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

	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

.

Let . Approximate using the 2-point difference formula with .

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

.

For fixed , , and finite difference formula, there is an that minimizes the error in the computed approximation to . Why?

	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 .

.

Use a 3-point centered difference formula with to approximate , where
. Identify the best 4 significant digit approximate below.

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

.

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?

	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

.

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

	A.
	B.
	C.
	D.

.

Let . Approximate using the 2-point difference formula with .

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

.

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

	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

.

Use Simpson's method () to approximate . Choose the best 4 significant digit answer below.

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

.

Use composite trapezoidal rule with to approximate . Choose the best 4 significant digit answer below.

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

.

How is the adaptive quadrature method adaptive?

	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.

.

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

	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

.

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

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

.

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

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

.

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

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

.

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?

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

.

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?

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

.

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

	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.

.

The machine epsilon is nearest to which of the following?

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

.

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

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

.

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?

	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.

.

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?

	A.
	B.
	C.
	D.
	E.

.

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

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

.

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 ____________.

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

.

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

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

.

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 ______________.

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

.

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?

	A.
	B.
	C.
	D.

.

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?

	A.
	B.
	C.
	D.

.

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

	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.

.

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

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

.

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

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

.

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

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

.

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

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

.

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

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

.

Which of the following is an initial value problem?

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

.

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

	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

.

What is the Euler slope?

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

.

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

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

.

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?

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

.

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

	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

.

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?

	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.

.

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

	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.

.

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 i, and .

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

.

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

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

.

Which of the following is an implicit method?

	A.
	B.
	C.
	D.

.

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

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

.

Consider the IVP . Approximate using Euler's method with . Select the best 3 significant digit answer below.

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

.

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

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

.

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

	A.
	B.
	C.
	D.

.

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

	A.
	B.
	C.
	D.

.

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

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

.

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

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

.

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

	A.
	B.
	C.
	D.

.

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

	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.

.

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

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

.

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

	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.

.

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

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

.

All single step IVP methods have a common iteration . If , what is the value of that would guarantee was on the solution curve?

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

.

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

	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 .

.

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) ___________.

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

.

Which of the following gives sufficient conditions for to have a unique solution?

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

.

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 ?

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

.

Which of the following is true for ?

	A.
	B.
	C.
	D.

.

Which of the following is true for ?

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

.

What is the minimum number of multiplications required to evaluate for arbitrary real ?

	A.
	B.
	C.
	D.

.

When dividing a polynomial of degree by the polynomial , the remainder will be a scalar. What choice below best describes this scalar?

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

.

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.

	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

.

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 ______________.

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

.

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

	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.

.

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

	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

.

What type of polynomial interpolator would be appropriate for the data ?

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

.

If is the Lagrange interpolator for the knots and , what is ?

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

.

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.

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

.

How many polynomials exist which interpolate the data where for ?

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

.

Suppose is the Lagrange interpolator for the knots with for . If is a polynomial of degree satisfying , for . What can be said of ?

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

.

If is the Lagrange interpolator for the knots and , what is ?

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

.

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

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

.

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?

	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

.

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?

	A.
	B.
	C.
	D.

.

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?

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

.

If is smooth enough in a neighborhood of , then the Taylor polynomial of degree n for at approximates with error_________.

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

.

What is the Taylor polynomial of degree 1 for at ?

	A.
	B.
	C.
	D.
	E.

.

What is the Taylor polynomial of degree 1 for at ?

	A.
	B.
	C.
	D.
	E.

.

The linear Taylor polynomial at for is . Use this Taylor polynomial to approximate . Select the best 5 significant digit answer to the Taylor approximation.

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

.

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

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

.

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

	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

.

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?

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

.

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]?

	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].

.

Which method cannot be used to find a root of ?

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

.

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?

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

.

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?

	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.

.

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)?

	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.

.

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

	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

.

Let and . Use Newton's method to find .
Identify the approximation below that is correct to 6 significant digits.

	A.
	B.
	C.
	D.
	E.

.

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

	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.

.

Which of the following is a Newton iteration to find ?

	A.
	B.
	C.
	D.

.

Which of the following is a Newton iteration to find ?

	A.
	B.
	C.
	D.

.

The secant method begins with two approximations, and , to a zero of . The next point, is found by doing which of the following?

	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

.

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

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

.

Let , and . Use the secant method to find . Identify the approximation below that is correct to 6 significant digits.

	A.
	B.
	C.
	D.
	E.

.