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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Calculate and graph L0, L1, L2, L3 with x0 = 0, x1 = 1, x2 = 2, x3 = 3 on common axes. Find p3(x) for the data (0, 1), (1, 0.765198), (2, 0.223891), (3, -0.260052) [values of the Bessel function
Overflow and underflow can sometimes be avoided by simple changes in a formula. Explain this in terms of √x2 + y2 = x√1 + (y/x)2 with x2 ≥ y2 and x so large that x2 would cause overflow. Invent
How did we use an interpolation polynomial in deriving Simpson’s rule?
What is adaptive integration? Why is it useful?
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f (0) = 1, f1 = f(1) = 0, f2 = f(2) = 1, f3 = f(3) = 0, k0 = 0, k3 = 6
Solve by fixed-point iteration and answer related questions where indicated. Show details.Prove that if g is continuous in a closed interval I and its range lies in I, then the equation x = g(x) has
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. Compute the integral J by Simpson’s rule with 2m = 8 and use the value and
Find the degree of the interpolation polynomial for the data (-4, 50), (-2, 18), (0, 2), (2, 2), (4, 18), using a difference table. Find the polynomial.
In what sense is Gauss integration optimal?
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f(0) = 2, f1 = f(1) = 3, f2 = f(2) = 8, f3 = f(3) = 12, k0 = k3 = 0
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Design a Newton iteration. Compute 3√7, x0 = 2.
Set up (14) for the data in Prob. 3 and compute Г(1.01), Г(1.03), Г(1.05) Data from Prob. 3Calculate the Lagrange polynomial P2(x) for the values Г(1.00) = 1.0000, Г(1.02) = 0.9888,
How did we obtain formulas for numeric differentiation?
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f(0) = 4, f1 = f(2) = 0, f2 = f(4) = 4, f3 = f(6) = 80, k0 = k3 = 0
Compute ln α - ln b with 6S arithmetic for α = 4.00000 and b = 3.99900(a) As given(b) From ln (α/b)
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. Find the smallest n in computing A (see Probs. 7 and 8) such that 5S-accuracy
Write 46.9028104, 0.000317399, 54/7, 890/3 in floating-point form with 5S (5 significant digits, properly rounded).
Compute (5.346 - 3.644)/(3.444 - 3.055) as given and then rounded step wise to 3S, 2S, 1S. Comment. (“Step wise” means rounding the rounded numbers, not the given ones.)
If a cubic spline is three times continuously differentiable (that is, it has continuous first, second, and third derivatives), show that it must be a single polynomial.
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Solve Prob. 5 by Newton’s method with x0 = 5, 4, 1, -3. Explain the result.Data from Prob. 5Solve by
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
Use p2(x) in (18) and the values of erf x, x = 0.2, 0.4, 0.6 in Table A4 of App. 5, compute erf 0.3 and the error. (4S-exact erf 0.3 = 0.3286).
Let 19.1 and 25.84 be correctly rounded. Find the shortest interval in which the sum s of the true (un rounded) numbers must lie.
Calculate 1/e = 0.367879 (6S) from the partial sums of 5–10 terms of the Maclaurin series(a) Of e-x with w = 1(b) Of ex with x = 1 and then taking the reciprocal. Which is more accurate?
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Find the smallest positive zero of P24 = (1 - x2)P"4 = 15/2 (-7x4 + 8x2 - 1)(a) By Newton’s method(b)
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
Write a program for the forward formula (14). Experiment on the increase of accuracy by successively adding terms. As data use values of some function of your choice for which your CAS gives the
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.x + ln x = 2, x0 = 2
What is the relative error of nα∼ in terms of that of α∼?
Show that can be obtained by the division algorithm
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.f = x3 - 5x + 3 = 0, x0 = 2, 0, -2
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
List 4–5 ideas that you feel are most important in this section. Arrange them in best logical order. Discuss them in a 2–3 page report.
Show that the relative error of α~2is about twice that of α~.
Solve x2 - 40x + 2 = 0 in two ways. Use 4S-arithmetic.
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Find the solution of cos x cosh x = 1 near x = 3/2π.
This simple but slowly convergent method for finding a solution of f(x) = 0 with continuous f is based on the intermediate value theorem, which states that if a continuous function f has opposite
Compute the solution of x4 = x + 0.1 near x = 0 by transforming the equation algebraically to the form x = g(x) and starting from x0 = 0.
Solve, using x0 and x1 as indicated:Prob 21, x0 = 1.0, x1 = 2.0Data from Prob. 21Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.f = x3 - 5x + 3 = 0, x0 =
Solve Prob. 25 by bisection (3S-accuracy).Data from Prob. 25Compute the solution of x4 = x + 0.1 near x = 0 by transforming the equation algebraically to the form x = g(x) and starting from x0 = 0.
Consider f(x) = x4 for x0 = 0, x1 = 0.2, x2 = 0.4, x3 = 0.6, x4 = 0.8. Calculate f'2 from (14a), (14b), (14c), (15). Determine the errors. Compare and comment.
In Prob. 26 Using ex < e (0 < x < 1), conclude that |In| ≤ e/(n + 1) → 0 as n → ∞. Solve the iteration formula for In-1 = (e - In)/n, start from I15 ≈ 0 and compute 4S values
Solve, using x0 and x1 as indicated:sin x = cot x, x0 = 1, x1 = 0.5
Find the cubic spline for the data f(0) = 0, f(1) = 0, f(2) = 4, k0 = 1, k2 = 5.
The 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,
Approximations of π = 3.14159265358979 · · · are 22/7 and 355/113. Determine the corresponding errors and relative errors to 3 significant digits.
Compare the methods in this section and problem set, discussing advantages and disadvantages in terms of examples of your own. No proofs, just motivations and ideas.
Compute π by Machin’s approximation 16 arctan (1/5) - 4 arctan (1/239) to 10S (which are correct). [In1986, D. H. Bailey (NASA Ames Research Center, Moffett Field, CA 94035) computed almost 30
Compute the integral of x3 from 0 to 1 by the trapezoidal rule with n = 5. What error bounds are obtained from (4) ? What is the actual error of the result?
Solve Prob. 32 by Gauss integration with n = 3 and n = 5.Data from Prob. 32Compute the integral of x3 from 0 to 1 by the trapezoidal rule with n = 5. What error bounds are obtained from (4) ? What is
Compute f"(0.2) for f(x) = x3(a) h = 0.2(b) h = 0.1.
Show the factorization and solve by Doolittle’s method. 4x1 + 5x2 = 1412x1 + 14x2 = 36
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare.
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[1 -3 8 0 -6 0]
Apply the power method without scaling (3 steps), using x0 = [1, 1]T or [1 1 1]T. Give Rayleigh quotients and error bounds. Show the details of your work.
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.(0, 2), (2, 0), (3, -2), (5, -3)
Solve graphically and explain geometrically. x1 - 4x2 = 20.13x1 + 5x2 = 5.9
Tridiagonalize. Show the details.
Show the factorization and solve by Doolittle’s method.2x1 + 9x2 = 823x1 - 5x2 = -62
When would you apply Gauss elimination and when Gauss–Seidel iteration?
Show the factorization and solve by Doolittle’s method. 5x1 + 4x2 + x3 = 6.810x1 + 9x2 + 4x3 = 17.610x1 + 13x2 + 15x3 = 38.4
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare.
What is pivoting? Why and how is it done?
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[0.2 0.6 -2.1 3.0]
Apply the power method without scaling (3 steps), using x0 = [1, 1]T or [1 1 1]T. Give Rayleigh quotients and error bounds. Show the details of your work.
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.(0, 1.8), (1, 1.6), (2, 1.1), (3,
Solve graphically and explain geometrically. 7.2x1 - 3.5x2 = 16.0-14.4x1 + 7.0x2 = 31.0
Show the factorization and solve by Doolittle’s method. 2x1 + x2 + 2x3 = 0-2x1 + 2x2 + x3 = 0 x1 + 2x2 - 2x3 = 18
What happens if you apply Gauss elimination to a system that has no solutions?
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[k2, 4k, k3], k > 4
Show the factorization and solve by Doolittle’s method. 3x1 + 9x2 + 6x3 = 4.618x1 + 48x2 + 39x3 = 27.2 9x1 - 27x2 + 42x3 = 9.0
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare.
What is Cholesky’s method? When would you apply it?
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[1 1 1 1 1]
Apply the power method (3 steps) with scaling, using x0 = [1 1 1]T or [1 1 1 1]T, as applicable. Give Rayleigh quotients and error bounds. Show the details of your
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?).
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 average speed
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
What do you know about the convergence of the Gauss–Seidel iteration?
Show the factorization and solve. 9x1 + 6x2 + 12x3 = 17.4 6x1 + 13x2 + 11x3 = 23.612x1 + 11x2 + 26x3 = 30.8
In Prob. 2, find T-T AT such that the radius of the Gerschgorin circle with center 5 is reduced by a factor 1/100. Data from Prob. 2 Find and sketch disks or intervals that contain the eigenvalues.
What is ill-conditioning? What is the condition number and its significance?
Apply the power method (3 steps) with scaling, using x0 = [1 1 1]T or [1 1 1 1]T, as applicable. Give Rayleigh quotients and error bounds. Show the details of your work.
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?).
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
Do three QR-steps to find approximations of the eigenvalues of: The matrix in the answer to Prob. 3 Data from Prob. 3 Tridiagonalize. Show the details.
Show the factorization and solve.4x1 + 6x2 + 8x3 = 06x1 + 34x2 + 52x3 = -1608x1 + 52x2 + 129x3 = -452
Explain the idea of least squares approximation.
Show the factorization and solve.0.01x1 + 0.03x3 = 0.14 0.16x2 + 0.08x3 = 0.160.03x1
If a symmetric n × n matrix A = [αjk] has been diagonalized except for small off-diagonal entries of size 10-5, what can you say about the eigenvalues?
What are eigenvalues of a matrix? Why are they important? Give typical examples.
Compute the matrix norm and the condition number corresponding to the l1-vector norm.
Prove that if x is an eigenvector, then δ = 0 in (2). Give two examples.
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?).
Fit a parabola (7) to the points (x, y). Check by sketching.(2, -3), (3, 0), (5, 1), (6, 0) (7, -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
Do three QR-steps to find approximations of the eigenvalues of:
Show the factorization and solve.4x1 + 2x3 = 1.5 4x2 + x3 = 4.02x1 + x2 + 2x3 = 2.5
How did we use similarity transformations of matrices in designing numeric methods?
Try to find out experimentally on what properties of a matrix the speed of decrease of off-diagonal entries in the QR-method depends. For this purpose write a program that first tridiagonalizes and

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