Review Taylor Series and Error Analysis Roots of Equations

Name: Review Taylor Series and Error Analysis Roots of Equations
Uploaded: 2017-07-13T10:27:05+00:00
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Description: Review Taylor Series and Error Analysis Roots of Equations

Review Taylor Series and Error Analysis Roots of Equations
Linear Algebraic Equations Optimization Numerical Differentiation and Integration Ordinary Differential Equations Partial Differential Equations Curve Fitting Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Taylor Series Lagrange remainder Numerical Methods Prof. Jinbo Bi
Through mean-value theorem, we can derive the Lagrange remainder for Taylor Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Roots of Equations Bracketing Methods Open Methods
Bisection Method False Position Method Open Methods Fixed Point Iteration Newton-Raphson Method Secant Method Roots of Polynomials Müller’s Method Bairstow’s Method Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Bisection Method Example: Use range of [202:204]
Root is in upper subinterval Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Bisection Method Use range of [203:204] Root is in lower subinterval
Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Fixed Point Iteration Example
Special attention Read Chap 6.1, 6.6 Fixed Point Iteration Example Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Newton-Raphson Method
Use tangent to guide you to the root Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Linear Algebraic Systems
Gaussian Elimination Forward Elimination Back Substitution LU Decomposition Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Gaussian Elimination Forward elimination Eliminate x1 from row 2
Multiply row 1 by a21/a11 Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Gaussian Elimination Eliminate x1 from row 2
Subtract row 1 from row 2 Eliminate x1 from all other rows in the same way Then eliminate x2 from rows 3-n and so on Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Gaussian Elimination Forward elimination
Back substitute to solve for x Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

LU Decomposition Substitute the factorization into the linear system
We have transformed the problem into two steps Factorize A into L and U Solve the two sub-problems LD = B UX = D Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

LU Decomposition Example Factorize A using forward elimination

LU Decomposition Example Numerical Methods Prof. Jinbo Bi Lecture 22
CSE, UConn

Optimization Methods One-dimensional unconstrained optimization
Golden-Section Quadratic Interpolation Newton’s Method Multidimensional unconstrained optimization Direct Methods Gradient Methods Constrained Optimization Linear Programming Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Golden-section search
Algorithm Pick two interior points in the interval using the golden ratio Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Two possibilities Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Example Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Newton’s Method Newton-Raphson could be used to find the root of an function When finding a function optimum, use the fact that we want to find the root of the derivative and apply Newton-Raphson Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Newton’s Method Example Numerical Methods Prof. Jinbo Bi Lecture 22
CSE, UConn

Quadratic interpolation
Special attention Quadratic interpolation Use a second order polynomial as an approximation of the function near the optimum Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Special attention Gradient Methods Given a starting point, use the gradient to tell you which direction to proceed The gradient gives you the largest slope out from the current position Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Numerical Integration
Newton-Cotes Trapezoidal Rule Simpson’s Rules (Special attention for unevenly distributed points) Romberg Integration Gauss Quadrature Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Newton-Cotes Formulas
Special attention Read Chap Newton-Cotes Formulas Trapezoidal Rule Simpson’s 1/3 Rule Simpson’s 3/8 Rule Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Integration of Equations
Romberg Integration Use two estimates of integration and then extrapolate to get a better estimate Special case where you always halve the interval - i.e. h2=h1/2 Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Romberg Integration Numerical Methods Prof. Jinbo Bi Lecture 22
CSE, UConn

Ordinary Differential Equations
Runge-Kutta Methods Euler’s Method Heun’s Method RK4 Multistep Methods Boundary Value Problems Eigenvalue Problems Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Euler’s Method Example: True: h=0.5 Numerical Methods Prof. Jinbo Bi
Lecture 22 Prof. Jinbo Bi CSE, UConn

Heun’s Method Local truncation error is O(h3) and global truncation error is O(h2) Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Heun’s Method Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Classic 4th-order R-K method
Special attention to ODE equation system Not only one equation Classic 4th-order R-K method Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Curve Fitting Least Squares Regression Interpolation
Fourier Approximation Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Polynomial Regression
Special attention Lecture note 19 Chap 17.1 Polynomial Regression An mth order polynomial will require that you solve a system of m+1 linear equations Standard error Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Newton (divided difference) Interpolation polynomials

Interpolation General Scheme for Divided Difference Coefficients

Interpolation Example:
Estimate ln 2 with data points at (1,0), (4, ) Linear interpolation Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Estimate ln 2 with data points at (1,0), (4, ), (5, ) Quadratic interpolation Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Estimate ln 2 with data points at (1,0), (4, ), (5, ), (6, ) Cubic interpolation Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Spline Interpolation Spline interpolation applies low-order polynomial to connect two neighboring points and uses it to interpolate between them. Typical Spline functions Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Cubic Spline Functions
This gives us n-1 equation with n-1 unknowns – the second derivatives Once we solve for the second derivatives, we can plug it into the Lagrange interpolating polynomial for second derivative to solve for the splines Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Example: (3,2.5), (4.5,1), (7,2.5), (9,0.5) At x=x1=4 Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

At x=x2=7 Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Solve the system of equations to find the second derivatives Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Cubic Spline Equations

Cubic Spline Equations
Substituting for other intervals Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Final Exam December 13 Friday, 10:30 AM~12:30 PM, ITE 119
Closed book, three cheat sheets (8.5x11in) allowed Office hours: December 12, 1-3pm, or by appointment TA December 10, 11am-12noon or by appointment Numerical Methods Lecture 22 Prof. Jinbo Bi CSE, UConn

Review Taylor Series and Error Analysis Roots of Equations

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