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ECIV 301 Programming & Graphics Numerical Methods for Engineers REVIEW III

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Topics Regression Analysis –Linear Regression –Linearized Regression –Polynomial Regression Numerical Integration –Newton Cotes –Trapezoidal Rule –Simpson Rules –Gaussian Quadrature

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Topics Numerical Differentiation –Finite Difference Forms ODE – Initial Value Problems –Runge Kutta Methods ODE – Boundary Value Problems –Finite Difference Method

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Regression Often we are faced with the problem… what value of y corresponds to x=0.935?

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Curve Fitting Question 2 : Is it possible to find a simple and convenient formula that represents data approximately ? e.g. Best Fit ? Approximation

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Experimental Measurements Strain Stress

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BEST FIT CRITERIA Strain y Stress Error at each Point

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Best Fit => Minimize Error Best Strategy

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Best Fit => Minimize Error Objective: What are the values of a o and a 1 that minimize ?

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Least Square Approximation In our case Since x i and y i are known from given data

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Least Square Approximation

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2 Eqtns 2 Unknowns

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Least Square Approximation

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Example

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Quantification of Error Average

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Quantification of Error Average

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Quantification of Error Average

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Quantification of Error Standard Deviation Shows Spread Around mean Value

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Quantification of Error

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“Standard Deviation” for Linear Regression

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Quantification of Error Better Representation Less Spread

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Quantification of Error Coefficient of Determination Correlation Coefficient

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Linearized Regression The Exponential Equation

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Linearized Regression The Power Equation

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Linearized Regression The Saturation-Growth-Rate Equation

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Polynomial Regression A Parabola is Preferable

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Polynomial Regression Minimize

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Polynomial Regression

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3 Eqtns 3 Unknowns

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Polynomial Regression Use any of the Methods we Learned

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Polynomial Regression With a 0, a 1, a 2 known the Total Error Standard Error Coefficient of Determination

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Polynomial Regression For Polynomial of Order m Standard Error Coefficient of Determination

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Numerical Integration & Differentiation

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Motivation

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AREA BETWEEN a AND b

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Motivation

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Calculate Derivative Given

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Motivation Given Calculate

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Think as Engineers!

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In Summary INTERPOLATE

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In Summary Newton-Cotes Formulas Replace a complicated function or tabulated data with an approximating function that is easy to integrate

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In Summary Also by piecewise approximation

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Closed/Open Forms CLOSEDOPEN

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Trapezoidal Rule Linear Interpolation

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Trapezoidal Rule Multiple Application

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xa=x o x1x1 x2x2 …x n-1 b=x n f(x)f(x 0 )f(x 1 )f(x 2 )f(x n-1 )f(x n )

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Simpson’s 1/3 Rule Quadratic Interpolation

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Simpson’s 3/8 Rule Cubic Interpolation

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Gauss Quadrature x1x1 x2x2

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General Case Gauss Method calculates pairs of wi, xi for the Integration limits -1,1 For Other Integration Limits Use Transformation

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Gauss Quadrature For x g =-1, x=a For x g =1, x=b

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Gauss Quadrature

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PointsWeighting Factors wi Function Arguments Error 2W0=1.0X0=-0.577350269 F (4) ( ) W1=1.0X1= 0.577350269 3W0=0.5555556X0=-0.77459669 F (6) ( ) W1=0.8888888X1=0.0 W2=0.5555556X2=0.77459669

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Gaussian Points PointsWeighting Factors wi Function Arguments Error 4W 0 =0.3478548X0=-0.861136312 F (8) ( ) W 1 =0.6521452X1=-339981044 W 2 =0.6521452X2=- 339981044 W 3 =0.3478548X3=0.861136312

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Gaussian Quadrature Not a good method if function is not available

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Fig 23.1 FORWARD FINITE DIFFERENCE

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Fig 23.2 BACKWARD FINITE DIFFERENCE

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Fig 23.3 CENTERED FINITE DIFFERENCE

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Data with Errors

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ODE IVP, BVP

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Pendulum W=mg Ordinary Differential Equation

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ODEs Non Linear Linearization Assume is small

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ODEs Second Order Systems of ODEs

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ODE

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ODE - OBJECTIVES Undetermined

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ODE- Objectives Initial Conditions

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ODE-Objectives Given Calculate

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Runge-Kutta Methods New Value = Old Value + Slope X Step Size

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Runge Kutta Methods Definition of yields different Runge-Kutta Methods

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Euler’s Method Let

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Sources of Error Truncation: Caused by discretization Local Truncation Propagated Truncation Roundoff: Limited number of significant digits

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Sources of Error Propagated Local

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Euler’s Method

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Heun’s Method PredictorCorrector 2-Steps

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Heun’s Method Predict Predictor-Corrector Solution in 2 steps Let

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Heun’s Method Correct Corrector Estimate Let

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Error in Heun’s Method

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The Mid-Point Method Remember: Definition of yields different Runge-Kutta Methods

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Mid-Point Method Predictor Corrector 2-Steps

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Mid-Point Method Predictor Predict Let

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Mid-Point Method Corrector Correct Estimate Let

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Runge Kutta – 2 nd Order

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Runge Kutta – 3rd Order

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Runge Kutta – 4th Order

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Boundary Value Problems

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Fig 23.3 CENTERED FINITE DIFFERENCE

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xoxo Boundary Value Problems x1x1 x2x2 x3x3 x n-1 xnxn...

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Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

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Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

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Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

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Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

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Boundary Value Problems Collect Equations: BOUNDARY CONDITIONS

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Example x1x1 x2x2 x3x3 x4x4

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