The document defines error as the difference between the true value and approximate value of a computed quantity. It provides examples of absolute error, which is the magnitude of the error, and relative error, which measures the error relative to the true value. There are two main sources of error - truncation error from using approximations, and rounding error from limitations of floating point representations. Truncation error is analyzed using examples of Taylor series approximations and numerical integration. Rounding error bounds are derived, showing the absolute error is bounded by 1/2 the least significant digit, while relative error is bounded by 1/2 times the number of significant digits.
This document contains solutions to problems related to semiconductor doping and carrier concentrations. Some key points:
1) It calculates intrinsic carrier concentration (ni) for silicon and gallium arsenide at different temperatures using the intrinsic carrier concentration equation.
2) It solves for temperature given intrinsic carrier concentration or doping concentration using the intrinsic carrier concentration equation.
3) It calculates ni, electron and hole concentrations for silicon and germanium at different temperatures.
4) It calculates electron and hole concentrations for n-type and p-type materials given doping concentrations and ni.
5) Other calculations include conductivity, current density, built-in potential, and more.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.
The document discusses various machine learning algorithms including polynomial regression, quadratic regression, radial basis functions, and robust regression. It provides mathematical formulas and visual examples to explain how each algorithm works. The key ideas are that polynomial regression fits nonlinear functions of inputs, quadratic regression extends linear regression by including quadratic terms, radial basis functions use kernel functions centered at data points to perform nonlinear regression, and robust regression aims to fit data robustly by down-weighting outliers.
This document provides solutions to quizzes from the textbook "Probability and Stochastic Processes" by Roy D. Yates and David J. Goodman. The solutions summarize the key concepts and formulas tested in each quiz question. MATLAB code is also provided to simulate some of the probabilistic experiments described in the textbook. Errors found in any quiz solutions will be corrected and posted online.
The document discusses methods for estimating solutions to polynomial equations, including:
1) Rewriting polynomial equations with simpler coefficients and putting them into standard form.
2) Using a graphing calculator to find estimates of zeros by graphing the equation and using the calculator's solver function.
3) Estimating zeros visually from a graphed polynomial equation by finding where it crosses the x-axis.
This document contains solutions to problems related to semiconductor doping and carrier concentrations. Some key points:
1) It calculates intrinsic carrier concentration (ni) for silicon and gallium arsenide at different temperatures using the intrinsic carrier concentration equation.
2) It solves for temperature given intrinsic carrier concentration or doping concentration using the intrinsic carrier concentration equation.
3) It calculates ni, electron and hole concentrations for silicon and germanium at different temperatures.
4) It calculates electron and hole concentrations for n-type and p-type materials given doping concentrations and ni.
5) Other calculations include conductivity, current density, built-in potential, and more.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.
The document discusses various machine learning algorithms including polynomial regression, quadratic regression, radial basis functions, and robust regression. It provides mathematical formulas and visual examples to explain how each algorithm works. The key ideas are that polynomial regression fits nonlinear functions of inputs, quadratic regression extends linear regression by including quadratic terms, radial basis functions use kernel functions centered at data points to perform nonlinear regression, and robust regression aims to fit data robustly by down-weighting outliers.
This document provides solutions to quizzes from the textbook "Probability and Stochastic Processes" by Roy D. Yates and David J. Goodman. The solutions summarize the key concepts and formulas tested in each quiz question. MATLAB code is also provided to simulate some of the probabilistic experiments described in the textbook. Errors found in any quiz solutions will be corrected and posted online.
The document discusses methods for estimating solutions to polynomial equations, including:
1) Rewriting polynomial equations with simpler coefficients and putting them into standard form.
2) Using a graphing calculator to find estimates of zeros by graphing the equation and using the calculator's solver function.
3) Estimating zeros visually from a graphed polynomial equation by finding where it crosses the x-axis.
This document provides an overview of partial derivatives, which are used to analyze functions with multiple variables. Key topics covered include:
- Definitions of limits, continuity, and partial derivatives for multivariable functions.
- Directional derivatives and the gradient, which describe the rate of change in a specified direction.
- The chain rule for partial derivatives, and implicit differentiation.
- Linearization and Taylor series approximations for multivariable functions.
- Finding local extrema and optimizing functions, using techniques like classifying critical points.
The document provides a review for a math final exam covering topics in linear programming including:
[1] Formulating linear programming problems and using the corner principle to solve them.
[2] Defining the dual linear programming problem and interpreting dual variable solutions.
[3] Using the simplex method, which involves constructing tableaus and performing pivotal operations, to solve linear programming problems in standard form.
The document discusses using Monte Carlo simulation to solve a partial differential equation (PDE) and using explicit time stepping to solve a different PDE. For the Monte Carlo method, as the number of random samples (M) increases, the approximation converges to the exact solution. For explicit time stepping, increasing the number of time and space steps (M and N) causes the error to diverge instead of converge due to exceeding memory cache capacity.
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
1. The document discusses place value systems for writing numbers up to 3000 and fractions up to 5/8.
2. It also covers solving quadratic equations of the form ax^2 + bx = c and calculating the volume of frustums of pyramids using formulas.
3. Historical topics discussed include the greatest pyramid built, the Pythagoreans, and Pythagoras' Theorem.
M A T H E M A T I C A L M E T H O D S J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains 8 sets of mathematical methods problems for an examination. Each set contains 8 multi-part problems related to topics like linear algebra, differential equations, interpolation, curve fitting, Fourier series, and more. The problems are intended for engineering students and test their understanding of key concepts and ability to apply various mathematical techniques to solve problems.
1. The document provides examples and explanations for solving various types of inequalities involving quadratic equations. It examines cases where a quadratic expression is less than, greater than or equal to zero.
2. Step-by-step workings are shown to arrive at the solution sets for each inequality. Roots of the auxiliary equations are used to determine boundaries for the ranges of values satisfying the inequalities.
3. Assumptions may be made in some cases to simplify the inequalities before determining the final solution sets. Multiple cases are considered to thoroughly address problems involving inequalities of quadratic expressions.
This document contains solutions to mathematics questions from the 2010 HSC exam in Australia. Question 1 involves solving equations, inequalities and finding derivatives. Question 2 involves finding derivatives of trigonometric functions. Question 3 involves vectors, gradients and parallel lines. Question 4 involves arithmetic progressions, integrals and area under curves. Question 5 involves volumes, surface areas, maxima and minima. Question 6 involves factorizing polynomials, discriminants and finding angles and areas of figures.
This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.
This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
1. Accuracy refers to how close measurements are to the true value, while precision refers to how close repeated measurements are to each other.
2. Measurements can be precise but not accurate if they are clustered around a value different from the true value, or accurate but not precise if widely dispersed around the true value.
3. Errors in measurement can be mistakes, systematic errors caused by natural or instrumental factors, or random errors that follow a normal distribution. The standard deviation describes the spread of random errors.
Elementary differential equations with boundary value problems solutionsHon Wa Wong
This document is the student solutions manual for elementary differential equations textbooks. It contains solutions to problems in chapters on topics like first order differential equations, linear and nonlinear second order equations, numerical methods, Laplace transforms, systems of differential equations, and boundary value problems. The solutions provided are concise and show the steps to reach the final answer.
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
1) This document contains a review of various algebra 2 concepts across 9 standards, including solving linear equations, quadratic equations, systems of equations, exponents, functions, and probability.
2) Several problems provide examples of solving systems of equations, factoring quadratic expressions, graphing quadratic and exponential functions, simplifying expressions with exponents, and calculating probabilities of independent and dependent events.
3) The review covers a wide range of algebra 2 topics to help students prepare for an upcoming benchmark exam.
Amth250 octave matlab some solutions (2)asghar123456
This document contains the solutions to 5 questions regarding numerical analysis techniques. Question 1 finds the zeros of a function graphically and numerically. Question 2 finds the millionth zero of tan(x)-x. Question 3 examines the convergence rates of Newton's method for various functions. Question 4 applies Newton's method to find the inverse of a function. Question 5 finds the maximum of a function using golden section search and parabolic interpolation.
This document provides a review of key algebra 1 concepts including equations of lines, solving various types of equations, factoring polynomials, expressions and equations involving perimeter, area, and geometry. Students are given examples to solve of each concept, including solving systems of equations, simplifying expressions, graphing lines, and determining equations of lines given points or other criteria. The review covers standard form, slope-intercept form, point-slope form of a line, solving linear and quadratic equations, factoring polynomials, perimeter, area, geometry relationships, graphing, and determining equations of lines from information provided.
Amth250 octave matlab some solutions (1)asghar123456
This document contains the solutions to 5 questions regarding numerical integration and differential equations. Question 1 involves numerically evaluating several integrals. Question 2 computes the Fresnel integrals. Question 3 uses Monte Carlo integration to estimate volumes. Question 4 examines the convergence and stability of the Euler method. Question 5 simulates the Lorenz system and demonstrates its sensitivity to initial conditions.
This document provides an introduction and overview of Matlab. It outlines what Matlab is, the main Matlab screen components, how to work with variables, arrays, matrices and perform indexing. It also covers basic arithmetic, relational and logical operators, different display facilities like plotting, and flow control structures like if/else statements and for loops. The document demonstrates how to use M-files to write scripts and user-defined functions in Matlab. It aims to introduce the key features and capabilities of the Matlab programming environment and language.
This document provides an overview of partial derivatives, which are used to analyze functions with multiple variables. Key topics covered include:
- Definitions of limits, continuity, and partial derivatives for multivariable functions.
- Directional derivatives and the gradient, which describe the rate of change in a specified direction.
- The chain rule for partial derivatives, and implicit differentiation.
- Linearization and Taylor series approximations for multivariable functions.
- Finding local extrema and optimizing functions, using techniques like classifying critical points.
The document provides a review for a math final exam covering topics in linear programming including:
[1] Formulating linear programming problems and using the corner principle to solve them.
[2] Defining the dual linear programming problem and interpreting dual variable solutions.
[3] Using the simplex method, which involves constructing tableaus and performing pivotal operations, to solve linear programming problems in standard form.
The document discusses using Monte Carlo simulation to solve a partial differential equation (PDE) and using explicit time stepping to solve a different PDE. For the Monte Carlo method, as the number of random samples (M) increases, the approximation converges to the exact solution. For explicit time stepping, increasing the number of time and space steps (M and N) causes the error to diverge instead of converge due to exceeding memory cache capacity.
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
1. The document discusses place value systems for writing numbers up to 3000 and fractions up to 5/8.
2. It also covers solving quadratic equations of the form ax^2 + bx = c and calculating the volume of frustums of pyramids using formulas.
3. Historical topics discussed include the greatest pyramid built, the Pythagoreans, and Pythagoras' Theorem.
M A T H E M A T I C A L M E T H O D S J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains 8 sets of mathematical methods problems for an examination. Each set contains 8 multi-part problems related to topics like linear algebra, differential equations, interpolation, curve fitting, Fourier series, and more. The problems are intended for engineering students and test their understanding of key concepts and ability to apply various mathematical techniques to solve problems.
1. The document provides examples and explanations for solving various types of inequalities involving quadratic equations. It examines cases where a quadratic expression is less than, greater than or equal to zero.
2. Step-by-step workings are shown to arrive at the solution sets for each inequality. Roots of the auxiliary equations are used to determine boundaries for the ranges of values satisfying the inequalities.
3. Assumptions may be made in some cases to simplify the inequalities before determining the final solution sets. Multiple cases are considered to thoroughly address problems involving inequalities of quadratic expressions.
This document contains solutions to mathematics questions from the 2010 HSC exam in Australia. Question 1 involves solving equations, inequalities and finding derivatives. Question 2 involves finding derivatives of trigonometric functions. Question 3 involves vectors, gradients and parallel lines. Question 4 involves arithmetic progressions, integrals and area under curves. Question 5 involves volumes, surface areas, maxima and minima. Question 6 involves factorizing polynomials, discriminants and finding angles and areas of figures.
This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.
This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
1. Accuracy refers to how close measurements are to the true value, while precision refers to how close repeated measurements are to each other.
2. Measurements can be precise but not accurate if they are clustered around a value different from the true value, or accurate but not precise if widely dispersed around the true value.
3. Errors in measurement can be mistakes, systematic errors caused by natural or instrumental factors, or random errors that follow a normal distribution. The standard deviation describes the spread of random errors.
Elementary differential equations with boundary value problems solutionsHon Wa Wong
This document is the student solutions manual for elementary differential equations textbooks. It contains solutions to problems in chapters on topics like first order differential equations, linear and nonlinear second order equations, numerical methods, Laplace transforms, systems of differential equations, and boundary value problems. The solutions provided are concise and show the steps to reach the final answer.
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
1) This document contains a review of various algebra 2 concepts across 9 standards, including solving linear equations, quadratic equations, systems of equations, exponents, functions, and probability.
2) Several problems provide examples of solving systems of equations, factoring quadratic expressions, graphing quadratic and exponential functions, simplifying expressions with exponents, and calculating probabilities of independent and dependent events.
3) The review covers a wide range of algebra 2 topics to help students prepare for an upcoming benchmark exam.
Amth250 octave matlab some solutions (2)asghar123456
This document contains the solutions to 5 questions regarding numerical analysis techniques. Question 1 finds the zeros of a function graphically and numerically. Question 2 finds the millionth zero of tan(x)-x. Question 3 examines the convergence rates of Newton's method for various functions. Question 4 applies Newton's method to find the inverse of a function. Question 5 finds the maximum of a function using golden section search and parabolic interpolation.
This document provides a review of key algebra 1 concepts including equations of lines, solving various types of equations, factoring polynomials, expressions and equations involving perimeter, area, and geometry. Students are given examples to solve of each concept, including solving systems of equations, simplifying expressions, graphing lines, and determining equations of lines given points or other criteria. The review covers standard form, slope-intercept form, point-slope form of a line, solving linear and quadratic equations, factoring polynomials, perimeter, area, geometry relationships, graphing, and determining equations of lines from information provided.
Amth250 octave matlab some solutions (1)asghar123456
This document contains the solutions to 5 questions regarding numerical integration and differential equations. Question 1 involves numerically evaluating several integrals. Question 2 computes the Fresnel integrals. Question 3 uses Monte Carlo integration to estimate volumes. Question 4 examines the convergence and stability of the Euler method. Question 5 simulates the Lorenz system and demonstrates its sensitivity to initial conditions.
This document provides an introduction and overview of Matlab. It outlines what Matlab is, the main Matlab screen components, how to work with variables, arrays, matrices and perform indexing. It also covers basic arithmetic, relational and logical operators, different display facilities like plotting, and flow control structures like if/else statements and for loops. The document demonstrates how to use M-files to write scripts and user-defined functions in Matlab. It aims to introduce the key features and capabilities of the Matlab programming environment and language.
This document provides an overview of the fundamentals of image processing. It begins with an introduction to key mathematical foundations, including vectors, matrices, vector spaces, bases, inner products, projections, and linear transforms. It then covers topics such as discrete-time signals and systems, linear time-invariant systems, sampling continuous signals to discrete and vice versa, digital filter design, image formation via lenses and sensors, point-wise and linear filtering operations, motion estimation, and useful tools like expectation-maximization and principal component analysis. The document serves as a guide to the core concepts and techniques in digital image processing.
This document discusses graphing and solving quadratic inequalities. It provides examples of graphing quadratic inequalities in two variables by drawing the parabola defined by the equation and shading the appropriate region based on the inequality symbol. It also discusses graphing systems of quadratic inequalities by identifying the common region where the individual graphs overlap. The document further explains how to solve quadratic inequalities in one variable either graphically by identifying the x-values where the parabola lies above or below the x-axis, or algebraically by finding the critical values and testing intervals. Examples are provided to illustrate both graphical and algebraic approaches.
The document discusses the least squares method and cubic fitting method. [1] It explains that least squares finds the best fit curve to a set of data points by minimizing the sum of the squared residuals. [2] Cubic fitting finds the smoothest curve that exactly fits the data points using a cubic polynomial function. [3] An example applies the cubic fitting method to bacterial growth data to determine the parameters for the best fitting cubic curve.
1. The solution to the system of equations y=2x and x/5 is (0,0).
2. The solutions to the systems of equations x+y=5 and 3x+2y-14=0 are (3,2) and the solutions to x=3 and y=6.5 are (3,6.5).
3. The system of equations representing spending $164 on books costing $15 each or $17 each can be expressed as a system of first degree equations in two variables.
This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.
This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval multiplied by the sub-interval width. The general formula sums these values over all sub-intervals divided by the number of intervals. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals and gets an approximate value of 1.26953125.
This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.
This document provides questions and solutions for a mock test on engineering concepts.
Q1 asks about matrix properties related to singularity. Q2 solves a linear system using Gaussian elimination and Cramer's rule. Q3 finds the eigenvalues and eigenvectors of a given matrix. The solutions show the step-by-step working and reasoning for each question.
1. The document provides information about precalculus chapter 2, which covers exponents and radicals, polynomials, factoring, and complex numbers.
2. Key topics include scientific notation, properties of exponents and radicals, adding/subtracting/multiplying polynomials, factoring polynomials, and performing operations with complex numbers.
3. Examples are provided for simplifying expressions with exponents, factoring trinomials and polynomials, using long division and synthetic division to divide polynomials, and solving quadratic equations with complex number solutions.
The document describes the Jacobi iterative method for solving systems of linear equations. It begins with an initial estimate for the solution variables, inserts them into the equations to get updated estimates, and repeats this process iteratively until the estimates converge to the desired solution. As an example, it applies the method to a set of 3 equations in 3 unknowns, showing the estimates after each iteration getting progressively closer to the exact solution obtained using Gaussian elimination. A Fortran program implementing the Jacobi method is also presented.
1) Find the limit of several functions as x approaches various values.
2) Complete a calculus test covering chapter 1 on limits and continuity.
3) The test contains 12 problems evaluating limits of functions both analytically and graphically.
This document outlines 6 questions for a math assignment on various interpolation techniques:
1. Use a degree 3 polynomial to estimate life expectancies in 3 years for 2 countries.
2. Fit an exponential function to 5 data points to determine coefficients.
3. Compare accuracy of interpolating a function using cubic spline, pchip cubic, and degree 5 polynomial.
4. Generate and analyze cubic spline and pchip interpolants, with derivatives, for another data set.
5. Find the least squares solution to an overdetermined system of linear equations from altitude measurements.
6. Determine the best fitting function - quadratic, power, or exponential - for another data set. Instructions are provided for including
1. This document contains 11 multi-part math problems involving systems of equations and inequalities. The problems cover topics such as solving systems graphically, algebraically, and determining if ordered pairs are solutions. They also involve word problems about ages, expenses, and splitting amounts into parts.
2. Key steps addressed include setting up tables of values, identifying line types, finding the solution set intersection, using substitution or elimination methods, stating yes or no for ordered pairs, and drawing graphs of solution sets for systems of inequalities.
3. The problems progress from simpler systems to more complex ones involving multiple equations or inequalities, requiring skills like algebraic manipulation, graphical analysis, and translating word problems into mathematical systems.
2. Definition
• The Error in a computed quantity is defined
as:
Error = True Value – Approximate Value
2
3. Examples:
a. True Value : phi = 3.14159265358979
Appr. Value : 22/7 = 3.14285714285714
Error = phi-22/7= -0.00126448926735
b. True Value : 12
Appr. Value: 11.78
Error = 12-11.78 = 0.22
c. True Value : 100
Appr. Value: 95.5
Error = 100-95.5 = 4.5
3
4. Kind of Error
• The Absolute Error is measure the
magnitude of the error
Ea Error
• The Relative Error is a measure of the error
in relation to the size of the true value
Ea
Er
True Value
4
5. Examples
• True value : 10 Ea = 10-9 = 1
Appr. Value : 9 Er = Ea/ 10 = 0.1
Ea 1, Er 0 .1
• True value : 1000 Ea = 1000-999= 1
Appr. Value : 999 Er =Ea/1000=0.001
Ea 1, Er 0 . 001
• True value : 250 Ea =250-240= 10
Appr. Value : 240 Er = Ea/250= 0.04
Ea 10 , Er 0 . 04 5
7. a. Truncation Error
• errors that result from using an approximation
in place of an exact mathematical procedure
7
8. Example of Truncation Error
Taking only a few terms of a Maclaurin series to
approximate e
x
2 3
x x x
e 1 x .......... ..........
2! 3!
If only 3 terms are used,
2
x x
Truncation Error e 1 x
2!
8
9. Example 1 —Maclaurin series
Calculate the value of e with an absolute
1 .2
relative approximate error of less than 1%.
2 3
1 .2 1 .2 1 .2
e 1 1 .2 .......... .........
2! 3!
n 1 .2
Ea %
e a
1 1 __ ___
2 2.2 1.2 54.545
3 2.92 0.72 24.658
4 3.208 0.288 8.9776
5 3.2944 0.0864 2.6226
6 3.3151 0.020736 0.62550
6 terms are required.
9
10. Example 2 —Differentiation
f (x x) f ( x)
Find for using
2
f ( 3) f ( x) x f ( x)
x
and x 0 .2
' f (3 0 .2 ) f (3)
f (3)
0 .2
2 2
f (3 .2 ) f (3) 3 .2 3 10 . 24 9 1 . 24
6 .2
0 .2 0 .2 0 .2 0 .2
The actual value is
' '
f ( x) 2 x, f (3) 2 3 6
Truncation error is then, 6 6 .2 0 .2
10
11. Example 3 — Integration
Use two rectangles of equal width to approximate
the area under the curve for
x over the interval [ 3,9 ]
2
f ( x)
y
90
9
y = x2 2
60
x dx
30 3
0 x
0 3 6 9 12
11
12. Integration example (cont.)
Choosing a width of 3, we have
9
2 2 2
x dx (x ) (6 3) (x ) (9 6)
x 3 x 6
3
2 2
(3 )3 ( 6 )3
27 108 135
Actual value is given by
9 9
3 3 3
2 x 9 3
x dx 234
3 3 3
3
Truncation error is then
234 135 99
12
13. b. Rounding Errors
• Round-off / Chopping Errors
• Recognize how floating point arithmetic operations
can introduce and amplify round-off errors
• What can be done to reduce the effect of round-off
errors
13
14. There are discrete points on the
number lines that can be
represented by our computer.
How about the space between ?
14
15. Implication of FP representations
• Only limited range of quantities may be
represented.
– Overflow and underflow
• Only a finite number of quantities within the range
may be represented.
– round-off errors or chopping errors
15
16. Round-off / Chopping Errors
(Error Bounds Analysis)
Let
z be a real number we want to represent in a computer, and
fl(z) be the representation of z in the computer.
What is the largest possible value of ?
z fl ( z )
z
i.e., in the worst case, how much data are we losing due to round-off or chopping
errors?
16
17. Chopping Errors (Error Bounds Analysis)
Suppose the mantissa can only support n digits.
e
z 0 .a1a 2 a n a n 1a n 2
, a1 0
e
fl ( z ) 0 .a1a 2 a n
Thus the absolute and relative chopping errors are
e e n
z fl ( z ) ( 0 .00 ... 0 a n 1 a n
2
) ( 0 .a n 1a n 2
)
n zeroes
e
z fl ( z ) ( 0 . 00 ... 0 a n 1 a n 2
)
e
z ( 0 .a 1a 2 a n a n 1a n 2
)
Suppose ß = 10 (base 10), what are the values of ai such that
the errors are the largest?
17
18. Chopping Errors (Error Bounds Analysis)
Because 0 .a n 1a n 2 a n 3
1
e n e n e n
z fl ( z ) 0 .a n 1a n 2
z fl ( z )
e
z fl ( z ) 0 . 00 ... 0 a n 1 a n 2
e
z 0 .a 1a 2 a n a n 1a n 2
e n
e
0 .a1a 2 a n a n 1a n 2
e n
e
0 .100000 a n 1 a n
2
n digits
e n e n
e n ( e 1) 1 n
z fl ( z ) 1 n
e 1 e
0 .1 z
18
19. Round-off Errors (Error Bounds Analysis)
e
z 0 .a 1 a 2 a n a n 1
, a1 0
1 ( sign ) base e exponent
e
( 0 .a 1 a 2 a n ) 0 an 1
2
fl ( z ) Round down
e
[( 0 .a 1 a 2 a n ) ( 0 ) ]
00 ...
. 01 an 1
n 2
Round up
fl(z) is the rounded value of z
19
20. Round-off Errors (Error Bounds Analysis)
Absolute error of fl(z)
When rounding down
e
z fl ( z ) 0 . 00 0 a n 1 a n a
2 n 3
e n
0 .a n 1 a n a
2 n 3
e n
z fl ( z ) 0 .a n 1 a n a
2 n 3
1 1 e n
an 1
(. a n 1
) z fl ( z )
2 2 2
Similarly, when rounding up
1
i.e., when an 1 z fl ( z )
e n
2 2
20
21. Round-off Errors (Error Bounds Analysis)
Relative error of fl(z)
1 e n
z fl ( z )
2
n
z fl ( z ) 1
e
z 2 z
n
1 e
because z (. a 1 a 2 )
2 (. a 1 a 2 )
n
1
because (. a 1 ) ( 0 . 1)
2 (. 1)
n
1
2
1 z fl ( z ) 1 1 n
z 2
21
22. Summary of Error Bounds Analysis
Chopping Errors Round-off errors
Absolute e n 1 e n
z fl ( z ) z fl ( z )
2
Relative z fl ( z ) 1 n
z fl ( z ) 1 1 n
z z 2
β base
n # of significant digits or # of digits in the mantissa
Regardless of chopping or round-off is used to round the numbers,
the absolute errors may increase as the numbers grow in magnitude
but the relative errors are bounded by the same magnitude.
22