MECE-2350 Final Review (Quizlet Edition)
(T/F) The parabolic interpolation method is based on fitting a parabola to three points on the function.
True
(T/F) The parabolic interpolation method is usually faster than the golden section method.
True
(T/F) The secant method requires two initial estimates, but they are not required to bracket the root.
True
(T?F) The finite difference method is based on converting continuous derivatives to algebraic approximations.
True
F'(i) = f(i+1)-f(i-1)/2*h
Central Formula
The second derivative of a function must be ________ at a minimum. negative positive
positive
(T/F) 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
(T/F) If an initial estimate is far from the root, open root-finding algorithms are more likely to converge.
False
Which of the following are initial value conditions? (a) y'(0) = 5, y(0) = 0 (b) y'(1) = 0, y(0) = 0 (c) y'(1) = -2, y(1) = 7 (d) y(10) = 0, y'(10) = 0 (e) y(10) = 0, y'(0) = 0
(a) (c) (d)
___ can separate both array elements and multiple commands.
, (comma)
(T/F) Matlab's fminbnd function can be used to find the minimum value of a function of a single independent variable.
True
(T/F) Only square matrices have determinants.
True
If we only have two function values, which type of method will result in the best approximation of the derivative? Forward Centered Backward
Centered
(T/F) The secant method is similar to the Newton-Raphson method, but uses an approximation of the derivative.
True
(T/F) The secant method requires two initial estimates of x, but does not require an analytical expression of the derivative.
True
(f^4(a)*h^4)/4!
Fifth term in Taylor series expansion
Describe the bisection method
Find midpoint using, m=(a+b)/2, check value if converged, does F(m) change sign, if so then for interval {a,m} (insert m into a). If F(m) is a positive value, (insert m into b) for interval {m,b}. Continue till F(m) gets close to zero
Order of convergence for Bisection Method? (First/Second-order)
First-order
(f^3(a)*h^3)/3!
Fourth term in Taylor series expansion
Error associated with a numerical solution of the problem
Truncation error
Which of the following are caused by using numerical methods to approximate an exact solution?
Truncation errors
Describe Local Error vs. Global Error
local error- refers to the error occurred over a single step Global error - total discrepancy due to past as well as present steps
Bracketing methods are based on the idea that if two function values, f1 and f2, have _______________ signs, there must be at least on root between them.
opposite
A parabola is a ____________ -order polynomial.
second
f'(i)=f(i)-f(i-1)/h
Backward formula
F'(i) = f(i+1)-f(i)/h
Forward Formula
Bracketing methods
Incremental Search Bisection False Position
(f''(a)*h^2)/2!
Third term in Taylor series expansion
(T/F) Cubic splines are always bad for interpolation
False
A straight line is a ______ order polynomial.
first
f(a)
First term in Taylor series expansion
Advantages of Newton Raphson compared to Secant Method
-Newton Raphson only requires one initial guess -Newton Raphson converges faster -Newton Raphson is more accurate
List ways that error associated with Taylor series can be eliminated
-Reducing the step size -Increasing the order of Taylor Series -Make the equation more linear
List characteristics that you would use to decide on the best numerical method to solve a problem
-number of initial guesses or starting point -Rate of convergence-Stability -Accuracy and precision
List two reasons to use and open method rather than a bracket method to solve a non-linear equation
-when multiple roots are present -when bracket for the roots is not present
How many bits are used to represent a number in Matlab?
64
(T/F) When solving initial value problems, accuracy is the same thing as stability.
False
Method(s) of determining the root that will always converge
Bisection or False Position
Matlab can solve which of the following? Only first ODEs Both first-order and higher-order ODEs
Both first-order and higher-order ODEs
(T/F) For the bisection algorithm, the number of iterations required to attain a particular error cannot be calculated in advance
False
(T/F) If the function values of a bracket have the same sign, there are no roots in the bracket.
False
(T/F) Inflection points and local maximums and minimums greatly increase the speed at which the Newton-Raphson method converges.
False
(T/F) Matlab's fminbnd function creates two matrices that can be used to create a 3D mesh plot of a function of two independent variables.
False
(T/F) Newton-Raphson is not an iterative method.
False
(T/F) Numerical integration can NOT be used to calculate the integral of discrete data points.
False
(T/F) The determinant of a matrix is a vector.
False
(T/F) The error for the "normal" accuracy versions of the forward, backward, and centered finite-difference approximations is O(h).
False
(T/F) The finite difference method is used to solve initial value problems.
False
(T/F) The incremental search method is typically very fast, even when employing a high degree of precision.
False
(T/F) The main advantage of the bisection algorithm is that it converges quickly.
False
(T/F) The main advantage of the modified secant method is that it does not require specifying a value of delta x.
False
(T/F) The main advantage of the secant method is that it is guaranteed to converge
False
(T/F) The only way to increase the accuracy of a numerical approximation of a derivative is to add more terms in the Taylor Series used to derive the approximation.
False
(T/F) The only way to increase the accuracy of a numerical approximation of a derivative is to decrease the step size h.
False
(T/F) The parabolic interpolation method is based on fitting a parabola to two points on the function.
False
What numerical integration method has an error that is proportional to the interval size to the fourth power?
Simpson's rule
Method of root finding that utilizes only one guess
Newton Raphson
Difference between regression and interpolation
Regression - 1)Represents general trend line designed to follow pattern not intersect at every point 2)Applied to data exhibiting significant error Interpolation 1)fit a curve that passes directly through all data points 2)Applied to very precise data
Number of Data points needed to derive a second order polynomial equation using regression vs. interpolation
Regression - Need 3 or more data points Interpolation - Need only 3
f'(a)*h
Second term in Taylor series expansion
Order of convergence for Newton's method? (First/Second-order)
Second-order
What numerical integration method fits a parabola?
Simpson's Rule
(T/F) Bracketing methods always require two initial estimates.
True
(T/F) Discontinuous functions can result in a bracket with opposite-signed function values having an even number of roots.
True
(T/F) Eigenvalues determine the type of response of a system (i.e. underdamped, critically damped, or overdamped).
True
(T/F) Higher accuracy versions of the forward, backward, and centered finite-difference approximations of a derivative are obtained by including more terms in the Taylor Series.
True
(T/F) If one row of a matrix consists entirely of zeros, then the determinant is zero.
True
(T/F) If one row of a matrix is a linear combination of two other rows, then the determinant is zero.
True
(T/F) The bisection method is a bracketing method.
True
(T/F) The determinant of a matrix is a scalar.
True
(T/F) The error for the "normal" accuracy versions of the forward and backward finite-difference approximations is O(h).
True
(T/F) The false position algorithm is identical to the bisection algorithm, except for the calculations for the estimated root location
True
(T/F) The false position algorithm is identical to the bisection algorithm, except for the calculations for the estimated root location.
True
(T/F) The goal of the golden section method is to reduce the range over which the search is conducted at each iteration.
True
(T/F) The main advantage of the bisection algorithm is that it always converges.
True
(T/F) The midpoint method is usually more accurate than Euler's method for the same step size.
True
Main difference between the bisection method and the secant method?
bisection = find , subdivide. Secant = use straight line based on latest values
In general, _________________ the step size, h, will decrease the error in a numerical approximation of a derivative.
decreasing
The composite Simpson's 3/8 Rule requires the number of segments to be [divisibility].
divisible by 3
In the golden section method, the distance from the lower endpoint to x1 is _________ the distance from the upper end point to x2. less than more than
less than
What is the numerical values of the analytic rate of convergence for Newton's method for the function f(x) - x^2-4 for the positive root?
f'(x) = 2x. f''(x) = 2. lambda = f''(x)/2f'(x) = 2/2*(4) = 1/4
How do you determine if a system of equations is singular
if the det(A) = 0
