Documents about Numerical Methods

  • 12 Pages

    lect_2

    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 ...

  • 6 Pages

    lect_9

    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 ...

  • 10 Pages

    lect_5

    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 ...

  • 12 Pages

    lect_8

    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 ...

  • 8 Pages

    lect_1

    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 ...