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I: Numerical Methods

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    The first part of this course consists of a concise introduction to numerical methods. We begin by learning how numbers are represented in the computer using the IEEE standard, and how this can result in round-off errors in numerical computations. We will then learn some fundamental numerical methods and their associated MATLAB functions. The numerical methods included are those used for root finding, integration, solving differential equations, solving systems of equations, finite difference methods, and interpolation.

    • 1: IEEE Arithmetic
      This page covers the representation of numbers in computers, highlighting key definitions like bits and bytes, and detailing the IEEE double precision format including its components: sign, exponent, and significand. It discusses round-off errors, especially with near-identical numbers and large subtractions, as well as special numbers like infinity and NaN.
    • 2: Root Finding
      This page explores numerical methods for solving equations without explicit solutions, covering the Bisection, Newton's, and Secant Methods, demonstrating them with examples like estimating √2. It addresses Newton's method in the complex plane for cube roots of unity, detailing convergence rates and orders. The Secant Method’s order of convergence is derived, revealing it converges faster than Bisection but slower than Newton, with an approximate order of 1.618, or the Golden Ratio.
    • 3: Integration
      This page covers numerical integration methods for calculating definite integrals, focusing on the midpoint, trapezoidal, and Simpson's rules derived from Taylor series expansions for error estimation. It discusses composite rules for multiple intervals and emphasizes adaptive integration and Richardson's extrapolation to enhance accuracy with varying data point spacing.
    • 4: Differential Equations
      This page covers numerical methods for solving ordinary differential equations (ODEs), including both initial value and boundary value problems. It introduces the Euler method, Modified Euler method, and various Runge-Kutta methods, emphasizing their error characteristics and formulation.
    • 5: Linear Algebra
      This page covers the numerical solutions to linear equations using Gaussian elimination and LU decomposition. It explains the method through examples, detailing the formation of an augmented matrix and row reduction to upper-triangular form, followed by backward substitution. LU decomposition is introduced, showing how to factor matrix \(A\) into lower and upper triangular matrices for efficient solving.
    • 6: Finite Difference Approximation
      This page covers numerical differentiation using finite difference approximations for solving partial differential equations. It explains finite difference formulas, central-difference methods, and provides an example of the Laplace equation discretized on a grid, which results in a Laplacian matrix. It also details the construction of this matrix in MATLAB, introducing a new function, sp_laplace_new.
    • 7: Iterative Methods
      This page covers iterative methods for solving systems of nonlinear equations, including Jacobi, Gauss-Seidel, and Successive Over Relaxation (SOR), highlighting their speed and simplicity. It introduces the red-black Gauss-Seidel method and briefly discusses Newton's method, emphasizing its derivation from a Taylor series and application to the Lorenz equations.
    • 8: Interpolation
      This page covers interpolation and extrapolation methods for functions, emphasizing the unique polynomial representation via \(n+1\) points. It details piecewise linear and cubic spline interpolations, highlighting the advantages of cubic splines for smoother curves through added constraints. The determination of coefficients for piecewise cubic polynomials is explained, along with boundary condition methods.
    • 9: Least-Squares Approximation
      This page discusses the least squares method for linear fitting of experimental data, aimed at minimizing the squared distances between data points and the fitted line \(y(x) = \alpha x + \beta\). It explains solving for the parameters \(\alpha\) and \(\beta\) by setting the partial derivatives of the residual sum to zero, leading to a system of linear equations. It also references MATLAB’s polyfit function for computing these values.

    Thumbnail: A three-dimensional surface plot of the unnormalized sinc function. (Public Domain; via Wikipedia)

    This page titled I: Numerical Methods is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Jeffrey R. Chasnov via source content that was edited to the style and standards of the LibreTexts platform.