Numerical differentiation formulas for a function f are typically found by doing which of the following?

Choose one answer.

	A. Finding a zero of f'
	B. Approximating f by a polynomial P and differentiating P
	C. Differentiating a Fourier transform of f and then applying an inverse transform
	D. Finding the slope of f at two randomly generated points
	E. None of the above

Question 2

Let

. Approximate

using the 2-point difference formula with

Choose one answer.

	A. 2.0
	B. 2.1
	C. 1.9
	D. 0.21
	E. 0.19

Question 3

For fixed

, and finite difference formula, there is an

that minimizes the error in the computed approximation to

. Why?

Choose one answer.

	A. Every difference formula has a truncation error formula that can be minimized.
	B. High degree interpolating polynomials oscillate too much.
	C. Underflows increase as .
	D. Overflows increase as .
	E. Rounding errors increase as .

Question 4

Use a 3-point centered difference formula with

to approximate

, where

. Identify the best 4 significant digit approximate below.

Choose one answer.

	A. 2.000
	B. 1.999
	C. 1.989
	D. 1.899

Question 5

If f(x) is only known at the uniformly spaced grid points

and a

approximation to

is desired at each

, then what combination of differentiation rules could be used?

Choose one answer.

	A. Centered difference formula at each
	B. 3-point difference formulas, forward difference at left endpoint, backward difference at right endpoint, and centered difference elsewhere
	C. Backward difference on left half of gridpoints, and forward difference on right half
	D. Alternating forward and backward differences, beginning with backward difference at

Question 6

What is the truncation error term for the 3-point centered difference formula for approximating

Choose one answer.

	A.
	B.
	C.
	D.

Question 7

Let

. Approximate

using the 2-point difference formula with

Choose one answer.

	A. 2.0
	B. 2.1
	C. 1.9
	D. 0.21
	E. 0.19

Question 8

Which of the following statements about composite Newton-Cotes quadrature rules is true?

Choose one answer.

	A. They use a Taylor polynomial for f'.
	B. They integrate a piecewise polynomial approximation to f.
	C. They are a type of Monte Carlo method.
	D. They require orthogonal polynomial integration.
	E. None of the above

Question 9

Use Simpson's method (

) to approximate

. Choose the best 4 significant digit answer below.

Choose one answer.

	A. 0.3750
	B. 0.5000
	C. 0.6667
	D. 0.3333
	E. 0.2500

Question 10

Use composite trapezoidal rule with

to approximate

. Choose the best 4 significant digit answer below.

Choose one answer.

	A. 0.3750
	B. 0.5000
	C. 0.3333
	D. 1.500
	E. 0.3125

Question 11

How is the adaptive quadrature method adaptive?

Choose one answer.

	A. Subdivides if the integral is too large.
	B. It subdivides if the derivative is too large.
	C. It subdivides if rounding error exceedes truncation error.
	D. It subdivides if error estimate is too large.

Question 12

Newton-Cotes quadrature rules for a function f are typically found by doing which of the following?

Choose one answer.

	A. Using a left sided Riemann sum
	B. Approximating f by a polynomial P and integrating P
	C. Integrating a Fourier transform of f and then applying an inverse transform
	D. Finding the average height of f at n randomly generated points
	E. None of the above

Question 13

In a floating point system with 3 decimal digits in the fractional part, what is the result of 7.234 + 5.784?

Choose one answer.

	A. 13.02
	B. 13.01
	C. 13.0
	D. 13.1
	E. None of the above

Question 14

What is the relative error in the 3 decimal digit floating point arithmetic computation (123 + 0.449)-122?

Choose one answer.

	A. 31%
	B. 0.449
	C. 44.9%
	D. 0
	E. 1.449

Question 15

What value results in the 3 decimal digit floating point arithmetic computation (123.3 + 0.4)-122.7?

Choose one answer.

	A. 1
	B. 0.4
	C. 0
	D. 0
	E. 1.3

Question 16

In a floating point system with t digits in the fractional part, if x and y share the same normalized exponent and the same g fractional digits, then how many digits of fl(x-y) are known to be correct?

Choose one answer.

	A. t-g
	B. g
	C. All t
	D. 0
	E. It depends on the compiler.

Question 17

In general, which of the following holds for the floating point addition of two or three floating point numbers whose sum does not overflow or underflow?

Choose one answer.

	A. Both commutative and associative
	B. Commutative but not associative
	C. Neither commutative nor associative
	D. Associative but not commutative

Question 18

Swamping and cancellation both describe a loss of information. Select the statement below that best describes their effects.

Choose one answer.

	A. Swamping does not violate the fundamental axiom of floating point arithmetic, but cancellation does.
	B. Cancellation loses precision, while swamping does not.
	C. Swamping loses precision, while cancellation does not.
	D. Swamping can only happen with multiplication and cancellation only with addition.

Question 19

The machine epsilon is nearest to which of the following?

Choose one answer.

	A. Underflow
	B. Overflow
	C. The distance between 1 and the nearest float to 1
	D. 1/Overflow

Question 20

Suppose x and y are floats and fl(x+y)=fl(x). What can be said of y?

Choose one answer.

	A. y underflowed.
	B. y=0.
	C. \|y\| is less than \|x\|*(machine epsilon).
	D. fl(0+y)=0.

Question 21

With a fixed length floating point word, adding a bit to the fractional part requires removing a bit from the exponent. Which of the following statements describes how this affects the set of floats?

Choose one answer.

	A. The floats are farther apart but have a larger range.
	B. The floats are nearer each other but have a smaller range.
	C. There are more floats.
	D. Underflow is smaller.
	E. The machine precision grows.

Question 22

Suppose a floating point system can represent K floats. If one bit is added to that floating point word, about how many floats can then be represented?

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 23

What can be said of any solution, x, to fl(1+x)=1?

Choose one answer.

	A. .
	B. .
	C. .
	D. .

Question 24

Fill in the blank. If it can be shown that the result of a computation is the exact answer to a nearby problem, then we say the computation is ____________.

Choose one answer.

	A. Highly accurate
	B. Rounding correct
	C. Backward stable
	D. Well conditioned
	E. Robust

Question 25

If x + y is smaller than both x and y, information in the sum x + y is lost. What is this called?

Choose one answer.

	A. Swamping
	B. Machine epsilon
	C. Truncation error
	D. Cancellation
	E. None of the above

Question 26

Fill in the blank. If a small change to a problem leads to a large change in its solution, then we say that the problem is ______________.

Choose one answer.

	A. Backward stable
	B. Well conditioned
	C. Robust
	D. Ill conditioned
	E. Highly accurate

Question 27

Problem

has input

and output

. A small perturbation

gives a new input

and a new output

. A relative condition number

for problem

should satisfy which of the following?

Choose one answer.

	A.
	B.
	C.
	D.

Question 28

Problem

has input

and output

. A small perturbation

gives a new input

and a new output

. An absolute condition number

for problem

should satisfy which of the following?

Choose one answer.

	A.
	B.
	C.
	D.

Question 29

Under what conditions on a problem and method are we guaranteed a good solution?

Choose one answer.

	A. Problem is stable and method is backward stable.
	B. Problem is stable and method is well conditioned.
	C. Problem is well conditioned and method is well conditioned.
	D. Problem is well conditioned and method is backward stable.

Question 30

What is an upper bound on the relative error between two neighboring floats called?

Choose one answer.

	A. Cancellation limit
	B. Machine epsilon
	C. Underflow
	D. Backward error
	E. None of the above

Question 31

In a floating point system with 3 decimal digits in the fractional part, what floating point number would be used to represent 0.0032419?

Choose one answer.

	A. 0.003
	B. 0.00324
	C. 0.003242
	D. 0.00
	E. None of the above

Question 32

In a floating point system with 3 decimal digits in the fractional part, what floating point number would be used to represent 142.32419?

Choose one answer.

	A. 142
	B. 142.324
	C. 142.3
	D. 142.32
	E. None of the above

Question 33

When a large magnitude and small magnitude float are added, information from the smaller number is lost. What is this called?

Choose one answer.

	A. Chopping
	B. Cancellation
	C. Truncation Error
	D. Swamping
	E. Tail Error

Question 34

The term Initial Value Problem is best defined in this course as which of the following?

Choose one answer.

	A. , where with initial value
	B.
	C. Given , find .
	D. Find such that .
	E.

Question 35

Which of the following is an initial value problem?

Choose one answer.

	A. .
	B. .
	C. .
	D. .

Question 36

A shooting method for a boundary value problem requires what 2 numerical methods?

Choose one answer.

	A. An interpolator and a numerical differentiation rule
	B. A quadrature rule and an IVP solver
	C. A root finder and a quadrature rule
	D. An IVP solver and a root finder

Question 37

What is the Euler slope?

Choose one answer.

	A.
	B.
	C.
	D.
	E. None of the above

Question 38

In theory, the error in Euler's method can be made arbitrarily small by varying h, why is this not the case in practice?

Choose one answer.

	A. Because of rounding errors
	B. Because of truncation errors
	C. As , the truncation errors explode.
	D. As , the truncation errors explode.

Question 39

When converting a scalar differential equation of order n into a system of first order differential equations, how many dimensions are needed to represent the solution vector?

Choose one answer.

	A.
	B.
	C.
	D.
	E. None of the above

Question 40

Why does an explicit 3-step method for IVP require the use of a single-step method?

Choose one answer.

	A. Because it averages 3 values in
	B. Because Euler's method is unstable
	C. Because the corrector needs a prediction
	D. Because a 3-step method needs 2 previous approximations to y
	E. Because a 3-step method needs 3 previous approximations to y

Question 41

An explicit 3 step method needs to use a single step method, but Euler's method is a bad choice. Why shouldn't one use Euler's method in this setting?

Choose one answer.

	A. Euler's method is unstable.
	B. Eulers method has local truncation error , which is lower than a 3 step method.
	C. Euler's method has large rounding errors.
	D. Euler's method is too slow for a 3 step method.
	E. Taylor methods are more general.

Question 42

In a multistep predictor-corrector method, what type of method is the corrector?

Choose one answer.

	A. The corrector is typically a higher order Runge Kutta method.
	B. The corrector is typically a low order Runge Kutta method.
	C. The corrector is typically an implicit method.
	D. The corrector is typically a higher order explicit multistep method.

Question 43

The Adams-Bashforth 2-step (AB2) method has the iteration

. Consider the IVP

. Select the best 4 significant digit approximation below to the AB2 value for

, taking

and

Choose one answer.

	A. 1.297
	B. 1.015
	C. 0.7650
	D. 1.547

Question 44

How many function evaluations are required per timestep for an Adams-Bashforth method of order

Choose one answer.

	A. 1
	B. 2
	C. 4
	D. 5
	E. 6

Question 45

Which of the following is an implicit method?

Choose one answer.

	A.
	B.
	C.
	D.

Question 46

Which of the following is a general purpose single step method?

Choose one answer.

	A. Taylor's method
	B. Adams-Bashforth's method
	C. Adams-Moulton's method
	D. Runge-Kutta's method

Question 47

Consider the IVP

. Approximate

using Euler's method with

. Select the best 3 significant digit answer below.

Choose one answer.

	A. 3.00
	B. 3.75
	C. 2.00
	D. 4.55
	E. 6.30

Question 48

Variable step-size methods all require which of the following?

Choose one answer.

	A. A good starting guess
	B. An error estimate
	C. Multistep method
	D. A list of allowable step sizes

Question 49

Choose the iteration below which represents the Runge-Kutta method called the midpoint method.

Choose one answer.

	A.
	B.
	C.
	D.

Question 50

Choose the iteration below which represents the Runge-Kutta method called the modified Euler or Heun's method.

Choose one answer.

	A.
	B.
	C.
	D.

Question 51

How many function evaluations are required per timestep for a Runge-Kutta method of order

Choose one answer.

	A. 1
	B. 2
	C. 4
	D. 5
	E. 6

Question 52

How many function evaluations are required per timestep for a Runge-Kutta method of order

Choose one answer.

	A. 1
	B. 2
	C. 4
	D. 5
	E. 6

Question 53

Consider the IVP

. Which of the following iterations correspond to the Taylor method of order 2 for this IVP?

Choose one answer.

	A.
	B.
	C.
	D.

Question 54

If an IVP is stiff, what restriction is imposed on the solver?

Choose one answer.

	A. Stiff IVP's require a small timestep.
	B. Stiff IVP's require predictor-corrector methods.
	C. Stiff IVP's require Taylor methods.
	D. Stiff IVP's require high order methods.

Question 55

What qualitative property should a method have when applied to the test problem

Choose one answer.

	A. .
	B. bounded for all .
	C. bounded for all .
	D. .

Question 56

Why are Taylor methods not as general purpose as Runge-Kutta methods?

Choose one answer.

	A. Taylor methods require the evaluation of for $k>1$.
	B. Taylor polynomials oscilate too much.
	C. Taylor methods replace derivatives with function evaluations.
	D. Taylor methods are not parallelizable because of nested function evaluations.

Question 57

Fill in the blank. Ignoring rounding errors, a successful IVP method with local truncation error of order k should have global error of order ________.

Choose one answer.

	A. k+1
	B. k/2
	C. 2k
	D. k-1

Question 58

All single step IVP methods have a common iteration

. If

, what is the value of

that would guarantee

was on the solution curve?

Choose one answer.

	A. .
	B. .
	C. .
	D. .

Question 59

What conditions on an initial value problem guarantee a unique solution?

Choose one answer.

	A. is smooth enough for all .
	B. is continuous in .
	C. is nonsingular.
	D. The initial is close enough to the solution.
	E. has a fixed point such that .

Question 60

Fill in the blank. A second order ordinary differential equation,

requires two extra conditions for uniqueness of solution. If these conditions are

and

, then the differential equation is called a(n) ___________.

Choose one answer.

	A. Partial differential equation
	B. Ill posed differential equation
	C. Side-condition differential equation
	D. Boundary value problem

Question 61

Which of the following gives sufficient conditions for

to have a unique solution?

Choose one answer.

	A. .
	B. continuous in , and exists in .
	C. continuous in , and continuous at .
	D. well-defined at the critical point .

Question 62

The Weierstrauss approximation theorem says that under certain conditions on

and for any

, there is a polynomial

, such that

for all

. Choose from below the weakest such conditions on

Choose one answer.

	A. has a power series representation on .
	B. is continuous on .
	C. has one continuous derivative on .
	D. has no poles on .

Question 63

Which of the following is true for

Choose one answer.

	A.
	B.
	C.
	D.

Question 64

Which of the following is true for

Choose one answer.

	A. for near .
	B. for near .
	C. for near .
	D. for near .

Question 65

What is the minimum number of multiplications required to evaluate

for arbitrary real

Choose one answer.

	A.
	B.
	C.
	D.

Question 66

When dividing a polynomial

of degree

by the polynomial

, the remainder will be a scalar. What choice below best describes this scalar?

Choose one answer.

	A. It has degree or degree .
	B. It is .
	C. It is .
	D. It is a root of .
	E. It is deflated.

Question 67

Let

be a zero of a polynomial

of degree n > 10, and suppose we have an initial approximation

such that both Newton's method and the secant method both converge, and an interval

of length about

containing

and for which bisection will converge to

. For this specialized case, list from fastest to slowest, the three methods.

Choose one answer.

	A. Newton's, secant, bisection
	B. Newton's, bisection, secant
	C. Secant, Newton's, bisection
	D. Secant, bisection, Newton's
	E. Bisection, Newton's, secant

Question 68

Fill in the blank. If

is a root of a polynomial

of degree

, then

is a polynomial of degree

. Finding the remaining roots of

by finding the roots of

is called ______________.

Choose one answer.

	A. The Chinese remainder method
	B. Degree slashing
	C. Synthetic division
	D. Deflation

Question 69

Why don't we use the quintic equation to find the zeros of a polynomial of degree 5 or less?

Choose one answer.

	A. Newton's method is more accurate.
	B. Extracting roots is very slow on a computer.
	C. It is pretend.
	D. Rounding errors and truncation errors work against each other.

Question 70

A Hermite interpolator is constructed for values and slopes at 4 knots. What can be said about the degree of this interpolator?

Choose one answer.

	A. Its degree is 4.
	B. Its degree is no more than 4.
	C. Its degree is no more than 7.
	D. Its degree must be more than 7.
	E. None of the above

Question 71

What type of polynomial interpolator would be appropriate for the data

Choose one answer.

	A. Hermite
	B. Taylor
	C. Lagrange
	D. Piecewise linear
	E. Quadratic spline

Question 72

is the Lagrange interpolator for the knots

and

, what is

Choose one answer.

	A. 2.6
	B. 2.8
	C. 3.9
	D. 1.5
	E. 2.9

Question 73

Suppose we approximate

over

using the Lagrange interpolator

for the nodes

and

. Use the Lagrange error term (remainder) to get an upper bound on the error.

Choose one answer.

	A.
	B.
	C.
	D.
	E. There is no error; it is well posed.

Question 74

How many polynomials exist which interpolate the data

where

for

Choose one answer.

	A. Only one
	B. Depends upon the knot positions
	C. n+1 Lagrange basis functions
	D. Infinitely many

Question 75

Suppose

is the Lagrange interpolator for the knots

with

for

. If

is a polynomial of degree

satisfying

, for

. What can be said of

Choose one answer.

	A.
	B.
	C.
	D. is a Hermite interpolator.
	E. None of the above

Question 76

is the Lagrange interpolator for the knots

and

, what is

Choose one answer.

	A. 4.1
	B. 3.8
	C. 3.9
	D. 3.5
	E. 2.9

Question 77

A Taylor polynomial is a special case of what type of interpolator?

Choose one answer.

	A. Osculating polynomial
	B. Lagrange interpolator
	C. Hermite interpolator
	D. Vandermonde interpolator
	E. None of the above

Question 78

Consider the Lagrange basis functions for the knots

, and

. How many basis functions are there for this set of knots, and what are their degrees?

Choose one answer.

	A. Three basis functions, each of degree 2
	B. Two basis functions, each of degree two
	C. Two basis functions, each of degree three
	D. Three basis functions, of degree 1, degree 2 and degree 3

Question 79

Let

be the value of the

derivative of a curve at the point

. Suppose we want to interpolate the data

and

. If a Vandermonde matrix is constructed for the interpolating polynomial, what is its size?

Choose one answer.

	A.
	B.
	C.
	D.

Question 80

Let

be the value of the

derivative of a curve at the point

. Suppose we want to interpolate the data

and

. In general, what is the degree of such an interpolating polynomial?

Choose one answer.

	A. 4
	B. 5
	C. 6
	D. 7

Question 81

is smooth enough in a neighborhood of

, then the Taylor polynomial of degree n for

approximates

with error_________.

Choose one answer.

	A.
	B.
	C.
	D.
	E. None of the above

Question 82

What is the Taylor polynomial of degree 1 for

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 83

What is the Taylor polynomial of degree 1 for

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 84

The linear Taylor polynomial at

for

. Use this Taylor polynomial to approximate

. Select the best 5 significant digit answer to the Taylor approximation.

Choose one answer.

	A. 10.149
	B. 10.150
	C. 5.5074
	D. 9.8500
	E. 10.001

Question 85

The set of polynomials of degree n over a field is a vector space of which dimension?

Choose one answer.

	A. n
	B. 2n
	C. n+2
	D. n/2
	E. None of the above

Question 86

The method of bisection can find a zero of a continuous function, if which of the following is true of the initial data?

Choose one answer.

	A. It is close to the correct answer.
	B. It gives the bisector of the zero.
	C. It is not complex.
	D. It gives function height and slope.
	E. None of the above

Question 87

Bisection is used on an initial interval [a,b], where b-a=2. How many iterations are required for the current interval to have length 1/16?

Choose one answer.

	A. 4
	B. 5
	C. 6
	D. 8
	E. 10

Question 88

If the method of bisection is used to find a zero of a continuous function f(x), and an initial interval [a,b] is given such that f(a)f(b) < 0, counting multiplicity, what can be said about the zeros of f in [a,b]?

Choose one answer.

	A. f has exactly one zero in [a,b].
	B. f has 0, 1, or infinitely many zeros in [a,b].
	C. f has an even number of zeros in [a,b].
	D. f has an odd number of zeros in [a,b].

Question 89

Which method cannot be used to find a root of

Choose one answer.

	A. Newton's method
	B. Bisection method
	C. Secant method
	D. Meuller's method
	E. False position

Question 90

Suppose it is known that the continuous function

has exactly 1 root strictly between 0 and 1, and it is not a double root. Which method can guarantee an approximate root with error less than 0.0001 in fewer than 14 function evaluations?

Choose one answer.

	A. Newton's method
	B. Secant method
	C. Bisection method
	D. None of the above

Question 91

Suppose it is known that the continuous function

has exactly 1 root strictly between 0 and 1, and it is a double root. An encrypted subroutine will return

given any

. Which of the following methods cannot be applied to this problem?

Choose one answer.

	A. Neither Newton's nor bisection can be applied here.
	B. Neither Newton's nor secant can be applied here.
	C. Bisection cannot be applied here.
	D. Newton's method cannot be applied here.

Question 92

Which of the following gives a rough estimate of convergence for bisection and Newton's method (in the case each converge at their respective order of convergence)?

Choose one answer.

	A. For each iteration, Newton's method adds 1 correct bit and bisection add about 0.5 correct bits.
	B. For each of the two iterations, Newton's method doubles number of correct bits and bisection adds 2 correct bits.
	C. For each iteration, Newton's method doubles the number of correct bits and bisection adds 1 correct bit.
	D. For each iteration, Newton's method adds two correct bits and bisection adds 1 correct bit.

Question 93

Suppose Newton's method begins with an initial approximation, x₀.The next approximation, x₁, is found by doing which of the following?

Choose one answer.

	A. Linearizing at and finding its root
	B. Bisecting the line from to
	C. Approximating with a Taylor polynomial of degree 2 at and finding its root
	D. Approximating with a Lagrange interpolator and finding its root
	E. None of the above

Question 94

Let

and

. Use Newton's method to find

.
Identify the approximation below that is correct to 6 significant digits.

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 95

What do we mean when we say Newton's method is not a general purpose method?

Choose one answer.

	A. It does not always converge.
	B. It may divide by zero.
	C. It requires the evaluation of f'.
	D. It may get stuck in a cycle.
	E. It requires complex arithmetic.

Question 96

Which of the following is a Newton iteration to find

Choose one answer.

	A.
	B.
	C.
	D.

Question 97

Which of the following is a Newton iteration to find

Choose one answer.

	A.
	B.
	C.
	D.

Question 98

The secant method begins with two approximations,

and

, to a zero of

. The next point,

is found by doing which of the following?

Choose one answer.

	A. Using the secant angle between , and
	B. Averaging the Newton method and the bisection method
	C. Approximating with a Taylor polynomial of degree 2 at and finding its root
	D. Finding the x-intercept of the line joining and
	E. None of the above

Question 99

The secant method requires how many function evaluations to complete k iterations?

Choose one answer.

	A. k-1
	B. k
	C. k+1
	D. k+2
	E. 2k

Question 100

Let

and

. Use the secant method to find

. Identify the approximation below that is correct to 6 significant digits.

Choose one answer.

	A.
	B.
	C.
	D.
	E.