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Condition |
free/or 0.5$ |
| m53758 | Factor the following matrices into the LU decomposition using the LU Factorization Algorithm with lii = 1 for all i.
a.
b.
c.
d. |
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| m53759 | Factor the following matrices into the LU decomposition using the LU Factorization Algorithm with lii = 1 for all i.
a.
b.
c.
d. |
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| m53776 | Find a bound for the error in Exercise 1 using the error formula, and compare this to the actual error.
In Exercise 1 |
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| m53777 | Find a bound for the error in Exercise 2 using the error formula, and compare this to the actual error.
In Exercise 2 |
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| m53801 | Find all Chebyshev rational approximations of degree 5 for f (x) = ex. Compare the results at xi = 0.2i, for i = 1, 2, 3, 4, 5, with those obtained in Exercises 3 and 4. |
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| m53809 | Find all the Chebyshev rational approximations of degree 2 for f (x) = e−x. Which give the best approximations to f (x) = e−x at x = 0.25, 0.5, and 1? |
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| m53810 | Find all the Chebyshev rational approximations of degree 3 for f (x) = cos x. Which give the best approximations to f (x) = cos x at x = π/4 and π/3? |
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| m53812 | Find all the zeros of f (x) = x2 +10 cos x by using the fixed-point iteration method for an appropriate iteration function g. Find the zeros accurate to within 10−4. |
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| m53836 | Find approximations to within 10−5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton s method and then reducing to polynomials of lower degree to determine any complex zeros.
a. f (x) = x4 + 5x3 − 9x2 − 85x − 136
b. f (x) = x4 − 2x3 − 12x2 + 16x − 40
c. f (x) = x4 + x3 + 3x2 + 2x + 2
d. f (x) = x5 + 11x4 − 21x3 − 10x2 − 21x − 5
e. f (x) = 16x4 + 88x3 + 159x2 + 76x − 240
f. f (x) = x4 − 4x2 − 3x + 5
g. f (x) = x4 − 2x3 − 4x2 + 4x + 4
h. f (x) = x3 − 7x2 + 14x - 6 |
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| m53837 | Find approximations to within 10−5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton s method and then reducing to polynomials of lower degree to determine any complex zeros.
a. f (x) = x4 + 5x3 − 9x2 − 85x − 136
b. f (x) = x4 − 2x3 − 12x2 + 16x − 40
c. f (x) = x4 + x3 + 3x2 + 2x + 2
d. f (x) = x5 + 11x4 − 21x3 − 10x2 − 21x − 5
e. f (x) = 16x4 + 88x3 + 159x2 + 76x − 240
f. f (x) = x4 − 4x2 − 3x + 5
g. f (x) = x4 − 2x3 − 4x2 + 4x + 4
h. f (x) = x3 − 7x2 + 14x - 6 |
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| m53862 | Find intervals containing solutions to the following equations.
a. x − 3−x = 0
b. 4x2 − ex = 0
c. x3 − 2x2 − 4x + 2 = 0
d. x3 + 4.001x2 + 4.002x + 1.101 = 0 |
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| m53863 | Find l∞ and l2 norms of the vectors.
a. x = (3,−4, 0, 3/2 )t
b. x = (2, 1,−3, 4)t
c. x = (sin k, cos k, 2k)t for a fixed positive integer k
d. x = (4/(k + 1), 2/k2, k2e−k)t for a fixed positive integer k |
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| m53865 | Find matrices A and B for which ρ(A+B) > ρ(A)+ρ(B). (This shows that ρ(A) cannot be a matrix norm.) |
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| m53866 | Find max a≤x≤b |f (x)| for the following functions and intervals
a. f (x) = (2 − ex + 2x)/3, [0, 1]
b. f (x) = (4x − 3)/(x2 − 2x), [0.5, 1]
c. f (x) = 2x cos(2x) − (x − 2)2, [2, 4]
d. f (x) = 1 + e−cos(x−1), [1, 2] |
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| m53876 | Find the approximations to within 10−4 to all the real zeros of the following polynomials using Newton s method.
a. f (x) = x3 − 2x2 − 5
b. f (x) = x3 + 3x2 − 1
c. f (x) = x3 − x − 1
d. f (x) = x4 + 2x2 − x − 3
e. f (x) = x3 + 4.001x2 + 4.002x + 1.101
f. f (x) = x5 − x4 + 2x3 − 3x2 + x - 4 |
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| m53877 | Find the approximations to within 10−4 to all the real zeros of the following polynomials using Newton s method.
a. f (x) = x3 − 2x2 − 5
b. f (x) = x3 + 3x2 − 1
c. f (x) = x3 − x − 1
d. f (x) = x4 + 2x2 − x − 3
e. f (x) = x3 + 4.001x2 + 4.002x + 1.101
f. f (x) = x5 − x4 + 2x3 − 3x2 + x - 4 |
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| m53915 | Find the Chebyshev rational approximation of degree 4 with n = m = 2 for f (x) = sin x. Compare the results at xi = 0.1i, for i = 0, 1, 2, 3, 4, 5, from this approximation with those obtained in Exercise 5 using a sixth-degree Padé approximation. |
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| m53916 | Find the complex eigenvalues and associated eigenvectors for the following matrices.
a.
b. |
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| m53917 | Find the complex eigenvalues and associated eigenvectors for the following matrices.
a.
b. |
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| m53934 | Find the eigenvalues and associated eigenvectors of the following 3 × 3 matrices. Is there a set of linearly independent eigenvectors?
a.
b.
c.
d. |
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