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This chapter discusses numerical approximation and error analysis in numerical methods. It defines error as the difference between the true value being sought and the approximate value obtained. There are two main sources of error: rounding error from representing values with a finite number of digits, and truncation error from using a finite number of terms to approximate infinite expressions. The concept of significant figures is also introduced to determine the precision of numerical methods.

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Numerical Method

This document discusses numerical methods and errors. It introduces that numerical methods provide approximate solutions rather than exact analytical solutions due to errors from measurements, algorithms, and output. Accuracy refers to how close an approximation is to the true value, while precision refers to the reproducibility of results. Significant figures indicate the precision of a number. True error, relative error, and percent error are defined to quantify the error between approximations and true values. Round-off errors from floating point representation on computers are also discussed.

Approximation and error

This document discusses approximation and round-off error in engineering. It defines approximation as using an inexact value when the exact value is unknown or difficult to obtain. Approximations introduce errors from measurements in the real world. There are two main types of errors - truncation error from dropping digits during approximations, and rounding error from representing numbers with a fixed number of significant figures. The absolute error is the difference between the true and approximate values, while relative error is the percentage difference between the absolute error and true value.

introduction to Numerical Analysis

This document provides an introduction and overview of numerical analysis. It begins by stating that numerical analysis aims to find approximate solutions to complex mathematical problems through repeated computational steps when analytical solutions are not available or practical. It then discusses that numerical analysis is important because it allows for the conversion of physical phenomena into mathematical models that can be solved through basic arithmetic operations. Finally, it explains that numerical analysis involves developing algorithms and numerical techniques to solve problems, implementing those techniques using computers, and analyzing errors in approximate solutions.

Error Finding in Numerical method

Numerical method errors analysis examines the difference between true and approximate values. Absolute error is the difference between true and approximate values, while relative error is the ratio of absolute error to true value. Percentage error is calculated by taking the absolute difference between true and approximate values, dividing by the absolute true value, and multiplying by 100. Examples are provided to demonstrate calculating absolute, relative, and percentage errors.

Numerical approximation and solution of equations

1. Numerical approximation involves finding approximate values that are close to the actual values of quantities. There are different types of errors that can occur due to approximation, such as truncation error and rounding error.
2. Accuracy refers to how close an approximate value is to the actual value, while precision describes how close repeated approximations are to each other. Greater accuracy means a lower absolute error, while greater precision means a lower standard deviation between repeated measurements.
3. For a numerical method, convergence means that repeated approximations get closer to the actual value with each iteration. Stability refers to the likelihood that a method will converge rather than diverge for a wide range of problems.

Linear regression

Machine learning is a type of artificial intelligence that allows systems to learn from data and improve automatically without being explicitly programmed. There are several types of machine learning algorithms, including supervised learning which uses labeled training data to predict outcomes, unsupervised learning which finds patterns in unlabeled data, and reinforcement learning which interacts with its environment to discover rewards or errors. Linear regression is an example machine learning model that fits a linear equation to describe the relationship between a dependent variable and one or more independent variables. It works by minimizing the residual sum of squares to find the coefficients that produce the best fitting line.

Numerical approximation

1) An approximation is an inexact representation of something that is still close enough to be useful as it may yield an accurate solution while reducing complexity.
2) Significant figures include certain digits that are accurate and one uncertain digit that has some error possibility.
3) When adding and subtracting numbers, the final result should be rounded to the same precision as the least precise initial value. When multiplying, dividing, or taking roots the result should have the same number of significant figures as the least precise number.

Outlier managment

Outlier Management, BASIC STATISTICS, Error, Accuracy, How to find Outliers, quartile, Data Management, Reporting and Evaluation, Communication & Corrective Action, Documentation,

Numerical Method

This document discusses numerical methods and errors. It introduces that numerical methods provide approximate solutions rather than exact analytical solutions due to errors from measurements, algorithms, and output. Accuracy refers to how close an approximation is to the true value, while precision refers to the reproducibility of results. Significant figures indicate the precision of a number. True error, relative error, and percent error are defined to quantify the error between approximations and true values. Round-off errors from floating point representation on computers are also discussed.

Approximation and error

This document discusses approximation and round-off error in engineering. It defines approximation as using an inexact value when the exact value is unknown or difficult to obtain. Approximations introduce errors from measurements in the real world. There are two main types of errors - truncation error from dropping digits during approximations, and rounding error from representing numbers with a fixed number of significant figures. The absolute error is the difference between the true and approximate values, while relative error is the percentage difference between the absolute error and true value.

introduction to Numerical Analysis

This document provides an introduction and overview of numerical analysis. It begins by stating that numerical analysis aims to find approximate solutions to complex mathematical problems through repeated computational steps when analytical solutions are not available or practical. It then discusses that numerical analysis is important because it allows for the conversion of physical phenomena into mathematical models that can be solved through basic arithmetic operations. Finally, it explains that numerical analysis involves developing algorithms and numerical techniques to solve problems, implementing those techniques using computers, and analyzing errors in approximate solutions.

Error Finding in Numerical method

Numerical method errors analysis examines the difference between true and approximate values. Absolute error is the difference between true and approximate values, while relative error is the ratio of absolute error to true value. Percentage error is calculated by taking the absolute difference between true and approximate values, dividing by the absolute true value, and multiplying by 100. Examples are provided to demonstrate calculating absolute, relative, and percentage errors.

Numerical approximation and solution of equations

1. Numerical approximation involves finding approximate values that are close to the actual values of quantities. There are different types of errors that can occur due to approximation, such as truncation error and rounding error.
2. Accuracy refers to how close an approximate value is to the actual value, while precision describes how close repeated approximations are to each other. Greater accuracy means a lower absolute error, while greater precision means a lower standard deviation between repeated measurements.
3. For a numerical method, convergence means that repeated approximations get closer to the actual value with each iteration. Stability refers to the likelihood that a method will converge rather than diverge for a wide range of problems.

Linear regression

Machine learning is a type of artificial intelligence that allows systems to learn from data and improve automatically without being explicitly programmed. There are several types of machine learning algorithms, including supervised learning which uses labeled training data to predict outcomes, unsupervised learning which finds patterns in unlabeled data, and reinforcement learning which interacts with its environment to discover rewards or errors. Linear regression is an example machine learning model that fits a linear equation to describe the relationship between a dependent variable and one or more independent variables. It works by minimizing the residual sum of squares to find the coefficients that produce the best fitting line.

Numerical approximation

1) An approximation is an inexact representation of something that is still close enough to be useful as it may yield an accurate solution while reducing complexity.
2) Significant figures include certain digits that are accurate and one uncertain digit that has some error possibility.
3) When adding and subtracting numbers, the final result should be rounded to the same precision as the least precise initial value. When multiplying, dividing, or taking roots the result should have the same number of significant figures as the least precise number.

Outlier managment

Outlier Management, BASIC STATISTICS, Error, Accuracy, How to find Outliers, quartile, Data Management, Reporting and Evaluation, Communication & Corrective Action, Documentation,

DATA SCIENCE - Outlier detection and treatment_ sachin pathania

This document discusses outlier detection and treatment using the interquartile range (IQR) method. It defines outliers as values that behave differently than other observations. As an example, it shows athlete performance increases where one athlete, Sam, had a decrease of -0.56m, making them an outlier. It then explains how IQR divides the data distribution into quartiles to identify outliers, with the lower bound set at Q1-1.5*IQR and upper bound at Q3+1.5*IQR. Outliers are values outside this range. Python code is provided to demonstrate outlier treatment using IQR.

mathematical model

The document presents an overview of mathematical models. It defines mathematical models as mathematical descriptions of real situations that make assumptions and simplifications about reality. There are three main types of models: linear, quadratic, and exponential models. The document discusses how to develop a mathematical model by comparing model predictions to real data. It provides an example of a differential equation model of the spread of a contagious flu.

Mathematical Modeling for Practical Problems

Mathematical modeling is an important step for developing many advanced technologies in various domains such as network security, data mining and etc… This lecture introduces a process that the speaker summarizes from his past practice of mathematical modeling and algorithmic solutions in IT industry, as an applied mathematician, algorithm specialist or software engineer , and even as an entrepreneur. A practical problem from DLP system will be used as an example for creating math models and providing algorithmic solutions.

10 advice for applying ml

To evaluate a machine learning model and prevent overfitting, the dataset should be split into a training set and test set. The model is trained on the training set and then evaluated on the test set. Cross-validation is used to select the best model by computing the cross-validation error JCV on different models - the model with the lowest JCV is chosen. Using the cross-validation data to analyze errors is preferred over the test data, to avoid developing features tailored specifically for the test set and not generalizing well.

Data Transformation

This document discusses data transformation in SPSS. It describes how to compute new variables using arithmetic, logical, and conditional expressions. It also explains how to recode the values of existing variables into new variables or categories using the recode command. Examples are provided to illustrate computing total scores, averages, increments with conditions, and recoding years of schooling into educational status categories.

Errors and statistics

This document discusses various topics in errors and statistics in analytical chemistry including:
- Classification of errors into systematic (determinate) and random (indeterminate)
- Definitions of accuracy and precision
- Computations for average deviation, reliability of results, and confidence intervals
- Methods for comparing results including Student's t-test and F-test
- Correlation, regression, and the equations for the line of best fit between a dependent and independent variable.

Or ppt,new

This document discusses methods of simulation and Monte Carlo simulation. It is authored by a group including Roy Thomas, Sam Scaria, Sonu Sebastian, and others. The document defines simulation as using a model of a real system to conduct experiments on a computer in order to describe, explain, and predict the behavior of the real system. Monte Carlo simulation is described as using probability and sampling to solve complicated equations. Key steps of Monte Carlo simulation include drawing a flow diagram, determining variable distributions, selecting random numbers, and applying mathematical functions to obtain solutions. Examples of applications include queuing problems, inventory problems, and risk analysis.

Basics mathematical modeling

1) Mathematical modeling seeks to formalize relationships between input and output variables in a system using mathematical language like differential equations.
2) There are different types of models including graphic, physical, analytical, numerical, and computational models.
3) Mathematical models contain dependent and independent variables, parameters, and functions that describe the forces in the system.
4) Thermal systems can be modeled using an energy balance equation relating changes in heat energy over time to the power input and power removed from the system. The temperature of a system can then be modeled using this energy balance relationship.

Application of Machine Learning in Agriculture

With the growing trend of machine learning, it is needless to say how machine learning can help reap benefits in agriculture. It will be boon for the farmer welfare.

Machine learning session6(decision trees random forrest)

Concepts include decision tree with its examples. Measures used for splitting in decision tree like gini index, entropy, information gain, pros and cons, validation. Basics of random forests with its example and uses.

Application's of Numerical Math in CSE

The document describes an assignment given to Md. Mehedi Hasan on the topic of applying numerical methods in computer science engineering. The assignment was given by five students and includes an index listing numerical methods to cover: error analysis, N-R method, interpolation, differentiation and max/min, curve fitting, and integration.

Zero to ECC in 30 Minutes: A primer on Elliptic Curve Cryptography (ECC)

Elliptic curve cryptography uses points on elliptic curves as a basis for public-key cryptography. The document discusses:
1) Elliptic curves take the form y^2 = x^3 + ax + b and have certain geometric properties that allow points on the curve to be combined via an "addition" operation.
2) To perform computations, points must have discrete coordinates from a finite field rather than continuous real numbers. Commonly the field of integers modulo a prime p is used.
3) The set of points on the curve forms an abelian group under the addition operation, with the point at infinity as the identity element. Certain points can generate all other points on the curve.

Monte carlo simulation

Monte Carlo simulation is a statistical technique that uses random numbers and probability to simulate real-world processes. It was developed in the 1940s by scientists working on nuclear weapons research. Monte Carlo simulation provides approximate solutions to problems by running simulations many times. It allows for sensitivity analysis and scenario analysis. Some examples include estimating pi by randomly generating points within a circle, and approximating integrals by treating the area under a curve as a target for random darts. The technique provides probabilistic results and allows modeling of correlated inputs.

Applications of numerical methods

The document discusses numerical methods and their applications. It provides definitions of numerical methods as procedures for solving problems with computable error estimates. Some common numerical methods are listed, including bisection, Newton-Raphson, iteration, and interpolation methods. Applications mentioned include root finding, profit/loss calculation, multidimensional root finding, and simulations. An example is given of using numerical methods for image deblurring. The document also discusses computational modeling, algorithm development and implementation, and limitations of computers in solving mathematical problems.

MT6702 Unit 2 Random Number Generation

Properties of Random Number, Random Number Generation Methods, Different Methods for Testing of Generated Random Numbers

Numerical Analysis and Epistemology of Information

The slides of my presentation at the workshop "Philosophical Aspects of Computer Science", European Centre for Living Technology, University “Ca’ Foscari”, Venice, March 2015.

Logic Development and Algorithm.

The document discusses algorithms and flowcharts for solving problems. It defines an algorithm as a set of sequential steps to solve a problem and notes that there are various techniques for specifying algorithms, including formally, informally, mathematically, or through graphical flowcharts. The document provides examples of algorithms to solve common problems and explains the properties and steps involved in algorithm development. It also describes flowcharts as a visual representation of an algorithm using standard symbols like ovals, rectangles, and diamonds to indicate starts/stops, processes, and decisions.

Calc 4.4a

The document discusses several topics related to calculus including:
1) Evaluating definite integrals using the fundamental theorem of calculus.
2) Finding the average value of a function over a closed interval using the mean value theorem.
3) Understanding that differentiation and integration are inverse processes where one deals with slopes and the other with areas.

numerical analysis

This document discusses sources of error in numerical calculations. It identifies two main types: round-off error, due to limitations in precision, and truncation error, due to approximations in numerical methods. Round-off error accumulates through repeated calculations and can dominate final results. Truncation error depends on how well the solution can be represented by the approximation. Care must be taken to evaluate errors and ensure results have enough significant figures to be meaningful for the problem.

Numerical Analysis And Linear Algebra

1. Numerical analysis provides approximate solutions to complex mathematical problems through repeated calculations. It is used when analytical solutions are not possible or too complex.
2. The document discusses the importance of numerical analysis in engineering and science for solving real-world problems. It also defines key concepts like errors, significant digits, and accuracy in numerical analysis.
3. Numerical methods allow finding approximate solutions to problems described by mathematical models through simple arithmetic operations. They are important when analytical solutions are not available.

Numerical Methods.pptx

A numerical method is an approximate computer method for solving a mathematical problem which often has no analytical solution.

Numerical approximation

1) An approximation is an inexact representation of something that is still close enough to be useful as it reduces complexity while still yielding an accurate solution.
2) When performing calculations with approximations, the number of significant figures in the final result should match the least precise initial value. For addition/subtraction, round to the same decimal places, and for multiplication/division/roots round to the same number of significant figures.
3) Numerical errors are the difference between the true and approximate values. Relative percent error is used to normalize the error by the true value when measuring differences in magnitude. Rounding and truncation errors contribute to the total numerical error.

DATA SCIENCE - Outlier detection and treatment_ sachin pathania

This document discusses outlier detection and treatment using the interquartile range (IQR) method. It defines outliers as values that behave differently than other observations. As an example, it shows athlete performance increases where one athlete, Sam, had a decrease of -0.56m, making them an outlier. It then explains how IQR divides the data distribution into quartiles to identify outliers, with the lower bound set at Q1-1.5*IQR and upper bound at Q3+1.5*IQR. Outliers are values outside this range. Python code is provided to demonstrate outlier treatment using IQR.

mathematical model

The document presents an overview of mathematical models. It defines mathematical models as mathematical descriptions of real situations that make assumptions and simplifications about reality. There are three main types of models: linear, quadratic, and exponential models. The document discusses how to develop a mathematical model by comparing model predictions to real data. It provides an example of a differential equation model of the spread of a contagious flu.

Mathematical Modeling for Practical Problems

Mathematical modeling is an important step for developing many advanced technologies in various domains such as network security, data mining and etc… This lecture introduces a process that the speaker summarizes from his past practice of mathematical modeling and algorithmic solutions in IT industry, as an applied mathematician, algorithm specialist or software engineer , and even as an entrepreneur. A practical problem from DLP system will be used as an example for creating math models and providing algorithmic solutions.

10 advice for applying ml

To evaluate a machine learning model and prevent overfitting, the dataset should be split into a training set and test set. The model is trained on the training set and then evaluated on the test set. Cross-validation is used to select the best model by computing the cross-validation error JCV on different models - the model with the lowest JCV is chosen. Using the cross-validation data to analyze errors is preferred over the test data, to avoid developing features tailored specifically for the test set and not generalizing well.

Data Transformation

This document discusses data transformation in SPSS. It describes how to compute new variables using arithmetic, logical, and conditional expressions. It also explains how to recode the values of existing variables into new variables or categories using the recode command. Examples are provided to illustrate computing total scores, averages, increments with conditions, and recoding years of schooling into educational status categories.

Errors and statistics

This document discusses various topics in errors and statistics in analytical chemistry including:
- Classification of errors into systematic (determinate) and random (indeterminate)
- Definitions of accuracy and precision
- Computations for average deviation, reliability of results, and confidence intervals
- Methods for comparing results including Student's t-test and F-test
- Correlation, regression, and the equations for the line of best fit between a dependent and independent variable.

Or ppt,new

This document discusses methods of simulation and Monte Carlo simulation. It is authored by a group including Roy Thomas, Sam Scaria, Sonu Sebastian, and others. The document defines simulation as using a model of a real system to conduct experiments on a computer in order to describe, explain, and predict the behavior of the real system. Monte Carlo simulation is described as using probability and sampling to solve complicated equations. Key steps of Monte Carlo simulation include drawing a flow diagram, determining variable distributions, selecting random numbers, and applying mathematical functions to obtain solutions. Examples of applications include queuing problems, inventory problems, and risk analysis.

Basics mathematical modeling

1) Mathematical modeling seeks to formalize relationships between input and output variables in a system using mathematical language like differential equations.
2) There are different types of models including graphic, physical, analytical, numerical, and computational models.
3) Mathematical models contain dependent and independent variables, parameters, and functions that describe the forces in the system.
4) Thermal systems can be modeled using an energy balance equation relating changes in heat energy over time to the power input and power removed from the system. The temperature of a system can then be modeled using this energy balance relationship.

Application of Machine Learning in Agriculture

With the growing trend of machine learning, it is needless to say how machine learning can help reap benefits in agriculture. It will be boon for the farmer welfare.

Machine learning session6(decision trees random forrest)

Concepts include decision tree with its examples. Measures used for splitting in decision tree like gini index, entropy, information gain, pros and cons, validation. Basics of random forests with its example and uses.

Application's of Numerical Math in CSE

The document describes an assignment given to Md. Mehedi Hasan on the topic of applying numerical methods in computer science engineering. The assignment was given by five students and includes an index listing numerical methods to cover: error analysis, N-R method, interpolation, differentiation and max/min, curve fitting, and integration.

Zero to ECC in 30 Minutes: A primer on Elliptic Curve Cryptography (ECC)

Elliptic curve cryptography uses points on elliptic curves as a basis for public-key cryptography. The document discusses:
1) Elliptic curves take the form y^2 = x^3 + ax + b and have certain geometric properties that allow points on the curve to be combined via an "addition" operation.
2) To perform computations, points must have discrete coordinates from a finite field rather than continuous real numbers. Commonly the field of integers modulo a prime p is used.
3) The set of points on the curve forms an abelian group under the addition operation, with the point at infinity as the identity element. Certain points can generate all other points on the curve.

Monte carlo simulation

Monte Carlo simulation is a statistical technique that uses random numbers and probability to simulate real-world processes. It was developed in the 1940s by scientists working on nuclear weapons research. Monte Carlo simulation provides approximate solutions to problems by running simulations many times. It allows for sensitivity analysis and scenario analysis. Some examples include estimating pi by randomly generating points within a circle, and approximating integrals by treating the area under a curve as a target for random darts. The technique provides probabilistic results and allows modeling of correlated inputs.

Applications of numerical methods

The document discusses numerical methods and their applications. It provides definitions of numerical methods as procedures for solving problems with computable error estimates. Some common numerical methods are listed, including bisection, Newton-Raphson, iteration, and interpolation methods. Applications mentioned include root finding, profit/loss calculation, multidimensional root finding, and simulations. An example is given of using numerical methods for image deblurring. The document also discusses computational modeling, algorithm development and implementation, and limitations of computers in solving mathematical problems.

MT6702 Unit 2 Random Number Generation

Properties of Random Number, Random Number Generation Methods, Different Methods for Testing of Generated Random Numbers

Numerical Analysis and Epistemology of Information

The slides of my presentation at the workshop "Philosophical Aspects of Computer Science", European Centre for Living Technology, University “Ca’ Foscari”, Venice, March 2015.

Logic Development and Algorithm.

The document discusses algorithms and flowcharts for solving problems. It defines an algorithm as a set of sequential steps to solve a problem and notes that there are various techniques for specifying algorithms, including formally, informally, mathematically, or through graphical flowcharts. The document provides examples of algorithms to solve common problems and explains the properties and steps involved in algorithm development. It also describes flowcharts as a visual representation of an algorithm using standard symbols like ovals, rectangles, and diamonds to indicate starts/stops, processes, and decisions.

Calc 4.4a

The document discusses several topics related to calculus including:
1) Evaluating definite integrals using the fundamental theorem of calculus.
2) Finding the average value of a function over a closed interval using the mean value theorem.
3) Understanding that differentiation and integration are inverse processes where one deals with slopes and the other with areas.

DATA SCIENCE - Outlier detection and treatment_ sachin pathania

DATA SCIENCE - Outlier detection and treatment_ sachin pathania

mathematical model

mathematical model

Mathematical Modeling for Practical Problems

Mathematical Modeling for Practical Problems

10 advice for applying ml

10 advice for applying ml

Data Transformation

Data Transformation

Errors and statistics

Errors and statistics

Or ppt,new

Or ppt,new

Basics mathematical modeling

Basics mathematical modeling

Application of Machine Learning in Agriculture

Application of Machine Learning in Agriculture

Machine learning session6(decision trees random forrest)

Machine learning session6(decision trees random forrest)

Application's of Numerical Math in CSE

Application's of Numerical Math in CSE

Zero to ECC in 30 Minutes: A primer on Elliptic Curve Cryptography (ECC)

Zero to ECC in 30 Minutes: A primer on Elliptic Curve Cryptography (ECC)

Monte carlo simulation

Monte carlo simulation

Applications of numerical methods

Applications of numerical methods

MT6702 Unit 2 Random Number Generation

MT6702 Unit 2 Random Number Generation

Numerical Analysis and Epistemology of Information

Numerical Analysis and Epistemology of Information

Logic Development and Algorithm.

Logic Development and Algorithm.

Calc 4.4a

Calc 4.4a

numerical analysis

This document discusses sources of error in numerical calculations. It identifies two main types: round-off error, due to limitations in precision, and truncation error, due to approximations in numerical methods. Round-off error accumulates through repeated calculations and can dominate final results. Truncation error depends on how well the solution can be represented by the approximation. Care must be taken to evaluate errors and ensure results have enough significant figures to be meaningful for the problem.

Numerical Analysis And Linear Algebra

1. Numerical analysis provides approximate solutions to complex mathematical problems through repeated calculations. It is used when analytical solutions are not possible or too complex.
2. The document discusses the importance of numerical analysis in engineering and science for solving real-world problems. It also defines key concepts like errors, significant digits, and accuracy in numerical analysis.
3. Numerical methods allow finding approximate solutions to problems described by mathematical models through simple arithmetic operations. They are important when analytical solutions are not available.

Numerical Methods.pptx

A numerical method is an approximate computer method for solving a mathematical problem which often has no analytical solution.

Numerical approximation

1) An approximation is an inexact representation of something that is still close enough to be useful as it reduces complexity while still yielding an accurate solution.
2) When performing calculations with approximations, the number of significant figures in the final result should match the least precise initial value. For addition/subtraction, round to the same decimal places, and for multiplication/division/roots round to the same number of significant figures.
3) Numerical errors are the difference between the true and approximate values. Relative percent error is used to normalize the error by the true value when measuring differences in magnitude. Rounding and truncation errors contribute to the total numerical error.

Numerical analysis using Scilab: Error analysis and propagation

This tutorial provides a collection of numerical examples and advises on error analysis and its propagation. All examples are developed in Scilab.

Approximation and error

This document discusses approximation and round-off error in engineering. It defines approximation as using an inexact value when the exact value is unknown or difficult to obtain. Approximations introduce errors from measurements in the real world. There are two main types of errors - truncation error from dropping digits during approximations, and rounding error from representing numbers with a fixed number of significant figures. The absolute error is the difference between the true and approximate values, while relative error is the percentage difference between the absolute error and true value. Understanding error is important for engineering applications that use numerical methods and measurements.

Course project solutions 2018

This document provides instructions and materials for a course project on solving problems using computer programming. It includes two problems - counting prime numbers below 10,000 and counting triangular numbers below 1,000,000. Algorithms are presented for both problems using pseudocode. Students are instructed to implement the algorithms in Scratch or another programming language. Sample Scratch and Python programs are included, along with testing to validate the outputs against known results. The document aims to help students learn programming skills through solving mathematical problems.

Ficha 1 errores

This document discusses measurement errors and how to calculate them. It defines absolute error as the difference between a measured value and the true value. Relative error is defined as the absolute error divided by the true value. Relative percent error converts the relative error to a percentage by multiplying by 100. Examples are given of calculating absolute error, relative error, and relative percent error for different measurements. The document also discusses representing numbers in scientific notation and rounding calculations involving measured values.

Measurements and error in experiments

Every measurement has some level of uncertainty that should be reported along with the measured value. There are several ways to determine the uncertainty, including making repeated measurements and reporting the standard deviation, or estimating uncertainty based on the precision of the measuring instrument. Accuracy refers to how close a measurement is to the true value, while precision refers to how reproducibly a measurement can be made. When reporting measurements, the number of significant figures should be based on the precision of the measuring device and include an estimated uncertainty.

Error(Computer Oriented Numerical and Statistical Method)

This document defines different types of errors that can occur in computational results, including inherent errors from inaccurate input data and numerical errors introduced during calculations. It discusses how inherent errors contain data errors from measurement limitations and conversion errors from computers' inability to store some numbers exactly. Numerical errors include round-off errors from fixed decimal representation and truncation errors from approximating infinite sums. Formulas are provided for absolute error, relative error, and percentage error to quantify accuracy. Examples demonstrate calculating these error measures.

Chapter 3.pptx

This document outlines the key topics in Analytical Chemistry I including significant figures, types of errors, propagation of uncertainty, and systematic vs random errors. It discusses how measurements have uncertainty and errors. There are two main types of errors - systematic errors which affect accuracy and can be discovered and corrected, and random errors which cannot be eliminated and have equal chances of being positive or negative. The document also describes how to calculate the propagation of uncertainty through calculations using addition, subtraction, multiplication, division and other operations. It emphasizes keeping extra digits in calculations to properly account for uncertainty.

What is algorithm

This document discusses algorithms and provides an example algorithm for adding three numbers and printing the sum. It defines an algorithm as a set of steps to solve a problem and lists characteristics of algorithms such as being clear, well-defined, finite, feasible, and language independent. The document also outlines a process for designing algorithms which includes defining the problem, developing a model, specifying the algorithm, designing it, checking correctness, analyzing it, implementing it, testing it, and documenting it. It then provides an example algorithm that takes three numbers as input, adds them together, stores the sum in a variable, and prints the result.

NUMERICAL APPROXIMATION

Numerical approximation methods provide alternative procedures to analytical methods for solving mathematical problems that are complex or do not have exact solutions. These methods use iteration to systematically approximate the true value of a variable. Accuracy refers to how close an approximation is to the actual value, while precision refers to the number of significant figures. For a numerical method to work well, it must converge, meaning the approximations get closer to the true value with more iterations, and be stable. The appropriate method depends on factors like the type of problem, complexity of the model, and characteristics of the numerical method itself such as speed of convergence and stability.

Laboratorios virtuales fisica mecanica

This document provides guidelines for a virtual laboratory on uncertainty in measurements in physics. It discusses key concepts like the factors that influence measurement uncertainty, such as instrument precision and accuracy. It also covers calculating absolute and relative error, and determining the number of significant figures in measurements. Combining uncertainties from multiple measurements is also addressed.

WEKA:Credibility Evaluating Whats Been Learned

- Training and test sets are used to measure classification success rates, with the test set being independent of the training set. The error rate on the training set is optimistic. Cross validation techniques like 10-fold stratified cross validation are used when data is limited.
- True success rates are predicted using properties of statistics and normal distributions. Confidence levels determine the range within which the true rate is expected to lie.
- Techniques like paired t-tests are used to statistically compare the performance of different algorithms or data mining methods. They determine if performance differences are statistically significant.

WEKA: Credibility Evaluating Whats Been Learned

This document discusses various techniques for evaluating machine learning models and comparing their performance, including:
- Measuring error rates on separate test and training sets to avoid overfitting
- Using techniques like cross-validation, bootstrapping, and holdout validation when data is limited
- Comparing algorithms using statistical tests like paired t-tests
- Accounting for costs of different prediction outcomes in evaluation and model training
- Visualizing performance using lift charts and ROC curves to compare models
- The Minimum Description Length principle for selecting the model that best compresses the data

Supervised Learning.pdf

This document discusses supervised learning. Supervised learning uses labeled training data to train models to predict outputs for new data. Examples given include weather prediction apps, spam filters, and Netflix recommendations. Supervised learning algorithms are selected based on whether the target variable is categorical or continuous. Classification algorithms are used when the target is categorical while regression is used for continuous targets. Common regression algorithms discussed include linear regression, logistic regression, ridge regression, lasso regression, and elastic net. Metrics for evaluating supervised learning models include accuracy, R-squared, adjusted R-squared, mean squared error, and coefficients/p-values. The document also covers challenges like overfitting and regularization techniques to address it.

Es272 ch2

This chapter discusses various types of errors that can occur in numerical analysis calculations, including:
- Round-off errors due to limitations in significant figures and binary representation in computers
- Truncation errors from using approximations instead of exact mathematical representations
- Error propagation when combining results with arithmetic operations
It also covers topics like accuracy vs precision, definitions of relative and absolute errors, floating point representation standards, and techniques to estimate errors like Taylor series expansions and machine epsilon values. The goal is to understand the sources and magnitudes of different errors to improve the reliability of numerical analysis methods.

AOA Week 01.ppt

The document discusses problem solving in computer science and algorithms. It defines an algorithm as a clearly defined set of steps to solve a problem. Key characteristics of algorithms are that they are unambiguous, have well-defined inputs and outputs, terminate in a finite number of steps, and are independent of programming languages. Examples of algorithms that find the largest number among three inputs and calculate a factorial are provided. The document also discusses sorting problems and examples of problems solved by algorithms like the human genome project, internet routing, and electronic commerce.

Important Classification and Regression Metrics.pptx

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Chapter 3

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.

Chapter 5

The document discusses two iterative methods for solving systems of linear equations:
1. The Jacobi method, which solves for each diagonal element using the previous iteration's values for other elements. It converges to the solution by iterating this process.
2. The Gauss-Seidel method, which sequentially updates elements using values from the current iteration, making it converge faster than the Jacobi method. Both methods decompose the matrix and iteratively solve for the unknowns until the solution converges.

Chapter 2

This chapter discusses numerical approximation and error analysis in numerical methods. It defines error as the difference between the true value being sought and the approximate value obtained. There are two main sources of error: rounding error from representing values with a finite number of digits, and truncation error from using a finite number of terms to approximate infinite expressions. The concept of significant figures is also introduced to determine the precision of numerical methods.

Chapter 4

This chapter discusses direct methods for solving systems of linear equations, including Gauss elimination, Gauss-Jordan elimination, and LU decomposition. It provides examples of using each method to solve systems and describes the steps involved, such as putting the matrix in echelon form and using row operations. LU decomposition involves decomposing the original matrix into lower and upper triangular matrices. The chapter concludes by outlining the steps to solve a system using LU decomposition.

Capitulo 4

El documento describe los conceptos básicos de las matrices y los sistemas de ecuaciones lineales, incluyendo la notación matricial, los tipos de matrices, la multiplicación y determinante de matrices, y métodos para resolver pequeños sistemas de ecuaciones como el método gráfico, la regla de Cramer y la eliminación de incógnitas.

Expocision

Este documento presenta una introducción a las matrices y los sistemas de ecuaciones lineales. Explica la notación matricial y los tipos de matrices. Luego describe métodos para multiplicar matrices y calcular determinantes. Finalmente, resume métodos analíticos para resolver sistemas de ecuaciones lineales pequeños, como el método gráfico, la regla de Cramer y la eliminación de incógnitas.

Expocision

Este documento presenta una introducción a las matrices y los sistemas de ecuaciones lineales. Explica la notación matricial y los tipos de matrices. Luego describe métodos para multiplicar matrices y calcular determinantes. Finalmente, resume métodos analíticos para resolver sistemas de ecuaciones lineales pequeños, como el método gráfico, la regla de Cramer y la eliminación de incógnitas.

Expocision

Este documento presenta una introducción a las matrices y los sistemas de ecuaciones lineales. Explica la notación matricial y los tipos de matrices. Luego describe métodos para multiplicar matrices y calcular determinantes. Finalmente, resume métodos analíticos para resolver sistemas de ecuaciones lineales pequeños, como el método gráfico, la regla de Cramer y la eliminación de incógnitas.

Chapter 1

Mathematical modeling is a process that uses mathematical concepts and language to describe and understand real-world phenomena. This involves formulating hypotheses about the relationships and rates of change between variables, which are then expressed through differential equations. Once a mathematical model is developed, the problem becomes solving these equations, which can be analyzed through various modeling methods to predict future behavior and understand the underlying processes.

Chapter 1

Mathematical modeling is a process that uses mathematical concepts and language to describe and understand real-world phenomena. This involves formulating hypotheses about the relationships and rates of change between variables, which are then expressed through differential equations. Once a mathematical model is developed, the problem becomes solving these equations, which can be analyzed through various modeling methods to predict future behavior and understand the underlying processes.

Chapter 3

Chapter 3

Chapter 5

Chapter 5

Chapter 2

Chapter 2

Chapter 4

Chapter 4

Capitulo 4

Capitulo 4

Expocision

Expocision

Expocision

Expocision

Expocision

Expocision

Expocision

Expocision

Chapter 1

Chapter 1

Chapter 1

Chapter 1

- 1. Chapter 2: Numerical Approximation By Erika Villarreal
- 2. Here comes your footer Page
- 4. Significant Figures Here comes your footer Page . . It is the concept that has been developed to describe formally the reliability of a numerical value. Significant digits of a number, are those that can be employed reliably to describe a quantity. For example, suppose you have an instrument where your meter brand: The instrument can handle two-digit accuracy. The third is estimated. So in general only have three significant digits for the instrument. It is important to note that the zeros are not always significant digits, these can be used to locate the decimal point, for example: a) 0.00001845 b) 0.0001845 c) 0.001845 d) 0.0000180 have four significant digits, where the number 1 is the first significant digit (digit significant main or most significant digit),8 is the second significant digit, 4 is the third significant digit and 5 is the fourth . has three significant digits, 1, 8 and 0
- 5. Significant Figures Here comes your footer Page The importance of the concept of significant figures in the study of numerical methods is mainly on two aspects 1. Criteria to determine the precision of a numerical method. Method is acceptable when it guarantees a certain number of significant figures in the result 2. Stop Criterion. As numerical methods are iterative techniques can be established that when it reaches a certain number of significant digits is sufficient condition to stop the method
- 6. Exactitude and precision Here comes your footer Page -Exactitude .- Whether the calculated value is close to the true value. -Precision.- Whereas numerical methods are iterative techniques, expresses how close an approximation or an estimate value with respect to approximations or previous iterations of the same. -Inexactitude .-. It's a systematic removal of real value to calculate. .- Imprecision or uncertainty. It is the measure of distance between them at the various approximations to a true value. By observing the above definitions, can be determined that the error associated with numerical methods to measure the degree of exactitude and precision of them.
- 7. Definition of error Here comes your footer Page Overall, the error of a numerical method is the difference between the true value being sought and the approximation obtained through a numerical technique Rounding error Truncation error
- 8. . Truncation error Here comes your footer Page When using the finite number of terms to calculate a value that requires an infinite number of terms. For example, an expression to accurately determine the value of the Euler number (base of natural logarithms) through a series of MacLaurin is: However, an approximation of that value can be obtained through finite expression: this finite expression is manageable computationally speaking, contrary to the formula set out in its infinite form..
- 9. . Rounding error Here comes your footer Page This is because a computer can only represent a finite number of terms. To express a quantity with an infinite decimal expansion, you have to do without most of them. For example, the number π = 3.14159265 ...., has a non-periodic infinite decimal expansion. Therefore, for purposes of calculation, only take some of their digits. This is accomplished through two strategies: 1. Rounding. It ignores a number of significant figures and make an adjustment on the final figure is not discarded: π ≈ 3.1416 2. Cutting or pruning: to forego a number of significant figures without making an adjustment to the latter figure does not discarded π ≈ 3.1415 In actuarial applications, science and engineering, we recommend rounding, since the cutting or pruning involves the loss of information.
- 10. Example Here comes your footer Page Consider the approximation of π ≈ 3.14159265. Perform cutting and rounding: a) Two significant digits. b) three significant digits. c) Four significant digits. d) Five significant digits. e) Six significant digits. f) Seven significant digits. g) Eight significant digits. Solution: The respective cutting and rounding to the respective number of significant digits, is summarized in the following table:
- 11. Here comes your footer Page Once established classification error (the two sources of error in numerical methods), we proceed to define the concepts of true absolute error, relative absolute error, approximate absolute error and approximate relative error estimate, all of them as a sum or a result of rounding and truncation errors. The following concepts can be used as error criteria for unemployment and measures of accuracy of numerical methods.
- 12. Remarks on tolerance t of a numerical method Here comes your footer Page If the following criterion is met, the result is correct to n significant digits: Example Consider the MacLaurin series for the determination of : Beginning with the first term and adding one term at a time, estimating the value . After adding each term, calculating the real and approximate relative error. The calculation ends until the absolute value of the approximate error is less than the t pre-determined criteria to ensure correct three significant digits
- 13. Solution Here comes your footer Page Consider the number 1.648721271 and the true value . If you want three correct significant digits, it must be n =3. Therefore: is guaranteed at least three significant digits correct, must be met: In this case, Namely The following table shows the development of exercise:
- 14. Solution Here comes your footer Page In the sixth iteration satisfies the criteria for tolerance, ea <t, given that the sixth iteration, ea = 0.0158, which is less than the predetermined tolerance: t = 0.05%. This will have to estimate is 1.648697917, with at least three correct significant digits: 1.648 697917.
- 15. Here comes your footer Page Software and Tools for Microsoft PowerPoint. The website with innovative solutions. Save time and money by automating your presentations. www.presentationpoint.com · Burden Richard L. & Faires J. Douglas, Análisis numérico . 2ª. ed., México, Grupo Editorial Iberoamérica, 1993. · Chapra Steven C. & Canale Raymond P., Métodos numéricos para ingenieros . 4ª. ed., México, McGraw-Hill, 2003. · Gerald Curtis F. & Wheatly Patrick O., Análisis numérico con aplicaciones . 6ª. ed., México, Prentice Hall, 2000. · Maron Melvin J. & López Robert J., Análisis numérico, con enfoque práctico . México, Editorial CECSA, 1995. · Mathews John H. & Fink Kurtis D., Métodos numéricos con MATLAB . 3ª. ed., España, Pearson-Prentice Hall, 2004. . http://www.acatlan.unam.mx/acatlecas/mn/MN_01.htm .http://webcache.googleusercontent.com/search?q=cache:9aM3JWGTWZ4J:bycase.blogspot.es/img/tarea1.doc+implicaciones+de+la+solución+analitica+en+metodos+numericos&cd=5&hl=es&ct=clnk&gl=co E-Mail: info@presentationpoint.com Bibliography