| № |
Condition |
free/or 0.5$ |
| m55368 | Repeat Exercise 3, using in Step 4 of Algorithm 12.4 the approximation
wi,1 = wi,0 + kg(xi), for each i = 1, . . . ,m − 1.
Repeat exercise 3 |
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| m55369 | Repeat Exercise 3 using instead the Padé approximation of degree 5 with n = 3 and m = 2. Compare the results at each xi with those computed in Exercise 3. |
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| m55370 | Repeat Exercise 3 using Simpson s rule and the results of Exercise 5.
In Exercise 3 |
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| m55371 | Repeat Exercise 3 using single-precision arithmetic on a computer.
In Exercise 3
a. Solve the linear system using Gaussian elimination with three-digit rounding arithmetic.
b. Solve the linear system using the conjugate gradient method with three-digit rounding arithmetic.
c. Does pivoting improve the answer in (a)?
d. Repeat part (b) using C−1 = D−1/2. Does this improve the answer in (b)? |
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| m55372 | Repeat Exercise 3 using the algorithm constructed in Exercise 5.
Repeat exercise 3 |
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| m55373 | Repeat Exercise 3 using the Cubic Spline Algorithm.
Repeat exercise 3 |
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| m55374 | Repeat Exercise 3 using the Jacobi method.
Repeat Exercise
Use the QR Algorithm to determine, to within 10−5, all the eigenvalues for the matrices given in Exercise 1.
In exercise
a.
b.
c.
d.
e.
f. |
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| m55375 | Repeat Exercise 3 using the Midpoint method.
In Exercise 3 |
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| m55376 | Repeat Exercise 3 using the Midpoint rule and the results of Exercise 9.
In Exercise 1 |
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| m55377 | Repeat Exercise 3 using the results of Exercise 7.
In Exercise 3
a. f (x) = x2 + 3x + 2, [0, 1];
b. f (x) = x3, [0, 2];
c. f (x) = 1/x, [1, 3];
d. f (x) = ex , [0, 2];
e. f (x) = 1/2 cos x + 1/3 sin 2x, [0, 1];
f. f (x) = x ln x, [1, 3]. |
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| m55378 | Repeat Exercise 3 using the Runge-Kutta method of order four.
In Exercise 3 |
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| m55379 | Repeat Exercise 3 with f (x, y) = −25π2 cos 5π/2 x cos 5π/2 y, using the Neumann boundary condition
∂u / ∂n (x, y) = 0. |
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| m55380 | Repeat Exercise 32, constructing three natural splines using Algorithm 3.4.
In Exercise 32
The upper portion of this noble beast is to be approximated using clamped cubic spline interpellants. The curve is drawn on a grid from which the table is constructed. Use Algorithm 3.5 to construct the three clamped cubic splines. |
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| m55381 | Repeat Exercise 3(a) and (b) using the extrapolation discussed in Example 2.
In exercise 3(a) and (b)
a. y" = −3y + 2y + 2x + 3, 0 ≤ x ≤ 1, y(0) = 2, y(1) = 1; use h = 0.1.
b. y" = −4x−1y + 2x−2y − 2x−2 ln x, 1≤ x ≤ 2, y(1) = −1 / 2, y(2) = ln 2; use h = 0.05.
Example 2
The results are listed in Table 11.4. The first extrapolation is
the second extrapolation is
and the final extrapolation is
Table 11.4 |
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| m55382 | Repeat Exercise 3(a) using extrapolation with h0 = 0.2, h1 = h0/2, and h2 = h0/4.
Repeat exercise 3(a)
a. ∂2u / ∂x2 + ∂2u / ∂y2 = 0, 0 < x < 1, 0 < y < 1;
u(x, 0) = 0, u(x, 1) = x,...............0≤ x ≤ 1;
u(0, y) = 0, u(1, y) = y, ...............0≤ y ≤ 1. |
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| m55383 | Repeat Exercise 4 using Heun s method.
In Exercise 4 |
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| m55384 | Repeat Exercise 4 using Simpson s rule and the results of Exercise 6.
In Exercise 4 |
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| m55385 | Repeat Exercise 4 using the initial approximations obtained as follows.
a. From 10.2(3c)
b. From 10.2(3d)
c. From 10.2(4c)
d. From 10.2(4d)
In exercise
Use the continuation method and the Runge-Kutta method of order four with N = 1 on the following nonlinear systems using x(0) = 0. Are the answers here comparable to Newton s method or are they suitable initial approximations for Newton s method?
a. x1 (1 − x1) + 4x2 = 12,
(x1 − 2)2 + (2x2 − 3)2 = 25.
Compare to 10.2(5c).
b. 5x21 − x22= 0,
x2 − 0.25(sin x1 + cos x2) = 0.
Compare to 10.2(5d).
c. 15x1 + x22 − 4x3 = 13,
x21 + 10x2 − x3 = 11.
x32 − 25x3 = − 22
Compare to 10.2(6c).
d. 10x1 − 2x22+ x2 − 2x3 − 5 = 0,
8x22+ 4x23 − 9 = 0.
8x2 x3 + 4 = 0
Compare to 10.2(6d). |
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| m55386 | Repeat Exercise 4 using the Midpoint method.
In Exercise 4 |
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| m55387 | Repeat Exercise 4 using the Midpoint rule and the results of Exercise 10.
In Exercise 2 |
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