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A comprehensive guide to mathematical modeling, and computation techniques for solving complex problems in various fields, from engineering to finance.
In mathematical modeling, a physical system is translated into mathematical expressions in order to be implemented in computers.
The Secant method is convergent for the function f(x) = 3x4 – x -3 whose initial values are present between x0 = 2 and x1 = 4
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.
For is the iteration may terminate if the difference between f (x) is already zero.
The characteristic polynomial formed from the matrix
For a two segment trapezoidal rule, it will use the points similar to the ones used by Simpson’s 1/3 rule.
For the function f(x) = ex-2 the value of f(x2) when x0 = -1 and x1 = 1 is f(x) = -0.5248
The tangent line is projected to approximate the root of the function where it crosses the
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
In the analysis of algorithm, this refers to the volume of memory.
The coefficients of the Newton's interpolating polynomial can also be expressed in terms of divided difference.
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:
Creating a mathematical model that will compute the following limit as will produce a result of
Another method called midpoint rule is an open type method numerical integration.
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.
Which of the following shows A - B ?
Secant method is categorized as bracketing method because it uses two points of the secant as initial values.
Simpson's rule is a numerical method that approximates the value of a definite integral by using third degree polynomials
Numerical integrations such as Trapezoidal and Simpson’s 1/3 rule should have intervals that are uniform.
The same algorithm can be represented in several different ways.
In giving initial values of x0 and x1, both of them should preferably be close to the solution.
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.
In the analysis of algorithm, this is the number of steps taken on any instance of size a.
Newton’s Method is ideal to function which is
The elements of both the coefficient matrix and determinant are the same, and so is their the mathematical concept.
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
The approximated root using the secant method lies within the two initial points which is used to project the secant line.
Rearranging rows are prohibited when evaluating the matrix if it is diagonally dominant.
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.
To measure the efficiency of the algorithm, the space and time efficiency are the important factors
The degree of the polynomial for the 10 sample values or data points is equal to 10.
The determinant of an identity matrix is equal to
The Cholesky factorization for the sample matrix given above is:
Which of the following is true about A x B?
Direct method for finding the eigenvalues is recommended since the calculation of zeros of a polynomial is numerically challenging if not unstable.
Power method is an iterative approach that can be employed to determine the largest or dominant eigenvalue
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.
Which of the following statements about algorithm is INCORRECT?
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?
The essential features of a physical system or process in mathematical terms should be carried out in the formulation of a mathematical model.
This is defined as the high level descriptions of instruction which is intended for human reading.
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
From the two data points(2,5) and (6, 11), using Lagrange polynomial method, the polynomial is Li(x) = 1.5x +2.
Choosing an initial guess which gives near f(x) = 0 is considered a good guess
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
Using Newton’s interpolation, with data given below compute for f(1.5) The solution of Y(1.5)=4.8
The matrix defined as is an upper triangular matrix
If the function, f(x) = cos3x was approximated by the polynomial, P(1.5) = -0.2, the amount of error approximately 0.05
For the function f(x) = ex-2 the next approximated value of the root when x0 = -1 and x1 = 1 is x2 = 0.98626.
A continuous function’s integral is approximated using either the trapezoidal or Simpson’s rule by translating the function into discrete form.
If matrix is A is positive definite then a11 > 0.
Using Lagrange interpolation, with data given below compute for f(1.5)
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.
The eigenvalues of A = have no rational values because of the zero element in the matrix.
If matrix A is invertible such that A−1 = L−TL−1 then matrix A can be decomposed using Cholesky’s method.
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.
Positive definite matrix can be efficiently solved using Cholesky decomposition.
Horner’s method which is a method for finding roots of a polynomial equation f(x) =0 is almost similar to Newton’s method.
The approximate integral of will give a value of 0.1786
Newton’s method is powerful in giving multiple roots of any differentiable function.
The point was added to the Simpson’s 1/3 Rule which gives a better approximation to the integral of the function.
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.
Only four points are needed in constructing a fourth order Newton Divided Difference polynomial,
In the analysis of algorithm, this refers to the number of steps to be taken in an algorithm.
Triangular matrices have their eigenvalues on the diagonal of the matrix therefore the eigenvalues of A are the diagonal elements.
A matrix which is denoted by a boldface lowercase letter and expressed as 1 x n matrix is
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
The approximate integral of
If the interval of the function is given as 0 to pi, for n = 6 segments, each node or segments will be
Using Lagrange interpolation, with data given below compute for f(1.5) the solution is f(1.5) = 4.8
Matrix has repeated eigenvalues
When the exact value is π = 3.14159265359 and the relative error is given as 0.001, the approximate value is ____________.
The eigenvalues that corresponds to the characteristic polynomial are λ2-4λ+3 are λ = 1 and λ = -3.
In Newton-Cotes integration methods, the nodes are uniformly distributed in [a, b] with x0 = a, xn = band the spacing h = (b – a) / n.
Which of the following statements is true for the function f(x) = -e x + sin x, when 0 is used as an initial value:
Thealgorithm of the Trapezoidalrule is described by
What is the next approximated root for the function f(x) = ex + sin(x) when x0 = 0?
In numerical differentiation, using a very small step size may increase the approximation error.
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.
Newton’s method is based on a truncated version of the Taylor series keeping only the first order terms.
The value of is
In the analysis of algorithm, this is minimum number of steps taken on any instance of size a.
Which of the following matrices can be represented by A = 5I? (where I is the identity matrix)
The goal in using Newton’s method is the When choosing an initial value, a good guess is :
The eigenvalues of A = are λ = 0 and λ = -3.
Which of the following correctly describes an algorithm?
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.
The determinant of the given matrix is equal to |D| = 27.
The relative error is related to the approximate value rather than to the exact value because the true value may not be known.
The determinant of the given matrix is?
Among the many applications of matrices are used in statistics, economics, physics, and engineering.
Which of the following is not true in finding the determinant of a matrix?
In the analysis of algorithm, this is maximum number of steps taken on any instance of size a.
First order differences is equivalent f[x0,x1] when i = 0.
If a matrix has its entire diagonal elements are positive, then the real parts of its eigenvalues are negative.
The factorization in Cholesky’s Method can be generated efficiently by recurrence relations.
The relative error is ______________ when the exact value is given by e = 2.718281828 and the approximate value is e a = 2.701.
Secant method replaces the tangent in Newton’s method to the slope of the function using two initial guesses.
Eigenvalues are used in the analysis of linear transformation such as scaling.
If matrix A gives the largest eigenvalue, it suggests that if A -1 exists, the smallest eigenvalue can be obtained through inverse power method.
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.
This is an informal and human readable description of an algorithm leaving many granular details of it.
If x2 is the approximated root in Secant method, it follows that; the value of f(x2) must be equal to 0.
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.
In employing Gauss-Seidel method, the most recent values should be used to substitute with the formula of finding x1, x2 and x3, respectively.
Simpson’s 1/3 rule uses a second degree polynomial formed by the two points of the original function.
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 _____.
In differentiation using numerical methods, one of the steps is interpolating the function by a polynomial p at suitable points.
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.
Gauss-Jordan method consists of guessing a value and then using a systematic method to obtain a refined estimate of the root.
For the function , its first derivative is f’(x) is
Approximating calculations which involve infinite value, most often used in series notations and in calculus doesn't introduce errors.
The absolute value of the ratio of is x0 = 0 is 8
Which of the following is true about A - B?
Another condition that must be satisfied is that the diagonal elements are all nonzero for the Gauss-Seidel method to be used:
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.
Using Newton’s interpolation, with data given below to compute for f(1.5) The second ordervalue is 1/3.
For the function f(x) =its first derivative is f’(x) is The first derivative of the function is f’(x) =
The eigenvectors of A = and are the same.
In using smaller integration interval for multiple segments, Trapezoidal method can reduce the approximation error better than Simpson’s 1/3 rule
For practical reasons, the absolute error is usually more meaningful than the relative error.
One of the advantages of using mathematical modeling is the ability to predict an output given a certain input.
The part where the derivative of the Newton-Raphson was replaced by the slope of the secant line in Secant method.
The commutative property also exist in matrices.
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
The Secant method is convergent for the function f(x) = 3x4 – x -3 whose initial values are present between x0 = 0 and x1 = -1
Matrix multiplication may proceed if the number of columns of the 1st matrix is equal to the number of rows of the 2nd one.
A matrix decomposition method that has an upper triangular matrix and an orthogonal matrix is referred to as the QR
The coefficient a0 is also equal to f(x0) in the Newton’s interpolating polynomial.
Using the Maclaurin series expansion of cos(x) , when x =
An additional benefit of the power method is that the corresponding eigenvector is obtained as a by-product of the method.
Reducing the equidistant points improves the approximation of the function, f(x) by the polynomial, P.
For both the Trapezoidal and Simpson’s 1/3 rule , using more strips will give better approximation of the curve.
One of the advantages of secant method over the Newton’s Method is the use of derivatives.
Interpreting the results graphically is one advantages of using software systems in numerical methods.
Which of the following is the first step in solving computational problems?
The given matrix above is a symmetric matrix.
The Average case occurs in linear search algorithmwhen item is the last element in the array.
Algorithms should have explicitly defined set of inputs and outputs
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
One of the advantages of Newton’s method is that its converges fast even if the initial guess was poorly chosen
In the case of the tridiagonal system strict diagonal dominance means simply that (with a0 = an = 0)
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.
If QT is the transpose of Q then QT Q = I or the identity matrix.
The complexity of sorting algorithm measures the running time as a function of the number n of items to be sorter.
Using Newton’s interpolation, with data given below to compute for f(1.5)
The inverse of A which is a 3 x 2 matrix is A -1 = 2 x 3 matrix.
A cubic polynomial can interpolate three points.
A mathematical model that will compute the following limit as will readily give _____________ answer.
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.
Numerical methods give more accurate results than analytic methods.
Best-case is the maximum number of steps taken on any instance of size of a variable.
Rate of growth of errors are needed to be identified, especially when dealing with iterative methods as this might affect the solution in general.
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.
It is possible to approximate a function by using values outside the data points which is known as the linear interpolation.
The slope of the secant line has nothing to do with the convergence of the Secant method.
Given the function f(x) = 0.75 + 1.1x, an exact value can be given instantly by Trapezoidal rule
The function f(x) = x3 -5 with initial guesses x0 and x1 would converge.
The determinant of the given matrix is D = |-5|.
Since matrices are used to represent properties of images, it follows that transformation of images may use eigenvalues and eigenvectors to do that.
The iteration in Secant method may terminate if the difference between two successive approximations equal to zero.
Simpson’s 1/3 rule is an example of an open type numerical integration method
Just like the Newton-Raphson method, the initial guesses affect the convergence of the Secant method.
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.
Each component of the new iterates in Gauss-Seidel method depends upon all previously computed components, the updates cannot be done simultaneously.
The largest eigenvalue of A−1 is the smallest eigenvalue of A in magnitude.
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.
Trapezoidal rule is a numerical method that approximates the value of a definite integral by using first degree polynomial
Which of the following shows A + B ?
If A is a 3 x 3 matrix and B is a 3x2 matrix , the statement “ A – B is not possible” is
In general, an n × n matrix will have a characteristic polynomial of degree of n+ 1, and its roots are the eigenvalues of A.
The iteration may terminate if the difference between approximated values of x is already zero.
It is impossible to find the complex roots of a polynomial function, using Newton’s Method” is:
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.
Roots of transcendental functions are easily approximated using Newton’s method provided that f’(x) ≠ 0.
Newton’s method and secant method has almost the same concept and are both fast.
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