MIT OpenCourseWare provides course materials for the free online course 6.094 Introduction to MATLAB taught in January 2009. The course covers topics like linear algebra, polynomials, optimization, differentiation and integration, and solving differential equations using MATLAB. Lecture 3 focuses on solving systems of linear equations, matrix operations, polynomial fitting to data, nonlinear root finding, function minimization, and numerical methods for differentiation, integration, and solving ordinary differential equations.
The document provides an introduction to machine learning. It discusses the author's path to becoming a data scientist and some key machine learning concepts, including:
- Required skills at different experience levels for machine learning roles
- Popular machine learning approaches like deep learning and reinforcement learning
- Common machine learning problems like one shot learning and imbalanced datasets
- How machine learning works by using tricks on data through parametric models and free parameters
- Key questions in machine learning like what to teach, how to teach, and to what entity
- Popular machine learning frameworks like TensorFlow that automate tasks
This document discusses Joseph-Louis Lagrange and interpolation. It provides:
1) A brief biography of Joseph-Louis Lagrange, an Italian mathematician who made significant contributions to calculus and probability.
2) A definition of interpolation as producing a function that matches given data points exactly and can be used to approximate values between points.
3) An explanation of Lagrange's interpolation formula for finding a polynomial that fits a set of data points, including an example of applying the formula.
Numerical method (curve fitting)
***TOPICS ARE****
Linear Regression
Multiple Linear Regression
Polynomial Regression
Example of Newton’s Interpolation Polynomial And example
Example of Newton’s Interpolation Polynomial And example
This document discusses various methods for polynomial interpolation of data points, including Newton's divided difference interpolating polynomials and Lagrange interpolation polynomials. It provides formulas and examples for linear, quadratic, and general polynomial interpolation using Newton's method. It also covers an example of using multiple linear regression with log-transformed data to develop a model relating fluid flow through a pipe to the pipe's diameter and slope.
The document discusses Lagrange interpolating polynomials, which provide an alternative way to write an nth order polynomial that passes through a given set of n+1 data points. The Lagrange interpolating polynomial is defined as the sum of nth order Lagrange coefficient polynomials multiplied by the y-values at each data point. This allows the interpolating polynomial to be determined directly from the data points using simple formulas, without solving systems of equations. An example demonstrates computing the 4th order Lagrange interpolating polynomial for voltage data from three different batteries over time.
The document discusses numerical computing and various interpolation techniques. Numerical computing involves solving complex mathematical problems using simple arithmetic operations by formulating models that can be solved numerically. The document then discusses nonlinear equations and various iterative methods to solve them, including bracketing methods like bisection and regula falsi, and open-end methods like Newton-Raphson and secant. It also discusses fixed point iteration. Finally, it covers interpolation techniques like Lagrange interpolation and Newton interpolation to estimate values of a function at intermediate points.
Tail Probabilities for Randomized Program Runtimes via Martingales for Higher...Satoshi Kura
The document presents an approach to overapproximating tail probabilities of randomized program runtimes using martingales for higher moments. It extends previous work that used ranking supermartingales to bound the expected runtime. The approach characterizes higher moments like E[T^2] as the least fixed point of monotone functions on complete lattices. This allows defining ranking supermartingales to bound higher moments simultaneously. Bounding higher moments enables using concentration inequalities to tighten the overapproximation of tail probabilities compared to prior work.
The document provides an introduction to machine learning. It discusses the author's path to becoming a data scientist and some key machine learning concepts, including:
- Required skills at different experience levels for machine learning roles
- Popular machine learning approaches like deep learning and reinforcement learning
- Common machine learning problems like one shot learning and imbalanced datasets
- How machine learning works by using tricks on data through parametric models and free parameters
- Key questions in machine learning like what to teach, how to teach, and to what entity
- Popular machine learning frameworks like TensorFlow that automate tasks
This document discusses Joseph-Louis Lagrange and interpolation. It provides:
1) A brief biography of Joseph-Louis Lagrange, an Italian mathematician who made significant contributions to calculus and probability.
2) A definition of interpolation as producing a function that matches given data points exactly and can be used to approximate values between points.
3) An explanation of Lagrange's interpolation formula for finding a polynomial that fits a set of data points, including an example of applying the formula.
Numerical method (curve fitting)
***TOPICS ARE****
Linear Regression
Multiple Linear Regression
Polynomial Regression
Example of Newton’s Interpolation Polynomial And example
Example of Newton’s Interpolation Polynomial And example
This document discusses various methods for polynomial interpolation of data points, including Newton's divided difference interpolating polynomials and Lagrange interpolation polynomials. It provides formulas and examples for linear, quadratic, and general polynomial interpolation using Newton's method. It also covers an example of using multiple linear regression with log-transformed data to develop a model relating fluid flow through a pipe to the pipe's diameter and slope.
The document discusses Lagrange interpolating polynomials, which provide an alternative way to write an nth order polynomial that passes through a given set of n+1 data points. The Lagrange interpolating polynomial is defined as the sum of nth order Lagrange coefficient polynomials multiplied by the y-values at each data point. This allows the interpolating polynomial to be determined directly from the data points using simple formulas, without solving systems of equations. An example demonstrates computing the 4th order Lagrange interpolating polynomial for voltage data from three different batteries over time.
The document discusses numerical computing and various interpolation techniques. Numerical computing involves solving complex mathematical problems using simple arithmetic operations by formulating models that can be solved numerically. The document then discusses nonlinear equations and various iterative methods to solve them, including bracketing methods like bisection and regula falsi, and open-end methods like Newton-Raphson and secant. It also discusses fixed point iteration. Finally, it covers interpolation techniques like Lagrange interpolation and Newton interpolation to estimate values of a function at intermediate points.
Tail Probabilities for Randomized Program Runtimes via Martingales for Higher...Satoshi Kura
The document presents an approach to overapproximating tail probabilities of randomized program runtimes using martingales for higher moments. It extends previous work that used ranking supermartingales to bound the expected runtime. The approach characterizes higher moments like E[T^2] as the least fixed point of monotone functions on complete lattices. This allows defining ranking supermartingales to bound higher moments simultaneously. Bounding higher moments enables using concentration inequalities to tighten the overapproximation of tail probabilities compared to prior work.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
The document discusses various algorithms that can be solved using backtracking. It begins by defining backtracking as a general algorithm design technique for problems that involve searching for solutions satisfying constraints. It then provides examples of problems that can be solved using backtracking, including the 8 queens problem, sum of subsets, graph coloring, and finding Hamiltonian cycles in a graph. For each problem, it outlines the key steps and provides pseudocode for the backtracking algorithm.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
The document defines and provides examples of various types of functions, including:
- Polynomial functions including constant, linear, and general polynomial functions.
- Rational functions defined as the ratio of two polynomial functions.
- Trigonometric functions including sine, cosine, and their inverses.
- Other common functions like absolute value, square root, exponential, logarithmic, floor, and ceiling functions.
It also defines properties of functions like being one-to-one, even, or odd and provides examples of each.
This document proposes a linear programming (LP) based approach for solving maximum a posteriori (MAP) estimation problems on factor graphs that contain multiple-degree non-indicator functions. It presents an existing LP method for problems with single-degree functions, then introduces a transformation to handle multiple-degree functions by introducing auxiliary variables. This allows applying the existing LP method. As an example, it applies this to maximum likelihood decoding for the Gaussian multiple access channel. Simulation results demonstrate the LP approach decodes correctly with polynomial complexity.
MATLAB - Aplication of Arrays and Matrices in Electrical SystemsShameer Ahmed Koya
This document discusses using matrices and linear algebra to analyze electrical systems. It explains how to represent sets of linear equations in matrix form as A*x=y and solve for x using the inverse of A. It also discusses calculating the rank of a matrix and condition number. Examples are given of using matrices to solve circuit problems involving node analysis, mesh analysis, and AC circuits. Solutions are found by taking the inverse of the matrix or using the backslash operator in MATLAB.
بررسی دو روش شناسایی سیستم های متغیر با زمان به همراه شبیه سازی و گزارشپروژه مارکت
بررسی دو روش شناسایی سیستم متغیر بازمان با جزییات و توضیحات ساده و گویا و توضیح روابط و علایم به طور کامل به همراه مثال و شبیه سازی در متلب و فایل های مرجع و مقالات مرجع
روش اول: شناسایی سیستم با پارامترهای متغیر با زمان با استفاده از Least Square
روش دوم: شناسایی سیستم دینامیکی با استفاده از شبکه های عصبی مصنوعی
مناسب برای ارائه به عنوان تحقیق یا پروژه درس شناسایی سیستم ها
به همراه شبیه سازی در متلب و گزارش 20 صفحه ای
https://www.prjmarket.com/product/%d8%a8%d8%b1%d8%b1%d8%b3%db%8c-%d8%af%d9%88-%d8%b1%d9%88%d8%b4-%d8%b4%d9%86%d8%a7%d8%b3%d8%a7%db%8c%db%8c-%d8%b3%db%8c%d8%b3%d8%aa%d9%85-%d9%87%d8%a7%db%8c-%d9%85%d8%aa%d8%ba%db%8c%d8%b1-%d8%a8%d8%a7/
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
The document discusses algorithm analysis and design. It begins with an introduction to analyzing algorithms including average case analysis and solving recurrences. It then provides definitions of algorithms both informal and formal. Key aspects of algorithm study and specification methods like pseudocode are outlined. Selection sort and tower of Hanoi problems are presented as examples and analyzed for time and space complexity. Average case analysis is discussed assuming all inputs are equally likely.
Recommendation System --Theory and PracticeKimikazu Kato
This document provides an overview of recommendation systems and collaborative filtering techniques. It discusses using matrix factorization to predict user ratings by representing users and items as vectors in a latent factor space. Optimization techniques like stochastic gradient descent can be used to learn the factorization from existing ratings. The document also notes challenges of sparsity and scale for practical systems and describes approaches like elastic net regularization and sparsification to address these.
Newton divided difference interpolationVISHAL DONGA
This document presents Newton's divided difference polynomial method of interpolation. It defines interpolation as finding the value of 'y' at an unspecified value of 'x' given a set of (x,y) data points. Newton's method uses divided differences to determine the coefficients of a polynomial that can be used to interpolate and estimate y-values between the given data points. The document includes an example of applying Newton's method to find the interpolating polynomial and estimate an unknown y-value for a given set of 5 (x,y) data points.
The branch and bound method searches a tree model of the solution space for discrete optimization problems. It uses bounding functions to prune subtrees that cannot contain optimal solutions, avoiding evaluating all possible solutions. The method generates the tree in a depth-first manner, and selects the next node to expand (the E-node) based on cost estimates. Common strategies are FIFO, LIFO, and least cost search. The traveling salesman problem can be solved using branch and bound by formulating the solution space as a tree and applying row and column reduction techniques to the cost matrix at each node to identify prunable branches.
This document contains lecture notes on sparse autoencoders. It begins with an introduction describing the limitations of supervised learning and the need for algorithms that can automatically learn feature representations from unlabeled data. The notes then state that sparse autoencoders are one approach to learn features from unlabeled data, and describe the organization of the rest of the notes. The notes will cover feedforward neural networks, backpropagation for supervised learning, autoencoders for unsupervised learning, and how sparse autoencoders are derived from these concepts.
The document provides information about Lagrangian interpolation, including:
1. It introduces Lagrangian interpolation as a method to find the value of a function at a discrete point using a polynomial that passes through known data points.
2. It gives the formula for the Lagrangian interpolating polynomial and provides an example of using it to find the velocity of a rocket at a certain time.
3. It discusses using higher order polynomials for interpolation, providing another example that calculates velocity using quadratic and cubic polynomials.
The document discusses various math and string classes in Java. It covers:
- Constructing objects using the new operator and passing parameters.
- Using the Random class to generate random numbers.
- Declaring constants using final and static final.
- Basic arithmetic, increment/decrement, and math methods.
- Creating and manipulating strings using methods like length(), substring(), and concatenation.
- Drawing shapes on a frame using Graphics2D methods in a JComponent's paintComponent method.
Path relinking for high dimensional continuous optimizationPatxi Gortázar
Una presentación de un algoritmo evolutivo para resolver problemas de optimización global que di en el Metaheuristics International Conference de 2011 en Udine, Italia.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
The document discusses dictionaries and different data structures used to implement them efficiently, including hashing. Dictionaries store key-value pairs and support fast retrieval of elements using keys. Hashing is described as an efficient solution to implement dictionaries, where keys are mapped to table slots using a hash function. Hashing provides expected O(1) time for search, insert and delete operations when the load factor is a constant. Collision resolution using chaining is explained, where elements hash to the same slot are stored in a linked list.
Here are the steps to solve this ODE problem:
1. Define the ODE function:
function dydt = odefun(t,y)
dydt = -t.*y/10;
end
2. Solve the ODE:
[t,y] = ode45(@odefun,[0 10],10);
3. Plot the result:
plot(t,y)
xlabel('t')
ylabel('y(t)')
This uses ode45 to solve the ODE dy/dt = -t*y/10 on the interval [0,10] with initial condition y(0)=10.
This document discusses various ways to work with polynomials in MATLAB. It covers representing polynomials as vectors, evaluating polynomials, finding roots, computing coefficients from roots, adding, subtracting, multiplying, dividing, taking derivatives and integrals of polynomials, fitting curves to data using polynomial regression, and interpolating values between data points using polynomials.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
The document discusses various algorithms that can be solved using backtracking. It begins by defining backtracking as a general algorithm design technique for problems that involve searching for solutions satisfying constraints. It then provides examples of problems that can be solved using backtracking, including the 8 queens problem, sum of subsets, graph coloring, and finding Hamiltonian cycles in a graph. For each problem, it outlines the key steps and provides pseudocode for the backtracking algorithm.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
The document defines and provides examples of various types of functions, including:
- Polynomial functions including constant, linear, and general polynomial functions.
- Rational functions defined as the ratio of two polynomial functions.
- Trigonometric functions including sine, cosine, and their inverses.
- Other common functions like absolute value, square root, exponential, logarithmic, floor, and ceiling functions.
It also defines properties of functions like being one-to-one, even, or odd and provides examples of each.
This document proposes a linear programming (LP) based approach for solving maximum a posteriori (MAP) estimation problems on factor graphs that contain multiple-degree non-indicator functions. It presents an existing LP method for problems with single-degree functions, then introduces a transformation to handle multiple-degree functions by introducing auxiliary variables. This allows applying the existing LP method. As an example, it applies this to maximum likelihood decoding for the Gaussian multiple access channel. Simulation results demonstrate the LP approach decodes correctly with polynomial complexity.
MATLAB - Aplication of Arrays and Matrices in Electrical SystemsShameer Ahmed Koya
This document discusses using matrices and linear algebra to analyze electrical systems. It explains how to represent sets of linear equations in matrix form as A*x=y and solve for x using the inverse of A. It also discusses calculating the rank of a matrix and condition number. Examples are given of using matrices to solve circuit problems involving node analysis, mesh analysis, and AC circuits. Solutions are found by taking the inverse of the matrix or using the backslash operator in MATLAB.
بررسی دو روش شناسایی سیستم های متغیر با زمان به همراه شبیه سازی و گزارشپروژه مارکت
بررسی دو روش شناسایی سیستم متغیر بازمان با جزییات و توضیحات ساده و گویا و توضیح روابط و علایم به طور کامل به همراه مثال و شبیه سازی در متلب و فایل های مرجع و مقالات مرجع
روش اول: شناسایی سیستم با پارامترهای متغیر با زمان با استفاده از Least Square
روش دوم: شناسایی سیستم دینامیکی با استفاده از شبکه های عصبی مصنوعی
مناسب برای ارائه به عنوان تحقیق یا پروژه درس شناسایی سیستم ها
به همراه شبیه سازی در متلب و گزارش 20 صفحه ای
https://www.prjmarket.com/product/%d8%a8%d8%b1%d8%b1%d8%b3%db%8c-%d8%af%d9%88-%d8%b1%d9%88%d8%b4-%d8%b4%d9%86%d8%a7%d8%b3%d8%a7%db%8c%db%8c-%d8%b3%db%8c%d8%b3%d8%aa%d9%85-%d9%87%d8%a7%db%8c-%d9%85%d8%aa%d8%ba%db%8c%d8%b1-%d8%a8%d8%a7/
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
The document discusses algorithm analysis and design. It begins with an introduction to analyzing algorithms including average case analysis and solving recurrences. It then provides definitions of algorithms both informal and formal. Key aspects of algorithm study and specification methods like pseudocode are outlined. Selection sort and tower of Hanoi problems are presented as examples and analyzed for time and space complexity. Average case analysis is discussed assuming all inputs are equally likely.
Recommendation System --Theory and PracticeKimikazu Kato
This document provides an overview of recommendation systems and collaborative filtering techniques. It discusses using matrix factorization to predict user ratings by representing users and items as vectors in a latent factor space. Optimization techniques like stochastic gradient descent can be used to learn the factorization from existing ratings. The document also notes challenges of sparsity and scale for practical systems and describes approaches like elastic net regularization and sparsification to address these.
Newton divided difference interpolationVISHAL DONGA
This document presents Newton's divided difference polynomial method of interpolation. It defines interpolation as finding the value of 'y' at an unspecified value of 'x' given a set of (x,y) data points. Newton's method uses divided differences to determine the coefficients of a polynomial that can be used to interpolate and estimate y-values between the given data points. The document includes an example of applying Newton's method to find the interpolating polynomial and estimate an unknown y-value for a given set of 5 (x,y) data points.
The branch and bound method searches a tree model of the solution space for discrete optimization problems. It uses bounding functions to prune subtrees that cannot contain optimal solutions, avoiding evaluating all possible solutions. The method generates the tree in a depth-first manner, and selects the next node to expand (the E-node) based on cost estimates. Common strategies are FIFO, LIFO, and least cost search. The traveling salesman problem can be solved using branch and bound by formulating the solution space as a tree and applying row and column reduction techniques to the cost matrix at each node to identify prunable branches.
This document contains lecture notes on sparse autoencoders. It begins with an introduction describing the limitations of supervised learning and the need for algorithms that can automatically learn feature representations from unlabeled data. The notes then state that sparse autoencoders are one approach to learn features from unlabeled data, and describe the organization of the rest of the notes. The notes will cover feedforward neural networks, backpropagation for supervised learning, autoencoders for unsupervised learning, and how sparse autoencoders are derived from these concepts.
The document provides information about Lagrangian interpolation, including:
1. It introduces Lagrangian interpolation as a method to find the value of a function at a discrete point using a polynomial that passes through known data points.
2. It gives the formula for the Lagrangian interpolating polynomial and provides an example of using it to find the velocity of a rocket at a certain time.
3. It discusses using higher order polynomials for interpolation, providing another example that calculates velocity using quadratic and cubic polynomials.
The document discusses various math and string classes in Java. It covers:
- Constructing objects using the new operator and passing parameters.
- Using the Random class to generate random numbers.
- Declaring constants using final and static final.
- Basic arithmetic, increment/decrement, and math methods.
- Creating and manipulating strings using methods like length(), substring(), and concatenation.
- Drawing shapes on a frame using Graphics2D methods in a JComponent's paintComponent method.
Path relinking for high dimensional continuous optimizationPatxi Gortázar
Una presentación de un algoritmo evolutivo para resolver problemas de optimización global que di en el Metaheuristics International Conference de 2011 en Udine, Italia.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
The document discusses dictionaries and different data structures used to implement them efficiently, including hashing. Dictionaries store key-value pairs and support fast retrieval of elements using keys. Hashing is described as an efficient solution to implement dictionaries, where keys are mapped to table slots using a hash function. Hashing provides expected O(1) time for search, insert and delete operations when the load factor is a constant. Collision resolution using chaining is explained, where elements hash to the same slot are stored in a linked list.
Here are the steps to solve this ODE problem:
1. Define the ODE function:
function dydt = odefun(t,y)
dydt = -t.*y/10;
end
2. Solve the ODE:
[t,y] = ode45(@odefun,[0 10],10);
3. Plot the result:
plot(t,y)
xlabel('t')
ylabel('y(t)')
This uses ode45 to solve the ODE dy/dt = -t*y/10 on the interval [0,10] with initial condition y(0)=10.
This document discusses various ways to work with polynomials in MATLAB. It covers representing polynomials as vectors, evaluating polynomials, finding roots, computing coefficients from roots, adding, subtracting, multiplying, dividing, taking derivatives and integrals of polynomials, fitting curves to data using polynomial regression, and interpolating values between data points using polynomials.
MatLab is a matrix-based program used for numeric computation and visualization. It contains functions for creating matrices and performing operations on them like addition, subtraction, multiplication, etc. It also allows plotting of functions and special matrices like magic squares, identity matrices, and Toeplitz matrices. Polynomial operations can also be performed such as addition, multiplication, differentiation and evaluation.
Here are the steps to solve this problem numerically in MATLAB:
1. Define the 2nd order ODE for the pendulum as two first order equations:
y1' = y2
y2' = -sin(y1)
2. Create an M-file function pendulum.m that returns the right hand side:
function dy = pendulum(t,y)
dy = [y(2); -sin(y(1))];
end
3. Use an ODE solver like ode45 to integrate from t=0 to t=6pi with initial conditions y1(0)=pi, y2(0)=0:
[t,y] = ode45
MATLAB can be used for calculation, plotting, image processing, and modeling. It allows defining variables and performing operations on vectors, arrays, and matrices. Key functions include plotting graphs, solving equations, performing calculus operations like derivatives and integrals, and applying built-in functions to arrays.
This document provides an introduction to MATLAB. It begins with an overview of the MATLAB environment and display windows. It then discusses getting help in MATLAB, variables, vectors, matrices, linear algebra, plotting, built-in functions, selection programming using if/else statements, M-files, user-defined functions, and specific topics. Key points covered include the MATLAB interface, basic programming constructs like variables and arrays, and tools for computation, visualization, and programming in MATLAB.
This document provides an introduction and overview of Matlab. It outlines what Matlab is, the main Matlab screen components, how to work with variables, arrays, matrices and perform indexing. It also covers basic arithmetic, relational and logical operators, different display facilities like plotting, and flow control structures like if/else statements and for loops. The document demonstrates how to use M-files to write scripts and user-defined functions in Matlab. It aims to introduce the key features and capabilities of the Matlab programming environment and language.
This document provides an overview of key algebra and functions concepts covered on the SAT, including operations on algebraic expressions, factoring, exponents, evaluating expressions, solving equations, inequalities, systems of equations, quadratic equations, rational expressions, direct and inverse variation, word problems, functions, and translations of functions. Key points are that the SAT will not include questions requiring the quadratic formula or complex numbers, and factoring may be required to solve or evaluate expressions even if a question does not explicitly say to factor.
This document provides an overview of key algebra and functions concepts covered on the SAT, including operations on algebraic expressions, factoring, exponents, evaluating expressions, solving equations, inequalities, systems of equations, quadratic equations, rational expressions, direct and inverse variation, word problems, functions, and translations of functions. Key points are that the SAT will not include solving quadratic equations using the quadratic formula or complex numbers. Factoring may be required to solve or evaluate expressions. Functions are represented using function notation and defined by their domain and range.
This document provides an overview of key algebra and functions concepts covered on the SAT, including operations on algebraic expressions, factoring, exponents, evaluating expressions, solving equations, inequalities, systems of equations, quadratic equations, rational expressions, direct and inverse variation, word problems, functions, and translations of functions. Key points are that the SAT will not include solving quadratic equations using the quadratic formula or complex numbers. Factoring may be required to solve or evaluate expressions. Functions are represented using function notation and defined by their domain and range.
This document provides an introduction to MATLAB. It discusses that MATLAB is a high-level language for technical computing where everything is a matrix and it is easy to perform linear algebra. It describes the MATLAB desktop interface and valid variable names. It also summarizes how to perform basic operations like addition, subtraction, multiplication, etc. on matrices and vectors. Finally, it outlines various matrix operations, statistical functions, random number generation, and plotting in MATLAB.
This document provides an introduction and overview of MATLAB (Matrix Laboratory), an interactive program for numerical computation and visualization. It discusses basic MATLAB commands and functions for creating variables and matrices, performing mathematical operations, plotting graphs, and working with polynomials.
The name MATLAB stands for MATrix LABoratory.MATLAB is a high-performance language for technical computing.
It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming.
These factor make MATLAB an excellent tool for teaching and research.
This document provides an introduction to MATLAB. It covers MATLAB basics like arithmetic, variables, vectors, matrices and built-in functions. It also discusses plotting graphs, programming in MATLAB by creating functions and scripts, and solving systems of linear equations. The document is compiled by Endalkachew Teshome from the Department of Mathematics at AMU for teaching MATLAB.
1. This document provides rules for taking derivatives of basic functions including:
- Constant functions: f'(x) = 0
- Power functions: f(x) = x^n, f'(x) = nx^(n-1)
2. It also outlines rules for derivatives of sums, differences, and constant multiples of functions.
3. Experiments are described using calculators and geometry software to demonstrate that the derivative of sin(x) is cos(x) and the derivative of e^x is also e^x.
This is the version of my 3D math talk that I used at CocoaConf Atlanta. This version includes the graphic representations of the different steps in implementing the shader.
Here are the steps to solve this problem in MATLAB:
1. Create a matrix A with the marks data:
A = [24 44 36 36;
52 57 68 76;
66 53 69 73;
85 40 86 72;
15 47 25 28;
79 72 82 91];
2. Define the column labels:
subjects = {'Math','Programming','Thermodynamics','Mechanics'};
3. Find the total marks of each subject:
totals = sum(A)
4. Find the average marks of each subject:
averages = mean(A)
5. Find the highest marks scored in each subject:
maxMarks = max
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This document provides an overview of the syllabus and assignments for an introductory Python programming course. The syllabus covers variables and types, functions, control flow, objects, modules, data structures, and two programming projects. Assignments include reading the textbook, writing simple print programs, installing Python, using the interpreter, and exercises on variables and basic operators.
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* Live demos with code snippets
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2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
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What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
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See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
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2. 6.094
Introduction to Programming in MATLAB®
Lecture 3 : Solving Equations and Curve Fitting
Sourav Dey
Danilo Šćepanović
Ankit Patel
Patrick Ho
IAP 2009
4. Systems of Linear Equations
• Given a system of linear equations MATLAB makes linear
x+2y-3z=5
algebra fun!
-3x-y+z=-8
x-y+z=0
• Construct matrices so the system is described by Ax=b
» A=[1 2 -3;-3 -1 1;1 -1 1];
» b=[5;-8;0];
• And solve with a single line of code!
» x=Ab;
x is a 3x1 vector containing the values of x, y, and z
• The will work with square or rectangular systems.
• Gives least squares solution for rectangular systems. Solution
depends on whether the system is over or underdetermined.
5. More Linear Algebra
• Given a matrix
» mat=[1 2 -3;-3 -1 1;1 -1 1];
• Calculate the rank of a matrix
» r=rank(mat);
the number of linearly independent rows or columns
• Calculate the determinant
» d=det(mat);
mat must be square
if determinant is nonzero, matrix is invertible
• Get the matrix inverse
» E=inv(mat);
if an equation is of the form A*x=b with A a square matrix,
x=Ab is the same as x=inv(A)*b
6. Matrix Decompositions
• MATLAB has built-in matrix decomposition methods
• The most common ones are
» [V,D]=eig(X)
Eigenvalue decomposition
» [U,S,V]=svd(X)
Singular value decomposition
» [Q,R]=qr(X)
QR decomposition
7. Exercise: Linear Algebra
• Solve the following systems of equations:
System 1:
x+4y=34
-3x+y=2
System 2:
2x-2y=4
-x+y=3
3x+4y = 2
8. Exercise: Linear Algebra
• Solve the following systems of equations:
System 1: » A=[1 4;-3 1];
x+4y=34 » b=[34;2];
-3x+y=2 » rank(A)
» x=inv(A)*b;
System 2: » A=[2 -2;-1 1;3 4];
2x-2y=4 » b=[4;3;2];
-x+y=3 » rank(A)
3x+4y = 2 rectangular matrix
» x1=Ab;
gives least squares solution
» A*x1
10. Polynomials
• Many functions can be well described by a high-order
polynomial
• MATLAB represents a polynomials by a vector of coefficients
if vector P describes a polynomial
– ax3+bx2+cx+d
P(1) P(2) P(3) P(4)
• P=[1 0 -2] represents the polynomial x2-2
• P=[2 0 0 0] represents the polynomial 2x3
11. Polynomial Operations
• P is a vector of length N+1 describing an N-th order polynomial
• To get the roots of a polynomial
» r=roots(P)
r is a vector of length N
• Can also get the polynomial from the roots
» P=poly(r)
r is a vector length N
• To evaluate a polynomial at a point
» y0=polyval(P,x0)
x0 is a single value; y0 is a single value
• To evaluate a polynomial at many points
» y=polyval(P,x)
x is a vector; y is a vector of the same size
12. Polynomial Fitting
• MATLAB makes it very easy to fit polynomials to data
• Given data vectors X=[-1 0 2] and Y=[0 -1 3]
» p2=polyfit(X,Y,2);
finds the best second order polynomial that fits the points
(-1,0),(0,-1), and (2,3)
see help polyfit for more information
» plot(X,Y,’o’, ‘MarkerSize’, 10);
» hold on;
» x = linspace(-2,2,1000);
» plot(x,polyval(p2,x), ‘r--’);
13. Exercise: Polynomial Fitting
• Evaluate x^2 over x=-4:0.1:4 and save it as y.
• Add random noise to these samples. Use randn. Plot the
noisy signal with . markers
• fit a 2nd degree polynomial to the noisy data
• plot the fitted polynomial on the same plot, using the same
x values and a red line
14. Exercise: Polynomial Fitting
• Evaluate x^2 over x=-4:0.1:4 and save it as y.
» x=-4:0.1:4;
» y=x.^2;
• Add random noise to these samples. Use randn. Plot the
noisy signal with . markers
» y=y+randn(size(y));
» plot(x,y,’.’);
• fit a 2nd degree polynomial to the noisy data
» [p]=polyfit(x,y,2);
• plot the fitted polynomial on the same plot, using the same
x values and a red line
» hold on;
» plot(x,polyval(p,x),’r’)
16. Nonlinear Root Finding
• Many real-world problems require us to solve f(x)=0
• Can use fzero to calculate roots for any arbitrary function
• fzero needs a function passed to it.
• We will see this more and more as we delve into solving
equations.
• Make a separate function file
» x=fzero('myfun',1)
» x=fzero(@myfun,1)
1 specifies a
point close to
the root
Courtesy of The MathWorks, Inc. Used with permission.
17. Minimizing a Function
• fminbnd: minimizing a function over a bounded interval
» x=fminbnd('myfun',-1,2);
myfun takes a scalar input and returns a scalar output
myfun(x) will be the minimum of myfun for -1≤x ≤ 2
• fminsearch: unconstrained interval
» x=fminsearch('myfun',.5)
finds the local minimum of myfun starting at x=0.5
18. Anonymous Functions
• You do not have to make a separate function file
• Instead, you can make an anonymous function
» x=fzero(@(x)(cos(exp(x))+x^2-1), 1 );
input function to evaluate
» x=fminbnd(@(x) (cos(exp(x))+x^2-1),-1,2);
19. Optimization Toolbox
• If you are familiar with optimization methods, use the
optimization toolbox
• Useful for larger, more structured optimization problems
• Sample functions (see help for more info)
» linprog
linear programming using interior point methods
» quadprog
quadratic programming solver
» fmincon
constrained nonlinear optimization
20. Exercise: Min-Finding
Find the minimum of the function f(x) =
cos(4*x).*sin(10*x).*exp(-abs(x)) over the range –pi to
pi. Use fminbnd. Is your answer really the minimum
over this range?
21. Exercise: Min-Finding
Find the minimum of the function f(x) =
cos(4*x).*sin(10*x).*exp(-abs(x)) over the range –pi to
pi. Use fminbnd. Is your answer really the minimum
over this range?
function y = myFun(x)
y=cos(4*x).*sin(10*x).*exp(-abs(x));
fminbnd(‘myFun’, -pi, pi);
23. Numerical Differentiation
1
• MATLAB can 'differentiate' numerically
0.8
» x=0:0.01:2*pi; 0.6
0.4
» y=sin(x); 0.2
» dydx=diff(y)./diff(x); 0
diff computes the first difference -0.2
-0.4
-0.6
• Can also operate on matrices -0.8
» mat=[1 3 5;4 8 6]; -1
0 100 200 300 400 500 600 700
» dm=diff(mat,1,2)
first difference of mat along the 2nd dimension, dm=[2 2;4 -2]
see help for more details
• 2D gradient
» [dx,dy]=gradient(mat);
24. Numerical Integration
• MATLAB contains common integration methods
• Adaptive Simpson's quadrature (input is a function)
» q=quad('derivFun',0,10);
q is the integral of the function derivFun from 0 to 10
» q2=quad(@sin,0,pi)
q2 is the integral of sin from 0 to pi
• Trapezoidal rule (input is a vector)
» x=0:0.01:pi;
» z=trapz(x,sin(x));
z is the integral of sin(x) from 0 to pi
» z2=trapz(x,sqrt(exp(x))./x)
x
z2 is the integral of e x from 0 to pi
26. ODE Solvers: Method
• Given a differential equation, the solution can be found by
integration:
Evaluate the derivative at a point and approximate by straight line
Errors accumulate!
Variable timestep can decrease the number of iterations
27. ODE Solvers: MATLAB
• MATLAB contains implementations of common ODE solvers
• Using the correct ODE solver can save you lots of time and
give more accurate results
» ode23
Low-order solver. Use when integrating over small intervals
or when accuracy is less important than speed
» ode45
High order (Runge-Kutta) solver. High accuracy and
reasonable speed. Most commonly used.
» ode15s
Stiff ODE solver (Gear's algorithm), use when the diff eq's
have time constants that vary by orders of magnitude
28. ODE Solvers: Standard Syntax
• To use standard options and variable time step
» [t,y]=ode45('myODE',[0,10],[1;0])
ODE integrator: Initial conditions
23, 45, 15s Time range
ODE function
• Inputs:
ODE function name (or anonymous function). This function
takes inputs (t,y), and returns dy/dt
Time interval: 2-element vector specifying initial and final
time
Initial conditions: column vector with an initial condition for
each ODE. This is the first input to the ODE function
• Outputs:
t contains the time points
y contains the corresponding values of the integrated fcn.
29. ODE Function
• The ODE function must return the value of the derivative at
a given time and function value
• Example: chemical reaction 10
Two equations
dA A B
= −10 A + 50 B
dt 50
dB
= 10 A − 50 B
dt
ODE file:
– y has [A;B]
– dydt has
[dA/dt;dB/dt]
Courtesy of The MathWorks, Inc.
Used with permission.
30. ODE Function: viewing results
• To solve and plot the ODEs on the previous slide:
» [t,y]=ode45('chem',[0 0.5],[0 1]);
assumes that only chemical B exists initially
» plot(t,y(:,1),'k','LineWidth',1.5);
» hold on;
» plot(t,y(:,2),'r','LineWidth',1.5);
» legend('A','B');
» xlabel('Time (s)');
» ylabel('Amount of chemical (g)');
» title('Chem reaction');
31. ODE Function: viewing results
• The code on the previous slide produces this figure
Chem reaction
1
A
0.9 B
0.8
0.7
Amount of chemical (g)
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (s)
32. Higher Order Equations
• Must make into a system of first-order equations to use
ODE solvers
• Nonlinear is OK!
• Pendulum example:
g
&&
θ+ sin (θ ) = 0
L
g
&&
θ =− sin (θ )
L
&
let θ = γ
g
γ& = − sin (θ )
L
v ⎡θ ⎤
x=⎢ ⎥
⎣γ ⎦
v &
dx ⎡θ ⎤
=⎢ ⎥
dt ⎣γ& ⎦
Courtesy of The MathWorks, Inc. Used with permission.
33. Plotting the Output
• We can solve for the position and velocity of the pendulum:
» [t,x]=ode45('pendulum',[0 10],[0.9*pi 0]);
assume pendulum is almost vertical (at top)
» plot(t,x(:,1));
» hold on;
» plot(t,x(:,2),'r');
» legend('Position','Velocity');
8
Position
Velocity
6
Velocity (m/s)
Position in terms of 4
angle (rad) 2
0
-2
-4
-6
-8
0 1 2 3 4 5 6 7 8 9 10
34. Plotting the Output
• Or we can plot in the phase plane:
» plot(x(:,1),x(:,2));
» xlabel('Position');
» yLabel('Velocity');
• The phase plane is just a plot of one variable versus the
other: 8
6
4
Velocity is greatest
2 when theta=0
Velocity
0
-2
Velocity=0 when -4
theta is the greatest -6
-8
-3 -2 -1 0 1 2 3
Position
35. ODE Solvers: Custom Options
• MATLAB's ODE solvers use a variable timestep
• Sometimes a fixed timestep is desirable
» [t,y]=ode45('chem',[0:0.001:0.5],[0 1]);
Specify the timestep by giving a vector of times
The function will be evaluated at the specified points
Fixed timestep is usually slower (if timestep is small) and
possibly inaccurate (if timestep is too large)
• You can customize the error tolerances using odeset
» options=odeset('RelTol',1e-6,'AbsTol',1e-10);
» [t,y]=ode45('chem',[0 0.5],[0 1],options);
This guarantees that the error at each step is less than
RelTol times the value at that step, and less than AbsTol
Decreasing error tolerance can considerably slow the solver
See doc odeset for a list of options you can customize
36. Exercise: ODE
• Use ODE45 to solve this differential equation on the range
t=[0 10], with initial condition y(0) = 10: dy/dt=-t*y/10.
Plot the result.
37. Exercise: ODE
• Use ODE45 to solve this differential equation on the range
t=[0 10], with initial condition y(0) = 10: dy/dt=-t*y/10.
Plot the result.
» function dydt=odefun(t,y)
» dydt=-t*y/10;
» [t,y]=ode45(‘odefun’,[0 10],10);
» plot(t,y);
38. End of Lecture 3
(1) Linear Algebra
(2) Polynomials
(3) Optimization
(4) Differentiation/Integration
(5) Differential Equations
We're almost done!
39. Issues with ODEs
• Stability and accuracy
if step size is too large, solutions might blow up
if step size is too small, requires a long time to solve
use odeset to control errors
– decrease error tolerances to get more accurate
results
– increase error tolerances to speed up computation
(beware of instability!)
• Main thing to remember about ODEs
Pick the most appropriate solver for your problem
If ode45 is taking too long, try ode15s