Math 136 Numerical Analysis Exam Review
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Questions and Answers

What is the main purpose of the Horner method in numerical analysis?

  • To solve a system of linear equations
  • To find the roots of a polynomial equation
  • To approximate the derivative of a function
  • To evaluate a polynomial at a given point (correct)

    • What is the error in the given Python code for the Horner method?

  • The function is not correctly importing the numpy library
  • The line 'b[lenp] = p[lenp]' is incorrect (correct)
  • The function is not returning the correct values
  • The loop should iterate from 0 to lenp-1 instead of lenp-2 to 0

    • What is the condition for the convergence of the Newton-Raphson method?

  • The initial guess should be close to the root
  • The function should have a simple root
  • The function should be continuously differentiable (correct)
  • The function should be a polynomial

    • What is the Fixed Point Theorem used for in numerical analysis?

    <p>Approximating the solution of a nonlinear equation</p> Signup and view all the answers

    What is the main difference between the Horner method and the Newton-Raphson method?

    <p>The Horner method is used for polynomial evaluation, while the Newton-Raphson method is used for root finding</p> Signup and view all the answers

    What is the role of the 'dtype' parameter in the 'np.zeros' function in the given Python code?

    <p>It specifies the data type of the array</p> Signup and view all the answers

    What is the purpose of the 'range' function in the given Python code?

    <p>To create a range of indices for the array</p> Signup and view all the answers

    What is the error in the given proof of the Fixed Point Theorem?

    <p>The proof assumes the existence of a fixed point without proving it</p> Signup and view all the answers

    What is the purpose of the 'complex' data type in the given Python code?

    <p>To represent complex numbers</p> Signup and view all the answers

    What is the main difference between the Newton-Raphson method and other root finding methods?

    <p>The Newton-Raphson method is an iterative method, while other methods are direct methods</p> Signup and view all the answers

    Study Notes

    Numerical Analysis Fundamentals

    • Algorithms are written in a specific programming language
    • Sources of errors include:
      • Truncation errors: replacing continuous models with discrete and finite processes for computability
      • Machine epsilon: property of machine representation and operations that affects the accuracy of calculations

    Scalar Root Finding Methods

    • Bisection Method:
      • Converges to a root if the initial interval contains the root
      • Does not depend on the initial iterate
    • First Order Approximations:
      • Chord, Secant, Regula-Falsi, Newton-Raphson, and Steffensen methods
      • Newton-Raphson method has a quadratic convergence rate
    • Fix Point Iterations:
      • Used to find roots of a function
      • Convergence depends on the initial iterate
    • Convergence of Root Finding Methods:
      • Depends on the initial iterate and the method used
      • Newton-Raphson method has a quadratic convergence rate

    Linear and Nonlinear Systems

    • Linear Systems:
      • Triangular Systems: can be solved using back-substitution
      • LU Factorization: a method for solving linear systems
      • Pivoting: a technique used to improve the accuracy of LU factorization
      • Jacobi Method: an iterative method for solving linear systems
      • Gauss-Seidel Method: an iterative method for solving linear systems
    • Nonlinear Systems:
      • Newton Method: an iterative method for solving nonlinear systems

    Error Analysis

    • Absolute Errors: measure the difference between the exact and approximate values
    • Relative Errors: measure the difference between the exact and approximate values relative to the exact value
    • Reduced number of function evaluations: preferred to reduce computational cost and improve efficiency

    Numerical Methods Implementation

    • Horner Method: an efficient method for evaluating polynomials
    • Python code implementation: requires careful handling of arrays and complex numbers

    Fixed Point Theorem

    • States that a fixed point iteration converges to a root if the initial iterate is close enough to the root
    • Proof involves showing that the sequence of iterates converges to the root

    Newton-Raphson Method

    • Converges to a simple root of a function if the initial iterate is close enough to the root
    • Quadratic convergence rate: the error decreases quadratically with each iteration

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    Math 136 Numerical Analysis Exam Reviewer PDF

    Description

    This quiz reviews numerical analysis concepts for Math 136 students at the University of the Philippines Baguio, covering topics such as codes, polynomials, and machine epsilon. It is designed to help students prepare for their first long examination.

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