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