Ch08 3
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The Runge-Kutta method approximates the solution to differential equations by taking a weighted average of the function values at the endpoints and midpoint of each interval. It has a local truncation error proportional to h^5, higher than previous methods. Adaptive Runge-Kutta methods allow the step size h to vary over the interval to better match the local error, improving efficiency over fixed step size Runge-Kutta.
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⎧
⎨
⎩
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cω
ω20−ω2
)
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cω
ω20−ω2
)
if ω0 < ω
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1 function LAB06ex1
2 omega0 = 2; c = 1; omega = 1.4;
3 param = [omega0,c,omega];
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c⃝2011 Stefania Tracogna, SoMSS, ASU 1
MATLAB sessions: Laboratory 6
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This document discusses using bootstrap methods to create confidence intervals for time series forecasts. It provides examples of time series data and introduces the AR(1) model. The document describes an algorithm for calculating a bootstrap confidence interval for forecasting from an AR(1) model. It then discusses a simulation study comparing empirical coverage rates of bootstrap confidence intervals under different parameters. Finally, it applies the bootstrap method to forecasting Gross National Product growth, comparing the results to a parametric approach.
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This document discusses using bootstrap methods to create confidence intervals for time series forecasts. It provides background on time series models and the autoregressive (AR) process. It then presents an algorithm for calculating a bootstrap confidence interval for forecasts from an AR(1) model. A simulation study compares coverage rates for bootstrap confidence intervals under different parameters. Finally, the method is applied to US Gross National Product data to forecast and construct confidence intervals.
Ch06 6
The document discusses the convolution integral and its relation to the Laplace transform. Specifically, it shows that the Laplace transform of the convolution of two functions f and g is equal to the product of the individual Laplace transforms, unlike the normal multiplication of functions. This defines a "generalized product" called the convolution. The document provides examples calculating convolutions and inverse Laplace transforms to demonstrate the convolution integral theorem.
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A successful maximum likelihood parameter estimation in skewed distributions ...
A successful maximum likelihood parameter estimation scheme using
the continuation method (homotopy method) is introduced. This
algorithm is particularly useful for the three-parameter skewed
distributions including thresholds. Such three-parameter
distributions are, for example, Weibull, log-normal, gamma and
inverse Gaussian distributions. As the proposed algorithm can almost
always obtain the local maximum likelihood estimates automatically,
it is of considerable practical value. The Monte Carlo simulation
study shows the effectiveness of the proposed method.
Comparison GUM versus GUM+1
This document discusses the use of Monte Carlo simulations to evaluate measurement uncertainty according to Supplement 1 to the Guide to the Expression of Uncertainty in Measurement (GUM+1). It provides examples of generating random numbers and using them in Monte Carlo simulations. One example evaluates the uncertainty in measuring the volume of a cylinder based on uncertainties in diameter and height measurements. The document concludes with an example of calibrating a thermometer and determining the uncertainties in the calibration curve parameters.
2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...
2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...The Statistical and Applied Mathematical Sciences Institute
This document discusses uncertainty propagation techniques for determining statistics of model outputs given uncertain model inputs. It covers analytic approaches for linear models, perturbation methods for nonlinear models, and direct sampling methods. It also discusses computing moments using stochastic spectral methods like stochastic Galerkin with polynomial chaos. The document provides an example of applying perturbation and sampling methods to a nonlinear oscillator model with uncertain parameters. It compares the results from both approaches to the true natural frequency. Finally, it discusses uncertainty quantification for a HIV model and the use of prediction intervals in nuclear power plant design.Similar to Ch08 3 (20)
Computer Controlled Systems (solutions manual). Astrom. 3rd edition 1997
Computer Controlled Systems (solutions manual). Astrom. 3rd edition 1997
Bayesian Experimental Design for Stochastic Kinetic Models
Bayesian Experimental Design for Stochastic Kinetic Models
MATLAB sessions Laboratory 6MAT 275 Laboratory 6Forced .docx
MATLAB sessions Laboratory 6MAT 275 Laboratory 6Forced .docx
Solucionário Introdução à mecânica dos Fluidos - Chapter 01
Solucionário Introdução à mecânica dos Fluidos - Chapter 01
Automatic Control Via Thermostats Of A Hyperbolic Stefan Problem With Memory
Automatic Control Via Thermostats Of A Hyperbolic Stefan Problem With Memory
Get Accurate and Reliable Statistics Assignment Help - Boost Your Grades!
Get Accurate and Reliable Statistics Assignment Help - Boost Your Grades!
A successful maximum likelihood parameter estimation in skewed distributions ...
A successful maximum likelihood parameter estimation in skewed distributions ...
2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...
2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...
More from Rendy Robert
Ch07 9
1) Nonhomogeneous linear systems generalize the theory of single nth order linear equations. They can be written in the form x' = P(t)x + g(t).
2) The general solution is the sum of the general solution to the homogeneous system x' = P(t)x and a particular solution to the nonhomogeneous system.
3) If the matrix P is constant and diagonalizable, the system can be transformed to uncoupled equations y' = Dy + h(t) via diagonalization, whose solutions give the solution to the original system.
Ch07 8
The document discusses repeated eigenvalues in systems of linear differential equations. If some eigenvalues are repeated, there may not be n linearly independent solutions of the form x = ξert. Additional solutions must be sought that are products of polynomials and exponential functions. For a double eigenvalue r, the first solution is x(1) = ξert, where ξ satisfies (A-rI)ξ = 0. The second solution has the form x(2) = ξtert + ηert, where η satisfies (A-rI)η = ξ and η is called a generalized eigenvector.
Ch07 7
The document discusses fundamental matrices and their properties. A fundamental matrix Ψ(t) is a matrix whose columns are fundamental solutions to the system x' = P(t)x. Ψ(t) satisfies the differential equation Ψ' = P(t)Ψ and is nonsingular. The general solution to the system can be written as x = Ψ(t)c, where c is a constant vector. For an initial value problem, the solution is x = Ψ(t)Ψ-1(t0)x0. The fundamental matrix Φ(t) corresponding to a set of fundamental solutions satisfying initial conditions is also discussed. Matrix exponential functions are introduced as the fundamental matrix
Ch07 6
The document discusses complex eigenvalues and eigenvectors for systems of linear differential equations. It shows that if the matrix A has complex conjugate eigenvalue pairs r1 and r2, then the corresponding eigenvectors and solutions will also be complex conjugates. This leads to real-valued fundamental solutions that can express the general solution. An example demonstrates these concepts, finding the complex eigenvalues and eigenvectors and expressing the general solution in terms of real-valued functions. Spiral points, centers, eigenvalues, and trajectory behaviors are also summarized.
Ch07 5
This document discusses homogeneous linear systems with constant coefficients. It begins by defining such a system as x' = Ax, where A is an n×n matrix of real constants. It then explains that the equilibrium solutions are found by solving Ax = 0, and stability is determined by the eigenvalues of A. Examples are provided to illustrate finding the direction field, eigenvalues/eigenvectors, general solution, and phase plane plots for specific 2D systems. Time plots of the solutions are also shown.
Ch07 4
This document discusses systems of first order linear differential equations. It introduces the general form of such a system as x' = P(t)x + g(t), where x is a vector and P and g are matrix-valued functions. Solutions to the system are vector-valued functions that satisfy the equations on some interval. The document presents theorems about determining whether a set of vector functions are fundamental solutions that span the general solution, using the Wronskian determinant. It also discusses representing solutions as linear combinations of fundamental solutions.
Ch07 3
The document summarizes key concepts related to systems of linear equations and linear algebra, including:
1) A system of n linear equations can be expressed in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector. If b = 0, the system is homogeneous, otherwise it is nonhomogeneous.
2) If the coefficient matrix A is nonsingular, the system Ax = b has a unique solution that can be found by computing x = A^-1b. If A is singular, the system may have no solution or infinitely many solutions.
3) A set of vectors is linearly dependent if there exist scalar multiples of
Ch07 2
This document provides an overview of key concepts in matrix theory, including definitions and examples of matrices, vectors, operations on matrices like transpose, conjugate, and multiplication, and properties like orthogonality and the identity matrix. Matrices are defined as rectangular arrays of elements arranged in rows and columns. Key matrix operations like addition, subtraction, and multiplication are introduced along with examples. Important matrix properties like the identity and zero matrices are also defined.
Ch07 1
This document introduces systems of first order linear differential equations. It defines such systems as having the general form where each derivative is a linear combination of the variables. It describes how higher order equations can be converted into equivalent first order systems by defining new variables for each derivative. The document states that for an initial value problem with a linear system, there exists a unique solution defined in some interval if the coefficients are continuous.
Ch06 5
This document discusses impulse functions and the Dirac delta function. It provides examples of how impulse functions can model phenomena that act over very short time intervals. The key points are:
1) Impulse functions can model forces or voltages that act over a short time interval. The total impulse is measured by integrating the function over all time.
2) The unit impulse function, or Dirac delta function δ, is defined such that its integral over any finite interval is 1.
3) The Laplace transform of the Dirac delta function is 1. This property allows impulse functions to be used to model initial conditions in differential equations.
Ch06 4
The document discusses solving differential equations with discontinuous forcing functions through examples. It summarizes the solution process for two initial value problems (IVPs). The solutions are composed of three separate IVPs corresponding to different time intervals defined by the forcing function. Jump discontinuities in the forcing function result in corresponding jump discontinuities in the highest derivative of the solution.
Ch06 3
This document discusses step functions and their Laplace transforms. It defines the unit step function uc(t) and negative step function -uty(c). Translating a function f(t) by c units results in the function g(t)=uc(t)f(t-c). The Laplace transform of uc(t) is 1/s. Translating f(t) corresponds to multiplying F(s) by e-cs in the Laplace domain. Several examples demonstrate finding the Laplace transform and inverse Laplace transform of functions involving step functions.
Ch06 2
The document discusses using the Laplace transform method to solve initial value problems for linear differential equations with constant coefficients. It shows that taking the Laplace transform of the differential equation transforms it into an algebraic equation, avoiding the need to separately solve homogeneous and nonhomogeneous parts. It also explains that determining the inverse Laplace transform to obtain the original function y(t) is the main difficulty, as it requires partial fraction decomposition and knowledge of Laplace transform pairs. Examples are provided to demonstrate solving initial value problems using this method.
Ch06 1
The document defines the Laplace transform and provides examples of its use.
The Laplace transform converts a function f(t) into another function F(s) through an integral transform. This allows problems involving discontinuous functions to be solved more easily. The transform is particularly useful for linear differential equations with constant coefficients. Examples show how to take the Laplace transform of basic functions like 1, e^at, and sin(at) and how the transform is linear.
Ch05 8
This document discusses Bessel's equation and its solutions. It begins by defining Bessel's equation of order ν. It then examines specific cases for ν = 0, 1/2, and 1. For each case it derives the recurrence relation and finds the first solution, which is a Bessel function of the first kind for that order. It also discusses the Bessel function of the second kind for order zero. The document provides graphs and discusses approximations of Bessel functions for large values of x.
Ch05 7
This document discusses series solutions near regular singular points of differential equations. It begins by deriving the recurrence relation for the series coefficients from substituting a power series solution into the differential equation. It then shows how to obtain the indicial equation and exponents at the singular point. Two series solutions are given corresponding to the two exponents. An example problem finds the singular points, exponents, and series solutions for a given third order differential equation.
Ch05 6
This document discusses solving second-order linear differential equations near a regular singular point, which is done by assuming power series solutions and obtaining recursion relations between the coefficients. Specifically, it provides an example of solving the differential equation ( ) 012 2=++′−′′ yxyxyx near the regular singular point x=0. Two linearly independent solutions are obtained in the forms of ( )( )∑∞=1!12753)1( nnnxaxy and ( )( )∑∞=12/1!12531)1( nnnxaxy , yielding the general solution ( )0),()()( 2211 >+= xxycxycxy .
Ch05 5
The document discusses solutions to Euler equations, which are differential equations of the form 0][ 2=+′+′′= yyxyxyL βα. It provides the general solutions based on whether the roots r1 and r2 of the characteristic equation are real/distinct, equal, or complex. For initial value problems, the constants in the general solution are determined using the given initial conditions. Near the singular point x=0, the qualitative behavior of the solutions depends on the nature of the roots r1 and r2.
Ch05 4
The document discusses regular singular points of differential equations. A point x0 is a regular singular point if the functions P, Q, and R are analytic at x0 or their limits exist and are finite as x approaches x0. Several examples are provided to illustrate regular singular points, including the Bessel and Legendre equations. An irregular singular point is defined as one where these limits do not exist. The behavior of solutions near regular singular points can be analyzed using power series methods.
Ch05 3
The document discusses series solutions to ordinary differential equations near ordinary points. It defines ordinary and singular points, and states that if the point is ordinary, then the solutions can be expressed as power series expansions with a radius of convergence at least as large as that of the coefficients' power series. Several examples are worked through to demonstrate finding the radius of convergence of the series solutions.
Ch08 3
- 1. Ch 8.3: The Runge-Kutta Method Consider the initial value problem y' = f (t, y), y(t0) = y0, with solution φ(t). We have seen that the local truncation errors for the Euler, backward Euler, and improved Euler methods are proportional to h2 , h2 , and h3 , respectively. In this section, we examine the Runge-Kutta method, whose local truncation error is proportional to h5 . As with the improved Euler approach, this method better approximates the integral introduced in Ch 8.1, where we had ( ) ( )( ) hfyy ttttftt tdttfdtt nnn nnnnnn t t t t n n n n += −+≅ =′ + ++ ∫∫ ++ 1 11 )(,)()( )(,)( 11 φφφ φφ
- 2. Runge-Kutta Method The Runge-Kutta formula approximates the integrand f (tn, φ(tn)) with a weighted average of its values at the two endpoints and at the midpoint. It is given by where Global truncation error is bounded by a constant times h4 for a finite interval, with local truncation error proportional to h5 . ( )43211 22 6 nnnnnn kkkk h yy ++++=+ ),( )2/,2/( )2/,2/( ),( 34 23 12 1 hkyhtfk hkyhtfk hkyhtfk ytfk nnnn nnnn nnnn nnn ++= ++= ++= =
- 3. Simpson’s Rule The Runge-Kutta formula is where If f(t, y) depends only on t and not on y, then we have which is Simpson’s rule for numerical integration. ( ) ( )[ ],)2/(4 6 1 htfhtftf h yy nnnnn ++++=−+ ( )43211 22 6 nnnnnn kkkk h yy ++++=+ ),( )2/,2/( )2/,2/( ),( 34 23 12 1 hkyhtfk hkyhtfk hkyhtfk ytfk nnnn nnnn nnnn nnn ++= ++= ++= =
- 4. Programming Outline: Runge-Kutta Method Step 1. Define f (t,y) Step 2. Input initial values t0 and y0 Step 3. Input step size h and number of steps n Step 4. Output t0 and y0 Step 5. For j from 1 to n do Step 6. k1 = f (t, y) k2 = f (t + 0.5*h, y + 0.5*h*k1) k3 = f (t + 0.5*h, y + 0.5*h*k2) k4 = f (t + h, y + h*k3) y = y + (h/6)*(k1 + 2*k2 + 2*k3 + k4) t = t + h Step 7. Output t and y Step 8. End
- 5. Example 1: Runge-Kutta Method (1 of 2) Recall our initial value problem To calculate y1 in the first step of the Runge-Kutta method for h = 0.2, we start with Thus .928.10)532.11,2.00( ,532.1;66.7)69.01,1.00( ,38.1;9.6)5.01,1.00( ,0.1;5)1,0( 04 0303 0202 0101 =++= ==++= ==++= === fk khfk khfk khfk ,1)0(,41 =+−=′ yyty ( ) 5016.2928.10)66.7(2)9.6(25 6 2.0 11 =++++=y
- 6. Example 1: Numerical Results (2 of 2) The Runge-Kutta method (h = .05) and the improved Euler method (h = .025) both require a total of 160 evaluations of f. However, we see that the Runge-Kutta is far more accurate. The Runge-Kutta method (h = .2) requires 40 evaluations of f and the improved Euler method requires 160 evaluations of f, and yet the accuracy at t = 2 is similar. 1)0(,41 =+−=′ yyty Improved Euler Runge- Kutta Runge- Kutta Runge- Kutta Improved Euler Runge - Kutta Runge - Kutta Runge - Kutta t h = 0.025 h = 0.2 h = 0.1 h = 0.05 Exact Rel Error h = 0.025 Rel Error h = 0.2 Rel Error h = 0.1 Rel Error h = 0.05 0.00 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.10 1.6079 1.6089 1.6090 1.6090 0.0684 0.0062 0.0000 0.20 2.5021 2.5016 2.5050 2.5053 2.5053 0.1277 0.1477 0.0120 0.0000 0.30 3.8223 3.8294 3.8301 3.8301 0.2037 0.0183 0.0000 0.40 5.7797 5.7776 5.7928 5.7941 5.7942 0.2503 0.2865 0.0242 0.0017 0.50 8.6849 8.7093 8.7118 8.7120 0.3111 0.0310 0.0023 1.00 64.4979 64.4416 64.8581 64.8949 64.8978 0.6162 0.7030 0.0612 0.0045 1.50 474.8340 478.8193 479.2267 479.2592 0.9233 0.0918 0.0068 2.00 3496.6702 3490.5574 3535.8667 3539.8804 3540.2001 1.2296 1.4023 0.1224 0.0090 E x a mple 2
- 7. Adaptive Runge-Kutta Methods The Runge-Kutta method with a fixed step size can suffer from widely varying local truncation errors. That is, a step size small enough to achieve satisfactory accuracy in some parts of the interval of interest may be smaller than necessary in other parts of the interval. Adaptive Runge-Kutta methods have been developed, resulting in a substantial gain in efficiency. Adaptive Runge-Kutta methods are a very powerful and efficient means of approximating numerically the solutions of a large class of initial value problems, and are widely available in commercial software packages.