PRESENTATION IS ABOUT SOLUTION OF DIFFERENTIAL EQUATION BY FOLLOWING THREE METHODS
1. VARIATION OF PARAMETER
2. CAUCHY S EQUATION
3. UNDETERMINED COEFFICIENT
AND BASIC FORMULAS AND SOLVED EXAMPLES ARE INCLUDED
Gaussian elimination and Gauss-Jordan elimination are methods for solving systems of linear equations by reducing the coefficients matrix to row-echelon form. Gauss-Jordan elimination transforms the matrix into an identity matrix by eliminating each variable in turn from all equations. Gauss-Jordan with pivoting chooses rows strategically to minimize rounding errors during calculations.
Parallel Numerical Methods for Ordinary Differential Equations: a SurveyUral-PDC
This document summarizes parallel numerical methods for solving ordinary differential equations (ODEs). It discusses two types of parallelism: across the system (space) and across the method (time). Predictor-corrector and Runge-Kutta methods are described that can exploit parallelism across time by parallelizing stages. Optimal Runge-Kutta methods use the minimum number of stages for a given order. Block methods solve ODE systems in parallel. Extrapolation and multiple shooting methods are also mentioned for parallelizing ODE solutions.
The document summarizes iterative methods for solving systems of linear equations, including the Jacobi method, Gauss-Seidel method, and Gauss-Seidel method with relaxation. The Jacobi method uses successive diagonalizations to drive the error to zero. The Gauss-Seidel method improves on Jacobi by using previous iterations to update components. Relaxation modifies Gauss-Seidel by taking a weighted average of current and previous iterations, allowing values between 0-1 for convergence or 1-2 for acceleration.
The document discusses diagonalization of matrices. Diagonalization involves finding an invertible matrix P such that P-1AP is a diagonal matrix. The procedure for diagonalizing a matrix A includes: (1) finding the eigenvalues of A, (2) finding linearly independent eigenvectors corresponding to the eigenvalues, (3) forming the matrix P with the eigenvectors as columns, and (4) showing that P-1AP is a diagonal matrix with the eigenvalues along the diagonal. An example demonstrates finding the eigenvectors of a 2x2 matrix A and constructing the matrix P to diagonalize A.
Math Geophysics-system of linear algebraic equationsAmin khalil
The document provides an overview of linear algebra concepts for mathematical geophysics, including:
- Definitions of equations, systems of linear algebraic equations, and the Gauss-Jordan reduction method.
- Types of systems include unique solution, no solution, and infinitely many solutions.
- Einstein summation convention simplifies tensor equations by implicitly summing over repeated indices.
- Gaussian elimination uses row operations to put a system of equations in row echelon form and then reduced row echelon form to solve for variables.
- Systems can have unique solutions, no solutions, or multiple solutions depending on the relationships between equations and variables.
The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.
Gaussian elimination and Gauss-Jordan elimination are methods for solving systems of linear equations by reducing the coefficients matrix to row-echelon form. Gauss-Jordan elimination transforms the matrix into an identity matrix by eliminating each variable in turn from all equations. Gauss-Jordan with pivoting chooses rows strategically to minimize rounding errors during calculations.
Parallel Numerical Methods for Ordinary Differential Equations: a SurveyUral-PDC
This document summarizes parallel numerical methods for solving ordinary differential equations (ODEs). It discusses two types of parallelism: across the system (space) and across the method (time). Predictor-corrector and Runge-Kutta methods are described that can exploit parallelism across time by parallelizing stages. Optimal Runge-Kutta methods use the minimum number of stages for a given order. Block methods solve ODE systems in parallel. Extrapolation and multiple shooting methods are also mentioned for parallelizing ODE solutions.
The document summarizes iterative methods for solving systems of linear equations, including the Jacobi method, Gauss-Seidel method, and Gauss-Seidel method with relaxation. The Jacobi method uses successive diagonalizations to drive the error to zero. The Gauss-Seidel method improves on Jacobi by using previous iterations to update components. Relaxation modifies Gauss-Seidel by taking a weighted average of current and previous iterations, allowing values between 0-1 for convergence or 1-2 for acceleration.
The document discusses diagonalization of matrices. Diagonalization involves finding an invertible matrix P such that P-1AP is a diagonal matrix. The procedure for diagonalizing a matrix A includes: (1) finding the eigenvalues of A, (2) finding linearly independent eigenvectors corresponding to the eigenvalues, (3) forming the matrix P with the eigenvectors as columns, and (4) showing that P-1AP is a diagonal matrix with the eigenvalues along the diagonal. An example demonstrates finding the eigenvectors of a 2x2 matrix A and constructing the matrix P to diagonalize A.
Math Geophysics-system of linear algebraic equationsAmin khalil
The document provides an overview of linear algebra concepts for mathematical geophysics, including:
- Definitions of equations, systems of linear algebraic equations, and the Gauss-Jordan reduction method.
- Types of systems include unique solution, no solution, and infinitely many solutions.
- Einstein summation convention simplifies tensor equations by implicitly summing over repeated indices.
- Gaussian elimination uses row operations to put a system of equations in row echelon form and then reduced row echelon form to solve for variables.
- Systems can have unique solutions, no solutions, or multiple solutions depending on the relationships between equations and variables.
The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.
Solution of equations for methods iterativosDUBAN CASTRO
This document discusses iterative methods for solving systems of equations. It describes the Jacobi method, which solves systems by iteratively updating solutions. It also describes Gauss-Seidel method, which improves on Jacobi by using previous updated solutions in the current iteration. Both methods are used to progressively calculate better approximations to the solution until reaching an acceptable level of accuracy.
Basic terminology description in convex optimizationVARUN KUMAR
This document provides an introduction to basic concepts in convex optimization including null space of a matrix, convex and affine sets, half spaces, hyperplanes, norms, norm balls, and practical applications of ellipses and ellipsoids. Key terms such as null space, convex and affine sets, half spaces, hyperplanes, various norms (l1, l2, l∞), norm balls, ellipses, and ellipsoids are defined. Graphical representations are also provided for concepts like convex vs affine sets and different norms. Practical applications of ellipses and ellipsoids to 2D and higher dimensional spaces are demonstrated through mathematical descriptions relating the concepts to norms.
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
1. This document discusses methods for solving linear algebraic equations and operations involving matrices. It covers topics such as matrix definitions, types of matrices, matrix operations, representing equations in matrix form, and methods for solving systems of linear equations including graphical methods, determinants, Cramer's rule, elimination, Gauss-Jordan, LU decomposition, and calculating the matrix inverse.
2. Key matrix operations include addition, multiplication, and rules for inverting a matrix. Methods for solving systems of equations include graphical techniques, determinants, Cramer's rule, elimination, Gauss, Gauss-Jordan, and LU decomposition.
3. LU decomposition involves writing a matrix as the product of a lower and upper triangular matrix, which can
This document defines key concepts related to quadratic functions and their graphs:
1) A quadratic function is a second-degree polynomial function of the form f(x) = ax^2 + bx + c, where a cannot be 0.
2) The graph of a quadratic function is called a parabola, which has a vertex and may open upward or downward depending on the sign of a.
3) The axis of symmetry is the vertical line through the vertex that divides the parabola into two equal parts.
Presentation on application of numerical method in our lifeManish Kumar Singh
This document discusses the application of numerical methods in real-life problems. It provides examples of using the bisection method to find the root of equations related to estimating ocean currents, modeling combustion flow, airflow patterns, and other applications. Specifically, it shows the steps to use the bisection method to estimate the depth at which a floating ball with given properties would be submerged. Over three iterations, it computes the estimated root, error, and number of significant digits estimated.
This document discusses three main topics: positive definite matrices, solving linear systems, and the least squares method.
Positive definite matrices are symmetric matrices where all eigenvalues are positive. Solving linear systems involves finding a single solution that satisfies two or more linear equations with the same variables.
The least squares method determines the line of best fit for a data set by minimizing the sum of the squared differences between the independent variable values and the dependent variable values predicted by the line or curve. It provides the closest approximate solution when a linear system has no exact solution.
This document discusses a technique for computing Euler angles (ψ, θ, φ) from a given rotation matrix. There are generally two possible solutions, except when the cosine of θ is 0, in which case there are an infinite number of solutions. The technique involves equating elements of the rotation matrix to those of a matrix representing rotation about each axis. This allows solving for the two possible values of θ, and the corresponding values of ψ and φ. Special cases when the cosine of θ is 0 require using different elements of the rotation matrix. Pseudocode demonstrates how to implement the technique to obtain the Euler angles from a given rotation matrix.
The document discusses iterative methods for solving systems of linear equations, including the Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. The Jacobi method works by rewriting the system in a form where the diagonal entries are isolated and computing successive approximations. The Gauss-Seidel method similarly computes approximations but uses the most recent values available at each step. Relaxation improves the Gauss-Seidel method's convergence by taking a weighted average of the current and previous iterations' results. Examples demonstrate applying the different methods to compute solutions.
This document solves 5 set theory laws:
1) Idempotence: A union A = A and A intersection A = A. Given A = {1,2,3,4}, A union A = {1,2,3,4} and A intersection A = {1,2,3,4}.
2) Associativity: (A union B) union C = A union (B union C). Given sets A, B, and C, both sides equal the union of all elements.
3) Commutativity: A union B = B union A and A intersection B = B intersection A. Given sets A and B, their union and intersection are symmetric.
4)
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
This document presents a formula for calculating the area of a regular n-sided polygon using the area and perimeter of an inscribed circle. It proves that the ratio of the polygon's perimeter to its area equals the ratio of the circle's perimeter to its area. The formula derived is that the area of the polygon equals the area of the circle multiplied by the polygon's perimeter divided by the circle's perimeter. This method provides an accurate and efficient way to calculate a regular polygon's area without needing to divide it into triangles first.
Definitions matrices y determinantes fula 2010 english subirHernanFula
The document discusses the history and properties of matrices. It describes how matrices were first introduced in 1850 and how their use has expanded. It then defines key matrix terms and concepts such as order, elements, types of matrices (e.g. triangular, diagonal), operations (e.g. addition, multiplication, inverse), and properties (e.g. of symmetric, banded and transpose matrices). It also provides examples of calculating the determinant of matrices using Sarrus' rule.
System of linear algebriac equations nsmRahul Narang
The document discusses systems of linear algebraic equations and methods for solving them numerically. It introduces systems of linear equations in matrix form Ax = b and describes elementary row operations that can transform the matrix A. It then explains Gaussian elimination and Gauss-Jordan elimination methods for solving systems of linear equations by transforming the augmented matrix into reduced row echelon form. Finally, it briefly describes Jacobi and Gauss-Seidel iterative methods as well as applications of linear algebra in computer science fields like statistical learning, image manipulation, and physics.
The document discusses solving linear differential equations with variable coefficients using power series representations. It begins by introducing series solution methods and properties of infinite series. It then discusses the Method of Frobenius for solving equations with variable coefficients that arise in cylindrical and spherical coordinate systems. The method involves finding the indicial equation to determine the index c, and then using the recurrence relation to determine the series coefficients an. An example is provided to illustrate the method where the roots of the indicial equation are distinct and not differing by an integer.
This document discusses rank-aware thresholding algorithms for compressed sensing. It begins by introducing compressed sensing and explaining how traditional linear algebra techniques cannot be used to recover sparse signals from undersampled measurements. It then describes how thresholding and rank-aware thresholding algorithms work by exploiting the sparsity of signals. The key points are that rank-aware thresholding outperforms standard thresholding by eliminating the "square-root bottleneck" and requires only O(k) measurements, versus O(k^2) for thresholding. Simulation results demonstrate this improvement. The document concludes by discussing modeling techniques to predict algorithm performance on very large problems that are impractical to simulate directly.
Bayesian Variable Selection in Linear Regression and A ComparisonAtilla YARDIMCI
In this study, Bayesian approaches, such as Zellner, Occam’s Window and Gibbs sampling, have been compared in terms of selecting the correct subset for the variable selection in a linear regression model. The aim of this comparison is to analyze Bayesian variable selection and the behavior of classical criteria by taking into consideration the different values of β and σ and prior expected levels.
Solution of equations for methods iterativosDUBAN CASTRO
This document discusses iterative methods for solving systems of equations. It describes the Jacobi method, which solves systems by iteratively updating solutions. It also describes Gauss-Seidel method, which improves on Jacobi by using previous updated solutions in the current iteration. Both methods are used to progressively calculate better approximations to the solution until reaching an acceptable level of accuracy.
Basic terminology description in convex optimizationVARUN KUMAR
This document provides an introduction to basic concepts in convex optimization including null space of a matrix, convex and affine sets, half spaces, hyperplanes, norms, norm balls, and practical applications of ellipses and ellipsoids. Key terms such as null space, convex and affine sets, half spaces, hyperplanes, various norms (l1, l2, l∞), norm balls, ellipses, and ellipsoids are defined. Graphical representations are also provided for concepts like convex vs affine sets and different norms. Practical applications of ellipses and ellipsoids to 2D and higher dimensional spaces are demonstrated through mathematical descriptions relating the concepts to norms.
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
1. This document discusses methods for solving linear algebraic equations and operations involving matrices. It covers topics such as matrix definitions, types of matrices, matrix operations, representing equations in matrix form, and methods for solving systems of linear equations including graphical methods, determinants, Cramer's rule, elimination, Gauss-Jordan, LU decomposition, and calculating the matrix inverse.
2. Key matrix operations include addition, multiplication, and rules for inverting a matrix. Methods for solving systems of equations include graphical techniques, determinants, Cramer's rule, elimination, Gauss, Gauss-Jordan, and LU decomposition.
3. LU decomposition involves writing a matrix as the product of a lower and upper triangular matrix, which can
This document defines key concepts related to quadratic functions and their graphs:
1) A quadratic function is a second-degree polynomial function of the form f(x) = ax^2 + bx + c, where a cannot be 0.
2) The graph of a quadratic function is called a parabola, which has a vertex and may open upward or downward depending on the sign of a.
3) The axis of symmetry is the vertical line through the vertex that divides the parabola into two equal parts.
Presentation on application of numerical method in our lifeManish Kumar Singh
This document discusses the application of numerical methods in real-life problems. It provides examples of using the bisection method to find the root of equations related to estimating ocean currents, modeling combustion flow, airflow patterns, and other applications. Specifically, it shows the steps to use the bisection method to estimate the depth at which a floating ball with given properties would be submerged. Over three iterations, it computes the estimated root, error, and number of significant digits estimated.
This document discusses three main topics: positive definite matrices, solving linear systems, and the least squares method.
Positive definite matrices are symmetric matrices where all eigenvalues are positive. Solving linear systems involves finding a single solution that satisfies two or more linear equations with the same variables.
The least squares method determines the line of best fit for a data set by minimizing the sum of the squared differences between the independent variable values and the dependent variable values predicted by the line or curve. It provides the closest approximate solution when a linear system has no exact solution.
This document discusses a technique for computing Euler angles (ψ, θ, φ) from a given rotation matrix. There are generally two possible solutions, except when the cosine of θ is 0, in which case there are an infinite number of solutions. The technique involves equating elements of the rotation matrix to those of a matrix representing rotation about each axis. This allows solving for the two possible values of θ, and the corresponding values of ψ and φ. Special cases when the cosine of θ is 0 require using different elements of the rotation matrix. Pseudocode demonstrates how to implement the technique to obtain the Euler angles from a given rotation matrix.
The document discusses iterative methods for solving systems of linear equations, including the Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. The Jacobi method works by rewriting the system in a form where the diagonal entries are isolated and computing successive approximations. The Gauss-Seidel method similarly computes approximations but uses the most recent values available at each step. Relaxation improves the Gauss-Seidel method's convergence by taking a weighted average of the current and previous iterations' results. Examples demonstrate applying the different methods to compute solutions.
This document solves 5 set theory laws:
1) Idempotence: A union A = A and A intersection A = A. Given A = {1,2,3,4}, A union A = {1,2,3,4} and A intersection A = {1,2,3,4}.
2) Associativity: (A union B) union C = A union (B union C). Given sets A, B, and C, both sides equal the union of all elements.
3) Commutativity: A union B = B union A and A intersection B = B intersection A. Given sets A and B, their union and intersection are symmetric.
4)
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
This document presents a formula for calculating the area of a regular n-sided polygon using the area and perimeter of an inscribed circle. It proves that the ratio of the polygon's perimeter to its area equals the ratio of the circle's perimeter to its area. The formula derived is that the area of the polygon equals the area of the circle multiplied by the polygon's perimeter divided by the circle's perimeter. This method provides an accurate and efficient way to calculate a regular polygon's area without needing to divide it into triangles first.
Definitions matrices y determinantes fula 2010 english subirHernanFula
The document discusses the history and properties of matrices. It describes how matrices were first introduced in 1850 and how their use has expanded. It then defines key matrix terms and concepts such as order, elements, types of matrices (e.g. triangular, diagonal), operations (e.g. addition, multiplication, inverse), and properties (e.g. of symmetric, banded and transpose matrices). It also provides examples of calculating the determinant of matrices using Sarrus' rule.
System of linear algebriac equations nsmRahul Narang
The document discusses systems of linear algebraic equations and methods for solving them numerically. It introduces systems of linear equations in matrix form Ax = b and describes elementary row operations that can transform the matrix A. It then explains Gaussian elimination and Gauss-Jordan elimination methods for solving systems of linear equations by transforming the augmented matrix into reduced row echelon form. Finally, it briefly describes Jacobi and Gauss-Seidel iterative methods as well as applications of linear algebra in computer science fields like statistical learning, image manipulation, and physics.
The document discusses solving linear differential equations with variable coefficients using power series representations. It begins by introducing series solution methods and properties of infinite series. It then discusses the Method of Frobenius for solving equations with variable coefficients that arise in cylindrical and spherical coordinate systems. The method involves finding the indicial equation to determine the index c, and then using the recurrence relation to determine the series coefficients an. An example is provided to illustrate the method where the roots of the indicial equation are distinct and not differing by an integer.
This document discusses rank-aware thresholding algorithms for compressed sensing. It begins by introducing compressed sensing and explaining how traditional linear algebra techniques cannot be used to recover sparse signals from undersampled measurements. It then describes how thresholding and rank-aware thresholding algorithms work by exploiting the sparsity of signals. The key points are that rank-aware thresholding outperforms standard thresholding by eliminating the "square-root bottleneck" and requires only O(k) measurements, versus O(k^2) for thresholding. Simulation results demonstrate this improvement. The document concludes by discussing modeling techniques to predict algorithm performance on very large problems that are impractical to simulate directly.
Bayesian Variable Selection in Linear Regression and A ComparisonAtilla YARDIMCI
In this study, Bayesian approaches, such as Zellner, Occam’s Window and Gibbs sampling, have been compared in terms of selecting the correct subset for the variable selection in a linear regression model. The aim of this comparison is to analyze Bayesian variable selection and the behavior of classical criteria by taking into consideration the different values of β and σ and prior expected levels.
This document summarizes part of a lecture on factor analysis from an machine learning course. It introduces the factor analysis model, which posits that observed data is generated by an underlying latent variable that is mapped to the observed space with noise. It describes the factor analysis model mathematically as a joint Gaussian distribution between the latent and observed variables. It also derives the E-step and M-step updates for performing maximum likelihood estimation of the factor analysis model parameters using EM algorithm.
This document provides information about Calculus 2, including lessons on indeterminate forms, Rolle's theorem, the mean value theorem, and differentiation of transcendental functions. It defines Rolle's theorem and the mean value theorem, provides examples of applying each, and discusses how Rolle's theorem can be used to find the value of c. It also defines inverse trigonometric functions and their derivatives. The document is for MATH 09 Calculus 2 and includes exercises for students to practice applying the theorems.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
This document provides information about solving systems of linear equations through various methods such as graphing, substitution, and elimination. It defines what a linear system is and explains the concepts of consistent and inconsistent systems. Graphing is discussed as a way to find the point where two lines intersect. The substitution and elimination methods are described step-by-step with examples shown of using each method to solve sample systems of equations. Additional topics covered include slope, matrix notation, and an example of using a matrix to perform a Hill cipher encryption on a short plaintext message.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
This document discusses eigenvalues, eigenvectors, and quadratic forms. It provides examples of how to:
- Find the eigenvalues and eigenvectors of a matrix by solving the characteristic equation.
- Express a quadratic form in terms of a matrix and change variables using an invertible matrix to diagonalize the quadratic form.
- Use orthogonal diagonalization to transform a quadratic form with cross-product terms into one without cross-product terms. Step-by-step solutions and explanations are provided for examples involving 2x2 and 3x3 matrices.
Given two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This is known as the division algorithm from elementary number theory.
This document provides an introduction to calculus by discussing pure versus applied mathematics. It then reviews basic mathematical concepts such as exponents, algebraic expressions, solving equations, inequalities, and sets that are used in numerical analysis. Finally, it discusses graphical representations of rectangular and polar coordinate systems and includes examples of converting between the two systems.
This document presents a summary of a talk on building a harmonic analytic theory for the Gaussian measure and the Ornstein-Uhlenbeck operator. It discusses how the Gaussian measure is non-doubling but satisfies a local doubling property. It introduces Gaussian cones and shows how they allow proving maximal function estimates for the Ornstein-Uhlenbeck semigroup in a similar way as for the heat semigroup. The talk outlines estimates for the Mehler kernel of the Ornstein-Uhlenbeck semigroup and combines them to obtain boundedness of the maximal function.
The document discusses the basic proportionality theorem in geometry. It states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, the lengths of the segments of those two sides will be divided in the same ratio. This is also known as Thales' theorem, after the Greek mathematician who discovered it. The document also provides proofs of both the theorem and its converse.
The document applies the variational iteration method (VIM) to solve linear and nonlinear ordinary differential equations (ODEs) with variable coefficients. It emphasizes the power of the method by using it to solve a variety of ODE models of different orders and coefficients. The document also uses VIM to solve four scientific models - the hybrid selection model, Thomas-Fermi equation, Kidder equation for unsteady gas flow through porous media, and the Riccati equation. The VIM provides efficient iterative approximations for both analytic solutions and numeric simulations of real-world applications in science and engineering.
Maximum likelihood estimation of regularisation parameters in inverse problem...Valentin De Bortoli
This document discusses an empirical Bayesian approach for estimating regularization parameters in inverse problems using maximum likelihood estimation. It proposes the Stochastic Optimization with Unadjusted Langevin (SOUL) algorithm, which uses Markov chain sampling to approximate gradients in a stochastic projected gradient descent scheme for optimizing the regularization parameter. The algorithm is shown to converge to the maximum likelihood estimate under certain conditions on the log-likelihood and prior distributions.
This document discusses eigen values, eigen vectors, and diagonalization of matrices. It defines eigen values as the roots of the characteristic equation of a matrix. Eigen vectors are non-zero vectors that satisfy AX=λX, where λ is the eigen value. Diagonalization is the process of transforming a matrix A into a diagonal matrix D using a similarity transformation with an invertible matrix P, such that D=P-1AP. The document provides examples to illustrate these concepts and lists various properties of eigen values and eigen vectors.
Application of the Monte-Carlo Method to Nonlinear Stochastic Optimization wi...SSA KPI
This document describes a method for solving nonlinear stochastic optimization problems with linear constraints using Monte Carlo estimators. The key aspects are:
1) An ε-feasible solution approach is used to avoid "jamming" or "zigzagging" when dealing with linear constraints.
2) The optimality of solutions is tested statistically using the asymptotic normality of Monte Carlo estimators.
3) The Monte Carlo sample size is adjusted iteratively based on the gradient estimate to decrease computational trials while maintaining solution accuracy.
4) Under certain conditions, the method is proven to converge almost surely to a stationary point of the optimization problem.
5) As an example, the method is applied to portfolio optimization with
The document discusses various numerical methods for finding the roots or zeros of equations, including closed and open methods. Closed methods like bisection and false position trap the root within a closed interval by repeatedly dividing the interval in half. Open methods like Newton-Raphson and secant methods use information about the nonlinear function to iteratively refine the estimated root without being restricted to an interval. The document also covers methods for equations with multiple roots like Muller's method.
The document provides information about numerical methods topics including:
1) Lagrange's interpolation formula for finding a polynomial that passes through given data points, either equally or unequally spaced. The formula uses divided differences to find the coefficients.
2) Newton's divided difference interpolation formula for unequal intervals that also uses divided differences.
3) The nature of divided differences - for a polynomial of degree n, the nth divided difference is constant.
4) Examples of evaluating divided differences and constructing divided difference tables are given.
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2. TOPICS :
1. METHOD OF VARIATION OF PARAMETERS
2. CAUCHY’S LINEAR EQUATION
3. METHOD OF UNDETERMINED COEFFICIENTS
3. METHOD OF VARIATION OF PARAMETERS :
This method can be used for finding the particular integral yp
yp =y1 ʃ R(x) dx + y2 ʃ R(x) dx + y3 ʃ R(x) dx + .......
Where y1, y2, y3,… are basis of the solution.
For y” m2 two roots y1, y2
W = |y1 y2 | ,
|y1’ y2’|
W1 = |0 y2 |
|1 y2’|
W2 = |y1 0 |
|y1’ 1 |
8. CAUCHY’S LINEAR EQUATION :
The ODE of the form ,
is called Cauchy Linear equation.
To convert the about equation into equation with constant coefficient, take
Where θ = d/ dz
13. METHOD OF UNDETERMINED COEFFICIENTS :
This method can be used to find particular integral only if linearly independent
derivatives of Q(x) are finite in number.
This restriction implies that Q(x) can only have the terms such as k, xn, eax, sin ax, cos
ax and combination of such terms where k and a are constant and n is a positive
integer.
However, when Q(x) = 1/x or tan x or sec x, etc. , this method fails, since each function
has an infinite number of linearly independent derivatives.
14. Some of the choices of the particular integrals are given below :
In the table A0, A1, A2,……..,An are coefficients to be determined. To obtain the
values of these coefficients, we use the fact that the particular integral satisfies the
given differential equation.