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