Numerical Methods Test #2

From which mathematical concept is the Newton-Raphson iteration formula derived?

 

Fourier series

 

Taylor series

 

Maclaurin series

 

Binomial series

Taylor Series

In the Newton-Raphson iteration formula, what does  represent?

 

The function value at x_n

 

The second-order derivative at x_n

 

The slope (first-order derivative) at x_n

 

The error at x_n

The slope (first-order derivative) at x_n

What is the intuition behind the correction f(xn)/f’(xn) term ?

 

If f(x_n) is large, take a small step

 

If slope is steep, take a large step

 

If f(x_n) is large, take a big step; if slope is steep, take a smaller step

 

The term is constant regardless of function values

If f(x_n) is large, take a big step; if slope is steep, take a smaller step

Which of the following is NOT a common stopping criterion for Newton-Raphson method?

 

Absolute error

 

Relative error

 

Residual

 

Number of function evaluations

Number of function evaluations

Why should a maximum iteration limit be included in the algorithm?

 

To speed up convergence

 

To prevent infinite loops

 

To improve accuracy

 

To reduce memory usage

To prevent infinite loops

In the geometric interpretation of Newton-Raphson, what does the next estimate xn+1 represent?

 

The point where the curve crosses the y-axis

 

The point where the tangent line at (x_n, f(x_n)) crosses the x-axis

 

The maximum point of the function

 

The minimum point of the function

The point where the tangent line at (x_n, f(x_n)) crosses the x-axis

Starting with x0=1.5 for x²-2=0, what is x1?

 

1.5

 

1.416666667

 

1.414215686

 

1.414213562

1.416666667

In the Python function newton_raphson_sqrt2(), how is the relative error calculated?

 

error = abs(x_next - xn) / abs(x_next)

 

error = abs(x_next - xn)

 

error = abs(f(x_next))

 

error = abs(x_next - xn) / abs(xn)

error = abs(x_next - xn) / abs(x_next)

What is the purpose of the max_iter parameter in the Newton-Raphson function?

 

To set the tolerance

 

To limit the number of iterations and prevent infinite loops

 

To determine the initial guess

 

To calculate the derivative

To limit the number of iterations and prevent infinite loops

What would happen if the initial guess for x²-2=0 was chosen as x0=0?

 

The method would converge to √2 quickly

 

The method would fail because f′(0) = 0 (division by zero)

 

The method would converge to −√2

 

The method would converge to 0

The method would fail because f′(0) = 0 (division by zero)

Starting with x0 = 3 for x³-20=0, what is x1?

 

3.00000

 

2.74074

 

2.71467

 

2.71442

2.74074

What is the relative error after the first iteration for x³-20=0 starting from x0=3?

 

0.96%

 

9.46%

 

0.009%

 

1.00

9.46%

What is the relative error after the third iteration for x³-20=0 starting from x0=3?

 

0.96%

 

9.46%

0.009%

0.0009%

0.009%

In the convergence analysis, what does εn = xn-x* represent?

 

The function value

 

The derivative value

 

The error or residual at iteration n

 

The step size

The error or residual at iteration n

What order of Taylor polynomial is used to establish quadratic convergence? 

 

First-order

 

Second-order

 

Third-order

 

Zero-order

Second-order

For ,f(x) = x² - 2 what is the asymptotic error constant C?

 

0.5

 

0.3535

 

1.0

 

0.7071

0.3535

What does it mean if f’’(x*)=0?

 

The method fails

 

Convergence is slower

 

Convergence can be even faster than quadratic

 

The derivative is zero

Convergence can be even faster than quadratic

What determines the speed of convergence in Newton-Raphson method?

 

The initial point only

 

The asymptotic error constant C

 

The maximum number of iterations

 

The tolerance value

The asymptotic error constant C

If is small, what does that imply about convergence?

 

Convergence is slower

 

Convergence is faster

 

Convergence is not affected

 

The method will fail

Convergence is faster

What does the rate of convergence describe?

 

The initial guess

 

How fast the sequence approaches x*

 

The number of iterations

 

The function value

How fast the sequence approaches x*

What is the key advantage of quadratic convergence mentioned in the slides?

 

Digits decrease each iteration

 

Digits double each iteration

 

Digits remain constant

 

Digits triple each iteration

Digits double each iteration

For the equation (x-1)³ + 0.512 = 0, what is the root?

x = 1.0

x = 0.2

x = 0.512

x = -0.8

x = 0.2

Starting with for X0=5 for f(x)=(x-1)³, what happens to the Newton-Raphson iterations?

 

They converge immediately to the root

 

They diverge and move toward the inflection point

 

They oscillate between two values

 

They converge to x = 5

They diverge and move toward the inflection point

What happens when f’(xn) = 0 in the Newton-Raphson formula?

 

The method converges faster

 

The method becomes more accurate

 

The next iterate cannot be computed (division by zero)

 

The method automatically adjusts

The next iterate cannot be computed (division by zero)

Why can near-zero derivatives cause convergence problems?

 

They make the correction term very small

 

They make the correction term very large

 

They have no effect on convergence

 

They guarantee convergence

They make the correction term very large

What oscillation problem can occur when starting near a local extrema?

 

The method converges to the extrema

 

The method may oscillate between two points in an infinite cycle

 

The method always converges to the correct root

 

The method stops immediately

The method may oscillate between two points in an infinite cycle

For f(x) = x², what type of convergence does Newton-Raphson method exhibit?

 

Quadratic

 

Linear

Superlinear

 

No convergence

Linear

What is the matrix form of a system of linear equations?

 

x=Ab

 

Ax=b

 

A+x=b

 

A−x=b

Ax=b

In civil engineering, what does the equation represent?

 

Stress-strain relationship

 

Global stiffness matrix relationship where K is stiffness, u is displacement, F is load

 

Moment-curvature relationship

 

Force equilibrium

Global stiffness matrix relationship where K is stiffness, u is displacement, F is load

By multiplying A and x directly

 

By scaling the column vectors of A by the components of x and adding them

 

By scaling the row vectors of A

 

By taking the dot product of rows

By scaling the column vectors of A by the components of x and adding them

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22

 

43

 

50

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