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Numerical Methods

Showing 151-225 of 256 answers

Since matrices are used to represent properties of images, it follows that transformation of images may use eigenvalues and eigenvectors to do that.
  • True Correct
  • False
Some elementary functions simply do not have antiderivatives that are_______________ where Trapezoidal and Simpson's 1/3 rule can be more useful
  • positive definite
  • rational
  • constants
  • elementary functions Correct
Suppose a computing machine can only display up to 4 decimal places. Assuming that the true value of π is 3.14159265359. Using an approximate value of πa =  3.1416  Calculate the absolute error and the relative error.
  • ɛa = 7346410 x 10-7, %ɛ = 2 3384 x 10 -6 Correct
  • ɛa = 728754 x 10-7, %ɛ = 22264 x 10 -6
  • ɛa = 74132 x 10-7, %ɛ = 28243 x 10 -6
  • ɛa = 71126 x 10-7, %ɛ = 28124 x 10 -6
Suppose we do not know that the true value of the root of f(x) = x3 -1 is 1. How many iterations will be used to get the true value suppose the initial value of x = 0?
  • 2
  • 1 Correct
  • 4
  • 3
The absolute error of the function f(x) =  e x when the the true value of f(x) = 2.718281828 compared to the approximated value of using the first five terms of the Maclaurin Series center at when x = 1,  c =0 is _____.
  • 883 x10-3
  • 995 x10-3 Correct
  • 667 x10-3
  • 335 x10-3
The absolute value of the ratio of   is x0 = 0 is 8
  • True Correct
  • False
The advantage of quadratic formula over the Newton's method is that
  • it can get multiple root
  • both "it can get complex root" and "it can get multiple root" are correct Correct
  • it can get complex root
  • neither "it can get complex root" nor "it can get multiple root" are correct
The approximate value of f(x) =  e x using the first five terms of the Maclaurin Series center at when x = 1,  c =0 is
  • 2708 Correct
  • 2525
  • 2803
  • 2656
The approximated root using the secant method lies within the two initial points which is used to project the secant line.
  • True Correct
  • False
The Average case occurs in linear search algorithm when item is the last element in the array
  • True
  • False Correct
The characteristic polynomial formed from the matrix is given as:  -λ3+3λ2-4λ+4
  • True Correct
  • False
The characteristic polynomial formed from the matrixis given as: -=CE=BB3+3=CE=BB2-4=CE=BB+4
  • True Correct
  • False
The characteristic polynomial involves the calculation of zeros. These values of the roots are the eigenvalues.
  • True Correct
  • False
The Cholesky factorization for the sample matrix given above is:
  • True Correct
  • False
The coefficient a0 is also equal to f(x0) in the Newton's interpolating polynomial
  • True Correct
  • False
The coefficients of the Newton's interpolating polynomial can also be expressed in terms of divided difference
  • True Correct
  • False
The column vector of the constant replaces the column of the unknown to find the determinant of the variable in question and divide it to the determinant of the coefficient matrix
  • True Correct
  • False
The commutative property also exist in matrices.
  • True
  • False Correct
The complexity of sorting algorithm measures the running time as a function of the number n of items to be sorter
  • True Correct
  • False
The degree of the polynomial for the 10 sample values or data points is equal to 10
  • True
  • False Correct
The determinant of an identity matrix is equal to 
  • 0
  • 1 Correct
  • Infinite
  • -1
The determinant of the given matrix is D = |-5|.
  • No correct answer Correct
  • 5
  • 4
  • 2
The determinant of the given matrix is equal to |D| = 27.
  • No correct answer Correct
  • 5
  • 4
  • 2
The eigenvalues of A =   are λ = 0 and λ = -3.
  • True Correct
  • False
The eigenvalues of A =  have no rational values because of the zero element in the matrix.
  • True Correct
  • False
The eigenvalues of A = are =CE=BB = 0 and =CE=BB = -3
  • True Correct
  • False
The eigenvalues that corresponds to the characteristic polynomial are =CE=BB2-4=CE=BB+3 are =CE=BB = 1 and =CE=BB = -3
  • True
  • False Correct
The eigenvalues that corresponds to the characteristic polynomial are λ2-4λ+3 are λ = 1 and λ = -3.
  • True Correct
  • False
The eigenvectors of A =  and  are the same.
  • True
  • False Correct
The elements of both the coefficient matrix and determinant are the same, and so is their the mathematical concept.
  • True
  • False Correct
The essential features of a physical system or process in mathematical terms should be carried out in the formulation of a mathematical model.
  • True Correct
  • False
The factorization in Cholesky’s Method can be generated efficiently by recurrence relations.
  • True Correct
  • False
The function f(x) = x3 -5 with initial guesses x0 and x1 would converge.
  • True Correct
  • False
The given matrix above is a symmetric matrix.
  • True Correct
  • False
The goal in using Newton's method is the When choosing an initial value, a good guess is :A value which when substituted to the function will give a near zero valueA value with f '(x) =E2=89=A0 0Always starting with 0
  • "A value which when substituted to the function will give a near zero value" and "A value with f '(x) =E2=89=A0 0" are correct Correct
  • All of the answers correct
  • "A value which when substituted to the function will give a near zero value" is correct
  • "A value which when substituted to the function will give a near zero value" and "Always starting with 0" are correct
The goal in using Newton’s method is the   When choosing an initial value, a good guess is : A value which when substituted to the function will give a near zero value A value with f ’(x) ≠ 0 Always starting with 0
  • "A value which when substituted to the function will give a near zero value" and "Always starting with 0" are correct
  • All of the answers correct Correct
  • "A value which when substituted to the function will give a near zero value" is correct
  • "A value which when substituted to the function will give a near zero value" and "A value with f ’(x) ≠ 0" are correct
The higher the degree of the resulting polynomial from Lagrange interpolating polynomial implies that
  • fewer data points are used in the interpolation
  • the lesser the oscillation it will exhibit between the data points
  • more data points are used in the interpolation Correct
  • may be a good predictor of the function between point
The inverse of A  which is a 3 x 2  matrix is A -1 = 2 x 3 matrix.
  • True
  • False Correct
The iteration in Secant method may terminate if the difference between two successive approximations equal to zero.
  • True Correct
  • False
The iteration may terminate if the difference between approximated values of x is already zero.
  • True Correct
  • False
The Lagrange interpolating polynomial is thepolynomial P(x)of degree ________ that passes through thenpoints
  • < n
  • < (n-1) Correct
  • n+1
  • n > 1
The largest eigenvalue of A−1  is the smallest eigenvalue of A in magnitude.
  • True Correct
  • False
The linear interpolant obtained from the Lagrange interpolating polynomial is of ________degree
  • 3
  • n-1 degree
  • 2
  • 1 Correct
The matrix defined as    is an upper triangular matrix
  • True
  • False Correct
The matrix shown below is a
  • 4 x 4 lower triangular matrix
  • 3 x 3 lower triangular matrix
  • 4 x 4 upper triangular matrix Correct
  • 3 x 3 upper triangular matrix
The part where the derivative of the Newton-Raphson was replaced by the slope of the secant line in Secant method.
  • True Correct
  • False
The relative error is  ______________ when the exact value is given by e = 2.718281828 and the approximate value is e a = 2.701.
  • 365763 x 10-3
  • 565763 x 10-3
  • 755763 x 10-3
  • 635763 x 10-3 Correct
The relative error is related to the approximate value rather than to the exact value because the true value may not be known.
  • True Correct
  • False
The same algorithm can be represented in several different ways
  • True Correct
  • False
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.
  • True Correct
  • False
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.
  • True Correct
  • False
The Secant method is convergent for the function f(x) = 3x4 – x -3 whose initial values are present between x0 = 0 and x1 = -1 
  • True Correct
  • False
The Secant method is convergent for the function f(x) = 3x4 – x -3 whose initial values are present between x0 = 2 and x1 = 4 
  • True Correct
  • False
The slope of the secant line has nothing to do with the convergence of the Secant method.
  • True Correct
  • False
The tangent line is projected to approximate the root of the function where it crosses the
  • origin Correct
  • x axis
  • y axis
  • function at its lowest point
The tri method is specially used for
  • all of the answers correct
  • banded matrix Correct
  • triangular matrix
  • square matrix
The value of  is 
  • a positive value
  • has an absolute value of less than 1
  • zero
  • a negative value Correct
The value of Dy is -7/ 5 for the matrix given-x-y+z=-22x+y=-3x+2z=1
  • True
  • False Correct
The value of the determinant is D= 5 for the matrix given-x-y+z=-22x+y=-3x+2z=1
  • True
  • False Correct
Thealgorithm of the Trapezoidalrule is described by>>h=(b-a)/n>>x=a sum=f(x)>>for i=1:n-1>>x=x+h>>sum=sum+2*f(x)>> end>>sum=sum+f(b)>>(b-a)*sum/(2*n)
  • True Correct
  • False
TheNewton polynomialis sometimes called the NDDP which stands for
  • Newton's divided differenceinterpolating polynomial Correct
  • Newton's defined derivative polynomial
  • Newton's discrete/difference polynomial
  • Newton's derivative/difference polynomial
There exists a minimization problem such that (i) assuming P = NP, there is no polynomial-time 1-approximation algorithm for the problem; and (ii) for any constant =C7=AB > 0, there is a polynomial-time (1 + =C7=AB)-approximation algorithm for the problem
  • True Correct
  • False
This is an informal and human readable description of an algorithm leaving many granular details of it
  • Process
  • Pseudocode Correct
  • Algorithm
  • Instruction
This is defined as the high level descriptions of instruction which is intended for human reading
  • Pseudocode Correct
  • Algorithm
  • Instruction
  • Process
This method involves creating an identity matrix at the end stage of the procedure
  • Cramer's Rule
  • Gauss-Seidel method
  • Gauss- Jordan method Correct
  • Gaussian elimination method
To measure the efficiency of the algorithm, the space and time efficiency are the important factors
  • True Correct
  • False
Trapezoidal rule is a numerical method that approximates the value of a definite integral by using first degree polynomial
  • True Correct
  • False
Triangular matrices have their eigenvalues on the diagonal of the matrix therefore the eigenvalues of A are the diagonal elements.
  • No correct answer
  • their eigenvalues on the first column of the matrix
  • their eigenvalues on the bottom of the matrix
  • their eigenvalues on the diagonal of the matrix Correct
Triangular matrix is a square matrix in which all of the elements on one side of the main diagonal are zero The remaining elements should have
  • same as the diagonals
  • may be zero or not Correct
  • no correct answer
  • a non-zero value
Using Lagrange interpolation, with data given below compute for f(15)x0 = 0 f(x0) = 2x1 = 3 f(x1) = 7the solution is f(15) = 45
  • True Correct
  • False
Using Lagrange interpolation, with data given below compute for f(15)x0 = 0 f(x0) = 2x1 = 3 f(x1) = 7x2 = 5 f(x2) = 9the solution is f(15) = 48
  • True Correct
  • False
Using Newton's interpolation, with data given below compute for f(15)x0 = 0 f(x0) = 2x1 = 3 f(x1) = 7x2 = 5 f(x2) = 9The solution of Y(15)=48
  • True Correct
  • False
Using Newton's interpolation, with data given below to compute for f(15)x0 = 0 f(x0) = 2x1 = 3 f(x1) = 7x2 = 5 f(x2) = 9The first order from x0 = 0 to x1 = 3 has a value of 48 which is similar to Lagrange
  • True
  • False Correct
Using Newton's interpolation, with data given below to compute for f(15)x0 = 0 f(x0) = 2x1 = 3 f(x1) = 7x2 = 5 f(x2) = 9The first order from x0 = 3 to x1 = 5 has a value of 1
  • True Correct
  • False
Using Newton's interpolation, with data given below to compute for f(15)x0 = 0 f(x0) = 2x1 = 3 f(x1) = 7x2 = 5 f(x2) = 9The second ordervalue is 1/3
  • True
  • False Correct
This course is taught by the mentor:
Professor Franzelino Abella

Franzelino Abella

Master of Arts in Mathematics Education specialization in College Teaching.

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