Numerical Methods Terms Questions

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Emmanuel

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問題一覧

What is the main purpose of the graphical method in root finding?

To visualize and approximate the location of the root

In the graphical method, the root of a function is identified as:

The point where the graph crosses the x-axis

Which of the following is a limitation of the graphical method in bracketing techniques?

It may not provide sufficient accuracy for precise computation

What is the primary condition for using the Bisection Method to solve for the root of an equation, f(x)=0?

The function must be continuous on the interval [a, b], and the function values at the endpoints, f(a) and f(b), must have opposite signs.

What happens to the interval [a, b] after each iteration in the Bisection Method?

It is halved

The Bisection Method is based on which of the following principles?

Intermediate Value Theorem

The Bisection Method continues until:

The error is less than a specified tolerance

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

Always converges if initial interval is correct

What is the order of convergence of the Bisection Method?

Linear

The False-Position Method uses which of the following to approximate the root?

The secant line between two points

What distinguishes the False-Position Method from the Bisection Method?

False-Position uses function values to weight the interval ends

What can be a disadvantage of the False-Position Method compared to Bisection?

One endpoint may remain fixed, slowing convergence

What is the order of convergence of the False-Position Method?

Linear

Which of the following is an advantage of Simple Fixed-Point Iteration?

It doesn't require derivatives

The Newton-Raphson method requires which of the following to be known?

Both f(x) and f'(x)

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

Quadratic

The Newton-Raphson method may fail to converge if:

The derivative is too small or zero at any iteration

Which of the following is a major advantage of the Newton–Raphson method?

It has a fast rate of convergence

If the initial guess is far from the actual root, Newton–Raphson’s solution:

May diverge or converge to a wrong root

What is a key difference between the Secant Method and the Newton-Raphson Method?

Secant method does not require derivative

A potential drawback of the Secant Method is:

May fail if f(x_n) = f(x_{n-1})

Which of the following numerical methods for root finding requires three initial approximations to begin the iteration?

Muller's Method

Muller's method is particularly useful for finding roots that are:

complex or multiple

Chebyshev's method for root finding has a rate of convergence of order:

Cubic

What is a main disadvantage of Chebyshev's method compared to Newton's method?

It requires calculating higher-order derivatives.

Aitken's Δ2 method is primarily used for:

Accelerating the convergence of a sequence.

Gauss Elimination transforms a system of equations into:

An upper triangular matrix

The last step in Gauss Elimination is known as:

Back substitution

The Gauss-Jordan method eliminates variables to form:

Reduced row echelon form (identity matrix)

Compared to Gauss Elimination, Gauss-Jordan requires:

No back substitution

For a symmetric, positive-definite matrix A, which decomposition method is the most computationally efficient?

Cholesky decomposition

A key characteristic of Crout's method for LU decomposition, A = LU, is that the diagonal elements of which matrix are set to 1?

Which of the following is a primary reason for using LU decomposition methods instead of direct Gauss elimination when solving a system of linear equations?

They are more efficient when solving for multiple right-hand side vectors b.

Which of these non-iterative methods is a special case of LU decomposition where A = LL^T?

Cholesky decomposition

For a system Ax=b, what is the first step in using Crout's method to find the solution?

Decompose matrix A into L and U matrices.

In the context of non-iterative methods, what is the purpose of pivoting?

To avoid division by zero and minimize rounding errors.

Which of the following describes a scenario where Cholesky decomposition is the most suitable method for solving a linear system Ax=b?

When A is symmetric and positive-definite.

Which of the following is a condition that must be met for Cholesky decomposition to be a valid method for a matrix A?

The matrix must be symmetric and positive-definite.

The LU decomposition of a matrix A can be a single solution for solving Ax=b. What is the primary reason why it is not a 'one-step' solution?

You must perform both forward and backward substitutions.

In the Jacobi method, to compute the next iteration values, we:

Use old values from the previous iteration only

The Jacobi method converges if the matrix is:

Diagonally dominant

The Gauss-Seidel method differs from Jacobi in that it:

Uses newly computed values immediately

For Gauss-Seidel to converge, the matrix should preferably be:

Diagonally dominant

Which of the following methods is non-iterative?

Gauss Elimination

Which method can be used efficiently for large, sparse systems?

Jacobi or Gauss-Seidel

Which of the following statements is TRUE regarding iterative and direct methods for solving linear systems?

Iterative methods are generally preferred for large sparse systems due to lower memory usage

Which numerical differentiation method is generally considered the most accurate for a given step size h, assuming the function is smooth and evaluated at three points?

Central difference

If the step size h is halved, the truncation error of the central difference approximation for the first derivative will be approximately:

Quartered

The primary source of error in numerical differentiation using finite difference formulas is:

Truncation error

For a given function, as the step size h approaches zero, the forward difference approximation for the first derivative will:

Approach the true value of the derivative.

When would you typically use a backward difference formula instead of a forward or central difference one?

At the end of a data set where there are no points available after the last data point.

What is the primary advantage of using the central difference method over the forward and backward difference methods?

It is more accurate for the same step size h.

The Trapezoidal Rule for numerical integration approximates the area under the curve by using:

Straight lines

Which of the following is an advantage of using the composite Trapezoidal Rule over the simple Trapezoidal Rule?

It provides a more accurate approximation for a given function.

Simpson's 1/3 Rule is derived by approximating the integrand with a:

Second-degree polynomial (quadratic)

A key condition for applying Simpson's 1/3 Rule to a composite interval is that the number of subintervals must be:

Even

What is the degree of a polynomial for which the Trapezoidal Rule gives an exact result?

Degree 1 (linear)

The main advantage of Gaussian Quadrature over Newton-Cotes formulas is that it:

Is more accurate for the same number of function evaluations.

The error in the Composite Trapezoidal Rule is inversely proportional to:

h^2

Which numerical integration method would be the most efficient for finding the integral of a high-degree polynomial function on a standard interval?

Gauss-Legendre Quadrature

Which of the following is a key difference between Newton-Cotes formulas and Gaussian Quadrature?

Newton-Cotes formulas require equally spaced nodes, while Gaussian Quadrature uses unequally spaced nodes.

Which of the following numerical methods for solving an initial value problem is classified as a single-step method?

Euler's method

What is the primary drawback of using Euler's method to solve a differential equation?

Its low accuracy, as the error is proportional to the step size.

The fourth-order Runge-Kutta (RK4) method improves upon Euler's method by:

Taking a weighted average of four different slope estimates.

Which of the following is an example of a predictor-corrector method?

Adams-Bashforth-Moulton method

An implicit multi-step method, such as the Adams-Moulton method, is generally preferred over an explicit one, such as the Adams-Bashforth method, because it is:

More accurate and more stable.

Which method would be the most suitable to use as a 'starter' for a multi-step method?

Runge-Kutta 4 (RK4) method

When applying Euler's method, if you halve the step size h, what would be the approximate effect on the global truncation error?

It would be halved.

Euler's method can be derived directly from the first two terms of a:

Taylor series expansion

Which of the following describes a 'stiff' ordinary differential equation?

Its solution contains components that decay at vastly different rates.

Which of these methods is generally the most suitable for solving a stiff differential equation?

An implicit method like the Backward Euler method

A predictor-corrector method works by:

Estimating the next value using an explicit formula and then refining that estimate using an implicit formula.

What is the primary trade-off when using a higher-order method (like RK4) instead of a lower-order method (like Euler's)?

Higher accuracy for more computational cost per step.

Which type of numerical ODE solver is most suitable for problems where the solution needs to be found at irregular, non-equally spaced time points?

Single-step methods

Which of the following is the fundamental principle behind Euler's method?

Approximating the solution curve with a series of tangent lines.

When compared to Euler's method, the Runge-Kutta methods are generally considered superior due to their:

Significantly higher accuracy for a given step size.

問題一覧

What is the main purpose of the graphical method in root finding?

To visualize and approximate the location of the root

In the graphical method, the root of a function is identified as:

The point where the graph crosses the x-axis

Which of the following is a limitation of the graphical method in bracketing techniques?

It may not provide sufficient accuracy for precise computation

What is the primary condition for using the Bisection Method to solve for the root of an equation, f(x)=0?

The function must be continuous on the interval [a, b], and the function values at the endpoints, f(a) and f(b), must have opposite signs.

What happens to the interval [a, b] after each iteration in the Bisection Method?

It is halved

The Bisection Method is based on which of the following principles?

Intermediate Value Theorem

The Bisection Method continues until:

The error is less than a specified tolerance

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

Always converges if initial interval is correct

What is the order of convergence of the Bisection Method?

Linear

The False-Position Method uses which of the following to approximate the root?

The secant line between two points

What distinguishes the False-Position Method from the Bisection Method?

False-Position uses function values to weight the interval ends

What can be a disadvantage of the False-Position Method compared to Bisection?

One endpoint may remain fixed, slowing convergence

What is the order of convergence of the False-Position Method?

Linear

Which of the following is an advantage of Simple Fixed-Point Iteration?

It doesn't require derivatives

The Newton-Raphson method requires which of the following to be known?

Both f(x) and f'(x)

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

Quadratic

The Newton-Raphson method may fail to converge if:

The derivative is too small or zero at any iteration

Which of the following is a major advantage of the Newton–Raphson method?

It has a fast rate of convergence

If the initial guess is far from the actual root, Newton–Raphson’s solution:

May diverge or converge to a wrong root

What is a key difference between the Secant Method and the Newton-Raphson Method?

Secant method does not require derivative

A potential drawback of the Secant Method is:

May fail if f(x_n) = f(x_{n-1})

Which of the following numerical methods for root finding requires three initial approximations to begin the iteration?

Muller's Method

Muller's method is particularly useful for finding roots that are:

complex or multiple

Chebyshev's method for root finding has a rate of convergence of order:

Cubic

What is a main disadvantage of Chebyshev's method compared to Newton's method?

It requires calculating higher-order derivatives.

Aitken's Δ2 method is primarily used for:

Accelerating the convergence of a sequence.

Gauss Elimination transforms a system of equations into:

An upper triangular matrix

The last step in Gauss Elimination is known as:

Back substitution

The Gauss-Jordan method eliminates variables to form:

Reduced row echelon form (identity matrix)

Compared to Gauss Elimination, Gauss-Jordan requires:

No back substitution

For a symmetric, positive-definite matrix A, which decomposition method is the most computationally efficient?

Cholesky decomposition

A key characteristic of Crout's method for LU decomposition, A = LU, is that the diagonal elements of which matrix are set to 1?

Which of the following is a primary reason for using LU decomposition methods instead of direct Gauss elimination when solving a system of linear equations?

They are more efficient when solving for multiple right-hand side vectors b.

Which of these non-iterative methods is a special case of LU decomposition where A = LL^T?

Cholesky decomposition

For a system Ax=b, what is the first step in using Crout's method to find the solution?

Decompose matrix A into L and U matrices.

In the context of non-iterative methods, what is the purpose of pivoting?

To avoid division by zero and minimize rounding errors.

Which of the following describes a scenario where Cholesky decomposition is the most suitable method for solving a linear system Ax=b?

When A is symmetric and positive-definite.

Which of the following is a condition that must be met for Cholesky decomposition to be a valid method for a matrix A?

The matrix must be symmetric and positive-definite.

The LU decomposition of a matrix A can be a single solution for solving Ax=b. What is the primary reason why it is not a 'one-step' solution?

You must perform both forward and backward substitutions.

In the Jacobi method, to compute the next iteration values, we:

Use old values from the previous iteration only

The Jacobi method converges if the matrix is:

Diagonally dominant

The Gauss-Seidel method differs from Jacobi in that it:

Uses newly computed values immediately

For Gauss-Seidel to converge, the matrix should preferably be:

Diagonally dominant

Which of the following methods is non-iterative?

Gauss Elimination

Which method can be used efficiently for large, sparse systems?

Jacobi or Gauss-Seidel

Which of the following statements is TRUE regarding iterative and direct methods for solving linear systems?

Iterative methods are generally preferred for large sparse systems due to lower memory usage

Which numerical differentiation method is generally considered the most accurate for a given step size h, assuming the function is smooth and evaluated at three points?

Central difference

If the step size h is halved, the truncation error of the central difference approximation for the first derivative will be approximately:

Quartered

The primary source of error in numerical differentiation using finite difference formulas is:

Truncation error

For a given function, as the step size h approaches zero, the forward difference approximation for the first derivative will:

Approach the true value of the derivative.

When would you typically use a backward difference formula instead of a forward or central difference one?

At the end of a data set where there are no points available after the last data point.

What is the primary advantage of using the central difference method over the forward and backward difference methods?

It is more accurate for the same step size h.

The Trapezoidal Rule for numerical integration approximates the area under the curve by using:

Straight lines

Which of the following is an advantage of using the composite Trapezoidal Rule over the simple Trapezoidal Rule?

It provides a more accurate approximation for a given function.

Simpson's 1/3 Rule is derived by approximating the integrand with a:

Second-degree polynomial (quadratic)

A key condition for applying Simpson's 1/3 Rule to a composite interval is that the number of subintervals must be:

Even

What is the degree of a polynomial for which the Trapezoidal Rule gives an exact result?

Degree 1 (linear)

The main advantage of Gaussian Quadrature over Newton-Cotes formulas is that it:

Is more accurate for the same number of function evaluations.

The error in the Composite Trapezoidal Rule is inversely proportional to:

h^2

Which numerical integration method would be the most efficient for finding the integral of a high-degree polynomial function on a standard interval?

Gauss-Legendre Quadrature

Which of the following is a key difference between Newton-Cotes formulas and Gaussian Quadrature?

Newton-Cotes formulas require equally spaced nodes, while Gaussian Quadrature uses unequally spaced nodes.

Which of the following numerical methods for solving an initial value problem is classified as a single-step method?

Euler's method

What is the primary drawback of using Euler's method to solve a differential equation?

Its low accuracy, as the error is proportional to the step size.

The fourth-order Runge-Kutta (RK4) method improves upon Euler's method by:

Taking a weighted average of four different slope estimates.

Which of the following is an example of a predictor-corrector method?

Adams-Bashforth-Moulton method

An implicit multi-step method, such as the Adams-Moulton method, is generally preferred over an explicit one, such as the Adams-Bashforth method, because it is:

More accurate and more stable.

Which method would be the most suitable to use as a 'starter' for a multi-step method?

Runge-Kutta 4 (RK4) method

When applying Euler's method, if you halve the step size h, what would be the approximate effect on the global truncation error?

It would be halved.

Euler's method can be derived directly from the first two terms of a:

Taylor series expansion

Which of the following describes a 'stiff' ordinary differential equation?

Its solution contains components that decay at vastly different rates.

Which of these methods is generally the most suitable for solving a stiff differential equation?

An implicit method like the Backward Euler method

A predictor-corrector method works by:

Estimating the next value using an explicit formula and then refining that estimate using an implicit formula.

What is the primary trade-off when using a higher-order method (like RK4) instead of a lower-order method (like Euler's)?

Higher accuracy for more computational cost per step.

Which type of numerical ODE solver is most suitable for problems where the solution needs to be found at irregular, non-equally spaced time points?

Single-step methods

Which of the following is the fundamental principle behind Euler's method?

Approximating the solution curve with a series of tangent lines.

When compared to Euler's method, the Runge-Kutta methods are generally considered superior due to their:

Significantly higher accuracy for a given step size.