Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization Numerical Differentiation and Integration Ordinary Differential.

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 MethodsLecture 22

Prof. Jinbo BiCSE, UConn

1

Taylor Series

• Lagrange remainder



2

Roots of Equations

• Bracketing 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



3

Bisection Method

• Example:

• Use range of [202:204]

• Root is in upper subinterval



4

Bisection Method

• Use range of [203:204]

• Root is in lower subintervalNumerical MethodsLecture 22


5

Fixed Point Iteration Example



6

Special attention

Read Chap 6.1, 6.6

Newton-Raphson Method

• Use tangent to guide you to the root



7

Linear Algebraic Systems

• Gaussian Elimination• Forward Elimination• Back Substitution

• LU Decomposition



8

Gaussian Elimination

• Forward elimination

• Eliminate x1 from row 2

• Multiply row 1 by a21/a11



9


• 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 onNumerical MethodsLecture 22


10


• Forward elimination

• Back substitute to solve for x



11

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



12

LU Decomposition

• Example

• Factorize A using forward elimination



13

LU Decomposition

• Example



14

LU Decomposition

• Example



15

LU Decomposition

• Example



16

Optimization Methods• One-dimensional unconstrained optimization

• Golden-Section

• Quadratic Interpolation

• Newton’s Method

• Multidimensional unconstrained optimization• Direct Methods

• Gradient Methods

• Constrained Optimization• Linear Programming



17

Golden-section search

• Algorithm• Pick two interior points in the interval using the

golden ratio



18


• Two possibilities



19


• Example



20




21




22

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



23

Newton’s Method

• Example



24

Newton’s Method

• Example



25

Quadratic interpolation

• Use a second order polynomial as an approximation of the function near the optimum



26

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



27

Special attention

Numerical Integration• Newton-Cotes

• Trapezoidal Rule• Simpson’s Rules (Special attention for

unevenly distributed points)

• Romberg Integration• Gauss Quadrature



28

Newton-Cotes Formulas• Trapezoidal Rule

• Simpson’s 1/3 Rule

• Simpson’s 3/8 Rule



29

Special attention

Read Chap 21.2-3

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



30

Romberg Integration



31

Ordinary Differential Equations• Runge-Kutta Methods

• Euler’s Method• Heun’s Method• RK4

• Multistep Methods• Boundary Value Problems• Eigenvalue Problems



32

Euler’s Method

• Example:

• True:• h=0.5



33

Heun’s Method

• Local truncation error is O(h3) and global truncation error is O(h2)



34

Heun’s Method



35

Classic 4th-order R-K method



36

Special attention to

ODE equation system

Not only one equation

Curve Fitting

• Least Squares Regression• Interpolation• Fourier Approximation



37

Polynomial Regression• An mth order polynomial will require that you

solve a system of m+1 linear equations

• Standard error



38

Special attention

Lecture note 19

Chap 17.1

Newton (divided difference) Interpolation polynomials



39

Newton (divided difference) Interpolation polynomials



40

Interpolation• General Scheme for Divided Difference

Coefficients



41

Interpolation

• General Scheme for Divided Difference Coefficients



42

Interpolation

• Example:• Estimate ln 2 with data points at (1,0),

(4,1.386294)• Linear interpolation



43

Interpolation


(4,1.386294), (5,1.609438)• Quadratic interpolation



44

Interpolation


(4,1.386294), (5,1.609438), (6,1.791759)• Cubic interpolation



45

Spline Interpolation

• Spline interpolation applies low-order polynomial to connect two neighboring points and uses it to interpolate between them.

• Typical Spline functions



46

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



47


• Example: (3,2.5), (4.5,1), (7,2.5), (9,0.5)• At x=x1=4



48


• At x=x2=7



49


• Solve the system of equations to find the second derivatives



50

Cubic Spline Equations



51

Cubic Spline Equations

• Substituting for other intervals



52

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



53

Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization Numerical Differentiation and Integration Ordinary Differential.

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