptual integrity. Control over the evolution of code is easier/safer when the code
base is smaller and able to be intuitively grasped.
• Accuracy of derivatives over finite difference methods. Single-iteration finite difference methods always include
a ∆x free variable that must be carefully chosen for each application. If ∆x is too small, then numerical errors
become large. If ∆x is too large, then mathematical errors become large. With autodiff, there are no free
5
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
6. variables to set and the accuracy of the answer is generally superior to finite difference methods even with the
best choice of ∆x.
4 Mathematics
In order for the usage of the autodiff library to make sense, a basic understanding of the mathematics will help.
4.1 Truncated Taylor Series
Basic calculus courses teach that a real analytic function f : D → R is one which can be expressed as a Taylor series
at a point x0 ∈ D ⊆ R:
f(x) = f(x0) + f0
(x0)(x − x0) +
f00
(x0)
2!
(x − x0)2
+
f000
(x0)
3!
(x − x0)3
+ · · ·
One way of thinking about this form is that given the value of an analytic function f(x0) and its derivatives
f0
(x0), f00
(x0), f000
(x0), ... evaluated at a point x0, then the value of the function f(x) can be obtained at any other
point x ∈ D using the above formula.
Let us make the substitution x = x0 + ε and rewrite the above equation to get:
f(x0 + ε) = f(x0) + f0
(x0)ε +
f00
(x0)
2!
ε2
+
f000
(x0)
3!
ε3
+ · · ·
Now consider ε as an abstract algebraic entity that never acquires a numeric value, much like one does in basic algebra
with variables like x or y. For example, we can still manipulate entities like xy and (1 + 2x + 3x2
) without having
to assign specific numbers to them.
Using this formula, autodiff goes in the other direction. Given a general formula/algorithm for calculating
f(x0 + ε), the derivatives are obtained from the coefficients of the powers of ε in the resulting computation. The
general coefficient for εn
is
f(n)
(x0)
n!
.
Thus to obtain f(n)
(x0), the coefficient of εn
is multiplied by n!.
4.1.1 Example
Apply the above technique to calculate the derivatives of f(x) = x4
at x0 = 2.
The first step is to evaluate f(x0 + ε) and simply go through the calculation/algorithm, treating ε as an abstract
algebraic entity:
f(x0 + ε) = f(2 + ε)
= (2 + ε)4
= 4 + 4ε + ε2
2
= 16 + 32ε + 24ε2
+ 8ε3
+ ε4
.
Equating the powers of ε from this result with the above ε-taylor expansion yields the following equalities:
f(2) = 16, f0
(2) = 32,
f00
(2)
2!
= 24,
f000
(2)
3!
= 8,
f(4)
(2)
4!
= 1,
f(5)
(2)
5!
= 0.
Multiplying both sides by the respective factorials gives
f(2) = 16, f0
(2) = 32, f00
(2) = 48, f000
(2) = 48, f(4)
(2) = 24, f(5)
(2) = 0.
These values can be directly confirmed by the power rule applied to f(x) = x4
.
6
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
7. 4.2 Arithmetic
What was essentially done above was to take a formula/algorithm for calculating f(x0) from a number x0, and
instead apply the same formula/algorithm to a polynomial x0 + ε. Intermediate steps operate on values of the form
x = x0 + x1ε + x2ε2
+ · · · + xN εN
and the final return value is of this polynomial form as well. In other words, the normal arithmetic operators
+, −, ×, ÷ applied to numbers x are instead applied to polynomials x. Through the overloading of C++ operators
and functions, floating point data types are replaced with data types that represent these polynomials. More
specifically, C++ types such as double are replaced with std::arraydouble,N+1, which hold the above N + 1
coefficients xi, and are wrapped in a class that overloads all of the arithmetic operators.
The logic of these arithmetic operators simply mirror that which is applied to polynomials. We’ll look at each of
the 4 arithmetic operators in detail.
4.2.1 Addition
The addition of polynomials x and y is done component-wise:
z = x + y
=
N
X
i=0
xiεi
!
+
N
X
i=0
yiεi
!
=
N
X
i=0
(xi + yi)εi
zi = xi + yi for i ∈ {0, 1, 2, ..., N}.
4.2.2 Subtraction
Subtraction follows the same form as addition:
z = x − y
=
N
X
i=0
xiεi
!
−
N
X
i=0
yiεi
!
=
N
X
i=0
(xi − yi)εi
zi = xi − yi for i ∈ {0, 1, 2, ..., N}.
4.2.3 Multiplication
Multiplication produces higher-order terms:
7
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
8. z = x × y
=
N
X
i=0
xiεi
! N
X
i=0
yiεi
!
= x0y0 + (x0y1 + x1y0)ε + (x0y2 + x1y1 + x2y0)ε2
+ · · · +
N
X
j=0
xjyN−j
εN
+ O εN+1
=
N
X
i=0
i
X
j=0
xjyi−jεi
+ O εN+1
zi =
i
X
j=0
xjyi−j for i ∈ {0, 1, 2, ..., N}.
In the case of multiplication, terms involving powers of ε greater than N, collectively denoted by O εN+1
, are
simply discarded. Fortunately, the values of zi for i ≤ N do not depend on any of these discarded terms, so there is
no loss of precision in the final answer. The only information that is lost are the values of higher order derivatives,
which we are not interested in anyway. If we were, then we would have simply chosen a larger value of N to begin
with.
4.2.4 Division
Division is not directly calculated as are the others. Instead, to find the components of z = x ÷ y we require that
x = y × z. This yields a recursive formula for the components zi:
xi =
i
X
j=0
yjzi−j
= y0zi +
i
X
j=1
yjzi−j
zi =
1
y0
xi −
i
X
j=1
yjzi−j
for i ∈ {0, 1, 2, ..., N}.
In the case of division, the values for zi must be calculated sequentially, since zi depends on the previously calculated
values z0, z1, ..., zi−1.
4.3 General Functions
Calling standard mathematical functions such as log(), cos(), etc. should return accurate higher order derivatives.
For example, exp(x) may be written internally as a specific 14th
-degree polynomial to approximate ex
when 0
x 1. This would mean that the 15th
derivative, and all higher order derivatives, would be 0, however we know that
d15
dx15 ex
= ex
. How should such functions whose derivatives are known be written to provide accurate higher order
derivatives? The answer again comes back to the function’s Taylor series.
To simplify notation, for a given polynomial x = x0 + x1ε + x2ε2
+ · · · + xN εN
define
xε = x1ε + x2ε2
+ · · · + xN εN
=
N
X
i=1
xiεi
.
This allows for a concise expression of a general function f of x:
8
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
9. f(x) = f(x0 + xε)
= f(x0) + f0
(x0)xε +
f00
(x0)
2!
x2
ε +
f000
(x0)
3!
x3
ε + · · · +
f(N)
(x0)
N!
xN
ε + O εN+1
=
N
X
i=0
f(i)
(x0)
i!
xi
ε + O εN+1
where ε has been substituted with xε in the ε-taylor series for f(x). This form gives a recipe for calculating f(x) in
general from regular numeric calculations f(x0), f0
(x0), f00
(x0), ... and successive powers of the epsilon terms xε.
For an application in which we are interested in up to N derivatives in x the data structure to hold this information
is an (N + 1)-element array v whose general element is
v[i] =
f(i)
(x0)
i!
for i ∈ {0, 1, 2, ..., N}.
4.4 Multiple Variables
In C++, the generalization to mixed partial derivatives with multiple independent variables is conveniently achieved
with recursion. To begin to see the recursive pattern, consider a two-variable function f(x, y). Since x and y are
independent, they require their own independent epsilons εx and εy, respectively.
Expand f(x, y) for x = x0 + εx:
f(x0 + εx, y) = f(x0, y) +
∂f
∂x
(x0, y)εx +
1
2!
∂2
f
∂x2
(x0, y)ε2
x +
1
3!
∂3
f
∂x3
(x0, y)ε3
x + · · · +
1
M!
∂M
f
∂xM
(x0, y)εM
x + O εM+1
x
=
M
X
i=0
1
i!
∂i
f
∂xi
(x0, y)εi
x + O εM+1
x
.
Next, expand f(x0 + εx, y) for y = y0 + εy:
f(x0 + εx, y0 + εy) =
N
X
j=0
1
j!
∂j
∂yj
M
X
i=0
εi
x
1
i!
∂i
f
∂xi
!
(x0, y0)εj
y + O εM+1
x
+ O εN+1
y
=
M
X
i=0
N
X
j=0
1
i!j!
∂i+j
f
∂xi∂yj
(x0, y0)εi
xεj
y + O εM+1
x
+ O εN+1
y
.
Similar to the single-variable case, for an application in which we are interested in up to M derivatives in x and
N derivatives in y, the data structure to hold this information is an (M + 1) × (N + 1) array v whose element at
(i, j) is
v[i][j] =
1
i!j!
∂i+j
f
∂xi∂yj
(x0, y0) for (i, j) ∈ {0, 1, 2, ..., M} × {0, 1, 2, ..., N}.
The generalization to additional independent variables follows the same pattern.
4.4.1 Declaring Multiple Variables
Internally, independent variables are represented by vectors within orthogonal vector spaces. Because of this, one
must be careful when declaring more than one independent variable so that they do not end up in parallel vector
spaces. This can easily be achieved by following one rule:
• When declaring more than one independent variable, call make ftuple() once and only once.
The tuple of values returned are independent. Though it is possible to achieve the same result with multiple calls to
make fvar, this is an easier and less error-prone method. See Section 2.2 for example usage.
9
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
10. 5 Writing Functions for Autodiff Compatibility
In this section, a general procedure is given for writing new, and transforming existing, C++ mathematical functions
for compatibility with autodiff.
There are 3 categories of functions that require different strategies:
1. Piecewise-rational functions. These are simply piecewise quotients of polynomials. All that is needed is to
turn the function parameters and return value into generic (template) types. This will then allow the function
to accept and return autodiff’s fvar types, thereby using autodiff’s overloaded arithmetic operators which
calculate the derivatives automatically.
2. Functions that call existing autodiff functions. This is the same as the previous, but may also include calls to
functions that are in the autodiff library. Examples: exp(), log(), tgamma(), etc.
3. New functions for which the derivatives can be calculated. This is the most general technique, as it allows for
the development of a function which do not fall into the previous two categories.
Functions written in any of these ways may then be added to the autodiff library.
5.1 Piecewise-Rational Functions
f(x) =
1
1 + x2
By simply writing this as a template function, autodiff can calculate derivatives for it:
#include boost/math/differentiation/autodiff.hpp
#include iostream
template typename T
T rational(T const x) {
return 1 / (1 + x * x);
}
int main() {
using namespace boost::math::differentiation;
auto const x = make_fvardouble, 10(0);
auto const y = rational(x);
std::cout std::setprecision(std::numeric_limitsdouble::digits10)
y.derivative(10) = y.derivative(10) std::endl;
return 0;
}
/*
Output:
y.derivative(10) = -3628800
*/
As simple as f(x) may seem, the derivatives can get increasingly complex as derivatives are taken. For example, the
10th
derivative has the form
f(10)
(x) = −3628800
1 − 55x2
+ 330x4
− 462x6
+ 165x8
− 11x10
(1 + x2)11
.
Derivatives of f(x) are useful, and in fact used, in calculating higher order derivatives for arctan(x) for instance,
since
arctan(n)
(x) =
d
dx
n−1
1
1 + x2
for 1 ≤ n.
10
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
11. 5.2 Functions That Call Existing Autodiff Functions
Many of the standard library math function are overloaded in autodiff. It is recommended to use argument-dependent
lookup (ADL) in order for functions to be written in a way that is general enough to accommodate standard types
(double) as well as autodiff types (fvar).
Example:
#include boost/math/constants/constants.hpp
#include cmath
using namespace boost::math::constants;
// Standard normal cumulative distribution function
template typename T
T Phi(T const x)
{
return 0.5 * std::erfc(-one_div_root_twoT() * x);
}
Though Phi(x) is general enough to handle the various fundamental floating point types, this will not work if x is
an autodiff fvar variable, since std::erfc does not include a specialization for fvar. The recommended solution is
to remove the namespace prefix std:: from erfc:
#include boost/math/constants/constants.hpp
#include boost/math/differentiation/autodiff.hpp
#include cmath
using namespace boost::math::constants;
// Standard normal cumulative distribution function
template typename T
T Phi(T const x)
{
using std::erfc;
return 0.5 * erfc(-one_div_root_twoT() * x);
}
In this form, when x is of type fvar, the C++ compiler will search for and find a function erfc within the same
namespace as fvar, which is in the autodiff library, via ADL. Because of the using-declaration, it will also call
std::erfc when x is a fundamental type such as double.
5.3 New Functions For Which The Derivatives Can Be Calculated
Mathematical functions which do not fall into the previous two categories can be constructed using autodiff helper
functions. This requires a separate function for calculating the derivatives. In case you are asking yourself what
good is an autodiff library if one needs to supply the derivatives, the answer is that the new function will fit in with
the rest of the autodiff library, thereby allowing for the creation of additional functions via all of the arithmetic
operators, plus function composition, which was not readily available without the library.
The example given here is for cos:
template typename RealType, size_t Order
fvarRealType, Order cos(fvarRealType, Order const cr) {
using std::cos;
using std::sin;
using root_type = typename fvarRealType, Order::root_type;
constexpr size_t order = fvarRealType, Order::order_sum;
root_type const d0 = cos(static_castroot_type(cr));
if constexpr (order == 0)
return fvarRealType, Order(d0);
else {
11
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
12. root_type const d1 = -sin(static_castroot_type(cr));
root_type const derivatives[4]{d0, d1, -d0, -d1};
return cr.apply_derivatives(order,
[derivatives](size_t i) { return derivatives[i 3]; });
}
}
This uses the helper function fvar::apply derivatives which takes two parameters:
1. The highest order derivative to be calculated.
2. A function that maps derivative order to derivative value.
The highest order derivative necessary to be calculated is generally equal to fvar::order sum. In the case of sin
and cos, the derivatives are cyclical with period 4. Thus it is sufficient to store only these 4 values into an array,
and take the derivative order modulo 4 as the index into this array.
A second helper function, not shown here, is apply coefficients. This is used the same as apply derivatives
except that the supplied function calculates coefficients instead of derivatives. The relationship between a coefficient
Cn and derivative Dn for derivative order n is
Cn =
Dn
n!
.
Internally, fvar holds coefficients rather than derivatives, so in case the coefficient values are more readily available
than the derivatives, it can save some unnecessary computation to use apply coefficients. See the definition of
atan for an example.
Both of these helper functions use Horner’s method when calculating the resulting polynomial fvar. This works
well when the derivatives are finite, but in cases where derivatives are infinite, this can quickly result in NaN values as
the computation progresses. In these cases, one can call non-Horner versions of both function which better “isolate”
infinite values so that they are less likely to evolve into NaN values.
The four helper functions available for constructing new autodiff functions from known coefficients/derivatives
are:
1. fvar::apply coefficients
2. fvar::apply coefficients nonhorner
3. fvar::apply derivatives
4. fvar::apply derivatives nonhorner
6 Function Writing Guidelines
At a high level there is one fairly simple principle, loosely and intuitively speaking, to writing functions for which
autodiff can effectively calculate derivatives:
Autodiff Function Principle (AFP)
A function whose branches in logic correspond to piecewise analytic calculations over non-singleton inter-
vals, with smooth transitions between the intervals, and is free of indeterminate forms in the calculated
value and higher order derivatives, will work fine with autodiff.
Stating this with greater mathematical rigor can be done. However what seems to be more practical, in this case, is
to give examples and categories of examples of what works, what doesn’t, and how to remedy some of the common
problems that may be encountered. That is the approach taken here.
6.1 Example 1: f(x) = max(0, x)
One potential implementation of f(x) = max(0, x) is:
12
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
13. templatetypename T
T f(const T x)
{
return 0 x ? x : 0;
}
Though this is consistent with Section 5, there are two problems with it:
1. f(nan) = 0. This problem is independent of autodiff, but is worth addressing anyway. If there is an indetermi-
nate form that arises within a calculation and is input into f, then it gets “covered up” by this implementation
leading to an unknowingly incorrect result. Better for functions in general to propagate NaN values, so that
the user knows something went wrong and doesn’t rely on an incorrect result, and likewise the developer can
track down where the NaN originated from and remedy it.
2. f0
(0) = 0 when autodiff is applied. This is because f returns 0 as a constant when x==0, wiping out any of
the derivatives (or sensitivities) that x was holding as an autodiff variable. Instead, let us apply the AFP and
identify the two intervals over which f is defined: (−∞, 0] ∪ (0, ∞). Though the function itself is not analytic
at x = 0, we can attempt somewhat to smooth out this transition point by averaging the calculation of f(x)
at x = 0 from each interval. If x 0 then the result is simply 0, and if 0 x then the result is x. The average
is 1
2 (0 + x) which will allow autodiff to calculate f0
(0) = 1
2 . This is a more reasonable answer.
A better implementation that resolves both issues is:
templatetypename T
T f(const T x)
{
if (x 0)
return 0;
else if (x == 0)
return 0.5*x;
else
return x;
}
6.2 Example 2: f(x) = sinc(x)
The definition of sinc : R → R is
sinc(x) =
(
1 if x = 0
sin(x)
x otherwise.
A potential implementation is:
templatetypename T
T sinc(const T x)
{
using std::sin;
return x == 0 ? 1 : sin(x) / x;
}
Though this is again consistent with Section 5, and returns correct non-derivative values, it returns a constant when
x==0 thereby losing all derivative information contained in x and contributions from sinc. For example, sinc00
(0) = −1
3 ,
however y.derivative(2) == 0 when y = sinc(make fvardouble,2(0)) using the above incorrect implementa-
tion. Applying the AFP, the intervals upon which separate branches of logic are applied are (−∞, 0) ∪ [0, 0] ∪ (0, ∞).
The violation occurs due to the singleton interval [0, 0], even though a constant function of 1 is technically analytic.
The remedy is to define a custom sinc overload and add it to the autodiff library. This has been done. Mathematically,
it is well-defined and free of indeterminate forms, as is the 3rd
expression in the equalities
1
x
sin(x) =
1
x
∞
X
n=0
(−1)n
(2n + 1)!
x2n+1
=
∞
X
n=0
(−1)n
(2n + 1)!
x2n
.
13
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
14. The autodiff library contains helper functions to help write function overloads when the derivatives of a function are
known. This is an advanced feature and documentation for this may be added at a later time.
For now, it is worth understanding the ways in which indeterminate forms can occur within a mathematical
calculation, and avoid them when possible by rewriting the function. Table 1 compares 3 types of indeterminate
forms. Assume the product a*b is a positive finite value.
f(x) =
a
x
× (bx2
) g(x) =
a
x
× (bx) h(x) =
a
x2
× (bx)
Mathematical
Limit
lim
x→0
f(x) = 0 lim
x→0
g(x) = ab lim
x→0
h(x) = ∞
Floating Point
Arithmetic
f(0) = inf*0 = nan g(0) = inf*0 = nan h(0) = inf*0 = nan
Table 1: Automatic differentiation does not compute limits. Indeterminate forms must be simplified manually.
(These cases are not meant to be exhaustive.)
Indeterminate forms result in NaN values within a calculation. Mathematically, if they occur at locally isolated
points, then we generally prefer the mathematical limit as the result, even if it is infinite. As demonstrated in Table 1,
depending upon the nature of the indeterminate form, the mathematical limit can be 0 (no matter the values of
a or b), or ab, or ∞, but these 3 cases cannot be distinguished by the floating point result of nan. Floating point
arithmetic does not perform limits (directly), and neither does the autodiff library. Thus it is up to the diligence of
the developer to keep a watchful eye over where indeterminate forms can arise.
6.3 Example 3: f(x) =
√
x and f0
(0) = ∞
When working with functions that have infinite higher order derivatives, this can very quickly result in nans in higher
order derivatives as the computation progresses, as inf-inf, inf/inf, and 0*inf result in nan. See Table 2 for an
example.
f(x) f(0) f0
(0) f00
(0) f000
(0)
sqrt(x) 0 inf -inf inf
sqr(sqrt(x)+1) 1 inf nan nan
x+2*sqrt(x)+1 1 inf -inf inf
Table 2: Indeterminate forms in higher order derivatives. sqr(x) == x*x.
Calling the autodiff-overloaded implementation of f(x) =
√
x at the value x==0 results in the 1st
row (after the
header row) of Table 2, as is mathematically correct. The 2nd
row shows f(x) = (
√
x + 1)2
resulting in nan values
for f00
(0) and all higher order derivatives. This is due to the internal arithmetic in which inf is added to -inf
during the squaring, resulting in a nan value for f00
(0) and all higher orders. This is typical of inf values in autodiff.
Where they show up, they are correct, however they can quickly introduce nan values into the computation upon the
addition of oppositely signed inf values, division by inf, or multiplication by 0. It is worth noting that the infection
of nan only spreads upward in the order of derivatives, since lower orders do not depend upon higher orders (which
is also why dropping higher order terms in an autodiff computation does not result in any loss of precision for lower
order terms.)
The resolution in this case is to manually perform the squaring in the computation, replacing the 2nd
row with
the 3rd
: f(x) = x + 2
√
x + 1. Though mathematically equivalent, it allows autodiff to avoid nan values since
√
x is
more “isolated” in the computation. That is, the inf values that unavoidably show up in the derivatives of sqrt(x)
for x==0 do not have the chance to interact with other inf values as with the squaring.
6.4 Summary
The AFP gives a high-level unified guiding principle for writing C++ template functions that autodiff can effectively
evaluate derivatives for.
Examples have been given to illustrate some common items to avoid doing:
1. It is not enough for functions to be piecewise continuous. On boundary points between intervals, consider
returning the average expression of both intervals, rather than just one of them. Example: max(0, x) at x = 0.
14
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
15. In cases where the limits from both sides must match, and they do not, then nan may be a more appropriate
value depending on the application.
2. Avoid returning individual constant values (e.g. sinc(0) = 1.) Values must be computed uniformly along with
other values in its local interval. If that is not possible, then the function must be overloaded to compute the
derivatives precisely using the helper functions from Section 5.3.
3. Avoid intermediate indeterminate values in both the value (sinc(x) at x = 0) and derivatives ((
√
x + 1)2
at
x = 0). Try to isolate expressions that may contain infinite values/derivatives so that they do not introduce
NaN values into the computation.
7 Acknowledgments
• Kedar Bhat — C++11 compatibility, Boost Special Functions compatibility testing, codecov integration, and
feedback.
• Nick Thompson — Initial feedback and help with Boost integration.
• John Maddock — Initial feedback and help with Boost integration.
References
[1] https://en.wikipedia.org/wiki/Automatic_differentiation
[2] Andreas Griewank, Andrea Walther. Evaluating Derivatives. SIAM, 2nd ed. 2008.
15
Copyright c Matthew Pulver 2018–2019. Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE 1 0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)