6/9/2021 Midterm Exam

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Midterm Exam

Question 1
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What is the next approximated root for the function f(x) = x + cos(x) when x0 = 0?

Select one:
a. -0.7391

b. -1

c. 0

d. 1

Clear my choice

Question 2
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For the given systems of linear equations, with initial values x1 =0;  x2 =0; x3 = 0. The next iterative value of x2 using Gauss-Seidel Method is
1.5. 

Select one:
True

False

Question 3
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If A−1 (if it exists) the eigenvalues of A−1 is 1/5, ¼ and ½ if matrix A  has eigenvalues 5, 4 and 2.

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True

False

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Question 4
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Which of the following statements is true for the function f(x) = -e x + sin x, when 0 is used as an initial value:

Select one:
a. Newton’s method cannot be used

b. Newton’s method is divergent

c. Newton’s method can be used since the function is “smooth”

d. Newton’s method is convergent

Clear my choice

Question 5
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Secant method replaces the tangent in Newton’s method to the slope of the function using two initial guesses.

Select one:
True

False

Question 6
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A Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose and can be decomposed
using Cholesky’s method.

Select one:
True

False

Question 7
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Triangular matrices have their eigenvalues on the diagonal of the matrix therefore the eigenvalues of A are the diagonal elements.

Select one:
True

False


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Question 8
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Roots of transcendental functions are easily approximated using Newton’s method provided that f’(x) ≠ 0.

Select one:
True

False

Question 9
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The tangent line is projected to approximate the root of the function where it crosses the

Select one:
a. function at its lowest point 

b. origin 

c. x axis 

d. y axis 

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Question 10
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Methods that uses a single initial value or two initial values that do not necessarily brackets the root where Newton’s method is categorized
are called

Select one:
a. Iterative method

b. Closed method

c. Open method

d. Non-bracketing method

Clear my choice

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Question 11
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The goal in using Newton’s method is the   When choosing an initial value, a good guess is :

1. A value which when substituted to the function will give a near zero value
2. A value with f ’(x) ≠ 0
3. Always starting with 0

Select one:
a. All of the answers correct

b. "A value which when substituted to the function will give a near zero value" and "Always starting with 0" are correct. 

c. "A value which when substituted to the function will give a near zero value" is correct 

d. "A value which when substituted to the function will give a near zero value" and "A value with f ’(x) ≠ 0" are correct. 

Clear my choice

Question 12
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The eigenvalues of A =  have no rational values because of the zero element in the matrix.

Select one:
True

False

Question 13
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The largest eigenvalue of A−1  is the smallest eigenvalue of A in magnitude.

Select one:
True

False

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Question 14
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In general, an n × n matrix will have a characteristic polynomial of degree of n+ 1, and its roots are the eigenvalues of A.

Select one:
True

False

Question 15
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If QT is the transpose of Q then QT Q = I or the identity matrix.

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True

False

Question 16
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Just like the Newton-Raphson method, the initial guesses affect the convergence of the Secant method.

Select one:
True

False

Question 17
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In the case of the tridiagonal system strict diagonal dominance means simply that (with a0 = an = 0) 

Select one:
True

False

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Question 18
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One of the advantages of secant method over the Newton’s Method is the use of derivatives.

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True

False

Question 19
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A square matrix is said to be diagonally dominant if for every row of the matrix, the magnitude of the diagonal entry in a row is less than or
equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row.

Select one:
True

False

Question 20
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The secant method can fail to find a root of a nonlinear function that has a small slope near the root assures the presence of the root.

Select one:
True

False

Question 21
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If x2 is the approximated root in Secant method, it follows that; the value of f(x2) must be equal to 0.

Select one:
True

False

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Question 22
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The secant method is almost similar to the concept of False-position in the bracketing methods as it uses two initial approximations
however secant doesn’t bother with the bracketing.

Select one:
True

False

Question 23
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If matrix A gives the largest eigenvalue, it suggests that if A -1 exists, the smallest eigenvalue can be obtained through inverse power
method.

Select one:
True

False

Question 24
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Eigenvalues are used in the analysis of linear transformation such as scaling.

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True

False

Question 25
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For the function f(x) = ex-2 the value of f(x2) when x0 = -1 and x1 = 1 is f(x) =  -0.5248

Select one:
True

False

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Question 26
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The factorization in Cholesky’s Method can be generated efficiently by recurrence relations.

Select one:
True

False

Question 27
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Given a matrix A , its QR -decomposition is an upper triangular matrix and an orthogonal matrix. An orthogonal matrix is a matrix whose
transpose is equivalent to its inverse.

Select one:
True

False

Question 28
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One of the advantages of Newton’s method is that its converges fast even if the initial guess was poorly chosen

Select one:
True

False

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Question 29
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Newton’s Method is ideal to function which is 

1. Differentiable also known as a “smooth” function

Select one:
a. "Differentiable also known as a “smooth” function" is correct 

b. Both of "Differentiable also known as a “smooth” function" and "Containing multiple roots." are correct 

c. Both of "Differentiable also known as a “smooth” function" and "Transcendental or that which cannot be expressed in finite number of
terms." are correct 

d. All of the answers correct 

Clear my choice

Question 30
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The iteration may terminate if the difference between approximated values of x is already zero.

Select one:
True

False

Question 31
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Newton’s method and secant method has almost the same concept and are both fast.

Select one:
True

False

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Question 32
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The Secant method is convergent for the function f(x) = 3x4 – x -3 whose initial values are present between x0 = 0 and x1 = -1 

Select one:
True

False

Question 33
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The absolute value of the ratio of   is x0 = 0 is 8

Select one:
True

False

Question 34
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An additional benefit of the power method is that the corresponding eigenvector is obtained as a by-product of the method.

Select one:
True

False

Question 35
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For  is the iteration may terminate if the difference between f (x)  is already zero.

Select one:
True

False

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Question 36
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The function f(x) = x3 -5 with initial guesses x0 and x1 would converge.

Select one:
True

False

Question 37
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The factorization in Cholesky’s Method can be generated efficiently by recurrence relations.

Select one:
True

False

Question 38
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The characteristic polynomial formed from the matrix

is given as:  -λ3+3λ2-4λ+4

Select one:
True

False

Question 39
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It is impossible to find the complex roots of a polynomial function, using Newton’s Method” is:

Select one:
True

False

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Question 40
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If matrix A is invertible such that A−1 = L−TL−1 then matrix A can be decomposed using Cholesky’s method.

Select one:
True

False

Question 41
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Direct method for finding the eigenvalues is recommended since the calculation of zeros of a polynomial is numerically challenging if not
unstable.

Select one:
True

False

Question 42
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What is the next approximated root for the function f(x) = ex + sin(x) when x0 = 0?

Select one:
a. Invalid initial value

b. 1

c. -0.5

d. 0.5

Clear my choice

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6/9/2021 Midterm Exam

Question 43
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The eigenvalues of A =   are λ = 0 and λ = -3.

Select one:
True

False

Question 44
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The iteration in Secant method may terminate if the difference between two successive approximations equal to zero.

Select one:
True

False

Question 45
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The approximated root using the secant method lies within the two initial points which is used to project the secant line.

Select one:
True

False

Question 46
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Choosing an initial guess which gives near f(x) = 0 is considered a good guess

Select one:
True

False

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Question 47
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If the magnitude of the diagonals is greater than the sum of the non-diagonals in the same row, then the matrix is not diagonally dominant.

Select one:
True

False

Question 48
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The Secant method is convergent for the function f(x) = 3x4 – x -3 whose initial values are present between x0 = 2 and x1 = 4 

Select one:
True

False

Question 49
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Which of the following  is the best initial guess for the function

f(x) = e x – sin x

Select one:
a. Π

b. 0

c. 1

d. 0.5

Clear my choice

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6/9/2021 Midterm Exam

Question 50
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For the function  f(x) =its first derivative is  f’(x) is   The first derivative of the function   is f’(x) = 

Select one:
True

False

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W10: Lesson 7 - Numerical Differentiation and Integration - Module ►

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