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free/or 0.5$ |
| m53530 | Determine the Padé approximation of degree 5 with n = 2 and m = 3 for f (x) = ex. Compare the results at xi = 0.2i, for i = 1, 2, 3, 4, 5, with those from the fifth Maclaurin polynomial. |
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| m53531 | Determine the Padé approximation of degree 6 with n = m = 3 for f (x) = sin x. Compare the results at xi = 0.1i, for i = 0, 1, . . . , 5, with the exact results and with the results of the sixth Maclaurin polynomial. |
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| m53532 | Determine the Padé approximations of degree 6 with (a) n = 2,m = 4 and (b) n = 4, m = 2 for f (x) = sin x. Compare the results at each xi to those obtained in Exercise 5. |
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| m53533 | Determine the singular values of the following matrices.
a.
b.
c.
d. |
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| m53534 | Determine the singular values of the following matrices.
a.
b.
c.
d. |
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| m53536 | Determine the trigonometric interpolating polynomial of degree 4 for f (x) = x(π −x) on the interval [−π, π] using:
a. Direct calculation;
b. The Fast Fourier Transform Algorithm. |
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| m53537 | Determine the trigonometric interpolating polynomial S2(x) of degree 2 on [−π, π] for the following functions, and graph f (x) − S2(x):
a. f (x) = π(x − π)
b. f (x) = x(π − x)
c. f (x) = |x|
d. |
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| m53539 | Determine the values of n and h required to approximate
To within 10−4 Use
a. Composite Trapezoidal rule.
b. Composite Simpson s rule.
c. Composite Midpoint rule. |
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| m53540 | Determine the values of n and h required to approximate
To within 10−5 and compute the approximation. Use
a. Composite Trapezoidal rule.
b. Composite Simpson s rule.
c. Composite Midpoint rule. |
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| m53558 | Determine which matrices in Exercise 1 are tri diagonal and positive definite. Repeat Exercise 1 for these matrices using the optimal choice of ω.
In Exercise 1
a. 3x1 − x2 + x3 = 1,
3x1 + 6x2 + 2x3 = 0,
3x1 + 3x2 + 7x3 = 4.
b. 10x1 − x2 = 9,
−x1 + 10x2 − 2x3 = 7,
− 2x2 + 10x3 = 6.
c. 10x1 + 5x2 = 6,
5x1 + 10x2 − 4x3 = 25,
− 4x2 + 8x3 − x4 = −11,
− x3 + 5x4 = −11.
d. 4x1 + x2 + x3 + x5 = 6,
−x1 − 3x2 + x3 + x4 = 6,
2x1 + x2 + 5x3 − x4 − x5 = 6,
−x1 − x2 − x3 + 4x4 = 6,
2x2 − x3 + x4 + 4x5 = 6 |
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| m53559 | Determine which matrices in Exercise 2 are tri diagonal and positive definite. Repeat Exercise 2 for these matrices using the optimal choice of ω.
In Exercise 2
a. 4x1 + x2 − x3 = 5,
−x1 + 3x2 + x3 = −4,
2x1 + 2x2 + 5x3 = 1.
b. −2x1+ x2 + 1/2 x3 = 4,
x1−2x2 - 1/2 x3 = −4,
x2 + 2x3 = 0.
c. 4x1 + x2 − x3 + x4 = −2,
x1 + 4x2 − x3 − x4 = −1,
−x1 − x2 + 5x3 + x4 = 0,
x1 − x2 + x3 + 3x4 = 1.
d. 4x1 − x2 = 0,
−x1 + 4x2 − x3 = 5,
− x2 + 4x3 = 0,
+ 4x4 − x5 = 6,
− x4 + 4x5 − x6 = −2,
− x5 + 4x6 = 6. |
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| m53560 | Determine which of the following matrices are (i) symmetric, (ii) singular, (iii) strictly diagonally dominant, (iv) positive definite.
a.
b.
c.
d. |
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| m53561 | Determine which of the following matrices are (i) symmetric, (ii) singular, (iii) strictly diagonally dominant, (iv) positive definite.
a.
b.
c.
d. |
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| m53562 | Determine which of the following matrices are nonsingular, and compute the inverse of these matrices:
a.
b.
c.
d. |
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| m53563 | Determine which of the following matrices are nonsingular, and compute the inverse of these matrices:
|
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| m53578 | Discuss consistency, stability, and convergence for the Implicit Trapezoidal method
wi+1 = wi + h/2 (f (ti+1,wi+1) + f (ti ,wi)) , for i = 0, 1, . . . , N − 1,
With w0 = α applied to the differential equation
y = f (t, y), a ≤ t ≤ b, y(a) = α. |
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| m53642 | Equations (1.2) and (1.3) in Section 1.2 give alternative formulas for the roots x1 and x2 of ax2 + bx + c = 0. Construct an algorithm with input a, b, c and output x1, x2 that computes the roots x1 and x2 (which may be equal or be complex conjugates) using the best formula for each root. |
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| m53745 | Express the following rational functions in continued-fraction form:
a.
b.
c.
d. |
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| m53755 | Extend Algorithms 3.4 and 3.5 to include as output the first and second derivatives of the spline at the nodes. |
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| m53756 | Extend Algorithms 3.4 and 3.5 to include as output the integral of the spline over the interval [x0, xn]. |
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