University of Florida, LECT 3421
Excerpt: ... CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 Due to poor attendance in lecture, I will be using the board for most of the rest of the year. I may not post any additional notes, so show up if you want the material. Numerical Methods Lecture 8 page 116 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 117 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 118 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 119 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 120 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 121 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 122 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 123 of 124 CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 8 page 124 of 124 ...
Stanford, CEE 262
Excerpt: ... CEE262C Lecture 6: Numerical methods for ODEs II Overview Graphical interpretation of ODE solvers Choosing the time step size Interval halving Stiff systems Implicit methods References: Kreyszig, 20.1-20.2 CEE262C Lecture 6: Numerical methods for ODEs II 1 Graphical interpretation of numerical methods CEE262C Lecture 6: Numerical methods for ODEs II 2 CEE262C Lecture 6: Numerical methods for ODEs II 3 Choosing the time step size CEE262C Lecture 6: Numerical methods for ODEs II 4 CEE262C Lecture 6: Numerical methods for ODEs II 5 Interval halving procedure CEE262C Lecture 6: Numerical methods for ODEs II 6 Example using interval halving Ref: http:/classes.cecs.ucf.edu/eel5891/klee/ CEE262C Lecture 6: Numerical methods for ODEs II 7 tank.m CEE262C Lecture 6: Numerical methods for ODEs II 8 CEE262C Lecture 6: Numerical methods for ODEs II 9 Stiff Systems CEE262C Lecture 6: Numerical methods for ODEs II 10 CEE262C Lecture 6: Numerical methods for ODEs II 11 CEE ...
Midwestern State University, EE 351
Excerpt: ... EE351 Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Date 1/12 1/14 1/16 1/19 1/21 1/23 1/26 1/28 1/30 2/2 2/4 2/6 2/9 2/11 2/13 2/16 2/18 2/20 2/23 2/25 2/27 3/1 3/3 3/5 3/8 3/10 3/12 3/15 3/17 3/19 3/22 3/24 3/26 3/29 3/31 4/2 4/5 4/7 4/9 4/12 4/14 4/16 4/19 4/21 4/23 4/26 4/28 4/30 Lecture 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Tentative Course Outline Topic Plane waves: Maxwell's Equations & Wave equation Plane waves Plane waves No classes Plane waves Plane waves Plane waves Planes waves Waves guides Waves guides Waves guides Waves guides Waves guides Waves guides Waves guides No Classes Plane waves & Waveguides Antennas Antennas Antennas Antennas Antennas Antennas Antennas Antennas Antennas Antennas Spring Break (No classes) Antennas Antennas Modern Topics Modern Topics Modern Topics Modern Topics Numerical Methods Numerical Methods Numerical Methods Numerical Methods Numerical Methods Numerical Methods Numerical Methods Numerical Methods Modern Topics & Numerical Methods Revi ...
San Diego State, MATH 693
Excerpt: ... blomgren@terminus.SDSU.EDU http:/terminus.SDSU.EDU $Id: lecture.tex,v 1.5 2008/01/22 17:52:41 blomgren Exp $ Numerical Methods for PDEs: Introduction p. 1/18 ...
San Diego State, MATH 693
Excerpt: ... Numerical Methods for PDEs: Introduction p. 2/18 Numerical Methods for PDEs: Introduction p. 1/18 Numerical Methods for PDEs: Introduction p. 3/18 Numerical Methods for PDEs: Introduction p. 4/18 Numerical Methods for PDEs: Introduction p. 5/18 ...
San Diego State, MATH 693
Excerpt: ... Numerical Methods for PDEs: Introduction p. 2/18 Numerical Methods for PDEs: Introduction p. 1/18 Numerical Methods for PDEs: Introduction p. 3/18 Numerical Methods for PDEs: Introduction p. 4/18 Numerical Methods for PDEs: Introduction p. 5/18 ...
Washington, ASTR 509
Excerpt: ... Astr 509: Astrophysics III: Stellar Dynamics Winter Quarter 2005, University of Washington, Zeljko Ivezi c Lecture 4: Potential Theory III The Milky Way Potential, Numerical Methods 1 The Milky Way Potential Already discussed in detail in Lecture 1. The most popular models are double exponential disk (thin and thick in the Z direction), with a power-law or logarithmic halo. In general, the potentials are constrained using the spatial distribution of stars, or the kinematic information (e.g. the rotation curve). Some recent good reviews: Bahcall (1986, ARA&A 24, 577) Gilmore, Wyse & Kuijken (1989, ARA&A 27, 555) Majewski (1993, ARA&A 31, 575) Freeman & Bland-Hawthorn (2002, ARA&A 40, 487) 2 Numerical Methods Numerical methods are most efficient for studying N-body gravitational systems: major results in galaxy formation and dynamical evolution, globular clusters, galaxy clusters, stability and evolution of planetary systems, etc. The key to success is accurate computation of the gravitational force t ...
University of Texas, CHE 348
Excerpt: ... Introduction to Numerical Analysis for Engineers Fundamentals of Digital Computing Digital Computer Models Convergence, accuracy and stability Number representation Arithmetic operations Recursion algorithms Error Analysis Error propagation numerical stability Error estimation Error cancellation Condition numbers 13.002 Numerical Methods for Engineers Lecture 2 Floating Number Representation m b e Mantissa Base Exponent Examples Decimal Binary Convention Decimal Binary General 13.002 Max mantissa Min mantissa Max exponent Min exponent Numerical Methods for Engineers Lecture 2 Error Analysis Number Representation Absolute Error Shift mantissa of largest number Relative Error Result has exponent of largest number Absolute Error Relative Error Unbounded Addition and Subtraction Multiplication and Division Relative Error 13.002 Numerical Methods for Engineers Bounded Lecture 2 Error Propagation Spherical Bessel Functions Differential Equation Solut ...
University of Texas, CHE 348
Excerpt: ... Introduction to Numerical Analysis for Engineers Mathews Interpolation Lagrange interpolation Triangular families Newtons iteration method Equidistant Interpolation 4.1-4.4 4.3 4.4 4.4 4.4 Numerical Differentiation Numerical Integration Error of numerical integration 6.1-6.2 7.1-7.3 13.002 Numerical Methods for Engineers Lecture 9 Numerical Differentiation f(x) n=1 Taylor Series First order h x 13.002 Numerical Methods for Engineers Lecture 9 Numerical Differentiation Second order f(x) n=2 h h x Second Derivatives n=2 n=3 13.002 Forward Difference Central Difference Numerical Methods for Engineers Lecture 9 Numerical Integration Lagrange Interpolation Equidistant Sampling f(x) Integration Weights (Cotes Numbers) Properties a b x 13.002 Numerical Methods for Engineers Lecture 9 Numerical Integration f(x) Trapezoidal Rule n=1 Simpsons Rule f(x) h x n=2 h h x 13.002 Numerical Methods for Engineers Lecture 9 Numerical Integr ...
University of Texas, CHE 348
Excerpt: ... Introduction to Numerical Analysis for Engineers Systems of Linear Equations Cramers Rule Gaussian Elimination Mathews 3.3-3.5 Numerical implementation 3.3-3.4 Numerical stability Partial Pivoting Equilibration Full Pivoting Multiple right hand sides Computation count LU factorization Error Analysis for Linear Systems Condition Number 3.5 3.4 Iterative Methods Special Matrices 13.002 Jacobis method Gauss-Seidel iteration Convergence 3.6 Numerical Methods for Engineers Lecture 5 Linear Systems of Equations Tri-diagonal Systems Forced Vibration of a String f(x,t) xi y(x,t) Finite Difference Discrete Difference Equations Matrix Form Harmonic excitation f(x,t) = f(x) cos( t) Differential Equation Boundary Conditions Tridiagonal Matrix Symmetric, positive definite: No pivoting needed 13.002 Numerical Methods for Engineers Lecture 5 Linear Systems of Equations Tri-diagonal Systems General Tri-diagonal Systems LU Facto ...
University of Florida, LECT 3421
Excerpt: ... CGN 3421 - Computer Methods Gurley Numerical Methods - Lecture 1 Topics: Numerical Methods Introduction WHY ARE WE STUDYING THIS? matrix methods curve fitting root finding optimization statistical analysis probability analysis integration / differentiation ordinary differential equations signal processing NUMERICAL METHODS (why?) Our creations are based on mathematical models of what occurs in nature. Structural response to wind/waves/earthquakes Airplane performance Chemical reactions Automobile suspensions Steel bridge deformation under heavy railroad loads Traffic patterns for infrastructure design Maximum capacity of piles holding up foundations Waste water treatment plants The betting line on Florida vs. Florida State Insurance premiums . These mathematical models of physical processes allow us to design and build new creations that account for the various stresses they will see. Without numerical methods (the creation and application of mathematical models), there would ...
San Diego State, MATH 693
Excerpt: ... blomgren@terminus.SDSU.EDU http:/terminus.SDSU.EDU $Id: lecture.tex,v 1.5 2008/01/22 17:52:41 blomgren Exp $ Numerical Methods for PDEs: Introduction p. 1/18 Numerical Methods for PDEs: Introduction p. 2/18 Numerical Methods for PDEs: Introduction p. 2/18 ...
Duke, CPS 258
Excerpt: ... Introduction to Numerical Methods for ODEs and PDEs Methods of Approximation Lecture 3: finite differences Lecture 4: finite elements Prevalent numerical methods in engineering and the sciences We will introduce in some detail the basic ideas associated with two classes of numerical methods Finite Difference Methods (in which the strong form of the boundary value problem, introduced in the model problems, is directly approximated using difference operators) Finite Element Methods (in which the weak form of the boundary value problem, derived through integral weighting of the BVP, is approximated instead) .while skipping a third class of methods which are quite prevalent Boundary Element Methods (BEM) Predominantly for linear problems; based on reciprocity theorems and Greens function solutions Finite Difference Methods Rely on direct approximation of governing differential equations, using numerical differentiation formulas Ordinary derivative approximations Forward differenc ...
University of Texas, CHE 348
Excerpt: ... Introduction to Numerical Analysis for Engineers Mathews Interpolation Lagrange interpolation Triangular families Newtons iteration method Equidistant Interpolation 4.1-4.4 4.3 4.4 4.4 4.4 Numerical Differentiation Numerical Integration Error of numerical integration 6.1-6.2 7.1-7.3 13.002 Numerical Methods for Engineers Lecture 8 Numerical Interpolation Given: Find for Purpose of numerical Interpolation 1. Compute intermediate values of a sampled function 2. Numerical differentiation foundation for Finite Difference and Finite Element methods 3. Numerical Integration 13.002 Numerical Methods for Engineers Lecture 8 Numerical Interpolation Polynomial Interpolation f(x) Polynomial Interpolation Coefficients: Linear System of Equations x Interpolation Interpolation function 13.002 Numerical Methods for Engineers Lecture 8 Numerical Interpolation Polynomial Interpolation Examples f(x) f(x) x x Linear Interpolation Quadratic Interpolation 13.002 N ...
University of Texas, CHE 348
Excerpt: ... Introduction to Numerical Analysis for Engineers Fundamentals of Digital Computing Digital Computer Models Convergence, accuracy and stability Number representation Arithmetic operations Recursion algorithms Error Analysis Error propagation numerical stability Error estimation Error cancellation Condition numbers 13.002 Numerical Methods for Engineers Lecture 1 Digital Computer Models Continuous Model Differential Equation Differentiation Integration Difference Equation w(x,t) Discrete Model Linear System of Equations x System of Equations Solving linear equations xn m n w(x,t) x Eigenvalue Problems Non-trivial Solutions n Root finding Accuracy and Stability => Convergence 13.002 Numerical Methods for Engineers Lecture 1 Floating Number Representation m b e Mantissa Base Exponent Examples Decimal Binary Convention Decimal Binary General 13.002 Max mantissa Min mantissa Max exponent Min exponent Numerical Methods for Engineers Lecture 1 Ari ...
University of Florida, LECT 3421
Excerpt: ... CGN 3421 - Computer Methods Gurley Numerical Methods - Lecture 1 Topics: Numerical Methods Introduction WHY ARE WE STUDYING THIS? matrix methods curve fitting root finding optimization statistical analysis probability analysis integration / differentiation ordinary differential equations signal processing NUMERICAL METHODS (why?) Our creations are based on mathematical models of what occurs in nature. Structural response to wind/waves/earthquakes Airplane performance Chemical reactions Automobile suspensions Steel bridge deformation under heavy railroad loads Traffic patterns for infrastructure design Maximum capacity of piles holding up foundations Waste water treatment plants The betting line on Florida vs. Florida State Insurance premiums . These mathematical models of physical processes allow us to design and build new creations that account for the various stresses they will see. Without numerical methods (the creation and application of mathematical ...
University of Florida, LECT 3421
Excerpt: ... CGN 3421 - Computer Methods Gurley Numerical Methods - Lecture 1 Topics: Numerical Methods Introduction WHY ARE WE STUDYING THIS? matrix methods curve fitting root finding optimization statistical analysis probability analysis integration / differentiation ordinary differential equations signal processing GUI design (graphical user interface) NUMERICAL METHODS (why?) Our creations are based on mathematical models of what occurs in nature. Structural response to wind/waves/earthquakes Airplane performance Chemical reactions Automobile suspensions Steel bridge deformation under heavy railroad loads Traffic patterns for infrastructure design Maximum capacity of piles holding up foundations Waste water treatment plants The betting line on Florida vs. Florida State Insurance premiums . These mathematical models of physical processes allow us to design and build new creations that account for the various stresses they will see. Without numerical methods (the cr ...
W. Alabama, SD 312
Excerpt: ... Boundary Value Problems Shooting Method Differential Equation y Boundary Conditions Shooting Method Initial value Problem Solve by Runge-Kutta a b x Shooting Iteration 13.002 Numerical Methods for Engineers Lecture 11 ...
University of Florida, LECT 3421
Excerpt: ... CGN 3421 - Computer Methods Gurley Numerical Methods - Lecture 1 Topics: Numerical Methods Introduction WHY ARE WE STUDYING THIS? matrix methods curve fitting root finding optimization statistical analysis probability analysis integration / differentiation ordinary differential equations signal processing GUI design (graphical user interface) NUMERICAL METHODS (why?) Our creations are based on mathematical models of what occurs in nature. Structural response to wind/waves/earthquakes Airplane performance Chemical reactions Automobile suspensions Steel bridge deformation under heavy railroad loads Traffic patterns for infrastructure design Maximum capacity of piles holding up foundations Waste water treatment plants The betting line on Florida vs. Florida State Insurance premiums . Well look at a brief introduction of each of the topics to be covered. Numerical Methods - Lecture 1 page 63 of 69 CGN 3421 - Computer Methods Gurley Matrix methods - s ...