Numerical Methods

Review all questions and their correct answers.

Question 1

For which of the following functions would a numerical method be necessary to find the root of f(x) = 0?

f(x) = 3x - 6
f(x) = x² - 4
f(x) = e⁻ˣ - x
f(x) = 2x

Question 2

What is the primary characteristic of a bracketing method for root finding?

It requires only one initial guess.
It requires two initial guesses that must trap the root.
It always converges quadratically.
It requires the derivative of the function.

Question 3

Which of the following is a major advantage of the Bisection method?

It converges very quickly.
It is guaranteed to converge if a root is bracketed.
It intelligently uses the function values to speed up.
It can find complex roots.

Question 4

The formula for the root estimate in the Bisection method is:

xᵣ = (xₗ + xᵤ) / 2
xᵣ = xᵤ - f(xᵤ)(xₗ - xᵤ) / (f(xₗ) - f(xᵤ))
xᵢ₊₁ = xᵢ - f(xᵢ) / f'(xᵢ)
xᵣ = xₗ - f(xₗ)(xᵤ - xₗ) / (f(xᵤ) - f(xₗ))

Question 5

In the Bisection method, if f(xₗ) * f(xᵣ) is positive, what is the next step?

The new upper bound is xᵤ = xᵣ.
The new lower bound is xₗ = xᵣ.
The root has been found.
The method has diverged.

Question 6

What is the core idea behind the False-Position method?

It approximates the function with a tangent line.
It always cuts the interval exactly in half.
It approximates the root by assuming the function is a straight line between the two bracketing points.
It uses three points to fit a parabola.

Question 7

What is a known disadvantage of the False-Position method?

It can diverge.
It is always slower than Bisection.
It requires the derivative of the function.
One of the bracketing points can get "stuck," slowing convergence.

Question 8

Newton-Raphson is an example of which type of method?

Bracketing method
Open method
Graphical method
Analytical method

Question 9

What is the formula for the Newton-Raphson method?

xᵣ = (xₗ + xᵤ) / 2
xᵢ₊₁ = xᵢ - f(xᵢ) / f'(xᵢ)
xᵢ₊₁ = xᵢ - f(xᵢ)(xᵢ₋₁ - xᵢ) / (f(xᵢ₋₁) - f(xᵢ))
xᵢ₊₁ = xᵢ + f(xᵢ) / f'(xᵢ)

Question 10

What is the main reason for the extremely fast (quadratic) convergence of the Newton-Raphson method?

It uses two initial guesses.
It uses the tangent line at the current guess to project to the next guess.
It brackets the root in every step.
It uses a finite difference to approximate the derivative.

Question 11

Under which condition will the Newton-Raphson method fail catastrophically?

If the initial guess is far from the root.
If the function is a straight line.
If an iteration lands on a point where the derivative is zero.
If the function has multiple roots.

Question 12

What is the primary advantage of the Secant method over the Newton-Raphson method?

It is more reliable and always converges.
It converges quadratically.
It does not require the analytical derivative of the function.
It only requires one initial guess.

Question 13

How does the Secant method approximate the derivative used in Newton-Raphson?

It assumes the derivative is always 1.
It uses the slope of the line passing through the two most recent guess points.
It calculates the derivative analytically.
It uses a parabolic fit.

Question 14

What is the stopping criterion for an iterative numerical method, where εₐ is the approximate relative error and εₛ is the tolerance?

εₐ > εₛ
εₐ = εₛ
εₐ < εₛ
εₐ = 0

Question 15

Muller's method is an extension of the Secant method that fits a __________ through three points.

Straight line
Tangent line
Cubic spline
Parabola

Question 16

What is a unique advantage of Muller's method?

It is the simplest method.
It always converges.
It can naturally find complex roots.
It does not require initial guesses.

Question 17

What is the primary purpose of Bairstow's method?

To find a single real root of any function.
To find all the roots (real and complex) of a polynomial.
To solve systems of linear equations.
To numerically integrate a function.

Question 18

Instead of finding individual roots, Bairstow's method works by finding __________ of the polynomial.

The derivative
The integral
Quadratic factors
Linear factors

Question 19

The difference between the true value and an approximation is known as:

Round-off Error
Truncation Error
True Error
Approximate Error

Question 20

Why is Approximate Relative Error used as a stopping criterion instead of True Error?

Because it is easier to calculate.
Because the true value is usually unknown in real problems.
Because it converges faster.
Because it is always more accurate.

Question 21

The formula |(Current Approx - Previous Approx) / Current Approx| * 100% calculates the:

True Error
True Relative Error
Approximate Error
Approximate Relative Error

Question 22

What type of error is caused by approximating a function with a finite number of terms from its Taylor series?

Round-off Error
Truncation Error
Human Error
Systematic Error

Question 23

What type of error is caused by a computer's finite ability to represent numbers?

Round-off Error
Truncation Error
Modeling Error
Discretization Error

Question 24

What is discretization?

The process of finding the exact analytical solution.
The process of converting a problem with continuous variables into one with discrete variables.
The process of removing errors from a calculation.
The process of finding the derivative of a function.

Question 25

In the context of discretization, what is the trade-off of using a smaller step size (h)?

Lower accuracy but faster computation.
Higher accuracy but slower computation.
Lower accuracy and slower computation.
Higher accuracy and faster computation.

Question 26

Numerical integration methods like the Trapezoidal rule are an application of:

Root finding
Discretization
Error analysis
Cramer's Rule

Question 27

Cramer's Rule is an analytical method used to solve what type of problem?

Finding roots of nonlinear equations
Systems of linear equations
Numerical integration
Differential equations

Question 28

For a system of equations Ax = b, Cramer's Rule states that xᵢ = |Aᵢ| / |A|. What does |A| represent?

The inverse of matrix A.
The determinant of matrix A.
The transpose of matrix A.
The matrix A with column i replaced by b.

Question 29

Under what condition does Cramer's Rule fail to find a unique solution?

When |A| = 1
When |A| > 0
When |A| < 0
When |A| = 0

Question 30

In Cramer's rule, how is the matrix Aᵢ formed?

By removing the i-th column from A.
By replacing the i-th column of A with the constant vector b.
By swapping the i-th column with the first column.
By taking the inverse of A.

Question 31

Given two initial guesses xₗ=2 and xᵤ=3 for f(x)=x²-5, what is the first root estimate (xᵣ) using the Bisection method?

2.5
2.25
2.75
2.0

Question 32

Comparing Bisection and False-Position, which statement is generally true?

Bisection is always faster.
False-Position is often faster but can sometimes stagnate.
Both require the function's derivative.
Both are open methods.

Question 33

A key difference between Open and Bracketing methods is that Open methods:

Are always more reliable.
Are always slower.
Can diverge depending on the initial guess.
Require the root to be trapped between initial guesses.

Question 34

Which method is also known as Regula Falsi?

Bisection
Newton-Raphson
Secant
False-Position

Question 35

What is the convergence rate of the Newton-Raphson method?

Linear
Quadratic
Super-linear
Logarithmic

Question 36

What is the convergence rate of the Bisection method?

Linear
Quadratic
Super-linear
Cubic

Question 37

Why is Cramer's Rule generally not used for large systems of linear equations?

It is not accurate enough.
It cannot handle more than 3 variables.
Calculating determinants is computationally very expensive for large matrices.
It does not provide an exact solution.

Question 38

Given a 3x3 system of equations, how many determinants must be calculated to solve it completely using Cramer's Rule?

Question 39

If the first iteration of a root-finding method gives an approximation of 10 and the second gives 8, what is the approximate relative error (εₐ)?

20%
25%
2%
12.5%

Question 40

If f(x) = x³ - 7x² + 14x - 6, what is f'(x), which would be required for the Newton-Raphson method?

3x² - 14x + 14
x⁴/4 - 7x³/3 + 7x² - 6x
3x² - 7x + 14
x³ - 7x²

Question 41

Which two methods are bracketing methods?

Bisection and Newton-Raphson
Secant and False-Position
Bisection and False-Position
Newton-Raphson and Secant

Question 42

Which two methods are open methods?

Bisection and Newton-Raphson
Secant and False-Position
Bisection and False-Position
Newton-Raphson and Secant

Question 43

A root of a function f(x) is the value of x for which:

f(x) = 1
f(x) is maximum
f(x) = 0
f(x) is minimum

Question 44

The term for when iterations move further away from the root is:

Convergence
Divergence
Tolerance
Discretization

Question 45

If f(x) = x³ - 7x² + 14x - 6, using Bisection with xₗ=0, xᵤ=1, the first midpoint xᵣ is 0.5. Since f(0) * f(0.5) < 0, the next interval is:

[0, 0.5]
[0.5, 1]
[0, 0.25]
[0.25, 0.5]

Question 46

The Secant method requires how many initial guesses?

Question 47

Which method would be a good choice if the function's derivative is very difficult or impossible to calculate?

Newton-Raphson
Secant
Bisection
Cramer's Rule

Question 48

In the example f(x) = e⁻ˣ - x, the root was found to be 0.56714. This means:

e⁻⁰⁵⁶⁷¹⁴ ≈ 0.56714
e⁰⁵⁶⁷¹⁴ ≈ 0.56714
e⁻⁰⁵⁶⁷¹⁴ ≈ 0
e⁻⁰⁵⁶⁷¹⁴ - 1 = 0

Question 49

The set of discrete points (x₀, x₁, ..., xₙ) used to approximate a continuous domain is called a:

Step Size
Grid or Mesh
Tolerance
Function

Question 50

What is a major advantage of Bairstow's method for finding polynomial roots?

It is the simplest method to implement by hand.
It finds complex conjugate pairs of roots directly.
It is a bracketing method and always converges.
It does not require any initial guesses.

Question 51

Given a system of linear equations, if the determinant of the coefficient matrix |A| is -3, what can you conclude?

The system has no solution.
The system has infinitely many solutions.
The system has a unique solution.
The system has 3 solutions.

Question 52

The main disadvantage of the Bisection method is that it:

Is unreliable.
Requires the derivative.
Is slow.
Can diverge.

Question 53

What is the primary difference in the algorithm between Bisection and False-Position?

The number of initial guesses required.
The formula used to calculate the next root estimate (xᵣ).
The way the next interval is chosen after finding xᵣ.
One is a bracketing method and the other is an open method.

Question 54

Which of these is NOT a root-finding method?

Bisection
Newton-Raphson
Cramer's Rule
Secant

Question 55

If a numerical method is described as having 'linear convergence', it means:

The error is reduced by a constant factor with each iteration.
The number of correct decimal places doubles with each iteration.
The method can only solve linear equations.
The function being solved must be a straight line.