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# Numerical Methods

A comprehensive guide to mathematical modeling, and computation techniques for solving complex problems in various fields, from engineering to finance.

## general

This is an informal and human readable description of an algorithm leaving many granular details of it.

• Algorithm
• Instruction
• Process
• Pseudocode

Trapezoidal rule is a numerical method that approximates the value of a definite integral by using first degree polynomial

• True
• False

The complexity of sorting algorithm measures the running time as a function of the number n of items to be sorter.

• True
• False

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

• True
• False

Which of the following matrices can be represented by A = 5I? (where I is the identity matrix)

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

• True
• False

A matrix which is denoted by a boldface lowercase letter and expressed as 1 x n matrix is

• row vector
• column vector
• not a matrix

Using Newton’s interpolation, with data given below to compute for f(1.5) The first order from x0 = 0 to x1 = 3 has a value of 4.8 which is similar to Lagrange.

• True
• False

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 ǫ > 0, there is a polynomial-time (1 + ǫ)-approximation algorithm for the problem.

• True
• False

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

• True
• False

Using the Maclaurin series expansion of cos(x) , when x =

• True
• False

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

• True
• False

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

• True
• False

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

• True
• False

In differentiation using numerical methods, one of the steps is interpolating the function by a polynomial p at suitable points.

• True
• False

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

• True
• False

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

• y axis
• origin
• function at its lowest point
• x axis

The characteristic polynomial formed from the matrix

• True
• False

The inverse of A which is a 3 x 2 matrix is A -1 = 2 x 3 matrix.

• True
• False

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

• True
• False

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

• True
• False

Creating a mathematical model that will compute the following limit as will produce a result of

• 0
• indeterminate
• infinite
• 1

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

• True
• False

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

• True
• False

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

• True
• False

Horner’s method which is a method for finding roots of a polynomial equation f(x) =0 is almost similar to Newton’s method.

• True
• False

When it comes to computer implementation, secant method may have disadvantage over the Newton-Raphson since Secant method depends on the previous approximation making it slower than the Newton Raphson

• True
• False

The determinant of the given matrix is?

• 4
• 5
• 2

Another condition that must be satisfied is that the diagonal elements are all nonzero for the Gauss-Seidel method to be used:

• True
• False

The degree of the polynomial for the 10 sample values or data points is equal to 10.

• True
• False

The point was added to the Simpson’s 1/3 Rule which gives a better approximation to the integral of the function.

• True
• False

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 _____.

• 9.95 x10-3
• 6.67 x10-3
• 8.83 x10-3
• 3.35 x10-3

Which of the following correctly describes an algorithm?

• All of these
• It is an efficient method that can be expressed within finite amount of time and space.
• An algorithm is a set of steps of operations to solve a problem performing calculation, data processing, and automated reasoning tasks.
• Algorithm is being used to determine the most practical kinds of solution to solve a problem.

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.

• 99.88 %
• 89.72 %
• 99.72 %
• 99.01 %

The elements of both the coefficient matrix and determinant are the same, and so is their the mathematical concept.

• True
• False

This is defined as the high level descriptions of instruction which is intended for human reading.

• Algorithm
• Instruction
• Pseudocode
• Process

Newton’s Method is ideal to function which is

• Both of "Differentiable also known as a “smooth” function" and "Containing multiple roots." are 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
• All of the answers correct
• "Differentiable also known as a “smooth” function" is correct

Which of the following is true about A x B?

• All of the answers correct.
• The resulting matrix A x B is a 3 x 2 matrix.
• Matrix multiplication is not possible.
• A x B is the same as B x A.

In creating a computer algorithm one important factor that should be considered is that the user would be prompted the values that are needed in solving.

• True
• False

Best-case is the maximum number of steps taken on any instance of size of a variable.

• True
• False

Given the function f(x) = 0.75 + 1.1x, an exact value can be given instantly by Trapezoidal rule

• True
• False

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

• True
• False

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

• In mathematical modeling, a physical system is translated into mathematical expressions in order to be implemented in computers.
• Select one:
• True
• False

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

• True
• 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
• False

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

• True
• False

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
• 4
• 1
• 3

While using two sets of points (x0,y0) and (x1,y1) a straight line is formed and could use the slope equation which follows that the first derivative can be approximated using the given values.

• True
• False

The slope of the secant line has nothing to do with the convergence of the Secant method.

• True
• False

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.

• True
• False

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

• True
• False

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

• average-case
• worst-case
• amortized
• best-case

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

• True
• False

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

• True
• False

The relative error is related to the approximate value rather than to the exact value because the true value may not be known.

• True
• False

The essential features of a physical system or process in mathematical terms should be carried out in the formulation of a mathematical model.

• True
• False

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

• True
• False

The commutative property also exist in matrices.

• True
• 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
• False

If A is a 3 x 3 matrix and B is a 3x2 matrix , the statement “ A – B is not possible” is

• True
• False

The eigenvectors of A = and are the same.

• True
• False

Numerical methods give more accurate results than analytic methods.

• True
• False

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.

• True
• False

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

• True
• False

The eigenvalues that corresponds to the characteristic polynomial are λ2-4λ+3 are λ = 1 and λ = -3.

• True
• False

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.

• True
• 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
• False

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

• True
• False

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 = 7.28754 x 10-7, %ɛ = 2.2264 x 10 -6
• ɛa = 7.4132 x 10-7, %ɛ = 2.8243 x 10 -6
• ɛa = 7.346410 x 10-7, %ɛ = 2. 3384 x 10 -6
• ɛa = 7.1126 x 10-7, %ɛ = 2.8124 x 10 -6

The value of is

• a negative value
• a positive value
• has an absolute value of less than 1
• zero

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

• True
• False

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

• True
• False

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

• True
• False

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

• True
• False

From the two data points(2,5) and (6, 11), using Lagrange polynomial method, the polynomial is Li(x) = 1.5x +2.

• True
• False

If a matrix has its entire diagonal elements are positive, then the real parts of its eigenvalues are negative.

• True
• False

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.

• True
• False

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

• Newton’s method is divergent
• Newton’s method can be used since the function is “smooth”
• Newton’s method is convergent
• Newton’s method cannot be used

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

• Space complexity
• Time complexity
• Process Complexity
• Step complexity

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

• In the analysis of algorithm, this refers to the volume of memory.
• Select one:
• Space complexity
• Time complexity
• Step complexity
• Process Complexity

Using Lagrange interpolation, with data given below compute for f(1.5)

• True
• False

For a two segment trapezoidal rule, it will use the points similar to the ones used by Simpson’s 1/3 rule.

• True
• False

A continuous function’s integral is approximated using either the trapezoidal or Simpson’s rule by translating the function into discrete form.

• True
• False

The determinant of the given matrix is equal to |D| = 27.

• True
• False

Which of the following statements about algorithm is INCORRECT?

• Algorithms must have a unique name
• Algorithms are well-ordered with unambiguous operations
• Algorithms can run for infinitely
• Algorithms should have explicitly defined set of inputs and outputs

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

• -0.5
• 1
• Invalid initial value
• 0.5

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

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

• True
• False

Gauss-Jordan method consists of guessing a value and then using a systematic method to obtain a refined estimate of the root.

• True
• False

The relative error is ______________ when the exact value is given by e = 2.718281828 and the approximate value is e a = 2.701.

• 7.55763 x 10-3
• 6.35763 x 10-3
• 3.65763 x 10-3
• 5.65763 x 10-3

First order differences is equivalent f[x0,x1] when i = 0.

• True
• 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.

• True
• 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
• False

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

• worst-case
• best-case
• amortized
• average-case

The Cholesky factorization for the sample matrix given above is:

• True
• False

If the interval of the function is given as 0 to pi, for n = 6 segments, each node or segments will be

• True
• False

For the function f(x) = ex-2 the next approximated value of the root when x0 = -1 and x1 = 1 is x2 = 0.98626.

• True
• 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
• False

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

• True
• False

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

• True
• False

To measure the efficiency of the algorithm, the space and time efficiency are the important factors

• True
• False

For the function , its first derivative is f’(x) is

• True
• False

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

• True
• 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
• False

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

• True
• False

Which of the following shows A - B ?

• Question text

For both the Trapezoidal and Simpson’s 1/3 rule , using more strips will give better approximation of the curve.

• True
• False

Using Newton’s interpolation, with data given below to compute for f(1.5) The second ordervalue is 1/3.

• True
• False

Positive definite matrix can be efficiently solved using Cholesky decomposition.

• True
• False

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

• True
• False

The part where the derivative of the Newton-Raphson was replaced by the slope of the secant line in Secant method.

• True
• False

Which of the following is the first step in solving computational problems?

• Specification of an Algorithm
• Development of a model
• Problem definition
• Designing an Algorithm

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
• Iterative method
• Closed method
• Non-bracketing method

Among the many applications of matrices are used in statistics, economics, physics, and engineering.

• True
• False

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
• False

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

• True
• False

A mathematical model that will compute the following limit as will readily give _____________ answer.

• indeterminate
• 0
• finite
• infinite

Using Lagrange interpolation, with data given below compute for f(1.5) the solution is f(1.5) = 4.8

• True
• 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
• False

An algorithm design technique is a general approach to solving problems algorithmically that is applicable to a variety of problems from different areas of computing.

• True
• False

The determinant of the given matrix is D = |-5|.

• True
• False

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

• True
• False

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

• worst-case
• amortized
• best-case
• average-case

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
• False

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

• True
• False

The coefficients of the Newton's interpolating polynomial can also be expressed in terms of divided difference.

• True
• False

The determinant of an identity matrix is equal to

• -1
• 0
• 1
• Infinite

The approximate integral of will give a value of 0.1786

• True
• 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 value A value with f ’(x) ≠ 0 Always starting with 0

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

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

• True
• False

A cubic polynomial can interpolate three points.

• True
• False

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

• True
• False

Matrix has repeated eigenvalues

• True
• 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
• False

Which of the following is true about A - B?

• All of the answers correct.
• The resulting matrix A - B is a 3 x 2 matrix.
• A - B is the same as B - A.
• Subtraction of matrices is not possible.

Using Newton’s interpolation, with data given below to compute for f(1.5)

• True
• False

Which of the following shows A + B ?

• Question text

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

• True
• False

Since matrices are used to represent properties of images, it follows that transformation of images may use eigenvalues and eigenvectors to do that.

• True
• False

The approximate integral of

• True
• False

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

• True
• False

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

• True
• False

The same algorithm can be represented in several different ways.

• True
• False

The matrix defined as is an upper triangular matrix

• True
• False

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

• True
• False

When the exact value is π = 3.14159265359 and the relative error is given as 0.001, the approximate value is ____________.

• 3.141706813
• 3.141890681
• 3.141906813
• 3.141606813

For practical reasons, the absolute error is usually more meaningful than the relative error.

• True
• False

The given matrix above is a symmetric matrix.

• True
• False

Approximating calculations which involve infinite value, most often used in series notations and in calculus doesn't introduce errors.

• True
• False

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

• True
• False

Thealgorithm of the Trapezoidalrule is described by

• True
• False

If there is a randomized algorithm that solves a decision problem in time t and outputs the correct answer with probability 0.5, then there is a randomized algorithm for the problem that runs in time Θ(t) and outputs the correct answer with probability at least 0.99.

• True
• False

Each component of the new iterates in Gauss-Seidel method depends upon all previously computed components, the updates cannot be done simultaneously.

• True
• False

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

• True
• 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

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.

• True
• False

A matrix decomposition method that has an upper triangular matrix and an orthogonal matrix is referred to as the QR

• True
• False

Using Newton’s interpolation, with data given below compute for f(1.5) The solution of Y(1.5)=4.8

• True
• False

The Average case occurs in linear search algorithmwhen item is the last element in the array.

• True
• False

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.

• True
• False

The coefficient a0 is also equal to f(x0) in the Newton’s interpolating polynomial.

• True
• False

Another method called midpoint rule is an open type method numerical integration.

• True
• False

Using Newton’s interpolation, with data given below to compute for f(1.5) The first order from x0 = 3 to x1 = 5 has a value of 1

• True
• 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 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.
• "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

If matrix is A is positive definite then a11 > 0.

• True
• False

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

• True
• False

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

• True
• False

If the function, f(x) = cos3x was approximated by the polynomial, P(1.5) = -0.2, the amount of error approximately 0.05

• True
• False

Algorithms should have explicitly defined set of inputs and outputs

• True
• False

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

• True
• False

For an interval of 0 to 1, a subinterval with a size of 0.2 will give n = 5 described as 5 segment Trapezoidal rule.

• True
• False

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

• 2.708
• 2.525
• 2.803
• 2.656

Which of the following is not true in finding the determinant of a matrix?

• It doesn’t work with non-square matrix.
• The operation between the matrices also takes an alternate sign of negative first then positive.
• Coefficients from systems of linear equations may form a matrix and could be used for finding the determinant.

For the given systems of linear equations, with initial values x1 =0; x2 =0; x3 = 0. The next iterative value of x1 using Gauss-Seidel Method is 1.3333.

• True
• False

From the two data points (1,4) and (3, 7), and (4,10) using Lagrange polynomial method, the polynomial is Li(x) = 0.5x2 -0.5x +2.

• True
• False

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

• True
• False