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advanced engineering mathematics

Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions

Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are givenDo Prob. 2 by RK, 5 steps, h = 0.2. Compare the errors.Data from Prob. 2y' = 2xy, y(0) = 1, h = 0.1
Solve the mixed boundary value problem for the Laplace equation ˆ‡2u = 0 in the rectangle in Fig. 458a (using the grid in Fig. 458b) and the boundary conditions ux= 0 on the left edge, ux= 3 on the right edge, u = x2on the lower edge, and u = x2- 1 on the upper edge. 13 23 бх u, = I. 12
Finer grid of 3 × 3 inner points. Solve Example 1, choosing h = 12/4 = 3 (instead of h = 12/3 = 4) and the same starting values.
Show that (12) gives the starting formula(where one can evaluate the integral numerically if necessary). In what case is this identical with (8)? (из+1,0 + и-1,0) + Иi,1 8(s) ds
Do 10 steps. Solve exactly. Compute the error. Show details.y' = 1/2π√1 - y2, y(0) = 0, h = 0.1
Solve by the Euler’s method. Graph the solution in the y1y2-plane. Calculate the errors.y'1 = -y1 + y2, y'2 = -y1 - y2, y1(0) = 0, y2(0) = 4, h = 0.2, 5 steps
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are giveny' = 2xy, y(0) = 1, h = 0.1
Using the present method, solve (1)–(4) with h = k = 0.2 for the given initial deflection and initial velocity 0 on the given t-interval.f(x) = x2 - x3, 0 ≤ t ≤ 2
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or more than one solution exists, give a reason.-47x1 + 4x2 - 7x3 = -11819x1 - 3x2 + 2x3 =
Fit a parabola (7) to the points (x, y). Check by sketching.t[hr] = Worker’s time on duty, y[sec] = His/her reaction time, (t, y) = (1, 2.0), (2, 1.78), (3, 1.90), (4, 2.35), (5, 2.70)
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[0 0 0 1 0]
Do 4 steps with scaling for the matrix in Prob. 19, starting for [1 1 1] and computing the Rayliegh quotients and error bounds.Data from Prob. 19Compute the inverse of: 15 20 10 20 35 15 15 10 90
Find and graph three circular disks that must contain all the eigenvalues of the matrix:In Prob. 20Data from Prob. 20Compute the inverse of: 5 1 6.
Find and graph three circular disks that must contain all the eigenvalues of the matrix:In Prob. 18Data from Prob. 18Compute the inverse of: Г2.0 0.1 3.3 1.6 4.4 0.5 0.3 -4.3 2.8
Fit and graph:A straight line to (-1, 0), (0, 2), (1, 2), (2, 3), (3, 3)
Compute the l1-, l2-, and l∞-norms of the vectors.[0 0 0 -1 0]T
Compute the l1-, l2-, and l∞-norms of the vectors.[0.2 -8.1 0.4 0 0 -1.3 2]T
(a) Vector norms in our text are equivalent, that is, they are related by double inequalities; for instance,Hence if for some x, one norm is large (or small), the other norm must also be large (or small). Thus in many investigations the particular choice of a norm is not essential. Prove (18).(b)
Show that κ(A) ≥ 1 for the matrix norms (10), (11), and κ(A) ≥ √n for the Frobenius norm (9).
Do 3 steps without scaling, starting from [1 1 1]T.0.2x1 + 4.0x2 - 0.4x3 = 32.00.5x1 - 0.2x2 + 2.5x3 = -5.17.5x1 + 0.1x2 - 1.5x3 = -12.7
Compute the norms (9), (10), (11) for the following (square) matrices. Comment on the reasons for greater or smaller differences among the three numbers. -k 2k -k k -2k -k -k 2k
Solve Ax = b1, Ax = b2. Compare the solutions and comment. Compute the condition number of A. 3.0 1.7 [4.7 4.7 A bị b2 |2.71 1.7 1.0 2.7 ||
Prove Theorem 2. Let A = B + C, B = diag(αjj), At = B + tC, and let t increase continuously from 0 to 1.
Compute the norms (9), (10), (11) for the following (square) matrices. Comment on the reasons for greater or smaller differences among the three numbers.The matrix in Prob. 10Data from Prob. 104x1 + 5x3 = 12.5 x1 + 6x2 + 2x3 = 18.58x1 + 2x2 + x3 =
(a) Find Î± and b such that Î±x1+ x2= b, x1+ x2= 3 has(i) A unique solution(ii) Infinitely many solutions(iii) No solutions.(b) Apply the Gauss elimination to the following two systems and compare the calculations step by step. Explain why the elimination fails if no
Verify (12) for the matrices in Probs. 9 and 10.Data from Prob. 9Compute the matrix norm and the condition number corresponding to the l1-vector norm. 2 4
Find the inverse by the Gauss–Jordan method, showing the details.In Prob. 9Data from Prob. 9Show the factorization and solve.0.01x1 + 0.03x3 = 0.14 0.16x2 + 0.08x3 = 0.160.03x1
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or more than one solution exists, give a reason.3.2x1 + 1.6x2
Do 5 steps, starting from x0 = [1 1 1]. Compare with the Gauss–Seidel iteration. Which of the two seems to converge faster? Show the details of your work.The system in Prob. 10Data from Prob. 104x1 + 5x3 = 12.5 x1 + 6x2 + 2x3
Compute the matrix norm and the condition number corresponding to the l1-vector norm. 5.25 21 10.5 10.5 5.25 4.2 5.25 4.2 3.5 5.25 4.2 3.5 3.
Use (4) to obtain an upper bound for the spectral radius:In Prob. 5Data from Prob. 5Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. ! 1 + i 2 -i 3 1- i
Find the inverse by the Gauss–Jordan method, showing the details.In Prob. 4Data from Prob. 4Show the factorization and solve by Doolittle’s method. 2x1 + x2 + 2x3 = 0-2x1 + 2x2 + x3 = 0 x1 + 2x2 - 2x3 = 18
The least squares approximation of a function f(x) on an interval Î± ‰¤ x ‰¤ b by a functionwhere y0(x), · · ·, ym(x) are given functions, requires the determination of the coefficients Î±0,· · ·,
Do 5 steps, starting from x0 = [1 1 1]. Compare with the Gauss–Seidel iteration. Which of the two seems to converge faster? Show the details of your work.The system in Prob. 4Data from Prob. 44x1 - x2 =
Compute the matrix norm and the condition number corresponding to the l1-vector norm. 0.01 0.01 1 0.01 0.01 1
Solve5x1 + x2 - 3x3 = 17 - 5x2 + 15x3 = -102x1 - 3x2 + 9x3 = 0
Solve 3x2 - 6x3 = 0 4x1 - x2 + 2x3 = 16-5x1 + 2x2 - 4x3 = -20
In Prob. 5, compute C(a) If you solve the first equation for x1, the second for x2, the third for x3, proving convergence(b) If you nonsensically solve the third equation for x1, the first for x2, the second for x3, proving divergence.Data from Prob. 5Do 5 steps, starting from x0 = [1
(a) Write a program for n Ã— n matrices that prints every step. Apply it to the (nonsymmetric!) matrix (20 steps), starting from [1 1 1]T.(b) Experiment in (a) with shifting. Which shift do you find optimal?(c). Considerand takeShow that for q = 0,
Compute the matrix norm and the condition number corresponding to the l1-vector norm. 6 5
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or more than one solution exists, give a reason.4x1 + 4x2 + 2x3 = 03x1 - x2 + 2x3 = 03x1 +
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?). Show the details.4x1 + 5x3 = 12.5 x1 + 6x2 + 2x3 =
Compute the matrix norm and the condition number corresponding to the l1-vector norm. [ 2.1 4.5 0.5 1.8
Do three QR-steps to find approximations of the eigenvalues of: 14.2 -0.1 -0.1 -6.3 0.2 0.2 2.1
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or more than one solution exists, give a reason. 5x1 + 3x2 + x3 = 2-4x2 +
Fit a parabola (7) to the points (x, y). Check by sketching.(-1, 5), (1, 3), (2, 4), (3, 8)
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?). Show the details.3x1 + 2x2 + x3 = 7 x1 + 3x2 + 2x3 = 42x1 + x2 +
Apply the power method (3 steps) with scaling, using x0= [1 1 1]Tor [1 1 1 1]T, as applicable. Give Rayleigh quotients and error bounds. Show the details of your work. 4 0 4 2. 2. 2.
By what integer factor can you at most reduce the Gerschgorin circle with center 3 in Prob. 6?Data from Prob. 6Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. -0.2 10 0.1 0.1 6. -0.2 3
Do three QR-steps to find approximations of the eigenvalues of:The matrix in the answer to Prob. 1Data from Prob. 1Tridiagonalize. Show the details. 0.98 0.04 0.44 0.04 0.56 0.40 0.44 0.80 0.40
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or more than one solution exists, give a reason. 25.38x1 - 15.48x2 = 30.60-14.10x1 +
Fit a straight line to the given points (x, y) by least squares. Show the details. Check your result by sketching the points and the line. Judge the goodness of fit.Estimate R from (i, U) = (2, 104), (4, 206), (6, 314), (10, 530).
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?). Show the details. x2 + 7x3 = 25.55x1 + x2
Apply the power method (3 steps) with scaling, using x0= [1 1 1]Tor [1 1 1 1]T, as applicable. Give Rayleigh quotients and error bounds. Show the details of your work. 4 2 3 2 3 4 7,
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. -0.2 10 0.1 0.1 6. -0.2 3
Crout€™s method factorizes where L is lower triangular and U is upper triangular with diagonal entries(a) Formulas. Obtain formulas for Crout€™s method similar to (4).(b) Examples. Solve Prob. 5 by Crout€™s method.(c) Factor the following matrix by the Doolittle,
Tridiagonalize. Show the details. 5 4 5 1 4 2 2. 4.
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or more than one solution exists, give a reason.6x1 + x2 = -34x1 - 2x2 = 6
Fit a straight line to the given points (x, y) by least squares. Show the details. Check your result by sketching the points and the line. Judge the goodness of fit.Estimate the spring modulus k from the force F [lb] and the elongation s [cm], where (F, s) = (1, 0.3), (2, 0.7), (4, 1.3), (6, 1.9),
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?). Show the details.4x1 - x2 =
Apply the power method without scaling (3 steps), using x0= [1, 1]Tor [1 1 1]T. Give Rayleigh quotients and error bounds. Show the details of your work. 3.6 -1.8 1.8 -1.8 2.8 -2.6 1.8 -2.6 2.8
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. 1 4 3 3 12
Tridiagonalize. Show the details. 1 1 1 1 1
Solve graphically and explain geometrically. -5.00x1 + 8.40x2 = 010.25x1 - 17.22x2 = 0
Fit a straight line to the given points (x, y) by least squares. Show the details. Check your result by sketching the points and the line. Judge the goodness of fit.How does the line in Prob. 1 change if you add a point far above it, say, (1, 3)? Guess first.Data from Prob. 1(0, 2), (2, 0), (3,
Show that for the system in Example 2 the Jacobi iteration diverges. Use eigenvalues.
Apply the power method without scaling (3 steps), using x0= [1, 1]Tor [1 1 1]T. Give Rayleigh quotients and error bounds. Show the details of your work. -3 -3 -1
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[4 -1 8]
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. 5 10-2 10-2 10-2 8. 10-2 10-2 10-2 9.
Prove that the series converges uniformly in the indicated region.
Find the Taylor series with center z0 and its radius of convergence.ez(z-2), z0 = 1
Compute sinh 0.4 from sinh 0, sinh 0.5 = 0.521, sinh 1.0 = 1.175 by quadratic interpolation.
Integrate by (11) with n = 5sin (x2) from 0 to 1.25
In Example 5 of the text, write down the difference table as needed for (18), then write (18) with general x and then with x = 0.56 to verify the answer in Example 5.
Compute p2(0.75) from the data in Prob. 9 and Newton’s divided difference formula (10).Data from problem 9Calculate the Lagrange polynomial p2(x) for the 5S-values f(0.25) = 0.27633,f(0.5) = 0.52050, f(1.0) = 0.84270 and from p2(x) an approximation of f(0.75) (= 0.71116).
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. See App. A3.1. They occur in optics.Si(1) by (2), n = 5, n = 10,
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.What happens in Prob. 15 for any other x0?Data from Prob. 15Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.f = 2x - cos x, x0 = 1. Compare with Prob. 3.Data from
Compute 1 - cos x with 6S arithmetic for x = 0.02(a) As given(b) By 2 sin2 1/2x (derive!).
Find the cubic spline g(x) for the given data with k0 and kn as given.Can you obtain the answer from that of Prob. 15? Data from in Problem 15 fo=f(0) = 2, f = f(2)= -2, f2 = f(4) = 2, f3 = f(6) = 78, ko = k3 = 0.
Evaluate the integralsby Simpson€™s rule with 2m as indicated, and compare with the exact value known from calculus.Integrate e-x from 0 to 2 by (7) with h = 1 and with h = 0.5. Give error bounds for the h = 0.5 value and an error estimate by (10). -2 dx .0.4 dx B = dx, J = хе 1 + x* х
Compute √x2 + 4 - 2 with 6S arithmetic for x = 0.001(a) As given and (b) From x2/√x2 + 4 + 2) (derive!).
Using (14), find f(1.25) by linear, by linear, quadratic, and cubic interpolation of the data (values of (40) in App. A31); 6S value and si(1.25) = 1.14645) f(1.0) = 0.94608, f(1.5) = 1.32468, f(2.0) = 1.60541, f(2.5) = 1.77852, and compute the errors. For the linear interpolation use f(1.0) and
Evaluate the integralsby Simpson€™s rule with 2m as indicated, and compare with the exact value known from calculus.J, 2m = 10 -2 dx .0.4 dx B = dx, J = хе 1 + x* х ||
Solve by fixed-point iteration and answer related questions where indicated. Show details.Let f(x) = x3 + 2x2 - 3x - 4 = 0. Write this as x = g(x), for g choosing (1) (x3 - f)1/3, (2) (x2 - 1/2f)1/2, (3) x + 1/3f, (4) (x3 - f)/x2, (5) (2x2 -f)/(2x), and (6) x - f/f' and in each case x0 = 1.5. Find
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f(0) = 1, f1 = f(2) = 9, f2 = f(4) = 41, f3 = f(6) = 41, k0 = 0, k3 = 12
Compute f'(0.2) for f(x) = x3(a) h = 0.2(b) h = 0.1. Compare the accuracy.
Compute the integral of cos (x2) from 0 to 1 by Simpson’s rule with 2m = 4.
Find the cubic spline q and the interpolation polynomial p for the data (0, 0), (1, 1), (2, 6), (3, 10), with q'(0) = 0, q'(3) = 0 and graph p and q on common axes.
Derive the formula in Prob. 29 from (14)Data from Prob. 29The derivative f'(x) can also be approximated in terms of first-order and higher order differences.Compute f'(0.4) in Prob. 27 from this formula, using differences up to and including first order, second order, third order, fourth order.Data
1 + 1/2 + 1/3 + · · · diverges. Is the same true for the corresponding series of computer numbers?
A €œfour-point formula€ for the derivative isApply it to f(x) = x4 with x1, · · ·, x4 as in Prob. 27, determine the error, and compare it with that in the case of (15).Data from Prob. 27Consider f(x) = x4 for x0 = 0, x1 = 0.2, x2 = 0.4, x3 = 0.6, x4 = 0.8.
Solve, using x0 and x1 as indicated:x = cos x, x0 = 0.5, x1 = 1
Integrating by parts, show that In = ∫10 exxn dx = e -nIn-1, I0 = e - 1.(a) Compute In, n = 0, · · ·, using 4S arithmetic, obtaining I8 = -3.906. Why is this nonsense? Why is the error so large?(b) Experiment in (a) with the number of digits k > 4. As you increase k, will the first negative
This method uses the trapezoidal rule and gains precision step wise by halving h and adding an error estimate. Do this for the integral of f(x) = e-xfrom x = 0 to x = 2 with TOL = 10-3, as follows.Apply the trapezoidal rule (2) with h = 2 (hence n = 1) to get an approximation J11. Halve h and use
Solve, using x0 and x1 as indicated:e-x - tan x = 0, x0 = 1, x1 = 0.7
Figure 430 shows the idea. We assume that f is continuous. We compute the x-intercept c0 of the line through (α0,f(α0)), (b0, f(b0)). If f(c0) = 0, we are done. If f(α0)f(c0) < 0 (as in Fig. 430), we set α1= α0, b1= c0 and repeat to get c1, etc. If f(α0)f(c0) > 0, then
Prove that any binary machine number has a finite decimal representation. Is the converse true?
Solve x2 - 100x + 1 = 0. Use 5S-arithmetic.
Integrate by (11) with n = 5cos x from 0 to 1/2π
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.At what time x (4S-accuracy only) will the processes governed by f1(x) = 100 (1 - e-0.2x) and f2(x) = 40e-0.01x reach the same temperature? Also find the latter.

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