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About 3307 results. 1294 free access solutions
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| m54633 | In Exercise 31 of Section 6.6, a symmetric matrix
was used to describe the average wing lengths of fruit flies that were offspring resulting from the mating of three mutants of the flies. The entry aij represents the average wing length of a fly that is the offspring of a male fly of type i and a female fly of type j.
a. Find the eigenvalues and associated eigenvectors of this matrix.
b. Is this matrix positive definite? |
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| m54638 | In Exercise 7 of Section 11.3, the deflection of a beam with supported ends subject to uniform loading was approximated. Using a more appropriate representation of curvature gives the differential equation
[1 + (w (x))2] −3/2w" (x) = S / EI w(x) + qx / 2EI (x − l), for 0 < x < l.
Approximate the deflection w(x) of the beam every 6 in., and compare the results to those of Exercise 7 of Section 11.3.
In exercise
The lead example of this chapter concerned the deflection of a beam with supported ends subject to uniform loading. The boundary-value problem governing this physical situation is
with boundary conditions w(0) = 0 and w(l) = 0.
Suppose the beam is a W10-type steel I-beam with the following characteristics: length l = 120 in., intensity of uniform load q = 100 lb/ft, modulus of elasticity E = 3.0×107 lb/in.2, stress at ends S = 1000 lb, and central moment of inertia I = 625 in.4. |
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| m54640 | In Exercise 9 the Frobenius norm of a matrix was defined. Show that for any n×n matrix A and vector x in Rn, ||Ax||2 ≤ ||A||F ||x||2. |
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| m54641 | In Exercise 9 we considered the problem of predicting the population in a predator-prey model.
Another problem of this type is concerned with two species competing for the same food supply. If the numbers of species alive at time t are denoted by x1(t) and x2(t), it is often assumed that, although the birth rate of each of the species is simply proportional to the number of species alive at that time, the death rate of each species depends on the population of both species. We will assume that the population of a particular pair of species is described by the equations
dx1(t)/dt = x1(t)[4 − 0.0003x1(t) − 0.0004x2(t)] and dx2(t)/dt =x2(t)[2 − 0.0002x1(t) − 0.0001x2(t)]. |
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| m54676 | In Section 3.6 we found that the parametric form (x(t), y(t)) of the cubic Hermite polynomials through (x(0), y(0)) = (x0, y0) and (x(1), y(1)) = (x1, y1) with guide points (x0+α0, y0+β0) and (x1−α1, y1−β1), respectively, are given by
x(t) = (2(x0 − x1) + (α0 + α1))t3 + (3(x1 − x0) − α1 − 2α0)t2 + α0t + x0,
And
y(t) = (2(y0 − y1) + (β0 + β1))t3 + (3(y1 − y0) − β1 − 2β0)t2 + β0t + y0.
The Bézier cubic polynomials have the form
x(t) = (2(x0 − x1) + 3(α0 + α1))t3 + (3(x1 − x0) − 3(α1 + 2α0))t2,+3α0t + x0
and
y(t) = (2(y0 − y1) + 3(β0 + β1))t3 + (3(y1 − y0) − 3(β1 + 2β0))t2 + 3β0t + y0.
a. Show that the matrix
Transforms the Hermite polynomial coefficients into the Bézier polynomial coefficients
b. Determine a matrix B that transforms the Bézier polynomial coefficients into the Hermite polynomial coefficients. |
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| m54709 | In the design of all-terrain vehicles, it is necessary to consider the failure of the vehicle when attempting to negotiate two types of obstacles. One type of failure is called hang-up failure and occurs when the vehicle attempts to cross an obstacle that causes the bottom of the vehicle to touch the ground. The other type of failure is called nose-in failure and occurs when the vehicle descends into a ditch and its nose touches the ground.
The accompanying figure, adapted from [Bek], shows the components associated with the nosein failure of a vehicle. In that reference it is shown that the maximum angle α that can be negotiated by a vehicle when β is the maximum angle at which hang-up failure does not occur satisfies the equation
A sin α cos α + B sin2 α − C cos α − E sin α = 0,
Where
A = l sin β1, B = l cos β1, C = (h + 0.5D) sin β1 − 0.5Dtan β1,
And E = (h + 0.5D) cos β1 − 0.5D.
a. It is stated that when l = 89 in., h = 49 in., D = 55 in., and β1 = 11.5◦, angle α is approximately 33◦. Verify this result.
b. Find α for the situation when l, h, and β1 are the same as in part (a) but D = 30 in. |
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| m54711 | In the discussion preceding Algorithm 8.3, an example for m = 4 was explained. Define vectors c, d, e, f, and y as
c = (c0, . . . , c7)t , d = (d0, . . . , d7)t , e = (e0, . . . , e7)t , f = (f0, . . . , f7)t , y = (y0, . . . , y7)t .
Find matrices A, B, C, and D so that c = Ad, d = Be, e = C f, and f = Dy. |
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| m54727 | In the lead example of this chapter, the linear system Aw = −0.04(ρ/p)λw must be solved for w and λ in order to approximate the eigenvalues λk of the Strum-Liouville system.
a. Find all four eigenvalues μ1, . . . , μ4 of the matrix
to within 10−5.
b. Approximate the eigenvalues λ1, . . . , λ4 of the system in terms of ρ and p. |
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| m54734 | In the previous exercise, all infected individuals remained in the population to spread the disease. |
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| m54743 | In the theory of the spread of contagious disease (see [Ba1] or [Ba2]), a relatively elementary differential equation can be used to predict the number of infective individuals in the population at any time, provided appropriate simplification assumptions are made. In particular, let us assume that all individuals in a fixed population have an equally likely chance of being infected and once infected remain in that state. Suppose x(t) denotes the number of susceptible individuals at time t and y(t) denotes the number of infective. It is reasonable to assume that the rate at which the number of infective changes is proportional to the product of x(t) and y(t) because the rate depends on both the number of infective and the number of susceptible present at that time. If the population is large enough to assume that x(t) and y(t) are continuous variables, the problem can be expressed
y (t) = kx(t)y(t),
Where k is a constant and x(t) + y(t) = m, the total population. This equation can be rewritten involving only y(t) as
y (t) = k(m − y(t))y(t).
a. Assuming that m = 100,000, y(0) = 1000, k = 2 × 10−6, and that time is measured in days, find an approximation to the number of infective individuals at the end of 30 days. |
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| m54794 | It can be shown (see, for example, [DaB], pp. 228-229) that if {pn}∞ n=0 are convergent Secant method approximations to p, the solution to f (x) = 0, then a constant C exists with |pn+1 − p| ≈ C |pn − p| |pn−1 − p| for sufficiently large values of n. Assume {pn} converges to p of order α, and show that α = (1 +√5)/2. (This implies that the order of convergence of the Secant method is approximately 1.62). |
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| m54799 | It is suspected that the high amounts of tannin in mature oak leaves inhibit the growth of the winter moth (Operophtera bromata L., Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree.
a. Use Lagrange interpolation to approximate the average weight curve for each sample.
b. Find an approximate maximum average weight for each sample by determining the maximum of the interpolating polynomial. |
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| m54800 | It is suspected that the high amounts of tannin in mature oak leave inhibit the growth of the winter moth (Operophtera bromata L., Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree.
a. Use a natural cubic spline to approximate the average weight curve for each sample.
b. Find an approximate maximum average weight for each sample by determining the maximum of the spline. |
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| m54804 | Jacobi s method for a symmetric matrix A is described by
A1 = A,
A2 = P1A1Pt1
and, in general,
Ai+1 = PiAiPti.
The matrix Ai+1 tends to a diagonal matrix, where Pi is a rotation matrix chosen to eliminate a large off-diagonal element in Ai. Suppose aj,k and ak,j are to be set to 0, where j ≠ k. If ajj ≠ akk, then
where
c = 2ajksgn(ajj − akk) and b = |ajj − akk|,
or if ajj = akk,
Pi)jj = (Pi)kk = √2 /2
And
(Pi)kj = − (Pi)jk =√2/2.
Develop an algorithm to implement Jacobi s method by setting a21 = 0. Then set a31, a32, a41, a42, a43, . . . , an,1, . . . , an,n−1 in turn to zero. This is repeated until a matrix Ak is computed with
sufficiently small. The eigenvalues of A can then be approximated by the diagonal entries of Ak. |
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| m54838 | Let A be a 3 × 3 matrix. Show that if is the matrix obtained from A using any of the operations
(E1) ↔ (E2), (E1) ↔ (E3), or (E2) ↔ (E3),
then det = −det A. |
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| m54839 | Let A be a given positive constant and g(x) = 2x − Ax2.
a. Show that if fixed-point iteration converges to a nonzero limit, then the limit is p = 1/A, so the inverse of a number can be found using only multiplications and subtractions.
b. Find an interval about 1/A for which fixed-point iteration converges, provided p0 is in that interval. |
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| m54840 | 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. |
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| m54842 | 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. |
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| m54844 | Let
a. Find limx→0 f (x).
b. Use four-digit rounding arithmetic to evaluate f (0.1).
c. Replace each trigonometric function with its third Maclaurin polynomial, and repeat part (b).
d. The actual value is f (0.1) = −1.99899998. Find the relative error for the values obtained in parts (b) and (c). |
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| m54855 | Let
And
Form the 16 × 16 matrix A in partitioned form,
Let b = (1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6)t.
a. Solve Ax = b using the conjugate gradient method with tolerance 0.05.
b. Solve Ax = b using the preconditioned conjugate gradient method with C−1 = D−1/2 and tolerance 0.05.
c. Is there any tolerance for which the methods of part (a) and part (b) require a different number of iterations? |
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