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Questions and Answers of Numerical Analysis

Repeat Exercise 3 using the Jacobi method.Repeat ExerciseUse the QR Algorithm to determine, to within 10ˆ’5, all the eigenvalues for the matrices given in Exercise 1.In
In the lead example of this chapter, the linear system Aw = ˆ’0.04(Ï/p)Î»w must be solved for w and Î» in order to approximate the eigenvalues
The (m ˆ’ 1) Ã— (m ˆ’ 1) tridiagonal matrixis involved in the Forward Difference method to solve the heat equation. For the stability of the method we need Ï(A) a. Î± = 1/4b. Î± =
The eigenvalues of the matrix A in Exercise 14 areCompare the approximations in Exercise 14 to the actual eigenvalues. Again, when is the method stable? In exercise a. Î± = 1/4 b.
Apply two iterations of the QR method without shifting to the following matrices.a.b. c. d.
Use the QR Algorithm to determine, to within 10ˆ’5, all the eigenvalues for the matrices given in Exercise 1.In exercisea.b. c. d. e. f.
Use the QR Algorithm to determine, to within 10ˆ’5, all the eigenvalues of the following matrices.a.b. c. d.
Use the Inverse Power method to determine, to within 10ˆ’5, the eigenvectors of the matrices in Exercise 1.In exercisea.b. c. d. e. f.
Use the Inverse Power method to determine, to within 10ˆ’5, the eigenvectors of the matrices in Exercise 2.In exercisea.b.c.d.
a. Show that the rotation matrixapplied to the vector x = (x1, x2)t has the geometric effect of rotating x through the angle Î¸ without changing its magnitude with respect to the l2 norm.b. Show
Let P be the rotation matrix with pii = pjj = cos Î¸ and pij = ˆ’pji = sin Î¸, for j
Show that the product of an upper triangular matrix (on the left) and an upper Hessenberg matrix produces an upper Hessenberg matrix.
Determine the singular values of the following matrices.a.b. c. d.
Suppose that the m × n matrix A has the singular value decomposition A = U SVt. Express the Nullity(A) in terms of Rank(S).
Suppose that the n × n matrix A has the singular value decomposition A = U SVt. Show that A−1 exists if and only if S−1 exists and find a singular value decomposition for A−1 when it exists.
Part (ii) of Theorem 9.26 states that Nullity (A) = Nullity (AtA). Is it also true that Nullity (A) = Nullity(AAt)?
Show that if A is an m × n matrix and P is an n × n orthogonal matrix, then PA has the same singular values as A.
Show that if A is an n × n nonsingular matrix with singular values s1, s2, ..., sn, then the l2 condition number of A is K2 (A) = (s1 / sn).
Use the result in Exercise 15 to determine the condition numbers of the nonsingular square matrices in Exercises 1 and 2. In exercise Show that if A is an n × n nonsingular matrix with singular
Given the dataa. Use the singular value decomposition technique to determine the least squares polynomial of degree 1. b. Use the singular value decomposition technique to determine the least squares
Given the dataa. Use the singular value decomposition technique to determine the least squares polynomial of degree 2. b. Use the singular value decomposition technique to determine the least squares
Determine the singular values of the following matrices.a.b. c. d.
Determine a singular value decomposition for the matrices in Exercise 1.In exercisea.b. c. d.
Determine a singular value decomposition for the matrices in Exercise 2.In exercisea.b. c. d.
Let A be the matrix given in Example 2. Show that (1, 2, 1)t, (1,−1, 1)t, and (−1, 0, 1)t are eigenvectors of AtA corresponding to, respectively, the eigenvalues λ1 = 5, λ2 = 2 and λ3 = 1.
Suppose that A is an m × n matrix A. Show that Rank(A) is the same as the Rank(At).
Suppose that A has the singular value decomposition A = U SVt. Determine, with justification a singular value decomposition of At.
Suppose that A has the singular value decomposition A = U SVt. Show that Rank(A) = Rank(S).
Repeat Exercise 8 using the Gauss-Seidel method.Repeat exercise
Show that a function F mapping D ⊂ Rn into Rn is continuous at x0 ∈ D precisely when, given any number ε > 0, a number δ > 0 can be found with property that for any vector norm
Let A be an n × n matrix and F be the function from Rn to Rn defined by F(x) = Ax. Use the result in Exercise 12 to show that F is continuous on Rn.
The nonlinear system −x1(x1 + 1) + 2x2 = 18, (x1 − 1)2 + (x2 − 6)2 = 25
The nonlinear systemx21 ˆ’ 10x1 + x22 + 8 = 0, x1x22 + x1 ˆ’ 10x2 + 8 = 0can be transformed into the fixed-point problema. Use Theorem 10.6 to show that G = (g1, g2)t mapping D
The nonlinear system 5x21 − x22 = 0, x2 − 0.25(sin x1 + cos x2) = 0
Use Theorem 10.6 to show that G: D ⊂ R3 → R3 has a unique fixed point in D. Apply functional iteration to approximate the solution to within 10−5, using the l∞ norm. a. D = {(x1, x2, x3)t |
Use functional iteration to find solutions to the following nonlinear systems, accurate to within 10−5, using the l∞ norm. a. x22+ x22− x1 = 0 x21 − x22 − x2 = 0. b. 3x21 − x22= 0, 3x1
Use the Gauss-Seidel method to approximate the fixed points in Exercise 7 to within 10ˆ’5, using the lˆž norm.In exercisea.D = {(x1, x2, x3)t | ˆ’1 ‰¤ xi
Use Newton's method with x(0) = 0 to compute x(2) for each of the following nonlinear systems.a.b. sin(4Ï€ x1 x2) ˆ’ 2x2 ˆ’ x1 = 0, c. x1(1 ˆ’ x1) + 4x2 =
What does Newton's method reduce to for the linear system Ax = b given by a11x1 + a12 x2 +· · ·+a1nxn = b1, a21x1 + a22 x2 +· · ·+a2nxn = b2, . . . an1x1 + an2x2 +· · ·+annxn = bn, where A
Show that when n = 1, Newton's method given by Eq. (10.9) reduces to the familiar Newton's method given by in Section 2.3.
The amount of pressure required to sink a large, heavy object in a soft homogeneous soil that lies above a hard base soil can be predicted by the amount of pressure required to sink smaller objects
In calculating the shape of a gravity-flow discharge chute that will minimize transit time of discharged granular particles, C. Chiarella, W. Charlton, and A.W. Roberts [CCR] solve the following
An interesting biological experiment (see [Schr2]) concerns the determination of the maximum water temperature, XM, at which various species of hydra can survive without shortened life expectancy.
Use Newton's method with x(0) = 0 to compute x(2) for each of the following nonlinear systems.a.b. x21 + x2 ˆ’ 37 = 0, x1 ˆ’ x22 ˆ’ 5 = 0, x1 + x2 + x3 ˆ’ 3 =
Use the graphing facilities of Maple to approximate solutions to the following nonlinear systems. a. b. sin(4πx1 x2) − 2x2 − x1 = 0, c. x1(1 − x1) + 4x2 = 12, (x1 − 2)2 + (2x2 − 3)2 =
Use the graphing facilities of Maple to approximate solutions to the following nonlinear systems within the given limits. a. b. c. d.
Use the answers obtained in Exercise 3 as initial approximations to Newton's method. Iterate until || x(k) ˆ’ x(kˆ’1)||ˆž In exercisea.b. sin(4Ï€x1 x2)
Use the answers obtained in Exercise 4 as initial approximations to Newton's method. Iterate until ||x(k) ˆ’ x(kˆ’1)||ˆž In exercisea.b. c. d.
Use Newton's method to find a solution to the following nonlinear systems in the given domain. Iterate until ||x(k) − x(k−1)||∞ < 10−6. a. 3x21 − x22 = 0, 3x1x22 − x31 − 1 = 0. Use x(0)
The nonlinear system E1: 4x1 − x2 + x3 = x1 x4,....................E2: −x1 + 3x2 − 2x3 = x2 x4, E3: x1 − 2x2 + 3x3 = x3 x4,........................E4: x21 + x22 + x23 = 1
Use Broyden's method with x(0) = 0 to compute x(2) for each of the following nonlinear systems.a.b. sin(4Ï€ x1 x2) ˆ’ 2x2 ˆ’ x1 = 0, c. 3x21 ˆ’ x22 = 0, 3x1
a. Use the result in Exercise 9 to show that if Aˆ’1 exists and x, y ˆˆ Rn, then (A + xyt)ˆ’1 exists if and only if ytAˆ’1x ‰ ˆ’1.b. By multiplying on the right by A + xyt, show
Use Broyden's method with x(0) = 0 to compute x(2) for each of the following nonlinear systems.a.b. x21 + x2 ˆ’ 37 = 0, x1 ˆ’ x22ˆ’ 5 = 0, x1 + x2 + x3 ˆ’ 3 =
Use Broyden's method to approximate solutions to the nonlinear systems in Exercise 1 using the following initial approximations x(0). a. (0, 0)t b. (0, 0)t c. (1, 1)t d. (2, 2)t
Use Broyden's method to approximate solutions to the nonlinear systems in Exercise 2 using the following initial approximations x(0). a. (1, 1, 1)t b. (2, 1,−1)t c. (−1,−2, 1)t d. (0, 0, 0)t
Use Broyden's method to approximate solutions to the following nonlinear systems. Iterate until || x(k) − x(k−1) || ∞ < 10−6. a. x1(1 − x1) + 4x2 = 12, (x1 − 2)2 + (2x2 − 3)2 = 25. b.
The nonlinear system 4x1 − x2 + x3 = x1 x4, − x1 + 3x2 − 2x3 = x2 x4, x1 − 2x2 + 3x3 = x3 x4, x21 + x22 + x23= 1
Show that if 0 ≠ y ∈ Rn and z ∈ Rn, then z = z1 + z2, where z1 = (ytz/ ||y||22)y is parallel to y and z2 is orthogonal to y.
Show that if u, v ∈ Rn, then det (I + uvt) = 1 + vtu.
Use the method of Steepest Descent with TOL = 0.05 to approximate the solutions of the following nonlinear systems. a. b. ln(x21 + x22) − sin(x1 x2) = ln 2 + ln π, ex1−x2 + cos (x1 x2) = 0. c.
Use the method of Steepest Descent with TOL = 0.05 to approximate the solutions of the following nonlinear systems. a. 15x1 + x22 − 4x3 = 13, x21 + 10x2 − x3 = 11, x32 − 25x3 = −22. b. 10x1
Use the results in Exercise 1 and Newton's method to approximate the solutions of the nonlinear systems in Exercise 1 to within 10−6. In exercise a. b. ln(x21 + x22) − sin(x1 x2) = ln 2 + ln
Use the results of Exercise 2 and Newton's method to approximate the solutions of the nonlinear systems in Exercise 2 to within 10−6. In exercise a. 15x1 + x22 − 4x3 = 13, x21 + 10x2 − x3 =
Use the method of Steepest Descent to approximate minima to within 0.005 for the following functions. a. g(x1, x2) = cos(x1 + x2) + sin x1 + cos x2 b. g(x1, x2) = 100(x21− x2)2 + (1 − x1)2 c.
a. Show that the quadratic polynomial P(α) = g1 + h1α + h3α (α − α2) interpolates the function h defined in (10.18): h(α) = g(x(0) − α∇g x(0)) at α = 0, α2, and α3. b. Show that a
The nonlinear system f1(x1, x2) = x21 − x22 + 2x2 = 0, f2(x1, x2) = 2x1 + x22− 6 = 0
Show that the continuation method and Euler's method with N = 1 gives the same result as Newton's method for the first iteration; that is, with x(0) = x(0) we always obtain x(1) = x(1).
Show that the homotopyG(λ, x) = F(x) − e−λ F(x(0))used in the continuation method with Euler's method and h = 1 also duplicates Newton's method for any x(0); that is, with x(0) = x(0), we have
Let the continuation method with the Runge-Kutta method of order four be abbreviated CMRK4. After completing Exercises 4, 5, 6, 7, 8, and 9, answer the following questions. a. Is CMRK4 with N = 1
Repeat Exercise 1 using the Runge-Kutta method of order four with N = 1. Repeat exercise The nonlinear system f1(x1, x2) = x21 − x22 + 2x2 = 0, f2(x1, x2) = 2x1 + x22− 6 = 0
Use the continuation method and Euler's method with N = 2 on the following nonlinear systems.a.b. sin(4Ï€x1 x2) ˆ’ 2x2 ˆ’ x1 = 0, c. d.
Use the continuation method and the Runge-Kutta method of order four with N = 1 on the following nonlinear systems using x(0) = 0. Are the answers here comparable to Newton's method or are they
Repeat Exercise 4 using the initial approximations obtained as follows. a. From 10.2(3c) b. From 10.2(3d) c. From 10.2(4c) d. From 10.2(4d) In exercise Use the continuation method and the Runge-Kutta
Use the continuation method and the Runge-Kutta method of order four with N = 1 on Exercise 7 of Section 10.2. Are the results as good as those obtained there?In exerciseUse the continuation method
Repeat Exercise 5 using N = 2.Repeat Exercise 4 using the initial approximations obtained as follows.a. From 10.2(3c)b. From 10.2(3d)c. From 10.2(4c)d. From 10.2(4d)In exerciseUse the continuation
Repeat Exercise 8 of Section 10.2 using the continuation method and the Runge-Kutta method of order four with N = 1. In exercise E1: 4x1 − x2 + x3 = x1 x4,....................E2: −x1 + 3x2 −
Show that the piecewise-linear basis functions {φi}ni =1 are linearly independent.
Show that the cubic spline basis functions {φi}n+1i=0 are linearly independent.
Show that the matrix given by the piecewise linear basis functions is positive definite.
Show that the matrix given by the cubic spline basis functions is positive definite.
Use the Piecewise Linear Algorithm to approximate the solution to the boundary-value problem − d / dx (xy') + 4y = 4x2 − 8x + 1, 0 ≤ x ≤ 1, y(0) = y(1) = 0 using x0 = 0, x1 = 0.4, x2 = 0.8,
Use the Piecewise Linear Algorithm to approximate the solutions to the following boundary-value problems, and compare the results to the actual solution: a. −x2y" − 2xy' + 2y = −4x2, 0 ≤ x
Use the Cubic Spline Algorithm with n = 3 to approximate the solution to each of the following boundary-value problems, and compare the results to the actual solutions given in Exercises 1 and 2: a.
Repeat Exercise 3 using the Cubic Spline Algorithm. Repeat exercise 3
Show that the boundary-value problem − d / dx (p(x)y') + q(x)y = f (x), 0≤ x ≤ 1, y(0) = α, y(1) = β, can be transformed by the change of variable z = y − βx − (1 − x)α into the
Use Exercise 6 and the Piecewise Linear Algorithm with n = 9 to approximate the solution to the boundary-value problem −y" + y = x, 0 ≤ x ≤ 1, y(0) = 1, y(1) = 1 + e−1. In exercise 6 Show
Repeat Exercise 7 using the Cubic Spline Algorithm. Repeat exercise 7
Show that the boundary-value problem − d / dx (p(x)y') + q(x)y = f (x), a ≤ x ≤ b, y(a) = α, y(b) = β, can be transformed into the form − d / dw (p(w)z') + q(w)z = F(w), 0≤ w ≤ 1, z(0)
The boundary-value problem y" = 4(y − x), 0 ≤ x ≤ 1, y(0) = 0, y(1) = 2,
The boundary-value problem y" = y' + 2y + cos x, 0≤ x ≤ π/2, y(0) = −0.3, y (π/2) = −0.1
Use the Linear Shooting method to approximate the solution to the following boundary-value problems. a. y" = −3y' + 2y + 2x + 3, 0 ≤ x ≤ 1, y(0) = 2, y(1) = 1; use h = 0.1. b. y" = −4x−1y'
Although q(x) < 0 in the following boundary-value problems, unique solutions exist and are given. Use the Linear Shooting Algorithm to approximate the solutions to the following problems, and compare
Use the Linear Shooting Algorithm to approximate the solution y = e−10x to the boundary-value problem Y" = 100y, 0 ≤ x ≤ 1, y(0) = 1, y(1) = e−10. Use h = 0.1 and 0.05.
Write the second-order initial-value problems (11.3) and (11.4) as first-order systems, and derive the equations necessary to solve the systems using the fourth-order Runge-Kutta method for systems.
Show that, under the hypothesis of Corollary 11.2, if y2 is the solution to y" = p(x)y' + q(x)y and y2(a) = y2(b) = 0, then y2 ≡ 0.
Consider the boundary-value problem Y" + y = 0, 0 ≤ x ≤ b, y(0) = 0, y(b) = B. Find choices for b and B so that the boundary-value problem has
Use the Nonlinear Shooting Algorithm with h = 0.25 to approximate the solution to the boundary-value problem Y" = 2y3, −1 ≤ x ≤ 0, y(−1) = 1/2, y(0) = 1/3. Compare your results to the actual
Use the Nonlinear Shooting method with TOL = 10−4 to approximate the solution to the following boundary-value problems. The actual solution is given for comparison to your results. a. y" =
Use the Nonlinear Shooting method with TOL = 10−4 to approximate the solution to the following boundary-value problems. The actual solution is given for comparison to your results. a. y" = y3 -yy',
a. Change Algorithm 11.2 to incorporate the Secant method instead of Newton's method. Use t0 = (β − α) / (b − a) and t1 = t0 + (β - y (b, t0)) / (b − a). b. Repeat Exercise 4(a) and 4(c)

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