Title: Default for OMT 252 New Title
Course: Numerical Analysis

Q: Which is more accurate for approximation of pi: 22/7 or 3.14?
- 22/7 (Try again)
- 3.14 (Try again)
- Both are accurate (Try again)
- both are equally accurate
A: 22/7

Q: Rounding off the number 32.68673 to 4 significant digits, we get the number
- 32.68 (Try again)
- 32.69 (Try again)
- 32.67
- 32.686
A: 32.69

Q: A polynomial of order n has  __________ zeros.
- n+1 (Try again)
- n (Try again)
- at least n (Try again)
- At most n
A: At most n

Q: Let h be the finite difference, then forward difference operator is defined by ____.
- f(x) = f(x+h)-f(x)
- f(x-h)-f(x)
- f(x)-h
- f(x) –f(x-h)
A: f(x) = f(x+h)-f(x)

Q: Find Δ (x + cos x).
- h+2sin(x+h/2) sin h/2
- h - 2sin(x+h/2) sin h/2
- h-2sin(x-h/2) sin h/2
- h+2sin(x-h/2).sin h/2
A: h+2sin(x+h/2) sin h/2

Q: The relationship between E and Δ is __________.
- E =1- Δ
- E =1 + Δ
- E = Δ -1 (E = 2Δ)
A: E =1 + Δ

Q: The highest order of polynomial integrand for which Simpson’s 1/3 rule of integration is exact is
- one
- two (t)
- three
- four
A: three

Q: The three node Gauss quadrature rule of integration is exact for integrating at most ________ order polynomials.
- Second
- Third
- fourth
- fifth
A: fifth

Q: The  estimates of  the definite  the integral of f(x) over the interval (a,b) using  1, 2, and 4 segment trapezoidal rule is given as 5.3460, 2.7708, and 1.7536, respectively. The best estimate of the integral you can find using Romberg integration is most nearly ____________
- 1.3355
- 1.3813
- 1.4145
- 1.9121
A: 1.3813

Q: If for a real continuous function  f,  we have f(a) f(b) < 0   then in the range of  [a,b] for there is (are)
- only one root
- at least one root
- at most one root
- n roots
A: at least one root

Q: The function f(x)= tan(x) has     ___________ root(s).
- one
- two
- an infinite number of
- zero
A: an infinite number of

Q: A function f has a root in the interval [a,b]. If we use the regula falsi procedure to find the root, the process
- will always converge to the root
- may or may not converge depending on the initial estimates
- will converge only when the derivative of the function is less than 1
- will always diverge
A: will always converge to the root

Q: Which of the following is NOT an elementary row operation?
- Multiplying a row by a non zero constant
- Exchanging rows
- adding a constant to the coefficients of a row.
- adding multiple of one row to another row
A: adding a constant to the coefficients of a row.

Q: Using [x_1   x_2   x_3] = [1   3   5] as the initial guess, the value of [x_1   x_2   x_3] after three iterations of Gauss-Seidal method is

 [Image]
- [-2.8333 -1.4333 -1.9727]
- [1.4959 -0.90464 -0.84914]
- [0.90666 -1.0115 -1.0242]
- [1.2148 -0.72060 -0.82451]
A: [-2.8333 -1.4333 -1.9727]

Q: A square matrix A is said to be diagonally dominant if the diagonal element in great row
- has a magnitude greater than the magnitude of each of the other elements in the row
- has a magnitude greater than the sum of the magnitudes of each ot the other elements in the same row
- has a magnitude greater than the magnitude of the sum of the other elements in the same row
- has a magnitude greater than the sum of the lements in the same row
A: has a magnitude greater than the sum of the magnitudes of each ot the other elements in the same row

Q: The dominant eigenvector of a square matrix is
- The eigenvector corresponding the eigenvalue of the matrix with the largest magnitude
- the eigenvector with the largest magnitude of the matrix
- The eigenvalue with the largest magnitude of the matrix
- the eigenvector corresponding to the dominant diagonal element
A: The eigenvector corresponding the eigenvalue of the matrix with the largest magnitude

Q: Write the number 234,967 to four significant figures.
- 2349
- 235,000
- 2350
- 234,900
A: 235,000

Q: The base of the natural logarithm e is estimated as 106/39. What is the relative error of this estimate?
- 0.000331
- -0.000331
- 0.01225
- 0.0001225
A: 0.0001225

Q: Which of these types of errors have the largest effect on numerical computations?
- Blunders
- Round-off errors
- Truncation errors
- Relative errors
A: Blunders

Q: Interpolation formula are based on the assumption that the data
- Can be expressed as a polynomial function
- have equally spaced nodes
- are linearly independent
- have unequally spaced nodes
A: Can be expressed as a polynomial function

Q: If (n+1) pairs of arguments and entries are given, Lagrange’s formula is
- A polynomial of degree n
- A polynomial of at least degree n
- A polynomial of at most degree n
- A polynomial of at least degree n+1
A: A polynomial of at most degree n

Q: One advantage of the Newton divided difference method over the Lagrange method of constructing interpolating polynomials is
- The Newton divided difference method gives more accurate results than the Lagrange method
- The Newton Divided difference method works better for unequally spaced nodes.
- Inclusion of an additional data point does not necessitate recalculation from the beginning
- The error bound in the Newton divided difference method is much less than that of the Lagrange method
A: Inclusion of an additional data point does not necessitate recalculation from the beginning

Q: The error bound of a numerical computation
- is greater or equal to the absolute error
- Is equal to the absolute error
- is greater or equal to the relative error
- is less or equal to the absolute error.
A: is greater or equal to the absolute error

Q: What would happen if we try to use an n-degree polynomial to interpolate n data points?
- This is the normal case. A unique polynomial can be determined.
- Such an interpolant generally exists, but it is not unique.
- A unique solution can still be determined, but it will be more oscillatory than using a n-1 degree polynomial.
- Such an interpolant will generally not exist.
A: Such an interpolant will generally not exist.

Q: Find the divided difference  f [2, 4, 9, 10] in f(x) = x^4 + x^2 + 1
- 25
- -9030
- 13
- 960
A: 25

Q: The second order Newton divided difference interpolation polynomial given here
p_2 (x) = b_0 + b_1 (x-15) + b_2 (x-15)(x-22)  passes through points (15, 24), (18, 37) and (22,25).  The value of b_1 is
- -1.048
- 0.1433
- 4.333
- 24
A: 4.333

Q: The relationship between central difference operator δ and the shift operator E is given by
- δ = E – E^-1
- δ = E + E^-1
- δ = E^1/2 + E^-1/2
- δ = E^1/2- E^-1/2
A: δ = E^1/2- E^-1/2

Q: The value of p in Newton’s backward difference formula is
- p = (x - x_0) ⁄ h
- p = x - x_n
- p = x - x_0
- p = (x – x_n) ⁄ h
A: p = (x – x_n) ⁄ h

Q: For the given distributed data find the value of Δ^3 y_0 .

x | 3.60 | 3.65 | 3.70 | 3.75
y | 36.598 | 38.475 | 40.447 | 42.521
- 0.095
- 0.007
- 1.872
- 0.123
A: 0.007

Q: Bessel’s interpolation formula is a rearrangement  of
- Everett Central Difference formula
- Gauss interpolation formulas
- Stirling Central difference formula
- Newton Forward Difference formula
A: Gauss interpolation formulas

Q: In the forward difference table corresponding to y = x^2, for x =1, 2, 3, 4, 5 and 6, determine Δ^2y_2.
- 0
- 1
- 2
- 4
A: 2

Q: A forward difference table has one entry in Δ^6y. How many entries will be in the column Δ^2y?
- 6
- 2
- 3
- 5
A: 5

Q: Are the operators ∇ and δ commutative?
- Yes
- No
- It depends on the function
- They are left commutative but not necessarily right commutative
A: Yes

Q: In interpolating using a finite difference table we use the Newton's backward difference formula when
- We want to interpolate near the end of a data set.
- We want to interpolate near the center of a data set.
- We want to interpolate near the beginning of a data set.
- We want to interpolate outside the data set.
A: We want to interpolate near the end of a data set.

Q: The following is the Everett interpolation formula:
- q > p
- q < p
- p < q + 1
- p + q = 1
A: p + q = 1

Q: Let D be the differential operator. Let x_0 = 1, h = 1 and f(x) = x^2 + x + 1. Determine  ∇ D f(x_3).
- 0
- 2
- 6
- 3
A: 2

Q: In regula falsi method, the first approximation is given by
- [Image: Q1 choice]
- [Image: Q1 choice 2]
- [Image: Q1 choice 3]
- [Image: Q1 choice 4]
A: [Image: Q1 choice]

Q: If f(a) < 0, f(b) > 0  and if x_0 Є (a,b)is first approximation with f(x_0) < 0  then in bisection method,
- x_0 is to be replaced by a
- a is to be replaced by x_0
- b is to be replaced by x_0
- x_0 is to be replaced by b
A: a is to be replaced by x_0

Q: For an equation like x^2= 0, a root exists at x = 0. The regula falsi method cannot be adopted to solve this equation in spite of the root existing at x=0 because the function f(x) =x^2
- is a polynomial
- has repeated roots at x= 0
- is always non-negative
- has a slope equal to zero at the root
A: is always non-negative

Q: The next iterative value of the root of x^2 – 4 = 0 using secant method, if the initial guesses are 3 and 4, is
- 2.2857
- 2.5000
- 3.5000
- 5.7143
A: 2.2857

Q: The Newton-Raphson iteration x_n + 1 = (x_n/2) + 3/(2x_n) can be used to solve the equation
- x^2 = 3
- x^3 = 3
- x^2 = 2
- x^3 = 2
A: x^2 = 3

Q: The root of the equation f (x) = 0 is found by using the secant method. Given one of the initial estimates is x_0 = 3 , f (3)  = 5, and the angle the secant line makes with the x-axis is 57° , the next estimate of the root, x_1, is
- -3.2470
- -0.2470
- 3.2470
- 6.2470
A: -0.2470

Q: The major difference between the secant method and the false position method is
- false-position method is not guaranteed to converge.
- secant method is guaranteed to converge.
- secant method requires the 2 initial guesses x_-1 and x_0 to satisfy f (x_-1 ) f(x_0) < 0
- false-position method requires the 2 initial guesses x_-1 and x_0 to satisfy f (x_-1 ) f(x_0) < 0
A: false-position method requires the 2 initial guesses x_-1 and x_0 to satisfy f (x_-1 ) f(x_0) < 0

Q: Newton-Raphson method is used to compute a root of the equation x^2 – 13 = 0 with 3.5 as the initial value. The approximation after one iteration i
- 3.575
- 3.676
- 3.667
- 3.607
A: 3.607

Q: The Newton method of finding the root of a multivariate function F has an iteration process of X_n+1 = X_n - H F(X_n). What is H?
- The Jacobian matrix of F
- The reciprocal of the Jacobian matrix of F
- The inverse of the Jacobian matrix of F
- The determinant o the Jacobian matrix of F
A: The inverse of the Jacobian matrix of F

Q: The Newton Multivariate method does not work if
- The Jacobian matrix of F is non singular.
- The derivative of F is equal to zero.
- The initial estimate is equal to the zero vector
- The determinant o the Jacobian matrix of F is zero
A: The Jacobian matrix of F is non singular.

Q: The three point Gauss quadrature formula is exact for polynomials of degree up to
- Three
- Five
- Six
- Four
A: Five

Q: By Romberg method, the value of  if f(0) = 0.25, f(1) = 0.20 and f(2) = 0.125 is
- 0.1833
- 0.1000
- 1.3917
- 1.3875
A: 1.3917

Q: The Simpson’s Rule for estimating the definite integral of a function is exact for polynomials of degree up to
- Three
- Five
- Two
- Four
A: Three

Q: Consider the integration formula

[Image: Question4.4 MCQ]

For what value(s) of α will this formula be exact for all polynomials of degree 1 or less?
- 1
- 0
- -1
- 1/2
A: 1

Q: The error bound of using the trapezoidal rule with two sub-intervals for calculating

[Image: Question 4.45 MCQ]

 is approximately
- 0.0208
- 0.0366
- -0.0366
- 0.2000
A: 0.0208

Q: In the Gauss elimination method for solving a system of linear algebraic equations, triangularization leads to
- Diagonal matrix
- Lower triangular matrix
- Upper triangular matrix
- Singular matrix
A: Upper triangular matrix

Q: If the dimensions of the coefficient matrix corresponding to system of linear equations is 3 × 3 then which of the following will be the dimensions of its decomposed matrices; ‘L’ and ‘U’? Select correct option:
- Dimensions of ‘L’ = 3 × 1, Dimensions of ‘U’ = 1 × 3 1. Dimensions of ‘L’ = 3 × 1, Dimensions of ‘U’ = 1 × 3
- Dimensions of ‘L’ = 3 × 2, Dimensions of ‘U’ = 2 × 3
- Dimensions of ‘L’ = 3 × 3, Dimensions of ‘U’ = 3 × 3
- Dimensions of ‘L’ = 3 × 4, Dimensions of ‘U’ = 4 × 3
A: Dimensions of ‘L’ = 3 × 3, Dimensions of ‘U’ = 3 × 3

Q: In the LU decomposition of the matrix  [Image: 5.3] if the diagonal elements of U are both 1, then the lower diagonal entry l_22 of L is
- 4
- 5
- 6
- 7
A: 5

Q: In Gaussian elimination method, original equations are transformed by using
- Column operations
- Row operations
- Mathematical operation
- Set operation
A: Row operations

Q: In the Gaussian Elimination procedure, the constants which are augmented to those of the coefficient matrix are
- lower constants
- right side constants
- Left side constants
- Upper side constants
A: right side constants

Q: Which of the following statements is true?
- Elementary row operation on an augmented matrix do not change the solution set of the associated linear system.
- Two matrices are equivalent if they have the same number of rows.
- If two linear systems have the same solution set, then they have the same augmented matrix.
- If two linear systems have the same coefficient matrix, then they have the same solution set.
A: Elementary row operation on an augmented matrix do not change the solution set of the associated linear system.

Q: If the dimensions of the coefficient matrix corresponding to system of linear equations is 3 × 3 then which of the following will be the dimensions of its decomposed matrices; ‘L’ and ‘U’? Select correct option:
- Dimensions of ‘L’ = 3 × 1, Dimensions of ‘U’ = 1 × 3
- Dimensions of ‘L’ = 3 × 2, Dimensions of ‘U’ = 2 × 3
- Dimensions of ‘L’ = 3 × 3, Dimensions of ‘U’ = 3 × 3
- Dimensions of ‘L’ = 3 × 4, Dimensions of ‘U’ = 4 × 3
A: Dimensions of ‘L’ = 3 × 3, Dimensions of ‘U’ = 3 × 3

Q: Which of the following row operations on a matrix may not be an elementary operation?
- Replace one row by the sum of itself and other two rows
- Multiply all entries of a row by a number.
- Interchange the first row and the last row.
- Replace one row by the difference of itself and another row.
A: Multiply all entries of a row by a number.

Q: If Gaussian elimination for a matrix equation Ax = b without pivoting encounters a division by zero  this means
- A is singular
- the system has multiple solutions
- the equation is ill-conditioned
- nothing can be concluded about A
A: A is singular

Q: In applying the Gauss elimination method to determine the determinant of a square matrix which of these operations is allowed?
- Multiplying a row by a non zero constant
- Adding a multiple of one row to another row
- Exchanging the positions of two rows
- Transposing the coefficient matrix
A: Adding a multiple of one row to another row

Q: The LU decomposition method is computationally more efficient than Gauss elimination for solving
- a single set of simultaneous linear equations.
- multiple sets of simultaneous linear equations with different coefficient matrices and the same right hand side vectors.
- multiple sets of simultaneous linear equations with the same coefficient matrix and different right hand side vectors.
- multiple sets of simultaneous linear equations.
A: multiple sets of simultaneous linear equations with the same coefficient matrix and different right hand side vectors.

Q: Consider the following system of linear equation: 3x + y – z = 10,   x + 5y + 2z = 18,  x + 4 y + 9z = 16. If  current approximation is x = 3.33,  y = 2.93,  z_  = 0.1, then the Gauss-Seidel method will give next approximation as x = 2.39, y = 3.08 and z =  ………..
- 0.1055
- 0.2102
- 0.1433
- 0.1934
A: 0.1433

Q: The power method for determining the dominant eigenvalue of a square matrix is applicable if the eigenvalues are ______________.
- Real and distinct
- Real and equal
- positive and distinct
- real and independent
A: Real and distinct

Q: The dominant eigenvector of a square matrix is
- The eigenvector with the largest magnitude out of all the eigenvectors of the matrix
- The eigenvector corresponding to the largest eigenvalue of the square matrix
- The eigenvalue with the largest magnitude of all the eigenvalues of the matrix
- The eigenvector corresponding the the eigenvalue with the largest magnitude.
A: The eigenvector corresponding the the eigenvalue with the largest magnitude.

Q: The eigenvectors  of the matrix [Image: capture 6.3]are
- 2 and -1.
- [Image: Capture 6.4a]
- [Image: capture 6.4b]
- 1 and -2
A: [Image: capture 6.4b]

Q: Using [1 3, 5]^T as initial guess, the value of (x, y, z) after three iterations in Gauss Seidel method for  12x +7y + 3z = 2; x +5y +z = -5; 2x +7y -11z = 6; is
- (1.4959,-0.9046,-08491)
- (0.9067, -1.0115, -1.0242)
- (1.0000, -1.0000, -1.0000)
- (1.4954, -0.9045,-1.3449)
A: (0.9067, -1.0115, -1.0242)

Q: Using [1 3, 5]^T as initial guess, the value of (x, y, z) after three iterations in Jacobi method for  12x +7y + 3z = 2; x +5y +z = -5; 2x +7y -11z = 6; is
- (1.4959,-0.9046,-08491)
- (0.9067, -1.0115, -1.0242)
- (1.0000, -1.0000, -1.0000)
- (1.4954, -0.9045,-1.3449)
A: (1.4954, -0.9045,-1.3449)

Q: Carry out three iterations of the power method to determine the dominant eigenvalue of the matrix , [Image: capture 6c]starting with the eigenvector estimate of  [1  1  1]^T.
- 3.4118
- 3.4142
- 2.0000
- 3.3333
A: 3.4118

Q: What is the use of the Raleigh coefficient in the power method for eigenvalues?
- It calculates the eigenvector corresponding to an eigenvalue
- It calculates the eigenvalue corresponding to an eigenvector
- It calculates the dominant eigenvalue in the power method
- It calculates the dominant eigenvector in the power method.
A: It calculates the eigenvalue corresponding to an eigenvector

Q: Which of the following method is a direct numerical method for solving a linear system of equations?
- LU factorization method
- The Cramer's Rule
- The Gauss Seidel Method
- The Jacobi method
A: LU factorization method

Q: Arrange the following system so that it is diagonally dominant.
                 a.            x – 2y + 4z = 20
                 b.            3x +y + z = 24
                 c.             -5x +7y +z = 21

                The arrangement will follow the following order:
- a, b, c
- b, c, a
- c, a, b
- b, a, c
A: b, c, a

Q: By Romberg method, the value of I_1 if f(0) = 0.25, f(1) = 0.20 and f(2) = 0.125 is
- 0.1833
- 0.1000
- 1.3917
- 1.3875
A: 1.3917

Q: Consider the integration formula

[Image: capture 3.4]

For what value(s) of α will this formula be exact for all polynomials of degree 1 or less?
- 1
- 0
- -2
- 1/2
A: 1

Q: The error bound of using the trapezoidal rule with two sub-interval for calculating

 [Image: capture 3.5]

        is approximately
- 0.0208
- 0.0366
- -0.0366
- 0.2000
A: 0.0208

Q: One of the advantage of the Gauss Quadrature method over the Simpson rule in estimating a definite integral is
- The Gauss Quadrature is exact for a polynomial of a higher degree than the Simpson’s rule for the same number of nodes.
- The Gauss Quadrature gives results of a better accuracy than the Simpson’s Rule for the same function.
- The Gauss Quadrature gives also works well when the nodes are not equally spaced as opposed tp Simpson’s Rule that needs equally spaced nodes.
- The Gauss Quadrature method requires fewer computations than the Simpson’s rule.
A: The Gauss Quadrature is exact for a polynomial of a higher degree than the Simpson’s rule for the same number of nodes.

Q: For a definite integral of any third order polynomial, the two-point Gauss quadrature rule will give the same results as the
- 1-segment trapezoidal rule
- 2-segment trapezoidal rule
- 3-segment trapezoidal rule
- Simpson's 1/3 rule
A: Simpson's 1/3 rule

Q: The one point Gauss Quadrature rule for estimating [Image: capture 8.0] is
- [Image: capture 8.1a]
- [Image: capture 8.1b]
- [Image: capture 8.1c]
- [Image: capture 8.1d]
A: [Image: capture 8.1b]

Q: The  estimates of [Image: capture 8.0]  using  1, 2, and 4 segment trapezoidal rule is given as 5.3460, 2.7708, and 1.7536, respectively. The best estimate of the integral you can find using Romberg integration is most nearly
- 1.3355
- 1.3813
- 1.4145
- 1.9124
A: 1.3813

Q: Using trapezoidal rule with  n =5 approximate the integral [Image: capture 10]
- 1.1501
- 1.1058
- 1.2500
- 0.8807
A: 1.1501