This document summarizes several numerical methods for finding roots of nonlinear equations or eigenvalues of matrices:
1) Bisection method, false position method, and secant method are iterative root-finding algorithms for nonlinear equations. They rely on checking the sign of the function at interval endpoints and successively narrowing the interval containing a root.
2) Newton's method and the power method are algorithms for finding roots or eigenvalues by using derivatives or matrix multiplication. Newton's method finds roots by iteratively computing the x-intercept of the tangent line. The power method finds the dominant eigenvalue by repeatedly multiplying a matrix by a vector.
3) Gerschgorin's circle theorem provides bounds on the locations of a
This document presents information on different interpolation methods including forward, backward, and central interpolation. It defines interpolation as finding values inside a known interval, while extrapolation finds values outside the interval. It discusses polynomial interpolation using linear, quadratic, and cubic polynomials. It also defines forward, backward, and central finite differences and difference operators. Tables are presented showing examples of applying first, second, third, and fourth differences using forward, backward, and central difference operators.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IVRai University
This document discusses finite difference and interpolation methods. It covers topics like finite differences, difference tables, Newton's forward and backward interpolation formulas, Stirling's interpolation formula, Newton's divided difference formula for unequal intervals, and Lagrange's divided difference formula for unequal intervals. Examples are provided to demonstrate calculating finite differences, constructing difference tables, and using interpolation formulas to estimate values between given data points.
numericai matmatic matlab uygulamalar ali abdullahAli Abdullah
The document discusses various interpolation methods including Newton's forward and backward interpolation methods. Newton's forward interpolation method uses forward difference operators to calculate interpolated values near the beginning of a data set. Newton's backward interpolation method uses backward difference operators to calculate interpolated values near the end of a data set. The document provides examples of applying Newton's forward and backward interpolation methods to calculate interpolated values using given data tables. It also discusses writing a MATLAB program to calculate interpolated values using a third degree polynomial interpolation.
The chapter discusses numerical methods for solving the 1D and 2D heat equation. Four methods are described for the 1D equation: Schmidt, Crank-Nicolson, iterative (Jacobi and Gauss-Seidel), and Du Fort-Frankel. The Schmidt method is explicit but conditionally stable, while Crank-Nicolson is implicit and unconditionally stable. Examples are solved using each method and compared to analytical solutions. The alternating direction explicit (ADE) method is described for the 2D equation.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-VRai University
This document describes numerical integration and differentiation techniques taught in a B.Tech Engineering Mathematics course. It covers the Trapezoidal, Simpson's 1/3 and 3/8 rules for numerical integration of functions. For numerical differentiation, it discusses Euler's method, Picard's method, and Taylor series for solving ordinary differential equations. Examples are provided to illustrate the application of these numerical methods to evaluate integrals and solve initial value problems.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
This document discusses numerical solutions of partial differential equations. It contains an introduction and four chapters:
1. Preliminaries - Defines basic concepts like differential equations, partial derivatives, order of a differential equation.
2. Partial Differential Equations of Second Order - Classifies second order PDEs and provides examples.
3. Parabolic Equations - Discusses explicit and implicit finite difference methods like Schmidt's method and Crank-Nicolson method to solve heat equation.
4. Hyperbolic Equations - Will discuss numerical methods to solve hyperbolic PDEs like the wave equation.
This document presents information on different interpolation methods including forward, backward, and central interpolation. It defines interpolation as finding values inside a known interval, while extrapolation finds values outside the interval. It discusses polynomial interpolation using linear, quadratic, and cubic polynomials. It also defines forward, backward, and central finite differences and difference operators. Tables are presented showing examples of applying first, second, third, and fourth differences using forward, backward, and central difference operators.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IVRai University
This document discusses finite difference and interpolation methods. It covers topics like finite differences, difference tables, Newton's forward and backward interpolation formulas, Stirling's interpolation formula, Newton's divided difference formula for unequal intervals, and Lagrange's divided difference formula for unequal intervals. Examples are provided to demonstrate calculating finite differences, constructing difference tables, and using interpolation formulas to estimate values between given data points.
numericai matmatic matlab uygulamalar ali abdullahAli Abdullah
The document discusses various interpolation methods including Newton's forward and backward interpolation methods. Newton's forward interpolation method uses forward difference operators to calculate interpolated values near the beginning of a data set. Newton's backward interpolation method uses backward difference operators to calculate interpolated values near the end of a data set. The document provides examples of applying Newton's forward and backward interpolation methods to calculate interpolated values using given data tables. It also discusses writing a MATLAB program to calculate interpolated values using a third degree polynomial interpolation.
The chapter discusses numerical methods for solving the 1D and 2D heat equation. Four methods are described for the 1D equation: Schmidt, Crank-Nicolson, iterative (Jacobi and Gauss-Seidel), and Du Fort-Frankel. The Schmidt method is explicit but conditionally stable, while Crank-Nicolson is implicit and unconditionally stable. Examples are solved using each method and compared to analytical solutions. The alternating direction explicit (ADE) method is described for the 2D equation.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-VRai University
This document describes numerical integration and differentiation techniques taught in a B.Tech Engineering Mathematics course. It covers the Trapezoidal, Simpson's 1/3 and 3/8 rules for numerical integration of functions. For numerical differentiation, it discusses Euler's method, Picard's method, and Taylor series for solving ordinary differential equations. Examples are provided to illustrate the application of these numerical methods to evaluate integrals and solve initial value problems.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
This document discusses numerical solutions of partial differential equations. It contains an introduction and four chapters:
1. Preliminaries - Defines basic concepts like differential equations, partial derivatives, order of a differential equation.
2. Partial Differential Equations of Second Order - Classifies second order PDEs and provides examples.
3. Parabolic Equations - Discusses explicit and implicit finite difference methods like Schmidt's method and Crank-Nicolson method to solve heat equation.
4. Hyperbolic Equations - Will discuss numerical methods to solve hyperbolic PDEs like the wave equation.
The document discusses numerical methods for solving ordinary differential equations (ODEs), including Taylor's series method and Picard's method. It provides examples of applying Taylor's series method to approximate solutions of first order ODEs at different values of x to 4-5 decimal places of accuracy. The examples given include solving ODEs with initial conditions and computing solutions at multiple x values by taking terms from the Taylor series expansion.
Study Material Numerical Solution of Odinary Differential EquationsMeenakshisundaram N
1. The document provides information about a numerical methods course for physics majors at Vivekananda College in Tiruvedakam West, including the reference textbook and details about Unit V on numerical solutions of ordinary differential equations.
2. It introduces the concept of using Taylor series approximations to find numerical solutions to differential equations, providing the general Taylor series expansion formula and explaining how to derive the terms needed to solve specific differential equations.
3. It gives examples of using the Taylor series method to solve sample ordinary differential equations, finding approximate values of y at increasing values of x to several decimal places.
This document provides an overview of numerical differentiation and integration methods. It discusses Newton's forward and backward difference formulas for computing derivatives, as well as Newton-Cote's formula, the trapezoidal rule, and Simpson's one-third and three-eighths rules for numerical integration. Examples of applying these methods to real-world problems are provided. The document also compares Simpson's one-third and three-eighths rules, noting their different assumptions about the polynomial order of the integrated function and requirements for the number of intervals.
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
The document discusses various mathematical methods for interpolation and solving equations including:
1) Bisection method, iteration method, and Newton-Raphson method for finding roots of equations.
2) Finite difference methods for numerical differentiation and interpolation using forward, backward, and central difference operators.
3) Newton's forward and backward interpolation formulas for equally spaced data using finite differences.
4) Gauss interpolation and Lagrange interpolation for unequally spaced data points.
This document discusses various methods of interpolation and numerical differentiation using divided differences and Newton's formulas. It introduces Lagrange interpolation for both equal and unequal intervals. Inverse interpolation and Newton's divided difference interpolation are also covered. Forward and backward difference formulas are presented for interpolation with equal intervals. Numerical differentiation can be performed by taking derivatives of the interpolation polynomial or using forward difference formulas to estimate derivatives at the data points.
AEM Integrating factor to orthogonal trajactoriesSukhvinder Singh
This document provides information about integrating factors and their use in solving differential equations. It discusses:
1) How to find integrating factors by inspection, including common differential forms.
2) Four rules for finding integrating factors for exact and homogeneous differential equations.
3) Using integrating factors to solve linear differential equations and the Bernoulli equation.
4) The concept of orthogonal trajectories and the working rule for finding the differential equation of orthogonal trajectories given a family of curves.
An example of finding the orthogonal trajectories of the curve y = x2 + c is provided.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IRai University
1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
The document describes three numerical methods for finding the roots or solutions of equations: the bisection method, Newton's method for single variable equations, and Newton's method for systems of nonlinear equations.
The bisection method works by repeatedly bisecting the interval within which a root is known to exist, narrowing in on the root through iterative halving. Newton's method approximates the function with its tangent line to find a better root estimate with each iteration. For systems of equations, Newton's method involves calculating the Jacobian matrix and solving a system of linear equations at each step to update the solution estimate. Examples are provided to illustrate each method.
1. The document discusses numerical methods for solving ordinary differential equations, including power series approximations, Taylor series, Euler's method, and the Runge-Kutta method.
2. It provides examples of using each of these methods to solve sample differential equations and compares the numerical solutions to exact solutions.
3. Truncation errors are defined as errors that result from using an approximation instead of an exact mathematical procedure.
This document discusses numerical methods for solving initial value problems for ordinary differential equations. It introduces the Taylor series method and Runge-Kutta method for solving initial value problems. Examples are provided to demonstrate solving first and second order differential equations using these two methods and to compare their results. Stability of numerical solutions is also discussed.
Linear differential equation with constant coefficientSanjay Singh
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
The document provides examples and explanations of concepts related to calculus including continuity, differentiability, limits, derivatives, and the mean value theorem. Some key points:
- It gives examples of determining if functions are continuous and differentiable at various points, including functions with absolute value.
- The mean value theorem is explained and examples are worked through showing a function satisfies the mean value theorem and finding the value of c.
- Numerous examples demonstrate calculating derivatives using rules like product, quotient, chain and implicit differentiation. Examples include derivatives of trigonometric, exponential and logarithmic functions.
- Implicit differentiation is used to find the equation of a tangent line to a curve at a given point.
The document discusses fractional calculus and fractional partial differential equations (FPDEs). It provides background on fractional calculus, including its origins in the late 17th century. It then discusses applications of FPDEs in fields like image processing and finance. The objective is to numerically solve two-sided FPDEs using finite difference methods. It introduces the Riemann-Liouville definition of fractional derivatives and the Grünwald definition for approximating fractional derivatives. It then discusses approximating one-sided and two-sided FPDEs using finite differences and analyzes the stability of the resulting schemes.
This document discusses numerical integration and interpolation formulas. It begins by explaining the general formula for numerical integration using equidistant values of a function f(x) between bounds a and b. It then derives Trapezoidal, Simpson's, and Weddle's rules by putting different values for n in the general formula. The document also discusses Newton's forward and backward interpolation formulas, Lagrange interpolation formula, and provides examples of their application. It concludes by comparing Lagrange and Newton interpolation and discussing uses of interpolation in computer science and engineering fields.
This document discusses Euler's method for solving ordinary differential equations numerically. It begins by considering the differential equation dy/dx = f(x,y), along with the initial condition y(x0) = y0. It then derives Euler's method by approximating the differential equation using the Taylor series expansion and neglecting higher order terms. The general step of Euler's method is given as yi+1 = yi + h*f(xi, yi), where h is the step size. Several examples are worked out applying Euler's method to solve initial value problems.
The document describes the Newton-Raphson method for finding the roots of nonlinear equations. It provides the derivation of the method, outlines the algorithm as a 3-step process, and gives an example of applying it to find the depth a floating ball submerges in water. The advantages are that it converges fast if it converges and requires only one initial guess. Drawbacks include potential issues with division by zero, root jumping, and oscillations near local extrema.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses fuzzy logic and fuzzy sets. It begins by explaining fuzzy logic is used to model imprecise concepts and dependencies using natural language terms. It then defines fuzzy variables, universes of discourse, and fuzzy sets which have membership functions assigning a degree of membership between 0 and 1. Operations on fuzzy sets like intersection, union, and complement are also covered. The document also discusses fuzzy rules, relations, and approximate reasoning using max-min inference.
The document provides examples of solving equations using numerical methods like the Newton Raphson method, Regula Falsi method, and Bisection method.
It contains 7 examples of finding the real roots of equations using the Newton Raphson method. 5 examples are given for finding real roots using the Regula Falsi method. 6 examples demonstrate finding approximate roots of equations using the Bisection method.
Two examples are given to find roots near 0.3 and 2.1 of the equation -4x=0 using the direct iteration or method of successive approximation. The document is presented by a group of 13 students from Yeshwantrao Chavan College of Engineering, guided by their professor.
The document discusses numerical methods for solving ordinary differential equations (ODEs), including Taylor's series method and Picard's method. It provides examples of applying Taylor's series method to approximate solutions of first order ODEs at different values of x to 4-5 decimal places of accuracy. The examples given include solving ODEs with initial conditions and computing solutions at multiple x values by taking terms from the Taylor series expansion.
Study Material Numerical Solution of Odinary Differential EquationsMeenakshisundaram N
1. The document provides information about a numerical methods course for physics majors at Vivekananda College in Tiruvedakam West, including the reference textbook and details about Unit V on numerical solutions of ordinary differential equations.
2. It introduces the concept of using Taylor series approximations to find numerical solutions to differential equations, providing the general Taylor series expansion formula and explaining how to derive the terms needed to solve specific differential equations.
3. It gives examples of using the Taylor series method to solve sample ordinary differential equations, finding approximate values of y at increasing values of x to several decimal places.
This document provides an overview of numerical differentiation and integration methods. It discusses Newton's forward and backward difference formulas for computing derivatives, as well as Newton-Cote's formula, the trapezoidal rule, and Simpson's one-third and three-eighths rules for numerical integration. Examples of applying these methods to real-world problems are provided. The document also compares Simpson's one-third and three-eighths rules, noting their different assumptions about the polynomial order of the integrated function and requirements for the number of intervals.
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
The document discusses various mathematical methods for interpolation and solving equations including:
1) Bisection method, iteration method, and Newton-Raphson method for finding roots of equations.
2) Finite difference methods for numerical differentiation and interpolation using forward, backward, and central difference operators.
3) Newton's forward and backward interpolation formulas for equally spaced data using finite differences.
4) Gauss interpolation and Lagrange interpolation for unequally spaced data points.
This document discusses various methods of interpolation and numerical differentiation using divided differences and Newton's formulas. It introduces Lagrange interpolation for both equal and unequal intervals. Inverse interpolation and Newton's divided difference interpolation are also covered. Forward and backward difference formulas are presented for interpolation with equal intervals. Numerical differentiation can be performed by taking derivatives of the interpolation polynomial or using forward difference formulas to estimate derivatives at the data points.
AEM Integrating factor to orthogonal trajactoriesSukhvinder Singh
This document provides information about integrating factors and their use in solving differential equations. It discusses:
1) How to find integrating factors by inspection, including common differential forms.
2) Four rules for finding integrating factors for exact and homogeneous differential equations.
3) Using integrating factors to solve linear differential equations and the Bernoulli equation.
4) The concept of orthogonal trajectories and the working rule for finding the differential equation of orthogonal trajectories given a family of curves.
An example of finding the orthogonal trajectories of the curve y = x2 + c is provided.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IRai University
1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
The document describes three numerical methods for finding the roots or solutions of equations: the bisection method, Newton's method for single variable equations, and Newton's method for systems of nonlinear equations.
The bisection method works by repeatedly bisecting the interval within which a root is known to exist, narrowing in on the root through iterative halving. Newton's method approximates the function with its tangent line to find a better root estimate with each iteration. For systems of equations, Newton's method involves calculating the Jacobian matrix and solving a system of linear equations at each step to update the solution estimate. Examples are provided to illustrate each method.
1. The document discusses numerical methods for solving ordinary differential equations, including power series approximations, Taylor series, Euler's method, and the Runge-Kutta method.
2. It provides examples of using each of these methods to solve sample differential equations and compares the numerical solutions to exact solutions.
3. Truncation errors are defined as errors that result from using an approximation instead of an exact mathematical procedure.
This document discusses numerical methods for solving initial value problems for ordinary differential equations. It introduces the Taylor series method and Runge-Kutta method for solving initial value problems. Examples are provided to demonstrate solving first and second order differential equations using these two methods and to compare their results. Stability of numerical solutions is also discussed.
Linear differential equation with constant coefficientSanjay Singh
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
The document provides examples and explanations of concepts related to calculus including continuity, differentiability, limits, derivatives, and the mean value theorem. Some key points:
- It gives examples of determining if functions are continuous and differentiable at various points, including functions with absolute value.
- The mean value theorem is explained and examples are worked through showing a function satisfies the mean value theorem and finding the value of c.
- Numerous examples demonstrate calculating derivatives using rules like product, quotient, chain and implicit differentiation. Examples include derivatives of trigonometric, exponential and logarithmic functions.
- Implicit differentiation is used to find the equation of a tangent line to a curve at a given point.
The document discusses fractional calculus and fractional partial differential equations (FPDEs). It provides background on fractional calculus, including its origins in the late 17th century. It then discusses applications of FPDEs in fields like image processing and finance. The objective is to numerically solve two-sided FPDEs using finite difference methods. It introduces the Riemann-Liouville definition of fractional derivatives and the Grünwald definition for approximating fractional derivatives. It then discusses approximating one-sided and two-sided FPDEs using finite differences and analyzes the stability of the resulting schemes.
This document discusses numerical integration and interpolation formulas. It begins by explaining the general formula for numerical integration using equidistant values of a function f(x) between bounds a and b. It then derives Trapezoidal, Simpson's, and Weddle's rules by putting different values for n in the general formula. The document also discusses Newton's forward and backward interpolation formulas, Lagrange interpolation formula, and provides examples of their application. It concludes by comparing Lagrange and Newton interpolation and discussing uses of interpolation in computer science and engineering fields.
This document discusses Euler's method for solving ordinary differential equations numerically. It begins by considering the differential equation dy/dx = f(x,y), along with the initial condition y(x0) = y0. It then derives Euler's method by approximating the differential equation using the Taylor series expansion and neglecting higher order terms. The general step of Euler's method is given as yi+1 = yi + h*f(xi, yi), where h is the step size. Several examples are worked out applying Euler's method to solve initial value problems.
The document describes the Newton-Raphson method for finding the roots of nonlinear equations. It provides the derivation of the method, outlines the algorithm as a 3-step process, and gives an example of applying it to find the depth a floating ball submerges in water. The advantages are that it converges fast if it converges and requires only one initial guess. Drawbacks include potential issues with division by zero, root jumping, and oscillations near local extrema.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses fuzzy logic and fuzzy sets. It begins by explaining fuzzy logic is used to model imprecise concepts and dependencies using natural language terms. It then defines fuzzy variables, universes of discourse, and fuzzy sets which have membership functions assigning a degree of membership between 0 and 1. Operations on fuzzy sets like intersection, union, and complement are also covered. The document also discusses fuzzy rules, relations, and approximate reasoning using max-min inference.
The document provides examples of solving equations using numerical methods like the Newton Raphson method, Regula Falsi method, and Bisection method.
It contains 7 examples of finding the real roots of equations using the Newton Raphson method. 5 examples are given for finding real roots using the Regula Falsi method. 6 examples demonstrate finding approximate roots of equations using the Bisection method.
Two examples are given to find roots near 0.3 and 2.1 of the equation -4x=0 using the direct iteration or method of successive approximation. The document is presented by a group of 13 students from Yeshwantrao Chavan College of Engineering, guided by their professor.
The document discusses properties of the normal distribution, including that it is bell-shaped and symmetrical, with mean, median and mode equal. It also discusses standard normal variates and how to use normal distribution tables to calculate probabilities. Several examples are provided, such as calculating the probability of electric bulb life exceeding a certain threshold, or the marks limit for promoting a certain percentage of students based on mean and standard deviation of exam scores.
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.
The bisection method is used to find the root of equations by repeatedly bisecting an interval and determining if the function value at the midpoint is positive or negative. The document provides examples of using the bisection method to find roots of equations like X^3-X-1, 4sinx-e^x, and X^2-4X-10. It shows calculating the function values at the endpoints of intervals, determining if the sign changes, bisecting the interval, and repeating until converging on the root.
The document discusses numerical methods for solving linear equations, including Gauss-Seidel, Jacobi, and Cholesky methods. It provides MATLAB code implementations of each method. It applies the different methods to example matrices and right-hand side vectors to compute the solutions and compares the performance of the methods. In particular, it finds that Gauss-Seidel diverges more quickly than Jacobi in some cases.
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.
Introduction to Artificial IntelligenceManoj Harsule
S: fuzzy relation defined on Y and Z.
To find the composite relation R o S on X and Z:
μR o S(x,z) = maxy [min(μR(x,y), μS(y,z))]
For each x and z, find the maximum membership grade obtained by considering all possible y values and taking the minimum of the membership grades of R and S.
This gives the generalized intersection-union definition of composition of fuzzy relations. It reduces to the usual composition rule when relations are crisp.
Statistics Assignment 1 HET551 – Design and Developm.docxrafaelaj1
Statistics Assignment 1
HET551 – Design and Development Project 1
Michael Allwright
Haddon O’Neill
Tuesday, 24 May 2011
1 Normal Approximation to the Binomial Distribution
This section of the assignment shows how a normal curve can be used to approximate the binomial distribution. This
section of the assignment was completed using a MATLAB function (shown in Listings 1) which would generate and
save plots of the various Binomial Distributions after normalisation, and then calculate the errors between the standard
normal curve and the binomial distribution.
The plots in Figures 1 and 2 show the binomial distribution for various n trials with probability p = 1
3
and p = 1
2
respectively. These binomial plots have been normalised so that they can be compared with the standard normal
distribution.
From these plots it can be seen that once the binomial distribution has been normalised, the normal approximation is
a good approach to estimating the binomial distribution. To determine its accuracy, the data in Table 1 shows the
evaluation of qn = P(bn ≥ µn + 2σn) for both the normal curve and binomial distribution.
qn = P(bn ≥ µn + 2σn) Calculation Error
n N(0, 1) B(n, 1
2
) B(n, 1
3
) B(n, 1
2
) B(n, 1
3
)
1 0.0228 0.0000 0.0000 -0.02278 -0.02278
2 0.0228 0.0000 0.0000 -0.02278 -0.02278
3 0.0228 0.0000 0.0370 -0.02278 0.01426
4 0.0228 0.0000 0.0123 -0.02278 -0.01043
5 0.0228 0.0313 0.0453 0.00847 0.02249
10 0.0228 0.0107 0.0197 -0.01203 -0.00312
20 0.0228 0.0207 0.0376 -0.00208 0.01486
30 0.0228 0.0214 0.0188 -0.00139 -0.00398
40 0.0228 0.0192 0.0214 -0.00354 -0.00134
50 0.0228 0.0164 0.0222 -0.00636 -0.00059
100 0.0228 0.0176 0.0276 -0.00518 0.00479
Table 1: Calculating the error of the normal approximation to the binomial for various n and p
2 Analytical investigation of the Exponential Distribution
For this part of the assignment the density function shown in Equation 1 was given.
f(x) = λe−λx for x ≥ 0 and λ ≥ 0 (1)
Before any calculations were attempted, the area under graph was checked to show that
´∞
−∞f(x) dx = 1. That is
that the total probability of all possible values was 1.
2.1 Derivation of CDF
To find the CDF of the given function, the function was integrated with 0 and x being the lower and upper bound
respectively. This derivation is shown in Equations 2 to 4.
CDF =
ˆ x
o
f(x) dx =
ˆ x
o
λe−λx dx (2)
2
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 0.33)
P
ro
b
a
b
ili
ty
(a) n = 1
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 0.67)
P
ro
b
a
b
ili
ty
(b) n = 2
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 1.00)
P
ro
b
a
b
ili
ty
(c) n = 3
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes.
This document contains a tutorial on applied statistics and mathematics in economics and business. It covers several examples:
1) A yogurt manufacturer calculating the probability that production exceeds 570 units and constructing a confidence interval between 490-510 units.
2) Estimating the probability of women working shift work between 0.6-0.7 based on a sample and the probability being over 0.5.
3) Calculating 90% and 99% confidence intervals for the mean income of a building society.
4) Estimating the proportion of voters preferring a candidate based on a sample and the 95% and 99% confidence intervals.
5) Testing hypotheses about the mean lifetime of light bulbs being different
This document contains a tutorial on applied statistics and mathematics in economics and business. It provides examples of calculating probabilities and confidence intervals for various statistical distributions and tests, including:
1) Calculating the probability that yogurt production exceeds a certain amount based on a normal distribution.
2) Estimating a confidence interval for the probability of women working shift work based on a binomial distribution.
3) Creating 90% and 99% confidence intervals for the mean production of a building society.
4) Estimating confidence intervals for the proportion of voters preferring a candidate based on a sample.
5) Performing one-tailed and two-tailed hypothesis tests on the mean lifetime of lightbulbs.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
This document discusses finite difference and interpolation methods. It defines finite differences of various orders (first, second, etc.) and describes forward, backward, and central difference tables. It also covers Newton's forward and backward interpolation formulas for unequal intervals using forward and backward differences. An example is provided to illustrate calculating interpolated values using these formulas.
Numerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature. ... A generalization of the trapezoidal rule is Romberg integration, which can yield accurate results for many fewer function evaluations.
The document discusses solving linear equations using numerical methods in MATLAB. It provides code implementations for Gauss-Seidel method, Jacobi method, and Cholesky method. It also includes routines for calculating the inverse of a 3x3 matrix, matrix norm, and forward substitution. Examples are given applying each method to different 3x3 matrices. Spectral radius is also calculated for some examples to analyze convergence properties.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
Two algorithms to accelerate training of back-propagation neural networksESCOM
This document proposes two algorithms to initialize the weights of neural networks to accelerate training. Algorithm I performs a step-by-step orthogonalization of the input matrices to drive them towards a diagonal form, aiming to place the network closer to the convergence points of the activation function. Algorithm II aims to jointly diagonalize the input matrices to also drive them towards a diagonal form. The algorithms are shown to significantly reduce training time compared to random initialization, though Algorithm I works best when the activation function has φ(0)>1.
Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1cideni
This document contains exercises related to functions and graphs. Exercise set 1.1 contains word problems involving various functional relationships and graphs. Exercise set 1.2 involves evaluating and sketching functions, determining domains and ranges, and identifying piecewise functions. Exercise set 1.3 involves selecting appropriate axis ranges and scales to graph functions over specified domains.
UHB 3022 / ULAB 3122 - Final Exam PaperAbdul Khaliq
This document contains instructions and situations for a final examination consisting of 3 parts assessing English for Workplace Communication skills. Part I involves writing a memo to staff about an upcoming team building program. Part II requires writing a letter to request details about a training program. Part III is writing a letter of complaint about poor maintenance services. The exam is 1 hour and 30 minutes long and covers a range of relevant workplace communication tasks.
SCSJ3203 - Theory Science Computer - Midterm PaperAbdul Khaliq
This document contains a midterm test for a Theory of Computer Science course at University Teknologi Malaysia. The test contains two parts - Part A consists of 10 true/false questions worth 10 marks total. Part B contains 10 subjective questions worth 90 marks total. The questions cover topics such as regular expressions, context-free grammars, language definitions, and string derivations from grammars. Students are instructed to answer all questions in the spaces provided in the test booklet.
SCSJ3553 - Artificial Intelligence Final Exam paper - UTMAbdul Khaliq
This document contains a 14-page AI exam with multiple choice, short answer, and structured questions. It tests knowledge of search techniques, knowledge representation, production systems, and other AI concepts. The exam is divided into sections on true/false questions, short explanations, and longer structured questions involving search algorithms, knowledge representation diagrams, and production systems examples.
The document provides instructions and situations for three tasks:
1) Write a letter to a local cafe asking for details about booking a table, menu options, payment methods, and discounts to organize a farewell party.
2) Write a memo to a colleague informing them that you will miss an upcoming department meeting because you will be attending a one-day training program.
3) Write a letter of complaint to a computer company regarding the wrong laptop model and incorrect price being sent, requesting the order be corrected.
ULAB3122 / UHS3022 - Final Exam Paper (2009)Abdul Khaliq
This document contains information about a final examination for the course "English for Workplace Communication" at Universiti Teknologi Malaysia (UTM). It provides details such as the course code, name, program, duration, date, and structure of the exam. It consists of two parts worth 10 and 20 marks respectively. It also contains reminders to students to follow exam rules and regulations, with warnings that academic misconduct could result in penalties such as suspension or expulsion.
The report summarizes the findings of a committee set up by the Ministry of Health to investigate organ pledging and donation trends in Malaysia over the years. It finds that while the number of organ pledges increases following public awareness campaigns, actual donations remain low due to families refusing to donate organs of deceased loved ones. The Chinese community contributes the most organ pledges compared to other ethnic groups. However, getting family consent remains the biggest challenge to increasing organ donations. The report makes recommendations to address issues of lack of awareness, cultural and religious sensitivities that discourage organ donation among Malaysians.
Kertas soalan orienteering memberikan soalan-soalan berkaitan konsep asas orienteering termasuk definisi, faedah, kaedah penggunaan peta dan kompas, teknik mencari lokasi, dan peralatan asas.
Dokumen ini menjelaskan konsep-konsep geografi militer dasar seperti titik utara, sistem mils, sudut grid magnet, dan arahan. Ia juga membahas perbedaan antara utara magnetik dan benar serta konversi antara derajat dan mils.
Dokumen ini menjelaskan tentang konsep kontor dan bentuk tanah serta kecerunannya. Kontor adalah garis bayangan pada permukaan tanah yang menunjukkan ketinggian relatif tanah. Selang tinggi antara kontor menunjukkan perbezaan ketinggian tanah. Bentuk kontor yang rapat menandakan cerun yang curam sedangkan kontor yang renggang menandakan cerun yang landai. Bukit, permatang dan anak bukit merupakan contoh bentuk tanah yang dijelaskan.
UICI 2022 - Bab 04 teknologi dalam islam (nota)Abdul Khaliq
Bab ini membahas tentang teknologi dalam Islam. Teknologi didefinisikan sebagai ilmu praktis atau teknik yang memungkinkan manusia membuat perubahan pada alam sekitar. Teknologi tidak dilarang dalam Islam asalkan tidak bertentangan dengan ajaran agama. Prinsip-prinsip utama teknologi Islam meliputi tauhid (ketauhidan kepada Allah), khilafah (manusia sebagai khalifah Allah di bumi), dan kemaslahat
UICI 2022 - Bab 02 perpindahan ilmu (nota)Abdul Khaliq
Dokumen tersebut membincangkan proses penyebaran ilmu dari dunia Islam ke dunia Barat melalui usaha penterjemahan besar-besaran dan pembukaan universiti di Barat. Faktor utama penyebaran ilmu tersebut adalah peranan pusat-pusat pembelajaran Islam di Sepanyol seperti Toledo serta aktiviti pelajar Eropah yang mempelajari ilmu di sana pada abad ke-12 dan 13 Masihi.
A intro uici 2022 sains teknologi dan manusiaAbdul Khaliq
Dokumen tersebut membahas kursus berjudul "Sains, Teknologi, & Manusia" yang mencakupi topik falsafah ilmu, sains Islam, teknologi, dan manusia dari perspektif Islam. Kursus ini bertujuan membantu pelajar memahami konsep-konsep tersebut dan mendiskusikan perbandingan pandangan Islam dan Barat mengenai topik-topik tersebut. Penilaian kursus ini terdiri dari tugas-tugas kelompok dan ujian
This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
How Barcodes Can Be Leveraged Within Odoo 17Celine George
In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
🔥🔥🔥🔥🔥🔥🔥🔥🔥
إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
3. Centre Limit Theorem
• Given an equation of f(x) with an interval of
[a,b], you need to determine whether there
exist at least a real root in that interval
• CLT said that if f(a) and f(b) have opposite sign
(one is –ve and another is +ve) then there
exist at least a real root in that interval
a b
f(a) +ve f(b) -ve
Prepared by Dr. Suhaila Mohamad Yusuf
4. Bisection Method
f(x) = x3 – 3x2 + 8x - 5 c = (a + b) / 2 [0,1] ℇ=0.005
i a b f(a) f(b) c f(c)
0 0 1 -5 1 0.5 -1.625
1These f(c) > ℇ then,
0.5are from 1the -1.625
given Calculated from this stop!
1 Is this < ℇ? Yes,
0.75 -0.266
2 new interval!
0.75 interval
1 -0.266 No, next iteration!
1 equation
0.875 0.373
3 0.75 0.875 -0.266 0.373 0.8125 0.056
4 0.75 0.813How to choose 0.056new 0.7815
-0.266 the -0.103
5 0.782 0.813 interval?
-0.103 0.056 0.7975 -0.021
6 0.798 0.813 -0.021 0.056 0.8055 0.020
7 0.798 0.806 -0.021 0.02 0.802 0.002
Make sure these parts
0 Repeat all the steps
0.5 1 This is the
CAREFULLY until f(c) < ℇ
have opposite sign!
-ve f(x) -ve f(x) +ve f(x) root!!
We need to take this c value. CLT said that f(a) and f(b)
How about another one? should have opposite sign!
Prepared by Dr. Suhaila Mohamad Yusuf
5. False Position Method
f(x) = x3 – 3x2 + 8x - 5 c = [af(b) - bf(a)] / [f(b) – f(a)] [0,1] ℇ=0.005
i a b f(a) f(b) c f(c)
0 0 1 -5 1 0.833 -1.625
1 0 0.8333 -5
These are from the given 0.162 Is this < ℇ? Yes,
0.807
Calculated from this0.029 stop!
2 0 0.807
interval -5 0.029 No, next iteration!
0.802
equation 0.004
This is the
Remember how to choose the root!!
interval value? What does the CLT
said about interval?
Prepared by Dr. Suhaila Mohamad Yusuf
6. Secant Method
f(x) = sin (x) + 3x – e3 xi+2 = [xif(xi+1) – xi+1f(xi)] / [f(xi+1) – f(xi)]
x0 = 1 , x1 = 0 ℇ=0.0005
i xi xi+1 xi+2 f(xi+2)
0 1 0 0.4710 0.2652
1 0 0.4710
ℇ? Yes, stop!
These are from the given Is this <0.0295
0.3723
Calculated from this
-0.0012
2 0.4710 0.3723
No, next iteration!
0.3599
values of x0 and x1 equation
3 0.3723 interval. Take the latest 2
New 0.3599 0.3604 0.0000
values as next interval
Continue iteration
until f(xi+2) < ℇ This is the
root!!
Prepared by Dr. Suhaila Mohamad Yusuf
7. Newton’s Method
f(x) = x3 – sin x Xn+1 = xn – [f(xn) / f’(xn)] x0 = 1 ℇ=0.0005
n xn f(xn) f’(xn)
0 1 0.15853 2.45970
1 0.93555 0.01392 2.03239
This is from the given Calculated from the
2 values of x
0.92870 0.00015 1.98858
| < ℇ? Yes, stop!
0 derivative of f(x)
Is |xn+1 – xn0.92862Calculated from this
3 -0.00001 1.98807
No, next iteration!
4 0.92862 equation
Continue iteration
This < ℇ
until |xn+1 – xn|is the
root!!
Prepared by Dr. Suhaila Mohamad Yusuf
11. Power Method
• Aims to find dominant eigenvalue
(largest value of eigenvalue)
1 2 1
v(0) = (0,0,1)T
A 1 0 1
ε = 0.001
4 4 5
Prepared by Dr. Suhaila Mohamad Yusuf
12. Power Method
1 2 1 v(0) = (0,0,1)T A * v Abs max
A 1 0 1 between
ε = 0.001
4 4 5 these values
k (v(k))T (Av(k))T mk+1
0 0 0 1 -1 1 5 5
1 -0.2 0.2 1 -0.8 0.8 3.4 3.4
These values
ǁv(k+1)-v(k)ǁ < ε ?
2 -0.235 0.235 1 0.765 -0.7653.12
Note that ǁv(k+1)-0.755 is a 3.04
v(k)ǁ
3.12
3 -0.245 0.245 1 -0.755 3.04
No, next iteration
4 -0.248 0.248 1 -0.752
divided by
difference of two vectors.3.016
0.752 3.016
that value
5 -0.249 0.2492 1 2-0.751 0.751 2 3.008 3.008
v u (v1 u1 ) (v2 u2 ) ... (vn un )
6 -0.250 0.250 1 -0.750 to have 3.000
0.750 3.000
v (1) 7 v ( 0) -0.250( 0.0.250 ) 2 (1 .2 0) 2 (1 1) 2 values
2 0 0 these
Eigenvector Eigenvalue
Prepared by Dr. Suhaila Mohamad Yusuf
13. Shifted Power Method
• Aims to find smallest eigenvalue and
intermediate eigenvalue
1 2 1
v(0) = (0,1,0)T
A 1 0 1 ε = 0.001
4 4 5 λ1 = 3.0
Prepared by Dr. Suhaila Mohamad Yusuf
14. Shifted Power Method
B A I
A 3 .0 I
1 2 1 1 0 0
1 0 1 3.0 0 1 0
4 4 5 0 0 1
1 2 1 3. 0 0 0
1 0 1 0 3. 0 0
4 4 5 0 0 3 .0
2 2 1
1 3 1
4 4 2
Prepared by Dr. Suhaila Mohamad Yusuf
15. Shifted Power Method
2 2 1 v(0) = (0,1,0)T B * v Abs max
B 1 3 1 between
ε = 0.001
4 4 2 these values
k (v(k))T (Bv(k))T mk+1
0 0 1 0 2 -3 -4 -4
1 -0.5 0.75 1 1.5 -1.75 -3 -3
2 -0.5 0.583 1.166 1
These values -1.249 -2.332 -2.332
ǁv(k+1)-v(k)ǁ < ε ? 0.517
3 -0.5 Note that ǁv(k+1)-v(k)ǁ is a
1 1.072 -1.108 -2.144 -2.144
No, 4next iteration
-0.5 0.508 differencedivided by
1 of two vectors.
1.034 -1.051 -2.068 -2.068
5 -0.5 0.504 1 that value
1.016 -1.024 -2.032 -2.032
v u6 (v1 u1 ) 2 0.502 u2 )1 ... tovhave n-1.012
-0.5 (v2 2
1.008 u )
( n 2
-2.016 -2.016
7 -0.5 0.501 1 these values
1.004 -1.006 -2.008 -2.008
8 -0.5 0.5 1 1.002 -1.003 -2.004 -2.004
9 -0.5 0.5 1 1.000 -1.000 -2.000 -2.000
10 -0.5 10 1 Eigenvector
Prepared by Dr. Suhaila Mohamad Yusuf Shifted Eigenvalue
16. Shifted Power Method
• λshifted = -2.0
• λ3 = λshifted + λ1 = -2.0 + 3.0 = 1.0
1 2 1
• Intermediate λ2 A 1 0 1
λ1 + λ2 + λ3 = a11 + a22 + a33
3.0 + λ2 + 1.0 = 1 + 0 + 5 4 4 5
λ2 = 2.0 Caution!!! Use the
original matrix, A.
Not the shifted
Prepared by Dr. Suhaila Mohamad Yusuf
matrix, B.
18. Interpolation Approximation
Least Square
Newton Forward
Difference
Newton Backward
Difference
Newton Divided
Difference May need table
re-arrangement
Langrage
Prepared by Dr. Suhaila Mohamad Yusuf
19. Newton Forward Difference
k 0 1 2 3 4 5
xk 1.0 1.2 1.4 1.6 1.8 2.0
yk 0.5000 0.4545 0.4167 0.3846 0.3571 0.3333
Find y(1.1)
k xk yk ∆yk ∆2yk ∆3yk ∆4yk ∆5yk x=1.0 is
x=1.1 0 1.0 0.5000 -0.0455 0.0077 -0.0020 0.0009 -0.0007 chosen
located as ref.
1 1.2 0.4545 -0.0378 0.0057 value
This -0.0011 0.0002
here point
2 1.4 0.4167 -0.0321 0.0046 -0.0009 Repeat until because
3 1.6 0.3846 -0.0275 0.0037 this
minus last column of higher
value degree
4 1.8 0.3571 -0.0238
5 2.0 0.3333 To get this
value
Prepared by Dr. Suhaila Mohamad Yusuf
20. Newton Forward Difference
• h = 1.2 – 1.0 = 0.2 and
r = (x – x0) / h = (1.1 – 1.0) / 0.2 = 0.5
r( r 1) 2 r( r 1)( r 2 ) 3
p5 ( x ) y 0 r y0 y0 y0
2! 3!
r( r 1)( r 2 )( r 3) 4 r( r 1)( r 2 )( r 3)( r 4 ) 5
y0 y0
4! 5!
( 0.5 )( 0.5 1)
p5 (1.1) 0.5000 ( 0.5 )( 0.0455) ( 0.0077)
2
( 0.5 )( 0.5 1)( 0.5 2 ) ( 0.5 )( 0.5 1)( 0.5 2 )( 0.5 3)
( 0.0020) ( 0.0009)
6 24
( 0.5 )( 0.5 1)( 0.5 2 )( 0.5 3)( 0.5 4 )
( 0.0007)
120
0.5000 0.02275 0.0009625 0.000125 0.0000352 0.0000191
0.4761 Prepared by Dr. Suhaila Mohamad Yusuf
21. Newton Backward Difference
k 0 1 2 3 4 5
xk 1.0 1.2 1.4 1.6 1.8 2.0
yk 0.5000 0.4545 0.4167 0.3846 0.3571 0.3333
Find y(1.9)
k xk yk ∇yk ∇2yk ∇3yk ∇4yk ∇5yk x=2.0 is
0 1.0 0.5000 chosen
This value as ref.
1 1.2 0.4545 -0.0455 point
2 1.4 0.4167 -0.0378 0.0077 this
minus because
3 1.6 0.3846 value
-0.0321 0.0057 -0.0020
of higher
To get this degree
x=1.9 4 1.8 0.3571 -0.0275 0.0046 -0.0011 0.0009
located
5 2.0 0.3333 value
-0.0238 0.0037 -0.0009 0.0002 -0.0007
here
Repeat until
last column
Prepared by Dr. Suhaila Mohamad Yusuf
22. Newton Backward Difference
• h = 1.2 – 1.0 = 0.2 and
r = (x – x0) / h = (1.9 – 2.0) / 0.2 = -0.5
r (r 1) 2 r (r 1)(r 2) 3
p5 ( x) y5 r y5 y5 y5
2! 3!
r (r 1)(r 2)(r 3) 4 r (r 1)(r 2)(r 3)(r 4) 5
y5 y5
4! 5!
( 0.5)( 0.5 1)
p5 (1.9) 0.3333 ( 0.5)( 0.0238) (0.0037)
2
( 0.5)( 0.5 1)( 0.5 2) ( 0.5)( 0.5 1)( 0.5 2)( 0.5 3)
( 0.0009) (0.0002)
6 24
( 0.5)( 0.5 1)( 0.5 2)( 0.5 3)( 0.5 4)
( 0.0007)
120
0.3333 0.0119 0.0004625 0.00005625 0.000007869 0.0000191
0.3448 Prepared by Dr. Suhaila Mohamad Yusuf
23. Newton Divided Difference 1st step,
k 0 1 2 3 4 mark the
See this col like
xk 1.0 1.6 2.5 3.0 3.2
number? this start
yk 0.5000 0.3846 0.2857 0.2500 0.2381 from
Find y(1.3) f[xk].
1 2 3 4 5
k xk f[xk] f1[xk] f2[xk] f3[xk] f4[xk] Go to col xk and
5000 0
0..3846 (0.0.1923)
01099 1 1.0 0.5000 -0.1923 0.0549 -0.0137 0.0032 count down the
1.6 1.0
25 col according to
1 2 1.6
1 0.3846 -0.1099 0.0275 -0.0066
0.2857 0.3846 number on top of
2.5 1.6 2 2 2.5
3 0.2857 -0.0714 0.0170 the col. Then the
3 3.0 0.2500 last value of x
-0.0595
minus with the
4 3.2 0.2381 first value of x.
To fill in this col, we
Big problem is ‘divide
know that lower
value – upper value with what?’
Prepared by Dr. Suhaila Mohamad Yusuf
24. Newton Divided Difference
• Interpolation Polynomial expression
p4 ( x ) y 0 f [ x 0 , x1 ]( x x 0 ) f [ x 0 , x1 , x 2 ]( x x 0 )( x x1 )
f [ x 0 , x1 , x 2 , x 3 ]( x x 0 )( x x1 )( x x 2 )
f [ x 0 , x1 , x 2 , x 3 , x 4 ]( x x 0 )( x x1 )( x x 2 )( x x 3 )
• Assign the value into the polynomial
expression
p 4 (1.3 ) 0.5 ( 0.1923)(1.3 1.0 ) 0.0549(1.3 1.0 )(1.3 1.6 )
( 0.0137)(1.3 1.0 )(1.3 1.6 )(1.3 2.5 )
0.0032(1.3 1.0 )(1.3 1.6 )(1.3 2.5 )(1.3 3.0 )
0.5 0.05769 0.004941 0.0014796 0.00058752
0.4353
Prepared by Dr. Suhaila Mohamad Yusuf
25. Newton Divided Difference
k 0 1 2 3 4
xk 1.0 1.6 2.5 3.0 3.2
yk 0.5000 0.3846 0.2857 0.2500 0.2381
Find y(2.8)
• Re-arrange the table then do as previous
• How to re-arrange table? 2.8 is in here
k 0 1
Value of x before 2Decending value of 4
3
xk and after 2.8 1.6
1.0 2.5 3.0
remaining x 3.2
yk 0.5000 0.3846 0.2857 0.2500 0.2381
k 0 1 2 3 4
xk
yk Prepared by Dr. Suhaila Mohamad Yusuf
26. Langrage
k 0 1 2 3 4
xk 1.0 1.6 2.5 3.0 3.2
yk 0.5000 0.3846 0.2857 0.2500 0.2381
Find y(1.3)
n
pn ( x ) L 0 ( x )y 0 L1( x )y 1 .......L n ( x )y n L i ( x )y i
i 0
4
p4 ( x ) L i ( x )y i
i 0
p4 ( x ) L 0 ( x )y 0 L 1( x )y 1 L 2 ( x )y 2 L 3 ( x )y 3 L 4 ( x )y 4
Prepared by Dr. Suhaila Mohamad Yusuf
27. Langrage
Because it is L0, x0 is nowhere to
• Calculate L0 be found up here
(1.3 x1 )(1.3 x 2 )(1.3 x 3 )(1.3 x 4 )
L 0 ( 1.3 )
( x 0 x1 )( x 0 x 2 )( x 0 x 3 )( x 0 x 4 )
(1.3 1.6 )(1.3 2.5 )(1.3 3.0 )(1.3 3.2 )
0.2936
(1.0 1.6 )(1.0 2.5 )(1.0 3.0 )(1.0 3.2 )
Because it is L0, x0 is deducted
• With the same with other x
method, calculate L1,
L2 ,L3 ,L4
Prepared by Dr. Suhaila Mohamad Yusuf
28. Langrage
4
p 4 (1.3 ) L i (1.3 )y i
i 0
0.2936( 0.5000) 0.9613 0.3846) 0.6152( 0.2857)
(
0.7329( 0.2500) 0.3726( 0.2381)
0.4353
Prepared by Dr. Suhaila Mohamad Yusuf
29. Least Square
• Determine the appropriate linear polynomial
expression, p(x) = a0 + a1x based on the following data:
• Determine f(2.3)
x 1 2 3 4 5
f(x) 0.50 1.40 2.00 2.50 3.10
s 0 s1 a 0 v0
s1 s 2 a 1 v1
Prepared by Dr. Suhaila Mohamad Yusuf
30. Least Square
xk 0 xk 1 xk 2 fk xk 0 f k xk 1 f k
1 1 1 0.5 0.5 0.5
1 2 4 1.4 1.4 2.8
1 3 9 2.0 2.0 6.0
1 4 16 2.5 2.5 10.0
1 5 25 3.1 3.1 15.5
5 15 55 - 9.5 34.8
s 0 s1 a 0 v0 5 15 a 0 9.5
s1 s 2 a 1 v1 15 55 a 1 34.8
Prepared by Dr. Suhaila Mohamad Yusuf
31. Least Square
5 15 a 0 9.5
15 55 a 1 34.8
• Solution, a0 = 0.01 dan a1 = 0.63
• Therefore, the polynomial expression is p(x) =
0.01x + 0.63
• To determine f(2.3):
p( 2.3) 0.01( 2.3) 0.63 0.653
f ( 2.3) p( 2.3) 0.653
Prepared by Dr. Suhaila Mohamad Yusuf
32. CHAPTER 9
NUMERICAL
DIFFERENTIATION
Prepared by Dr. Suhaila Mohamad Yusuf
33. Forward Backward Forward Backward Central Forward Central
Difference Difference Difference Difference Difference Difference Difference
2-point 3-point 5-point
Formula Formula Formula
FIRST
DERIVATIVE
f‘(x)
SECOND
DERIVATIVE
f‘’(x)
NUMERICAL
INTEGRATION 3-point
Formula 5-point
Formula
Central
Difference Central
Difference
Prepared by Dr. Suhaila Mohamad Yusuf
34. SORRY!!! I DIDN’T PREPARE
ANYTHING. THIS CHAPTER IS TOO
EASY. MAKE SURE YOU KNOW
WHICH FORMULA TO USE..
Prepared by Dr. Suhaila Mohamad Yusuf
35. CHAPTER 10
NUMERICAL
INTEGRATION
Prepared by Dr. Suhaila Mohamad Yusuf
37. Trapezoidal Rule
• Approximate the following integral using the
Trapezoidal rule with h=0.5
4 x
dx
1
x 4
b a 4 1
N 6
h 05
.
Prepared by Dr. Suhaila Mohamad Yusuf
38. Trapezoidal Rule
4 x 4
xi dx f ( x) dx
i xi f ( xi ) 1
x 4 1
xi 4 h
f f6 2 f1 f2 f3 f4 f5
0 1.0 0.4472 2 0
1 1.5 0.6396 0.5
1.8614 2 4.8486
2 2.0 0.8165 2
3 2.5 0.9806 2.8896
4 3.0 1.1339
5 3.5 1.2780
6 4.0 1.4142
Total 1.8614 4.8486
prepared by Razana Alwee
1st and last In-between
values values
Prepared by Dr. Suhaila Mohamad Yusuf
39. Simpson’s 1/3 Rule
• Approximate the following integral using the
Trapezoidal rule with h=0.5
4 x
dx
1
x 4
b a 4 1
N 6
h 05
.
Prepared by Dr. Suhaila Mohamad Yusuf
40. Simpson’s 1/3 Rule 3 2
4 x h
dx f0 f6 4 f 2i 1 2 f 2i
1
x 4 3 i 1 i 1
0.5
xi 1.8614 4(2.8982) 2(1.9504)
i xi fi f ( xi ) 3
xi 4
0 1.0 0.4472 2.8925
1 1.5 0.6396
2 2.0 0.8165
3 2.5 0.9806
4 3.0 1.1339
5 3.5 1.2780
6 4.0 1.4142
Total 1.8614 2.8982 1.9504
1st and last Odd Even
Prepared by Dr. Suhaila Mohamad Yusuf
values column column
41. Simpson’s 3/8 Rule
• Approximate the following integral using the
Trapezoidal rule with h=0.25
4 x
dx
1
x 4
b a 4 1
N 12
h 0.25
Prepared by Dr. Suhaila Mohamad Yusuf
42. Simpson’s 3/8 Rule
4 3
4 x 3h
dx f0 f12 3 f 3i 2 f 3i 1 2 f 3i
1
x 4 8 i 1 i 1
xi
i xi fi f ( xi )
xi 4 3(0.25)
0 1.00 0.4472
1.8614 3(7.7191) 2(2.9174)
8
1 1.25 0.5455
2 1.50 0.6396 2.8925
3 1.75 0.7298
4 2.00 0.8165
5 2.25 0.9000
6 2.50 0.9806
7 2.75 1.0586
8 3.00 1.1339
9 3.25 1.2070
10 3.50 1.2780
11 3.75 1.3470
12 4.00 1.4142
Total 1.8614 7.7191 2.9174
‘Ganda
1st and Remaining
3’ (3,6,9)
last values values Prepared by Dr. Suhaila Mohamad Yusuf
43. Romberg Integration
Formula for Romberg Table
i hi Ri,1 Ri,2 Ri,3
1 b a h1
R1,1 f0 f1
2
2 1
h1
2
3 1
h2
2
1 2 i 2
4 j 1 Ri , j 1 Ri 1, j 1
Ri ,1 Ri hi f 2k Ri , j
2
1,1 1
k 1
1 4j 1
1
Prepared by Dr. Suhaila Mohamad Yusuf
44. Romberg Integration
• Use Romberg integration to approximate
4 x
dx
1
x 4
• Compute the Romberg table until
|Ri,j Ri,j-1|<0.0005
Prepared by Dr. Suhaila Mohamad Yusuf
45. Romberg Integration
How to calculate
4 x these?
dx b a 4 1
1 N 1
x 4 From the h here, we h 3
can know the N
h1 b a 4 1 3 (number of segments)
Integration starts from Therefore, we only have
1 to 4 and only has 1 f0 and f1, the first node
h1 and the last node
R1,1 f 0 f1 segment
2 1 4
h=3
3
0.4472 1.4142 f1
2 f0
4 1 4 4
2.7921. 1
1 4
0.4772 1
4 4
1.4142
Find the f0 = f(1) and f1
= f(4) using this f(x)by Dr. Suhaila Mohamad Yusuf
Prepared
46. Romberg Integration
i hi Ri,1 Ri,2 Ri,3
1 3 2.7921
2 1.5 h2 is half of h1
Now we need
3 to fill in the 2nd
row
This value needs to be
Fill in the Rombergcalculated using other
Table with answers that
you’ve got previously
formula
Prepared by Dr. Suhaila Mohamad Yusuf
47. Romberg Integration
How to calculate
4 x this?
dx Remember!! b a 4 1
1 N 2
x Every time you calculatewe R ,
4 From the h here, h 1.5
can know the N new xx
1 your f , f , ...(number ofbe changed!!
fx will segments)
h2 h1 1.5 0 1
2
Draw diagram to be safe!! , we have three
Integration starts from Therefore
1 1
1 to 4 and has 2 nodes f0 ,f1, and f2,
R2,1 R1,1 h1 f 2k
2 k 1
1
segments
1
R1,1 h1 ( f1 ) 1 h = 1.5 2.5 4
2
1 f0 f1 f2
R1,1 h1 ( f (2.5))
2 4 2 .5
1 0.9806
1
2.7921 3 0.9806 2 .5 4
2
2.8670 Find the f1 = f(2.5) using
Prepared by Dr. Suhaila Mohamad Yusuf this f(x)
48. Romberg Integration
• Calculate R2,2 using another formula
4 R2,1 R1,1 4 2.8670 2.7921
R2 , 2 2.8920
3 3
• Is |Ri,j Ri,j-1|<0.0005? Only compare with R in
the same row, not the same column
R2, 2 R2,1 2.8920 2.8670 0.025 0.0005
Prepared by Dr. Suhaila Mohamad Yusuf
49. Romberg Integration
i hi Ri,1 Ri,2 Ri,3
1 3 2.7921
2 1.5 2.8670 2.8920
3 Fill in0.75Romberg half ofwith answers that
the h3 is Table h2
you’ve got previously
Now we need
to fill in the 3rd
This value row to be
needs
calculated using other
formula
Prepared by Dr. Suhaila Mohamad Yusuf
50. Romberg Integration
How to calculate
4 x this?
dx b a 4 1
1 N 4
x 4 From the h here, we h 0.75
can know the N
1 (number of segments)
h3 h2 0.75
2 Integration starts from Therefore, we have five
1 2 1 to 4 and has 4 nodes f0 ,f1, f2 ,f3 and f4,
R3,1 R2,1 h2 f 2 k 1 segments
2 k 1
1 1 1.75 2.5 3.25 4
R2,1 h2 ( f1 f 3 )
2 h = 0.75
1 f0 f1 f2 f3 f4
R2,1 h2 ( f (1.75) f (3.25)) 4 1.75
2 0.7298
4 3.25
1
1.2070
1 1.75 4 1
3.25 4
2.8670 1.5 0.7298 1.2070
2 Find the f1 = f(1.75) and
2.8861 Prepared by Dr. Suhaila Mohamad Yusuf f3 = f(3.25) using this f(x)
51. Romberg Integration
• Calculate R3,2 using another formula
4 R3,1 R2,1 4 2.8861 2.8670
R3, 2 2.8925
3 3
• Is |Ri,j Ri,j-1|<0.0005? Only compare with R in
the same row, not the same column
R3, 2 R3,1 2.8925 2.8861 0.0064 0.0005
Prepared by Dr. Suhaila Mohamad Yusuf
52. Romberg Integration
• Calculate R3,3 using another formula
4 R3, 2 R2, 2 16 2.8925 2.8920
R3,3 2.8925
15 15
• Is |Ri,j Ri,j-1|<0.0005? Only compare with R in
the same row, not the same column
R3,3 R3, 2 2.8925 2.8925 0.0000 0.0005
Stop Iteration!!
Prepared by Dr. Suhaila Mohamad Yusuf
53. Romberg Integration
i hi Ri,1 Ri,2 Ri,3
1 3 2.7921
2 You 1.5
don’t need to generate
2.8670 2.8920
Trapezoidal Table!
3 0.75 2.8861 2.8925 2.8925
Just follow my way..
The solution of
Fill in the Romberg Table with answers that
integration is here
you’ve got previously
Prepared by Dr. Suhaila Mohamad Yusuf