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Chapter 01 Sample Exam
- (10 points) Given the quadratic equation 0. 987 x^2 + 11. 2 x + 0.246 = 0. Find the best approx- imation to each of the two solutions using 3 digit chopping arithmetic and the appropriate equations for x 1 and x 2.
- (10 points) Given the quadratic equation 0. 987 x^2 − 11. 2 x − 0 .246 = 0. Find the best approx- imation to each of the two solutions using 3 digit rounding arithmetic and the appropriate formulas.
- (15 points) Let x 0 = 0.5. Given
f (x) = − 2 e−x^ + 1/ 4 x^4 −
x^5 + 2x f ′(x) = 2e−x^ + x^3 −
x^4 + 2
f ′′(x) = − 2 e−x^ + 3x^2 −
x^3 f ′′′(x) = 2e−x^ + 6x −
x^2
f (4)(x) = − 2 e−x^ + 6 − x f (5)(x) = 2e−x^ − 1 f (6)(x) = − 2 e−x
(a) (5 points) Find the Taylor Polynomial, T 3 (x), of degree at most 3 for f (x) expanded about x 0. (b) (5 points) Give the general error formula for f (x) − T 3 (x) for any x. (c) (5 points) Find the absolute error in using T 3 (0.65) to approximate f (0.65).
- (10 points) Let x 0 = 0. Given
f (x) = − 2 e−x^ + 1/ 4 x^4 −
x^5 + 2x f ′(x) = 2e−x^ + x^3 −
x^4 + 2
f ′′(x) = − 2 e−x^ + 3x^2 −
x^3 f ′′′(x) = 2e−x^ + 6x −
x^2
f (4)(x) = − 2 e−x^ + 6 − x f (5)(x) = 2e−x^ − 1 f (6)(x) = − 2 e−x
(a) (5 points) Find the Taylor Polynomial, T 3 (x), of degree at most 3 for f (x) expanded about x 0. (b) (5 points) Use the error formula to find a bound for the absolute error in approximating f (0.65) with T 3 (0.65).
- (10 points) Let f (x) = x^3 − e−x, x 0 = 0.5.
(a) (5 points) Find the Taylor Polynomial, T 2 (x), of degree at most 2 for f (x) expanded about x 0. (b) (5 points) Evaluate T 2 (0.8) and compute the actual error |f (0.8) − T 2 (0.8)|
Chapter 02 Sample Exam
- (10 points) The equation f (x) = x^2 − 2 ex^ = 0 has a solution in the interval [-1,1].
(a) (5 points) With p 0 = −1 and p 1 = 1 calculate p 2 using the Secant method. (b) (5 points) With p 2 from part (a) calculate p 3 using Newton’s method.
- (15 points) The equation f (x) = 2 − x^2 sin x = 0 has a solution in the interval [-1,2].
(a) (5 points) Verify that the Bisection method can be applied to the function f (x) on [-1,2]. (b) (5 points) Using the error formula for the Bisection method find the number of iterations needed for accuracy 0.000001. Do not do the Bisection calculations. (c) (5 points) Compute p 3 for the Bisection method.
- (15 points) The following refer to the fixed-point problem
(a) (5 points) State the theorem which gives conditions for a fixed-point sequence to converge to a unique fixed point.
(b) (5 points) Given g(x) =
2 − x^3 + 2x 3 , use the theorem to show that the fixed-point se- quence will converge to the unique fixed-point of g for any p 0 in [-1,1.1]. (c) (5 points) With p 0 = 0.5 generate p 3.
- (10 points) Suppose the function f (x) has a unique zero p in the interval [a, b]. Further, suppose f ′′(x) exists and is continuous on the interval [a,b]. (a) (5 points) Under what conditions will Newton’s Method give a quadratically convergent sequence to p? (b) (5 points) Define quadratic convergence.
- (10 points) Let g(x) = 2 − x^3 + 2x 3
on the interval [-1, 1.1]. Let the initial value be 0 and compute the result of 2 iterations of Stefffensen’s Method to approximate the solution of x = g(x).
Chapter 04 Sample Exam
- (10 points) Let f (x) =
cos x 1 + x^3
. Approximate f ′(0.9) using the three point centered difference formula with h = 0.2.
- (15 points) Let f (x) = x ln x + x^4
(a) (5 points) Approximate I =
1 f^ (x)dx^ using Composite Simpsons rule with^ n^ = 4. (b) (5 points) Find the smallest upper bound for the absolute error using the error formula. (c) (5 points) Find the values of n and h required for an error of at most 0.00001?
- (10 points) The Composite Trapezoidal Rule applied to the integral I =
∫ (^) b a f^ (x)dx^ gives the error E = −
b − a 12 h^3 f ′′(μ). Suppose f ′′(x) =
2 + 2x − ex 3 , a = 0. 51 , b = 1.0. What values of n and h should be used to approximate I to within 0.00001?
- (15 points) Let h = 0.2. Given
f (x) = − 2 e−x^ + 1/ 4 x^4 −
x^5 + 2x f ′(x) = 2e−x^ + x^3 −
x^4 + 2
f ′′(x) = − 2 e−x^ + 3x^2 −
x^3 f ′′′(x) = 2e−x^ + 6x −
x^2
f (4)(x) = − 2 e−x^ + 6 − x f (5)(x) = 2e−x^ − 1 f (6)(x) = − 2 e−x
(a) (5 points) Approximate f ′(0.65) using the three point centered difference formula. (b) (5 points) Give the general form of the error formula for the five point centered difference formula. (c) (5 points) Give the error formula for part (a).
- (10 points) Consider the integral I =
0 f^ (x)dx^ for the function
f (x) = − 2 e−x^ + 1/ 4 x^4 −
x^5 + 2x f ′(x) = 2e−x^ + x^3 −
x^4 + 2
f ′′(x) = − 2 e−x^ + 3x^2 −
x^3 f ′′′(x) = 2e−x^ + 6x −
x^2
f (4)(x) = − 2 e−x^ + 6 − x f (5)(x) = 2e−x^ − 1 f (6)(x) = − 2 e−x
(a) (5 points) Approximate I using the Composite trapezoidal rule with n = 4. (b) (5 points) Find a bound for the absolute error using the error formula.
- (10 points) Suppose a numerical procedure is available to approximate a number N with the approximation N (h), where the parameter of the procedure is h > 0. Further, suppose there is an error expansion N = N (h) + h + 2h^3. For a given h we have calculated N (h), N ( h 2 ) and N ( h 4 ). Give the formulas to obtain the best possible approximation to N using what has already been calculated along with extrapolation.
Chapter 05 Sample Exam
- (20 points) The Initial Value Problem
y′^ = 1 +
y t , 1 ≤ t ≤ 2 , y(1) = 2
has the solution y(t) = t ln t + 2t (a) (5 points) Approximate y(1.2) using Euler’s Method with h = 0.1. (b) (5 points) Approximate y(1.13) using the results of part (a) and linear interpolation. (c) (5 points) Approximate y(1.2) using the Taylor’s Method or Order 2 with h = 0.1. (d) (5 points) Find a bound for the error in part (a) given by |y(1.2) − w 2 |using the error formula for Euler’s Method from Chapter 5 of the textbook.
- (20 points) Show the initial value problem
y′^ = y − t cos ty, 0 ≤ t ≤ 2 , y(0) = 2
has a unique solution and is also well-posed.
- (20 points) Given the Initial Value Problem y′^ = f (t, y), a ≤ t ≤ b, y(a) = α and an integer N , let h = b−Na and ti = a + ih, i = 0, 1 ,... , N − 1. The Trapezoidal method gives the approximations wi to y(ti) defined by wi+1 = wi + 12 h(f (ti+1, wi+1) + f (ti, wi)) for i = 0, 1 ,... , N − 1. (a) (5 points) Is the method a one-step or two-step method? (b) (5 points) Is the method explicit or implicit? Why? (c) (5 points) Given the initial value problem y′^ = t − y + 2, 0 ≤ t ≤ 1 , y(0) = 3 Let N = 2 and generate w 2 to approximate y(1) using Euler’s method. (d) (5 points) Using Eulers method as a predictor method and the Trapezoidal method as a corrector method, generate another approximation w 2 for (1).
- (20 points) For the initial value problem y′^ = f (t, y), a ≤ t ≤ b, y(a) = α the Adams-Bashfoth three -step explicit method is given by
w 0 = α, w 1 = α 1 , w 2 = α 2 , wi+1 = wi + h 12 (23f (ti, wi) − 16 f (ti− 1 , wi− 1 ) + 5f (ti− 2 , wi− 2 ))
for i = 2, 3 ,... , N − 1, the Adams-Moulton two-step implicit method is given by
w 0 = α, w 1 = α 1 , wi+1 = wi + h 12
(5f (ti+1, wi+1) + 8f (ti, wi) − f (ti− 1 , wi− 1 )),
for i = 1, 2 ,... , N − 1, and a Runge-Kutta method of order three is given by
w 0 = α, wi+1 = wi +
h 4 (f (ti, wi) + 3(f (ti +
2 h 3 , wi +
2 h 3 f (ti +
h 3 , wi +
h 3 f (ti, wi)))))
for i = 1, 2 ,... , N − 1.
Chapter 06 Sample Exam
- (10 points) Solve the linear system { 0. 810 x 1 + 0. 211 x 2 = 1. 52 − 1. 03 x 1 + 0. 713 x 2 = − 0. 512
using 3 digit rounding arithmetic and Gaussian elimination.
- (10 points) Solve the linear system { 0. 810 x 1 + 0. 211 x 2 = 1. 52 − 1. 03 x 1 + 0. 713 x 2 = − 0. 512
using 3 digit chopping arithmetic and Gaussian elimination.
- (10 points) Solve the linear system { 0. 211 x 1 + 0. 811 x 2 = 1. 52 1. 71 x 1 − 1. 06 x 2 = − 0. 512
using 3 digit rounding arithmetic and Gaussian elimination with partial pivoting.
- (10 points) Solve the linear system
{
- 211 x 1 + 0. 811 x 2 = 1. 52
- 71 x 1 − 1. 06 x 2 = − 0. 512
using 3 digit chopping arithmetic and Gaussian elimination with partial pivoting.
- (10 points) Let A be an m by n matrix, B be an n by m matrix, x and y be column vectors of dimension n, and z be a column vector of dimension m. What are the following (BE SPECIFIC)?
(a) Ax (b) ztA (c) AtA (d) AAt^ (e) ytBx (f ) ytx (g) xyt^ (h) ztAx
- (15 points) Suppose the linear system Ax = b has S 1 = 22, S 2 = 17, S 3 = 21 and after one elimination step the system becomes
14 x 1 − 22 x 2 + 3 x 3 = 41 13 x 2 − 16 x 3 = 53 − 15 x 2 + 15x 3 = 33
What row and/or column interchange must be done to put the next pivot element into the a 22 position using (a) (5 points) Partial Pivoting
Numerical Analysis 10E Chapter 06 Sample Exam - Page 2 of ??
(b) (5 points) Scaled Partial Pivoting (c) (5 points) Total or Complete Pivoting
- (10 points) Given the linear system { 14 x + 15y = 48 21 x − 23 y = − 2
(a) (5 points) What is the pivot element if scaled partial pivoting is to be used to solve the linear system? (b) (5 points) What is the pivot element if complete or total pivoting is to be used to solve the system?
- (10 points) Let A = LU where
L =
(^) and U =
(a) (5 points) Solve the linear system Ax = b where b = (0, − 2 , 2) using the given factorization (b) (5 points) Using the factorization only, what is the determinant of A?
- (10 points) Which of the properties: nonsingular, singular, symmetric, strictly diagonally dom- inant, positive definite apply to the following matrixes. There may be more than one property applying to a matrix.
(a)
(^) (b)
(^) (c)
(^) (d)
- (15 points) Let A =
(^) and b =
(a) (5 points) Factor the matrix A using your choice of factorizations. (b) (5 points) Using the factorization obtained in (a) solve the system Ax = b. (c) (5 points) Using the factorization obtained in (a) compute the determinant of A.
Numerical Analysis 10E Chapter 07 Sample Exam - Page 2 of 2
(a) (5 points) Find ||A||∞ (b) (5 points) Find ρ(A). (c) (5 points) Find an eigenvector of A corresponding to the eigenvalue for which |λ| = ρ(A). (d) (5 points) List all properties of A.
Chapter 08 Sample Exam
- (10 points) Given the data {(1. 0 , 3 .6), (1. 1 , 3 .1), (1. 3 , 2 .4), (1. 8 , 0 .4)} find the linear least squares approximation ax + b and, also, compute the least squares error E(a, b)^2
- (10 points) Given the data {(xj , yj ), j = 1,... M }. Find the least squares approximation for the form φ(x) = ax + b sin (16x), that is, find the normal equations. Apply your approximation to the data in Problem 1 and, also, compute the least squares error.
- (10 points) Given the data {(xj , yj ), j = 1,... M }. Find the least squares approximation for the form φ(x) = ax + bx , that is, find the normal equations. Apply your approximation to the data in Problem 1 and, also, compute the least squares error.
- (10 points) Find the Pade’ approximation of the form
r(x) = p 0 + p 1 x + p 2 x^2 1 + q 1 x
for the function f (x) = 1 −
x +
x^2 −
x^3 +
x^4 −
x^5
- (10 points) Find the Pade’ approximation of the form
r(x) =
p 0 + p 1 x + p 2 x^2 1 + q 1 x + q 2 x^2
for the function f (x) = ln (1 + x)
- (10 points) Given the data xi 1. 0 1. 1 1. 3 1. 8 yi 3. 6 3. 1 2. 4 0. 4
find the best choice of a and b in the least squares sense to approximate the data with a function of the form
y =
a x
Find the error in the approximation.
- (15 points) Given the data
x 1.0 1.2 1.3 1. y 1.90 2.65 3.12 4.
(a) (5 points) Find the linear least squares approximation to the data and find its error. (b) (5 points) Find the discrete least squares approximation of degree two to the data and find its error.
Chapter 09 Sample Exam
- (15 points) Let
A =
(a) (5 points) Find the eigenvalues and eigenvectors of A. (b) (5 points) Prove that the eigenvectors are linearly independent. (c) (5 points) Find a matrix P such that P −^1 AP is a diagonal matrix. Is your matrix or- thogonal? If not, how can it be made orthogonal?
- (10 points) Let
A =
Perform two iterations of the Power Method on A.
- (10 points) Let
A =
(a) (5 points) Show that the QR decomposition of A is
Q =
2 5
1 5
and R =
(b) (5 points) Perform the first iteration of the unshifted QR Method on A
- (10 points) Let
A =
Find the singular values of A.
- (15 points) Let x = (1, 2 , 1)t, y = (2, − 1 , 1)t, z = (− 1 , 0 , 0)t.
(a) (5 points) Show that S = {x, y, z} is a linearly independent set. (b) (5 points) Use the Gram-Schmidt procedure to find a linearly independent set of vectors using S. (c) (5 points) Find a set of orthonormal vectors from the set in part (b).
Chapter 10 Sample Exam
- (15 points) Given G(x 1 , x 2 ) = (g 1 (x 1 , x 2 ), g 2 (x 1 x 2 ))t^ = (
4 + x^22 − 1 ,
1 + x 1 − 1)t^ is defined on the set D = {(x 1 , x 2 )| 1 ≤ x 1 ≤ 1. 5 , 0 ≤ x 2 ≤ 1 }. (a) (5 points) Show that G is continuous on D. (b) (5 points) Show that G has a unique fixed-point on D. (c) (5 points) Show that the fixed-point sequence {x(k)} defined by x(k)^ = G(x(k−1)) for k = 1, 2 , 3 ,... will converge to the unique fixed-point in D provided the initial vector x(0) is in D.
- (10 points) Using x(0)^ = (1. 25 , 0 .5)t^ compute x(4)^ using fixed-point iteration on the fixed point problem in question 1 part (c).
- (10 points) Repeat Question 2 using the Gauss-Seidel method. Which of the two methods appears to give faster convergence? Why?
- (10 points) The fixed-point problem in Question 1 is a means to solve the nonlinear system
(x 1 + 1)^2 − x^22 = 4 (x 2 + 1)^2 − x^21 = 1
Describe in detail how to compute x(1)^ using Newton;s method.
- (10 points) Given the nonlinear system
x^21 + x 1 x 2 + 4x 2 − 8 = 0 (x 1 − 2)^2 + (2x 2 + 3)^2 − 5 = 0
Let P (0)^ = (1. 0 , 1 .0)t^ be the starting value. Find P (1)^ using Newton’s Method.
- (10 points) Given the nonlinear system
x^31 + x^21 x 2 − x 1 x 3 + 6 = 0 ex^1 + ex^2 − x 3 = 0 x^22 − 2 x 1 x 3 − 4 = 0
and x(0)^ =
Give the steps required to compute x(1)^ using Newtons method but do not do the computations.
Chapter 12 Sample Exam
- (10 points) Consider Poisson’s Equation
∂^2 ∂x^2 u(x, y) +
∂^2
∂y^2 u(x, y) = x^2 + y^2 + 4, 0 < x < 1 , 0 < y < 1
with boundary conditions
u(x, 0) = x^2 , u(x, 1) = x^2 + 1, 0 ≤ x ≤ 1 u(0, y) = y^2 , u(x, 1) = y^2 + 1, 0 ≤ y ≤ 1
Derive the linear system for the finite-difference approximation of the solution of the partial differential equation with h = k = 13.
- (10 points) Consider the Heat Equation
∂ ∂t
u(x, t) −
π^2
∂^2
∂x^2
u(x, t) = 0, 0 < x < 1 , 0 < t
with boundary conditions u(x, t) = 0, u(1, t) = 0, t > 0 and initial conditions u(x, 0) = sin (πx), 0 < x < 1. Use the Forward Difference Method with h = 14 and k = 12 to approximate u(h, 2 k), u(2h, 2 k), u(3h, 2 k).
- (10 points) Consider the Heat Equation
∂ ∂t u(x, t) −
π^2
∂^2
∂x^2 u(x, y) = 0, 0 < x < 1 , 0 < t
with boundary conditions u(x, t) = 0, u(1, t) = 0, t > 0 and initial conditions u(x, 0) = sin (πx), 0 < x < 1. Use the Backward Difference Method with h = 14 and k = 12 to approximate u(h, 2 k), u(2h, 2 k), u(3h, 2 k).
- (10 points) Given the solution to the heat equation of Questions 2 and 3 is u(x, t) = e−t^ sin (πx). Which of the approximations of Questions 2 and 3 are better?
- (10 points) Consider the Wave equation
∂^2 ∂t^2 u(x, t) −
∂^2
∂x^2 u(x, t) = 0, 0 < x < π, 0 < t
with boundary conditions u(0, t) = 0, u(π, t) = 0, t > 0 and initial conditions u(x, 0) = sin (2x), 0 < x < π and
∂t
u(x, 0) = 0, 0 < x < π. Use the finite difference method with h = k = π 4 to approximate u(h, 3 k), u(2h, 3 k), u(3h, 3 k)
