Numerical Methods MATH-6330 exam answers

In employing Gauss-Seidel method, the most recent values should be used to substitute with the formula of finding x1, x2 and x3, respectively.

True Correct
False

In floating point addition, where the exponent of the smaller number must match that of the larger number making 3.141516 x 101 and 2.125 x 102 expressed in 3 digit precision as 0.314 x 102 and 2.13 x 102

True Correct
False

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

n+1 degree
n x n degree
n-1 degree
n degree Correct

In general, an n =C3=97 n matrix will have a characteristic polynomial of degree of n+ 1, and its roots are the eigenvalues of A

True
False Correct

In giving initial values of x0 and x1, both of them should preferably be close to the solution.

True Correct
False

In mathematical modeling, a physical system is translated into mathematical expressions in order to be implemented in computers.

True Correct
False

In matrix multiplication if the number of columns of the 1st matrix is equal to the number of rows of the 2nd one,

matrix multiplication may proceed Correct
matrix multiplication cannot be performed
swap matrices before proceeding
No correct answer

In matrix multiplication, a matrix of n x k by k x m size the resulting matrix would have a dimension of

k x k
n x k
m x k
n x m Correct

In most cases, __________ is suitable to linear behaving functions and that polynomial interpolation is suitable to non-linear behaving function

direct methods
iterative methods
linear interpolation Correct
non-linear interpolation

In Newton-Cotes integration methods, the nodes are uniformly distributed in [a, b] with x0 = a, xn = band the spacing h = (b - a) / n

True Correct
False

In numerical differentiation, using a very small step size may increase the approximation error

True Correct
False

In numerical integration, when both the end points of the interval of integration are used as nodes in the methods, the methods are called closed type methods

True Correct
False

In performing Gauss-Jordan method, although there is no specific procedure to perform this, it is more convenient to

No correct answer
Eliminate the right hand side of the equation one by one
Eliminate all elements below and above at once Correct
Eliminate the diagonals one at a time

In root finding algorithms, the requirement of initial values

may have one or two initial values Correct
no correct answer
should have two initial values
should have one initial value

In solving systems of linear equations using Gaussian elimination method, if the coefficient of the first variable in the first linear equation is zero and the rest are non-zero:

It is not an issue in Gaussian elimination method just like the Cramer's rule
Gaussian elimination is not possible
Swap the linear equations first to proceed Correct
No correct answer

In the analysis of algorithm, this is maximum number of steps taken on any instance of size a

amortized
average-case
best-case
worst-case Correct

In the analysis of algorithm, this is minimum number of steps taken on any instance of size a

average-case
worst-case
amortized
best-case Correct

In the analysis of algorithm, this is the number of steps taken on any instance of size a

average-case Correct
worst-case
best-case
amortized

In the analysis of algorithm, this refers to the number of steps to be taken in an algorithm

Time complexity Correct
Step complexity
Process Complexity
Space complexity

In the analysis of algorithm, this refers to the volume of memory

Time complexity
Step complexity
Process Complexity
Space complexity Correct

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

True Correct
False

In the factorization A = LU the matrix L is lower triangular and the matrix U is upper triangular, it is called Crout factorization when

Diagond is dominant
L is unit lower triangular Correct
No answer correct
U is unit upper triangular

In the factorization A = LU the matrix L is lower triangular and the matrix U is upper triangular, it is called Doolittle factorization when

No correct answer
U is unit upper triangular Correct
L is unit lower triangular
Diagonal is dominant

In using Gauss-Jordan method if a matrix has at least one zero row with non-zero right hand side, the solution

No correct answer
Doesn't exist Correct
Is Zero
Is infinite

In using Gaussian elimination method, the system of linear equations is transformed into upper triangular form where the values in the upper part of the diagonal are 0s

True
False Correct

In using smaller integration interval for multiple segments, Trapezoidal method can reduce the approximation error better than Simpson's 1/3 rule

True
False Correct

In what method can all of the eigenvalues of a matrix are found simultaneously?

Inverse power method
Direct method
QR method Correct
Power method

Interpreting the results graphically is one advantages of using software systems in numerical methods.

True Correct
False

Inverse of a matrix is possible for

Any matrix
No correct answer
Square matrix Correct
Banded matrix

It is a feature which is essential to a system design especially when dealing with changes in computation

accuracy
adaptability Correct
simplicity
robustness

It is a process for estimating values that lie between known data points

Interconnection
Approximation
Interpolation Correct
Convolution

It is an extension of the Trapezoidal rule This time, it uses three points that would touch the curve of the original function

RK4
Heun's method
Ralston's formula
Simpson's 1/3 rule Correct

It is an iterative approach that can be employed to determine the largest or dominant eigenvalue It has the additional benefit that the corresponding eigenvector is obtained as a by-product of the method

Inverse power method
Direct method
QR method
Power method Correct

It is determined using the expression f(x) = 0

root Correct
integral
no correct answer
derivative

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

True Correct
False

It is possible to approximate a function by using values outside the data points which is known as the linear interpolation

True
False Correct

It is the approximate computation of integral usingnumericaltechniques

indefinite integral
numerical integration Correct
definite integral
numerical differentiation

It iterative technique for solving a square system of n linear equations with unknown x, where Ax=b

Gauss Seidel method Correct
Cramer's rule
Gaussian Elimination method
No correct answer

Just like the Gaussian elimination method, the Gauss-Jordan method involves forward and backward substitution

True Correct
False

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

True Correct
False

LU decomposition can be viewed as the matrix form of Gaussian elimination

True Correct
False

Matrix has repeated eigenvalues

True Correct
False

Matrix multiplication may proceed if the number of columns of the 1st matrix is equal to the number of rows of the 2nd one.

True Correct
False

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

Open method Correct
Non-bracketing method
Closed method
Iterative method

Newton's Method is ideal to function which isDifferentiable also known as a "smooth" functionTranscendental or that which cannot be expressed in finite number of termsContaining multiple roots

Both of "Differentiable also known as a =E2=80=9Csmooth=E2=80=9D function" and "Containing multiple roots" are correct
"Differentiable also known as a =E2=80=9Csmooth=E2=80=9D function" is correct
Both of "Differentiable also known as a =E2=80=9Csmooth=E2=80=9D function" and "Transcendental or that which cannot be expressed in finite number of terms" are correct Correct
All of the answers correct

Newton’s method also known as the Newton-Raphson iteration is that, suppose at point xi of the function, there is a tangent at that point. This point is assumed to be:

the derivative of the function
the lower limit of the interval
the root of the function Correct
the upper limit of the interval

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

True Correct
False

Newton’s method is based on a truncated version of the Taylor series keeping only the first order terms.

True Correct
False

Newton’s Method is ideal to function which is Differentiable also known as a “smooth” function Transcendental or that which cannot be expressed in finite number of terms. Containing multiple roots

All of the answers correct Correct
"Differentiable also known as a “smooth” function" is correct
Both of "Differentiable also known as a “smooth” function" and "Transcendental or that which cannot be expressed in finite number of terms" are correct
Both of "Differentiable also known as a “smooth” function" and "Containing multiple roots" are correct

Newton’s method is powerful in giving multiple roots of any differentiable function.

True Correct
False

Numerical integrations such as Trapezoidal and Simpson's 1/3 rule should have intervals that are uniform

True Correct
False

Numerical methods give more accurate results than analytic methods.

True
False Correct

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

True Correct
False

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

True Correct
False

One of the advantages of using mathematical modeling is the ability to predict an output given a certain input.

True Correct
False

One of the disadvantages of this method is that it can jump to a value away from the rooth if the slope is small

Bisection
False position
Secant Method Correct
Newton-Raphson iteration

Only four points are needed in constructing a fourth order Newton Divided Difference polynomial,

True
False Correct

Perform floating point addition of 3.1 x 10 -1 and 12.25 x 10 1. If only 3 significant figures are allowed for mantissa, determine the percent accuracy of the result.

9972 % Correct
9901 %
9988 %
8972 %

Positive definite matrix can be efficiently solved using Cholesky decomposition.

True Correct
False

Power method is an iterative approach that can be employed to determine the largest or dominant eigenvalue

True Correct
False

Rate of growth of errors are needed to be identified, especially when dealing with iterative methods as this might affect the solution in general.

True Correct
False

Rearranging rows are prohibited when evaluating the matrix if it is diagonally dominant.

True Correct
False

Reducing the equidistant points improves the approximation of the function, f(x) by the polynomial, P

True
False Correct

Roots of transcendental functions are easily approximated using Newton's method provided that f'(x) =E2=89=A0 0

True Correct
False

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

True Correct
False

Secant method is categorized as bracketing method because it uses two points of the secant as initial values.

True Correct
False

Secant method is nearly as fast as the Newton-Raphson method and ensures convergence rather than the latter.

True Correct
False

Secant method is usually the best option if the function doesn’t have an exact formula but just a pair of x and y values.

True Correct
False

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

True Correct
False

Sequential algorithm is an algorithm which can be executed a piece at a time on many different processing devices, and then combined together again at the end to get the correct result

True
False Correct

Simpson's 1/3 rule is an example of an open type numerical integration method

True
False Correct

Simpson's 1/3 rule is an extension of the Trapezoidal rule This time, it uses _______ that would touch the curve of the original function

two points
two to three points
three points Correct
four points

Simpson's 1/3 rule uses a second degree polynomial formed by the two points of the original function

True
False Correct

Simpson's rule is a numerical method that approximates the value of a definite integral by using third degree polynomials

True
False Correct

Since Cramer's rule employ determinants, a system of linear equations with 3 unknowns even with 3 equations cannot be used if the constants are non-zero

True
False Correct

Numerical Methods

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