| № |
Condition |
free/or 0.5$ |
| m54856 | Let
And
Show that A1 is not convergent, but A2 is convergent. |
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| m54860 | Let λ be an eigenvalue of the n × n matrix A and x = 0 be an associated eigenvector.
a. Show that λ is also an eigenvalue of At.
b. Show that for any integer k ≥ 1, λk is an eigenvalue of Ak with eigenvector x.
c. Show that if A−1 exists, then 1/λ is an eigenvalue of A−1 with eigenvector x.
d. Generalize parts (b) and (c) to (A−1)k for integers k ≥ 2.
e. Given the polynomial q(x) = q0 + q1x + · · · + qkxk , define q(A) to be the matrix q(A) = q0I + q1A+· · ·+qkAk . Show that q(λ) is an eigenvalue of q(A) with eigenvector x.
f. Let α ≠ λ be given. Show that if A − αI is nonsingular, then 1/(λ − α) is an eigenvalue of (A − αI)−1 with eigenvector x. |
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| m54870 | Let E(h) = hM/2 + δ/h.
a. For the initial-value problem
y = −y + 1, 0 ≤ t ≤ 1, y(0) = 0,
Compute the value of h to minimize E(h). Assume δ = 5 × 10−(n+1) if you will be using n-digit arithmetic in part (c).
b. For the optimal h computed in part (a), use Eq. (5.13) to compute the minimal error obtainable.
c. Compare the actual error obtained using h = 0.1 and h = 0.01 to the minimal error in part (b).
Can you explain the results? |
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| m54874 | Let f be defined by
a. Investigate the continuity of the derivatives of f .
b. Use the Composite Trapezoidal rule with n = 6 to approximate
and estimate the error using the error bound.
c. Use the Composite Simpson s rule with n = 6 to approximate
Are the results more accurate than in part (b)? |
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| m54875 | Let f be defined on [a, b], and let the nodes a = x0 < x1 < x2 = b be given. A quadratic spline interpolating function S consists of the quadratic polynomial
S0(x) = a0 + b0(x − x0) + c0(x − x0)2 on [x0, x1]
And the quadratic polynomial
S1(x) = a1 + b1(x − x1) + c1(x − x1)2 on [x1, x2],
Such that
i. S(x0) = f (x0), S(x1) = f (x1), and S(x2) = f (x2),
ii. S ∈ c1[x0, x2]. |
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| m54877 | Let f ∈ C2 [a, b], and let the nodes a = x0 < x1 < · · · < xn = b be given. Derive an error estimate similar to that in Theorem 3.13 for the piecewise linear interpolating function F. Use this estimate to derive error bounds for Exercise 21.
In Exercise 21
Given the partition x0 = 0, x1 = 0.05, and x2 = 0.1 of [0, 0.1], find the piecewise linear interpolating function F for f (x) = e2x. Approximate
F(x) dx, and compare the results to the actual value. |
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| m54878 | Let f ∈ C[a, b], and let p be in the open interval (a, b).
a. Suppose f (p) ≠ 0. Show that a δ > 0 exists with f (x) ≠ 0, for all x in [p − δ, p + δ], with
[p − δ, p + δ] a subset of [a, b].
b. Suppose f (p) = 0 and k > 0 is given. Show that a δ > 0 exists with |f (x)| ≤ k, for all x in
[p − δ, p + δ], with [p − δ, p + δ] a subset of [a, b]. |
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| m54879 | Let f ∈ C[a, b] be a function whose derivative exists on (a, b). Suppose f is to be evaluated at x0 in (a, b), but instead of computing the actual value f (x0), the approximate value, (x0), is the actual value of f at x0 + , that is, (x0) = f (x0 + ().
a. Use the Mean Value Theorem 1.8 to estimate the absolute error |f (x0) − (x0)| and the relative error |f (x0) − (x0)|/|f (x0)|, assuming f (x0) ≠ 0.
b. If ( = 5 × 10−6 and x0 = 1, find bounds for the absolute and relative errors for
i. f (x) = ex
ii. f (x) = sin x
c. Repeat part (b) with ( = (5 × 10−6)x0 and x0 = 10. |
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| m54880 | Let f (x) = 2 tan x − sec 2x, for 2 ≤ x ≤ 4. Determine the discrete least squares trigonometric polynomials Sn(x), using the values of n and m as follows, and compute the error in each case.
a. n = 3, m = 6
b. n = 4, m = 6 |
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| m54881 | Let f (x) = 2x cos(2x) − (x − 2)2 and x0 = 0.
a. Find the third Taylor polynomial P3(x), and use it to approximate f (0.4).
b. Use the error formula in Taylor s Theorem to find an upper bound for the error |f (0.4) −P3 (0.4)|. Compute the actual error.
c. Find the fourth Taylor polynomial P4(x), and use it to approximate f (0.4).
d. Use the error formula in Taylor s Theorem to find an upper bound for the error |f (0.4) − P4 (0.4)|. Compute the actual error. |
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| m54882 | Let f (x) = 33x+1 − 7 · 52x.
a. Use the Maple commands solve and f solve to try to find all roots of f .
b. Plot f (x) to find initial approximations to roots of f.
c. Use Newton s method to find roots of f to within 10−16. |
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| m54883 | Let f (x) = 33x+1 − 7 . 52x.
a. Use the Maple commands solve and f solve to try to find all roots of f .
b. Plot f (x) to find initial approximations to roots of f.
c. Use Newton s method to find roots of f to within 10−16. |
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| m54884 | Let f (x) = 3xex − e2x.
a. Approximate f (1.03) by the Hermite interpolating polynomial of degree at most three using x0 = 1 and x1 = 1.05. Compare the actual error to the error bound.
b. Repeat (a) with the Hermite interpolating polynomial of degree at most five, using x0 = 1, x1 = 1.05, and x2 = 1.07. |
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| m54885 | Let
f (x) = (ex − e−x)/x.
a. Find limx→0 (ex − e−x)/x.
b. Use three-digit rounding arithmetic to evaluate f (0.1).
c. Replace each exponential function with its third Maclaurin polynomial, and repeat part (b).
d. The actual value is f (0.1) = 2.003335000. Find the relative error for the values obtained in parts (b) and (c). |
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| m54886 | Let f (x) = ex, for 0 ≤ x ≤ 2.
a. Approximate f (0.25) using linear interpolation with x0 = 0 and x1 = 0.5.
b. Approximate f (0.75) using linear interpolation with x0 = 0.5 and x1 = 1.
c. Approximate f (0.25) and f (0.75) by using the second interpolating polynomial with x0 = 0, x1 = 1, and x2 = 2
d. Which approximations are better and why? |
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| m54887 | Let f (x) = ex/2 sin(x/3). Use Maple to determine the following.
a. The third Maclaurin polynomial P3(x).
b. f (4)(x) and a bound for the error |f (x) − P3(x)| on [0, 1]. |
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| m54888 | Let f (x) = ln(x2 + 2). Use Maple to determine the following.
a. The Taylor polynomial P3(x) for f expanded about x0 = 1.
b. The maximum error |f (x) − P3(x)|, for 0 ≤ x ≤ 1.
c. The Maclaurin polynomial 3(x) for f.
d. The maximum error |f (x) − 3(x)|, for 0 ≤ x ≤ 1. |
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| m54889 | Let f (x) = (x − 1)10, p = 1, and pn = 1 + 1/n. Show that |f (pn)| < 10−3 whenever n > 1 but that |p − pn| < 10−3 requires that n > 1000. |
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| m54890 | Let f (x) = x2 − 6. With p0 = 3 and p1 = 2, find p3.
a. Use the Secant method.
b. Use the method of False Position.
c. Which of a. or b. is closer to √6? |
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| m54891 | Let f (x) = (x+2)(x+1)2x(x −1)3(x −2). To which zero of f does the Bisection method converge when applied on the following intervals?
a. [−1.5, 2.5]
b. [−0.5, 2.4]
c. [−0.5, 3]
d. [−3,−0.5] |
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