Advanced Numerical Methods Quiz

Choose the correct statement regarding the representation of integers in computers.

• Integers are always represented as floating-point numbers in computers.

• Integer representation is always accurate, with no rounding errors.

• Integers are typically represented in binary format, with a fixed number of bits.

• Integers are represented in base-10 to match human-readable formats.

Describe what happens in floating-point arithmetic when there is a loss of significance.

• It leads to more precise results and reduces errors in calculations.

• The precision of the calculation decreases due to the cancellation of significant digits.

• It helps in speeding up calculations by reducing the size of the numbers.

• Loss of significance is a technique to ensure exact results in numerical methods.

Define the term error propagation in numerical computations.

• Error propagation refers to how errors from one step in a computation affect subsequent steps.

• Error propagation refers to the elimination of errors through successive steps.

• Error propagation is the process of reducing errors in final calculations.

• Error propagation has no effect on the accuracy of numerical methods.

Identify the method used to estimate errors in numerical computations.

• Linear algebra

• Computational error estimation

• Graphical analysis

• Binary search

List some of the factors that contribute to the convergence of sequences in numerical methods.

• The size of the numbers in the sequence.

• The nature of the function being approximated and the step size used in the method.

• The color of the sequence elements.

• The order of calculation and the specific tools used.

Choose the correct statement regarding the existence and uniqueness of the interpolating polynomial.

• There is always exactly one interpolating polynomial for a given set of points.

• The interpolating polynomial is only unique if the points are not repeated.

• There is no uniqueness in the interpolating polynomial.

• The interpolating polynomial exists only for equally spaced points.

Describe the role of the divided difference table in constructing an interpolating polynomial.

• It is used to find the roots of the polynomial.

• It helps in computing the coefficients of the Newton form of the interpolating polynomial.

• It is only necessary for linear interpolation.

• It is used for calculating integrals of polynomials.

Define the term error of the interpolating polynomial.

• The error of the interpolating polynomial is the difference between the true function and the polynomial at the interpolation points.

• The error of the interpolating polynomial is always zero for equally spaced points.

• The error is independent of the degree of the polynomial.

• The error is calculated by comparing the polynomial to the derivative of the function.

Identify the situation where interpolation in a function table is performed based on equally spaced points.

• When the data points follow a non-linear pattern.

• When the function values are known at equally spaced intervals and an interpolation is needed between them.

• When the data points are randomly distributed and not in any particular order.

• When the function is continuous and differentiable.

List the primary components required to construct the divided difference table.

• The function values, the data points, and the step size.

• The data points and their corresponding derivatives.

• The function values and the corresponding differences of function values at different points.

• The data points and the interval length.