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About 3307 results. 1294 free access solutions
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| m52992 | 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. One approach to the solution of this problem uses a weighted least squares fit of the form f (x) = y = a/(x − b)c to a collection of experimental data. The x-values of the data refer to water temperature. The constant b is the asymptote of the graph of f and as such is an approximation to XM.
a. Show that choosing a, b, and c to minimize
reduces to solving the nonlinear system
b. Solve the nonlinear system for the species with the following data. Use the weights wi = ln yi.
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| m52996 | An n × n matrix A is called nilpotent if an integer m exists with Am = On. Show that if λ is an eigenvalue of a nilpotent matrix, then λ = 0. |
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| m53005 | Apply the Backward Euler method to the differential equations given in Exercise 1. Use Newton s method to solve for wi+1.
In Exercise 1 |
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| m53006 | Apply the Backward Euler method to the differential equations given in Exercise 2. Use Newton s method to solve for wi+1.
In Exercise 2 |
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| m53007 | Apply the extrapolation process described in Example 1 to determine N3(h), an approximation to f (x0), for the following functions and step sizes.
a. f (x) = ln x, x0 = 1.0, h = 0.4
b. f (x) = x + ex , x0 = 0.0, h = 0.4
c. f (x) = 2x sin x, x0 = 1.05, h = 0.4
d. f (x) = x3 cos x, x0 = 2.3, h = 0.4 |
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| m53008 | Apply two iterations of the QR method without shifting to the following matrices.
a.
b.
c.
d.
e.
f. |
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| m53009 | Apply two iterations of the QR method without shifting to the following matrices.
a.
b.
c.
d. |
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| m53011 | Approximate solutions to the following linear systems Ax = b to within 10−5 in the l∞ norm.
(i)
and
b = (1.902207, 1.051143, 1.175689, 3.480083, 0.819600,−0.264419, − 0.412789, 1.175689, 0.913337,−0.150209,−0.264419, 1.051143,1.966694, 0.913337, 0.819600, 1.902207)t
(ii)
and
b = (1, 0,−1, 0, 2, 1, 0,−1, 0, 2, 1, 0,−1, 0, 2, 1, 0,−1, 0, 2, 1, 0,−1, 0, 2)t
(iii)
And bi = 1.5i − 6, for each i = 1, 2, . . . , 40
a. Use the Jacobi method,
b. Use the Gauss-Seidel method,
c. Use the SOR method with ω = 1.3 in (i), ω = 1.2 in (ii), and ω = 1.1 in (iii).
d. Use the conjugate gradient method and preconditioning with C−1 = D−1/2. |
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| m53012 | Approximate the following integrals using formulas (4.25) through (4.32). Are the accuracies of the approximations consistent with the error formulas? Which of parts (d) and (e) give the better approximation? |
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| m53013 | Approximate the following integrals using Gaussian quadrature with n = 2, and compare your results to the exact values of the integrals. |
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| m53014 | Approximate the following integrals using Gaussian quadrature with n = 3, and compare your results to the exact values of the integrals. |
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| m53015 | Approximate the following integrals using Gaussian quadrature with n = 4, and compare your results to the exact values of the integrals. |
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| m53016 | Approximate the following integrals using Gaussian quadrature with n = 5, and compare your results to the exact values of the integrals. |
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| m53017 | Approximate the following integrals using the Trapezoidal rule. |
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| m53018 | Approximate the following integrals using the Trapezoidal rule. |
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| m53019 | Approximate the solution to the following partial differential equation using the Backward-Difference method.
∂u / ∂t − ∂2u / ∂x2 = 0, 0 < x < 2, 0 < t;
u(0, t) = u(2, t) = 0, 0 < t, u(x, 0) = sin π / 2 x, 0≤ x ≤ 2.
Use m = 4, T = 0.1, and N = 2, and compare your results to the actual solution u(x, t) = e−(π2/4)t sin π / 2 x. |
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| m53020 | Approximate the solution to the following partial differential equation using the Backward-Difference method.
∂u / ∂t - 1 / 16 ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t;
u(0, t) = u(1, t) = 0, 0 < t, u(x, 0) = 2 sin 2πx, 0≤ x ≤ 1.
Use m = 3, T = 0.1, and N = 2, and compare your results to the actual solution u(x, t) = 2e−(π2/4)t sin 2πx. |
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| m53021 | Approximate the solution to the partial differential equation
subject to the Dirichlet boundary condition
u(x, y) = 0,
using the Finite-Element Algorithm 12.5 with the elements given in the accompanying figure. Compare the approximate solution to the actual solution,
u(x, y) = sin 5π / 2 x sin 5π / 2 y,
at the interior vertices and at the points (0.125, 0.125), (0.125, 0.25), (0.25, 0.125), and (0.25, 0.25).
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| m53022 | Approximate the solution to the wave equation
∂2u / ∂t2 − ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t;
u(0, t) = u(1, t) = 0, 0 < t,
u(x, 0) = sin πx, 0≤ x ≤ 1,
∂u / ∂t (x, 0) = 0, 0 ≤ x ≤ 1,
using the Finite-Difference Algorithm 12.4 with m = 4, N = 4, and T = 1.0. Compare your results at t = 1.0 to the actual solution u(x, t) = cos πt sin πx. |
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| m53023 | Approximate the solution to the wave equation
∂2u / ∂t2 - 1/16π2 ∂2u/∂x2 = 0, 0 < x < 0.5, 0 < t;
u(0, t) = u(0.5, t) = 0, 0 < t,
u(x, 0) = 0, 0 ≤ x ≤ 0.5,
∂u / ∂t (x, 0) = sin 4πx, 0≤ x ≤ 0.5,
using the Finite-Difference Algorithm 12.4 with m = 4, N = 4 and T = 0.5. Compare your results at t = 0.5 to the actual solution u(x, t) = sin t sin 4πx. |
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