Numerical Methods
Given a system of equations Ax=b, if det(A)=0, then there is a unique solution. T/F
False
Adaptive Quadrature adjusts to use a smaller step size in areas where the function changes rapidly, and a larger step size where the function changes slowly. T/F
True
Bracketing methods always converge, but usually do so more slowly than open methods. (T/F)
True
Quadrature is another name for integration. T/F
True
The bisection method is a bracketing method. (T/F)
True
The determinant of a matrix is a scalar. True/False
True
The main disadvantage of the false position method is that it is one-sided. (T/F)
True
For the bisection algorithm, the number of iterations required to attain a particular error cannot be calculated in advance. (T/F)
True
If A and B are of the same size, then det(A*B)=det(A)*det(B). T/F
True
If an initial estimate is far from the root, open root-finding algorithms are more likely to converge. (T/F)
True
Ill-conditioned systems result from equations that correspond to parallel lines, coincident lines, or lines with nearly the same slope (nearly parallel). True/False
True
In the Newton-Raphson method, the new estimate at x_{i+1} x i + 1 is the intersection of the tangent line and the x-axis. (T/F)
True
Initial value problems have all necessary conditions given at a single value of the independent variable. True/False
True
One way to increase the accuracy of Newton-Cotes integration techniques is to increase the number of segments. True/False
True
Only square matrices have determinants. True/False
True
Second derivative (and higher derivative) finite-difference approximations are obtained by including more terms in the Taylor Series. True/False
True
The numerical approximations of the derivatives discussed in class were derived from the Taylor Series. True/False
True
The primary problem with stiff systems is efficiency. True/False
True
In this class, if the determinant of a matrix is zero, the inverse of the matrix is not defined. True/False
True
Increasing the damping decreases the magnitude of the response at the resonance frequency. True/False
True
Tangent points in a function can cause an opposite-valued bracket to contain an even number of roots. (T/F)
True
The method of undetermined coefficients is an approach to finding the particular solution to certain nonhomogenous ordinary differential equations. True/False
True
The trapezoid rule estimates the area under a function by the product of the width of the interval and the average value of the function. True/False
True
Two applications where numerical integration is used are integrating discrete data points and when it is difficult to obtain a closed form solution. True/False
True
Inflection points and local maximums and minimums greatly increase the speed at which the Newton-Raphson method converges. (T/F) True False
False
Linear interpolation can ALWAYS be used with the shooting method to guarantee the correct solution on the third "shot". True/False
False
MATLAB's built-in fzero function is based on the Newton-Raphson method. (T/F)
False
Multiple applications of the predictor and corrector steps will not improve the accuracy of the solution. True/False
False
The Midpoint method is a first order Runge-Kutta method. True/False
False
The error for the lower accuracy versions of the forward, backward, and centered finite difference approximations is O(h2). True/False
False
The main advantage of the secant method is that it is guaranteed to converge. (T/F)
False
The principle of superposition works even if the equations are non-linear. True/False
False
The secant method and the modified secant method require the same number of initial guesses. (T/F) True False
False
If one row of a matrix is a linear combination of two other rows, then the determinant is zero. True/False
True
If two rows of a matrix are identical, the determinant is zero. T/F
True
In the equation below, h is the step size. y_{i+1}=y_i+phi*h True/False
True
MATLAB can solve problems without closed-form solutions. (T/F)
True
Naïve Gauss Elimination fails if the pivot element is zero. T/F
True
Romberg's method can be used to combine two O(h2) estimates of the integral to obtain a single O(h4) estimate. True/False
True
Romberg's method is a Newton-Cotes method True/False
True
Simpson's 3/8 Rule approximates the function by a cubic polynomial . True/False
True
The Hybrid Simpson's approach can be used to obtain the integral over an interval divided into n=5 segments. True/False
True
The equal sign, "=", is used to assign data to a variable. (T/F)
True
The false position method determines its next guess by drawing a straight line between the function evaluated at the two brackets and seeing where it intersects the x-axis. (T/F)
True
The trapezoid rule estimates the area under a function by the product of the width of the interval and the average value of the function. T/F
True
Two applications where numerical integration is used are integrating discrete data points and when it is difficult to obtain a closed form solution. T/F
True
Which of the following are caused by using numerical methods to approximate an exact solution? Truncation Errors or Round-off errors
Truncation Errors
Which command will generate a vector of linearly spaced data between 1 and 5 with 50 points? linspace(1,50,5) linspace(1,5,50) linspace(1,5) linspace(5,1)
linspace(1,5,50)
Bracketing methods are based on the idea that if two function values, f1,f2, have opposite signs, there must be at least one root between them.
opposite
Given that: w = 2, X = 5, y = 3 and Z = 3 B 3 = B 2 sqrt(B 1) B 1 = w/ ( X + Z ) B 2 = y ⋅ w . Input your answer to 4 decimal places.
3
How many inputs are in the following function? Input your answer as a whole number. [a, b] = quiz1(c,d,e,f,g)
5
What is the order of the following system of ODEs? 3y''=4y''' 3+7y+14y''=0 Input your answer as a whole number.
5
The secant method is similar to the Newton-Raphson method, but...
Answer 1: it does not require the derivative of the function to be known
Boundary value problems have all the necessary conditions given at a single value of the independent variable. True/False
False
The composite Simpson's 1/3 Rule requires the number of segments to be [divisibility]. odd/even/div. by 3
even
A computer scientist is attempting to automate the process of cooking hamburgers. They are using MATLAB to analyze some data, and need a function file that does not have any output variables. Click on the dropdown menu to complete the header. function flippingBurgers(burgerTemperature, juiceIndex, condimentArray)
function
Which one of the following functions can be used to find the list of the lower bound and upper bound that bracket a root? incsearch/midpoint/newtraph/bisect
incsearch