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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 (C)</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 (B)</p> Signup and view all the answers

    What is the purpose of LU factorization of a matrix?

    <p>To solve systems of linear equations (B)</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 (D)</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 (B)</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 (D)</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 (A)</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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