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Untitled Quiz

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Questions and Answers

What is the primary reason why polynomials are used to approximate functions in numerical analysis?

  • Because they can be exactly represented in computer arithmetic
  • Because they are always differentiable
  • Because they are always monotonic
  • Because they can be used to model a wide range of functions (correct)

    • What is the machine epsilon (eps) in computer arithmetic?

  • The largest positive number that can be subtracted from 1 and still be distinguished from 1 in computer arithmetic
  • The smallest positive number that can be added to 1 and still be distinguished from 1 in computer arithmetic (correct)
  • The largest positive number that can be represented in computer arithmetic
  • The smallest positive number that can be represented in computer arithmetic

    • Which of the following is a characteristic of truncation errors?

  • They occur when continuous models are replaced by discrete and finite processes (correct)
  • They occur when the number of iterations is increased
  • They occur when discrete models are replaced by continuous processes
  • They occur when the number of iterations is decreased

    • What is the main advantage of using iterative methods over substitution methods?

    <p>Iterative methods require fewer function evaluations</p> Signup and view all the answers

    Which of the following is a root finding method that is guaranteed to converge to a root of a polynomial?

    <p>Bisection method</p> Signup and view all the answers

    What is the purpose of LU factorization of a matrix?

    <p>To solve systems of linear equations</p> Signup and view all the answers

    What is the main difference between absolute errors and relative errors?

    <p>Absolute errors are used to measure the difference between the exact and approximate values, while relative errors are used to measure the ratio of the difference to the exact value</p> Signup and view all the answers

    Which of the following statements is true about the Secant method?

    <p>The Secant method is a root finding method that uses an iterative formula</p> Signup and view all the answers

    What is the main disadvantage of using polynomial approximation to model a function?

    <p>Polynomials can be sensitive to the choice of degree and coefficients</p> Signup and view all the answers

    Which of the following is a characteristic of the Jacobi method?

    <p>The Jacobi method uses the diagonal of the coefficient matrix as the iteration matrix</p> Signup and view all the answers

    Study Notes

    Numerical Analysis

    Fundamentals

    • The course covers algorithms, sources of errors, machine representation and operations (machine epsilon, cancellation error)

    Scalar Root Finding Methods

    • Bisection Method
    • First Order Approximations (Chord, Secant, Regula-Falsi, Newton-Raphson, Steffensen)
    • Fix Point Iterations
    • Convergence of Root Finding Methods
    • Zeros of Polynomials (Deflation, Newton-Horner)

    Linear and Nonlinear Systems

    • Linear Systems (Triangular Systems, LU Factorization and Pivoting, Jacobi Method, Gauss-Seidel Method)
    • Nonlinear Systems (Newton Method)

    System of Equations

    • Solving systems using substitution method
    • Jacobi method to find the solution after 3 iterates

    Python Code Correction

    • Given a Horner method code, identify and correct errors for a smooth implementation
    • Explain the corrections made

    Mathematical Theorems

    • Fixed Point Theorem: statement and proof
    • Newton-Raphson method convergence to a root x∗ for any x0 in [a, b]

    True or False Statements

    • Codes are algorithms written in a specific programming language: TRUE
    • Polynomials of degree n have exactly n real zeros: FALSE
    • Machine epsilon eps has the property that for any ϵ < eps, the numbers 1 and 1 + ϵ are the same in computer arithmetic: TRUE
    • Truncation errors are obtained when continuous models are replaced by discrete and finite processes for computability: TRUE
    • And more...

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