MAE284 Final UAH
Bracketing methods always require two initial estimates.
True
If two rows of a matrix are interchanged, the determinant changes sign.
True
Tangent points in a function can cause an opposite-value bracket to contain an even number of roots.
True
Huens method is a first order Runge Kutta method
False
Newton-Raphson is not an iterative method.
False
The main advantage of the Newton-Raphson method is that it always converges.
False
The only way to increase the accuracy of a numerical approximation of a derivative is to decrease the step size h.
False
The shooting method is used to solve initial value problems
False
Using an explicit method on a stiff system will, in general, take less time to run than an implicit method.
False
The Secant method is similar to the _____ method. Check all that apply:
Newthon-Raphson False Position
Bracketing methods are based on the idea that if two functions values, f1 and f2, have ________ signs, there must be at least one root between them.
Opposite
An upper triangular matrix is an echelon matrix.
True
Eulers is a first order Runge Kutta method.
True
If A and B are the same size, then det(A*B)=det(A)*det(B).
True
If one row of a matrix consists of entirely of zeros, then the determinant is zero.
True
If two rows of a matrix are identical, the determinant is zero.
True
One way to increase the accuracy of Newton-Cotes integration techniques is to increase the number of segments.
True
Only square matrices have determinants.
True
Partial differential equations have more than one independent variable.
True
The Gauss Quadrature formulas we discussed in class are called Gauss-Legendre formulas.
True
Simpson's 3/8 Rule approximates the function by a _____.
Cubic Polynomial
The Secant method and the Modified Secant method require the same number of initial guesses.
False
The determinant of ill conditioned systems is zero or nearly zero.
False
MATLAB can solve problems without closed-form solutions.
True
MATLAB's ode45 function is an adaptive algorithm
True
ODE stands for Ordinary Differential Equation.
True
Two applications where numerical integration is used are integrating discrete data points and when it is difficult or impossible to obtain an analytic solution.
True
When it converges, the Newton-Raphson method does so very quickly.
True
For small step size h, decreasing the step size by half will result in less error than using a more accurate version (e.g using an O(h2) method instead of an O(h) method)
False
For the bisection algorithm, the number of iterations required to attain a particular error cannot b calculated in advance.
False
For the bisection algorithm, the number of iterations required to attain a particular error cannot be calculated in advance.
False
The error for the lower accuracy versions of the forward, backward, AND CENTERED finite difference approximations is O(h).
False
What is the first step if you are asked to find all real roots of a function?
Graph the function
The Trapezoidal rule approximates the function by a _____.
Straight Line
Double Integration can be thought of as calculating the volume under a surface defined by a function of two independent variables.
True
Eigenvalues are the solutions to the equation: | A-(lambda)*I | = 0
True
Eigenvectors are related to the mode shape.
True
For C=[1 2;3 4] C./2 will divide each element in C by 2.
True
For the fourth iteration in the Newton-Raphson method, you will use the estimated root location from the third iteration as the new initial point.
True
Gauss Quadrature is a class of techniques for evaluating the area under a straight line by joining the function values at any two points on a curve (rather than the function at the endpoints)
True
Given a system of equations Ax=b, if det(A)=/ 0 and the number of equations equals the number of unknowns, there is a unique solution.
True
If the error of an integration technique is proportional to h2, we say the error is of order h2, which we write as O(h2)
True
In mathematics, a root (a zero) of a function f(x) is a value for x that produces f(x)=0.
True
MATLAB functions for stiff systems use Implicit methods
True
The secant method requires two initial estimates of x, but does not require an analytical expression of the derivative.
True
The secant method requires two initial estimates, but they are not required to bracket the root.
True
The shooting method is based on guessing a value for one of the initial conditions and using an initial value algorithm.
True
Which of the following commands generates an array of equally spaced elements?
x=0:0.1:10
A student wants to skip the next three MAE284 labs, and is writing some MATLAB code to see what that will do to their GPA. They need to name a variable, but for some reason can't remember which names are valid. Of the potential variable names below, which one is valid?
GPA_no_skipping
The False Position method determines its next guess by drawing a straight line between the function elvaluated at the two brackets and seeing where it intersects the x-axis.
True
The Trapezoid Rule can be used to calculate the integral of a function or data with unequal-width segments.
True
The Trapezoid rule can be derived from the method of undetermined coefficients.
True
The Trapezoid rule can be used to calculate the integral of a function or data with unequal-width segments.
True
The Trapezoidal rule can be used to obtain the integral over an interval divided into n=5 segments.
True
The error for the lower accuracy versions of the forward and backward finite-difference approximations is O(h)
True
The function name MUST be the same as the filename
True
The goal of the golden section method is to reduce the range over which the search is conducted at each iteration.
True
The golden section method helps to minimize computational costs in finding a maximum/minimum.
True
The golden section search interval is reduced at every iteration by comparing the function values at x1 and x2, the interior points.
True
The key to Gauss Quadrature is to balance the positive and negative errors.
True
The main disadvantage of the Newton-Raphson method is that it may not converge.
True
The method of elimination involves eliminating unknowns by combining equations.
True
The method of undetermined coefficients is a method to calculate the integral of a function.
True
The only difference between Euler's, Huen's, and the Midpoint methods for solving ODEs is the increment function.
True
The parabolic interpolation method is usually faster than the golden section method.
True
The roots of a function f(x) are located at the points where the function crosses the x-axis.
True
Anonymous functions need to be defined before being used.
True
A straight line is a _______ order polynomial.
First
In the _______ method, the interval is divided in half and the function is evaluated at the midpoint.
Bisection
Which command can be used to calculate the factorial of m?
factorial(m)
The incremental search method is typically very fast, even when employing a high degree of precision.
False
The main advantage of the bisection algorithm is that it converges quickly.
False
The main advantage of the modified secant method is that it does not require specifying a value for Delta(x).
False
The main advantage of the secant method is that it is guaranteed to converge.
False
The main disadvantage of the False Position method is that it can diverge.
False
A NASA intern is writing a function with a single input in MATLAB to help detect sunspots automatically, but unfortunately they never payed attention in numerical methods and can't remember how. Help them out by selecting the appropriate component from the dropdown menu.
(telescopeData)
can separate both array elements and multiple commands.
, comma
How many points would we need to determine a double integral using Simpsons 3/8 rule?
16
Simpsons 3/8 Rule will result in zero error for a ______. Select all that apply.
1st 2nd 3rd order polynomial
A single application of the Trapezoidal Rule requires computing the value of the function at how many points?
2
How many inputs are in the following function? [a,b] = quiz1(c,d,e,f,g)
5
What will be the output of the following MATLAB code? x=[7 6 5 4 3 2 1]; x(3)
5
Suppose you are given a matrix A, which commands will calculate the inverse of A?
A^(-1) A^-1 inv(A)
In general, which of the following will take more time to calculate the solution for a stiff system?
An explicit adaptive algorithm (e.g. ode45)
The principle of superposition works even if the equations are non-linear.
False
When solving the Initial Value Problems, accuracy is the same thing as stability.
False
Select the Bracketing method(s). Check all that apply:
False Position Bisection
Which of the following is the simplest Runge-Kutta method?
Euler
Adaptive Quadrature uses more segments in regions where the function changes slowly.
False
All mathematical problems have a closed-form solution (i.e. they can be solved for a variable without having to make guesses)
False
Eigenvalues are related to the mode shape.
False
Every eigenvector is associated with two eigenvalues
False
For the third iteration in the Newton-Raphson method, you will use the estimated root location from the first iteration as the new initial point.
False
If the determinant of a matrix is zero, the matrix is NOT singular.
False
If the function values of a bracket have the same sign, there are guaranteed to be no roots in the bracket.
False
Inflection points and local maximums and minimums greatly increase the speed at which the Newton-Raphson method converges.
False
MATLAB's built-in fzero function is based of the Newton-Raphson method.
False
Numerical differentiation is not particularly susceptible to errors in data.
False
Partial differential equations have a single independent variable.
False
Romberg's method is NOT a Newton-Cotes method.
False
Simpsons 1/3 Rule can be used to calculate the integral of a function or data with unequal-width segments
False
The Bisection method is most effective when there are multiple roots inside a bracket.
False
The determinant of the 2x2 matrix [a b;c d] is ab-cd.
False
The error for the "normal' accuracy versions of the forward, backward, and centered finite difference approximations is O(h2)
False
The golden section method divides the interval into two sections
False
Select bracketing methods. Check all that apply.
Incremental Search Bisection
Which of the following are types of Boundary Conditions we discussed in class.
Neumann Gauss
Select all the root-finding methods that require only one initial guess. Check all that apply:
Newton-Raphson Modified Secant
The False Position algorithm is identical to the bisection algorithm, except for the calculations for the estimated root location.
True
The second derivative of a function must be __________ at a minimum.
Positive
Suppresses outputs to the command window and separates rows in an array.
Semicolon
What are the two methods we have discussed that are used to solve boundary value problems?
Shooting Method Finite Difference
The false position method is based on the false assumption that the function can be approximated by a(n) ________ ?
Straight Line
Simpson's 3/8 Rule approximates the function by a _____ order polynomial.
Third
Romberg Integration involves multiple applications of which rule?
Trapezoid
Another name for the False Position method is the regula-falsi method.
True
Bracketing methods always converge, but usually do so more slowly than open methods.
True
Discontinuous functions can result in a bracket with opposite-signed function values having an even number of roots.
True
Gauss Elimination is the systematic process of removing unknowns using forward elimination followed by back substitution to obtain the unknown values.
True
If one row of a matrix is a linear combination of two other rows, then the determinant is zero.
True
In the Newton-Raphson method, the new estimate at x(i+1) is the intersection of the tangent line and the x-axis.
True
In this class, if the determinant of a matrix is zero, the inverse of the matrix is not defined.
True
Increasing the damping decreases the magnitude of the response at the resonance frequency
True
MATLAB's built-in fzero function uses a combination of a search algorithm, bisection, and inverse quadratic interpolation.
True
MATLAB's fminbnd function can be used to find the minimum value of a function of a single independent variable
True
MATLAB's fminbnd function uses both the golden section and parabolic interpolation methods.
True
MATLAB's fminsearch function can be used to find the minimum value of a function of two or more independent variables.
True
MATLAB's mesh function can be used to visualize a function of two independent variables.
True
MATLAB's odeset function can be used to set various parameters for the ode45 function.
True
MATLABS's gradient function takes a vector of length n and returns a vector of length n. The first element is a forward difference while the last element is a backwards difference. The remaining elements are central differences.
True
Naive Gauss elimination can fail due to division by zero.
True
Numerical Integration can NOT be used to calculate the integral of the discrete data points
True
One disadvantage of the Newton-Raphson method is that it requires the user to analytically find the derivative of the function.
True
One way to increase the accuracy of Newton-Cotes integration techniques is to increase the order of the approximating polynomial.
True
Propagated truncation error can result from using a large step size.
True
Romberg's method is a Newton-Cotes method.
True
Tangent points in a function can cause an opposite-valued bracket to contain an even number of roots.
True
The "stencil" size of a finite-difference approximation to a derivative is the number of function values required.
True
The Bisection method is a bracketing method.
True
Given two vectors a and b, which comand would you use to calculate the cross product of a and b?
cross(a,b)
The incremental search method uses input values spaced at _______ spaced intervals.
evenly
A student is attempting to create a function file named quiz1_quiestion4.m After three tries, they type a function header that work correctly. All four attempts are listed below. Which one of the listed function headers is error-free?
function [d, e] = quiz1_question4(a, b, c)
Suppose you are given the following function: [a,b]=mae284(p,q,r,s) If the user does not specify the input value for 's', which line of code can be used to define a default value of s=100 within the function?
if nargin < 4, s=100; end
Given the following function. function xr = newtraph(m) Which line of code can be used as a "check" to ensure that the user specifies an input, and if not set the input to 1.
if nargin<1 || isempty(m), m=1; end
Which of the following functions can be used to find the number of roots of a function between a specified lower and upper bound.
incsearch
Which of the following methods can be used to find the list of the lower bound and upper bound that bracket a root?
incsearch
The secant method is similar to the Newton-Raphson method, but....
it does not require the derivative of the function to be known