Master Polynomial Interpolation: Enhance Your Numerical Analysis Skills

Question 1

Given a set of n distinct data points {(x_i, y_i)} where i = 1 to n, which of the following represents the Lagrange interpolating polynomial of degree n-1 for these data points?

Accepted Answer

P(x) = ∑y_i L_i(x)

Answer

P(x) = ∑x_i L_i(y)

Answer

P(x) = ∏(x - x_i)

Question 2

The Lagrange interpolating polynomial, given data points (x1, y1), (x2, y2), ..., (xn, yn), takes the form:

Accepted Answer

P(x) = Σ[yi * (x - xi) / (xi - xj)] for all i ≠ j

Answer

P(x) = Σ[xi * (y - yi) / (xi - xj)] for all i ≠ j

Answer

P(x) = Σ[yi * (xi - x) / (xi - xj)] for all i ≠ j

Answer

P(x) = Σ[xi * (y - yi) / (xi - xi)] for all i ≠ j

Question 3

Which statement is FALSE regarding the Lagrange interpolating polynomial?

Accepted Answer

It minimizes interpolation error over the entire interval.

Answer

It passes through all provided data points.

Answer

Its degree can be at most n-1, where n represents the number of data points.

Answer

It is a unique polynomial.

Question 4

In Lagrange interpolation, the error term R(x) is defined as:

Accepted Answer

R(x) = f(x) - P(x)

Answer

R(x) = f'(x) - P'(x)

Answer

R(x) = f''(x) - P''(x)

Answer

R(x) = f'''(x) - P'''(x)

Question 5

The order of convergence for the Lagrange interpolating polynomial is:

Accepted Answer

O(h^2)

Answer

O(h)

Answer

O(h^3)

Answer

O(h^4)

Question 6

Given a set of data points (x_i, y_i), the Lagrange form of the interpolating polynomial can be expressed as:

Accepted Answer

L(x) = Σ [y_i * Π(x - x_j) / (x_i - x_j)]

Answer

N(x) = Σ [c_i * x^i]

Answer

L(x) = y_0 + y_1 * (x - x_0) + y_2 * (x - x_0) * (x - x_1) + ...

Answer

N(x) = Σ [y_i * Π(x - x_j) / (x_i - x_j)]

Question 7

If a polynomial of degree n interpolates n + 1 distinct points, then it is referred to as:

Accepted Answer

Exact interpolation

Answer

Natural spline

Answer

Hermite interpolation

Answer

Univariate polynomial

Question 8

The divided difference table is used in:

Accepted Answer

Newton's divided difference form

Answer

Lagrange's interpolation formula

Answer

Spline interpolation

Answer

Gaussian quadrature

Question 9

The degree of the interpolating polynomial that yields the best approximation to a function f(x) is:

Accepted Answer

Less than or equal to the number of data points used

Answer

Equal to the number of data points used

Answer

Greater than the number of data points used

Answer

Dependent on the spacing of the data points

Question 10

Which of the following is NOT a property of Newton's interpolating polynomial?

Accepted Answer

It is independent of the order in which the data points are taken.

Answer

It passes through all the given data points.

Answer

It is unique for a given set of data points.

Answer

Its degree is one less than the number of data points.

Question 11

To determine the order of convergence of polynomial interpolation, the following limit is used:

Accepted Answer

lim (n->∞) ||f(x) - P_n(x)|| / ||f(x) - P_(n+1)(x)||

Answer

lim (n->∞) ||f(x) - P_n(x)|| / ||f'(x) - P'_n(x)||

Answer

lim (n->∞) ||f'(x) - P'_n(x)|| / ||f''(x) - P''_n(x)||

Answer

lim (n->∞) ||f''(x) - P''_n(x)|| / ||f'''(x) - P'''_n(x)||

Question 12

Which of the following is the most suitable interpolation method for data that exhibits sharp changes or discontinuities?

Accepted Answer

Spline interpolation

Answer

Polynomial interpolation

Answer

Trigonometric interpolation

Answer

Rational interpolation

Question 13

Determine the formula for the Newton interpolating polynomial of degree n for data points (x1, y1), (x2, y2), ..., (xn, yn).

Accepted Answer

P(x) = y1 + y2(x - x1) + y3(x - x1)(x - x2) + ... + yn(x - x1)(x - x2)...(x - xn-1)

Answer

P(x) = L1(x) + L2(x) + ... + Ln(x)

Answer

P(x) = (y1 - y2)/(x1 - x2)

Question 14

Which form of polynomial interpolation constructs the interpolating polynomial using the divided difference table?

Accepted Answer

Newton interpolation

Answer

Lagrange interpolation

Answer

Linear interpolation

Answer

Quadratic interpolation

Question 15

Define the divided difference of order k for a function f(x) at nodes x0, x1, ..., xk.

Accepted Answer

The difference between consecutive divided differences of order k-1 divided by the node spacing

Answer

The sum of function values at the nodes divided by the product of the node spacing

Answer

The derivative of the function at the nodes divided by the product of the node spacing

Answer

The integral of the function over the interval [x0, xk] divided by the product of the node spacing

Question 16

In the Newton interpolation formula, what is the expression for the error term?

Accepted Answer

R_n(x) = f^(n+1)(c) * (x - x0)(x - x1)...(x - xn)/n!

Answer

R_n(x) = f^n(c) * (x - x0)(x - x1)...(x - xn)/n!

Answer

R_n(x) = f'(c) * (x - x0)(x - x1)...(x - xn)/n!

Answer

R_n(x) = f(c) * (x - x0)(x - x1)...(x - xn)/n!

Question 17

What is the error term for the Lagrange interpolating polynomial?

Accepted Answer

f(x) - p(x) = f^(n+1)(c) * (x - x_0)(x - x_1)...(x - x_n)/((n+1)!)

Answer

f(x) - p(x) = f^(n+1)(c) * (x - x_1)(x - x_2)...(x - x_n)/((n)!

Answer

f(x) - p(x) = f^(n)(c) * (x - x_0)(x - x_1)...(x - x_n)/((n)!)

Answer

f(x) - p(x) = f^(n)(c) * (x - x_1)(x - x_2)...(x - x_n)/((n+1)!)

Question 18

Which form of the interpolating polynomial is the most efficient for evaluating the polynomial at a given point?

Accepted Answer

Newton form

Answer

Lagrange form

Answer

Both Lagrange and Newton forms are equally efficient

Answer

Neither Lagrange nor Newton form is efficient

Question 19

Which of the following can be used to find the divided differences for Newton interpolation?

Accepted Answer

Any of the above

Answer

Forward differences

Answer

Backward differences

Answer

Central differences

Question 20

What is the main advantage of using polynomial interpolation?

Accepted Answer

It provides an approximate value of the function at any point

Answer

It can be used to solve differential equations

Answer

It can be used to find roots of equations

Answer

It can be used to integrate functions

Question 21

Given the data points {(0,2), (1,3), (2,6)}, determine the Lagrange interpolating polynomial.

Accepted Answer

2 + x + x^2

Answer

2 - x + x^2

Answer

1 + 2x + x^3

Answer

1 - 2x - x^2

Question 22

State the error bound for the Lagrange interpolating polynomial.

Accepted Answer

f(x) - P_n(x) <= (M/(n+1)) * |x - x_0| * |x - x_1| * ... * |x - x_n|, where M is the maximum value of the (n+1)th derivative of f(x) on the interval containing the data points.

Answer

f(x) - P_n(x) <= (M/(n+1)^2) * |x - x_0| * |x - x_1| * ... * |x - x_n|

Answer

f(x) - P_n(x) <= (M/(2n+1)) * |x - x_0| * |x - x_1| * ... * |x - x_n|

Question 23

Indicate the order of convergence for polynomial interpolation.

Accepted Answer

n+1

Answer

n

Answer

2n

Answer

n^2

Question 24

Identify which of the following is NOT an advantage of polynomial interpolation.

Accepted Answer

It can accurately approximate functions with sharp corners or discontinuities.

Answer

It is straightforward to evaluate and implement.

Answer

It can provide precise approximations for smooth functions.

Question 25

In which scenario is polynomial interpolation most appropriate?

Accepted Answer

Approximating a smooth function over a closed interval where the function values are known at discrete points.

Answer

Approximating a function with sharp corners or discontinuities.

Question 26

Which method for constructing interpolating polynomials utilizes divided differences?

Accepted Answer

Newton's method

Answer

Lagrange's method

Answer

Gregory-Newton backward-difference formula

Answer

Gregory-Newton forward-difference formula

Question 27

Which of the following is the formula for Lagrange interpolation?

Accepted Answer

P(x) = ∑[yᵢ * ∏(x - xⱼ) / (xᵢ - xⱼ)]

Answer

P(x) = ∑[yᵢ * (x - xᵢ)] / ∑[xᵢ * (x - xᵢ)]

Answer

P(x) = ∑[yᵢ * ∏(x + xⱼ) / (xᵢ + xⱼ)]

Answer

P(x) = ∑[yᵢ * (x + xᵢ)] / ∑[xᵢ * (x + xᵢ)]

Question 28

What is the error bound for polynomial interpolation?

Accepted Answer

E(x) ≤ (M * |x - x₀|ⁿ) / (n!)

Answer

E(x) ≤ (M * |x - x₀|ⁿ) / (n)

Answer

E(x) ≤ (M * |x - x₀|) / (n)

Question 29

Which of the following is NOT a property of polynomial interpolation?

Accepted Answer

The error between the interpolating polynomial and the true function is always zero.

Answer

The interpolating polynomial passes through all the given data points.

Answer

The degree of the interpolating polynomial is always less than or equal to the number of data points.

Question 30

Which of the following is an application of polynomial interpolation?

Accepted Answer

Approximating the value of a function at a given point

Answer

Solving partial differential equations

Answer

Finding the roots of a non-polynomial equation

Question 31

What is the purpose of divided differences in polynomial interpolation?

Accepted Answer

To simplify the calculation of the coefficients of the interpolating polynomial

Answer

To improve the accuracy of the interpolating polynomial

Answer

To reduce the computational cost of interpolation

Question 32

What is the condition for the existence and uniqueness of a polynomial interpolant?

Accepted Answer

The data points must be distinct

Answer

The interpolating polynomial must be continuous

Answer

The interpolating polynomial must be differentiable

Answer

The degree of the polynomial must be less than the number of data points

Question 33

Consider the following table of values for f(x):
| x | f(x) |
| --- | --- |
| 0 | 1 |
| 1 | 2 |
| 2 | 5 |
| 3 | 10 |
Use Lagrange's interpolation formula to find the interpolating polynomial.

Accepted Answer

P(x) = 1 + x + 2x^2

Answer

P(x) = 2 + x + x^2

Answer

P(x) = 1 + 2x + x^2

Answer

P(x) = 1 + x + 3x^2

Question 34

Which of the following Lagrange basis polynomials is correct for the interpolation points x_0, x_1, ..., x_n?

Accepted Answer

L_i(x) = ∏_{j=0, j≠i}^n (x - x_j) / ∏_{j=0, j≠i}^n (x_i - x_j)

Answer

L_i(x) = ∏_{j=0, j≠i}^n (x + x_j) / ∏_{j=0, j≠i}^n (x_i + x_j)

Answer

L_i(x) = ∑_{j=0, j≠i}^n (x - x_j) * ∏_{k=0, k≠j}^n (x_i - x_k)

Question 35

State the error term for polynomial interpolation of degree n.

Accepted Answer

f(x) - p_n(x) = (f^(n+1)(c) / (n+1)!) * (x - x_0)(x - x_1)...(x - x_n), where c is in [min(x_0, x_1, ..., x_n), max(x_0, x_1, ..., x_n)]

Answer

f(x) - p_n(x) = (f^(n)(c) / n!) * (x - x_0)(x - x_1)...(x - x_n), where c is in [min(x_0, x_1, ..., x_n), max(x_0, x_1, ..., x_n)]

Question 36

Which of the following is an advantage of using divided differences in polynomial interpolation?

Accepted Answer

Divided differences allow for efficient evaluation of the interpolating polynomial at new points.

Answer

Divided differences are numerically unstable.

Answer

Divided differences can only be used for interpolating polynomials of degree 1.

Question 37

Define the divided difference of order 2 for x_0, x_1, and x_2.

Accepted Answer

[x_2, x_1, x_0] = ([x_2, x_1] - [x_1, x_0]) / (x_2 - x_0)

Answer

[x_2, x_1, x_0] = ([x_2, x_1] + [x_1, x_0]) / (x_2 + x_0)

Answer

[x_2, x_1, x_0] = ([x_2, x_1] * [x_1, x_0]) / (x_2 * x_0)

Question 38

What is the degree of the interpolating polynomial that passes through n distinct points?

Accepted Answer

n-1

Answer

n/2

Answer

n

Answer

n+1

Question 39

Which of the following is a correct form of Newton's divided difference formula?

Accepted Answer

f[x_0, x_1, ..., x_n] = (f[x_1, ..., x_n] - f[x_0, ..., x_{n-1}]) / (x_n - x_0)

Answer

f[x_0, x_1, ..., x_n] = (f[x_1, ..., x_n] + f[x_0, ..., x_{n-1}]) / (x_n + x_0)

Answer

f[x_0, x_1, ..., x_n] = (f[x_1, ..., x_n] * f[x_0, ..., x_{n-1}]) / (x_n * x_0)