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About 3307 results. 1294 free access solutions
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free/or 0.5$ |
| m54892 | Let f (x) = x3.
a. Find the second Taylor polynomial P2(x) about x0 = 0.
b. Find R2 (0.5) and the actual error in using P2 (0.5) to approximate f (0.5).
c. Repeat part (a) using x0 = 1.
d. Repeat part (b) using the polynomial from part (c). |
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| m54894 | Let
Find all values of α and β for which
a. A is singular.
b. A is strictly diagonally dominant.
c. A is symmetric.
d. A is positive definite. |
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| m54895 | Let
Find all values of α for which
a. A is singular.
b. A is strictly diagonally dominant.
c. A is symmetric.
d. A is positive definite. |
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| m54918 | Let g ∈ C1 [a, b] and p be in (a, b) with g( p) = p and |g ( p)| > 1. Show that there exists aδ > 0 such that if 0 < |p0 − p| < δ, then |p0 − p| < |p1 − p| . Thus, no matter how close the initial approximation p0 is to p, the next iterate p1 is farther away, so the fixed-point iteration does not converge if p0 ≠ p. |
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| m54921 | Let i0, i1, . . . , in be a rearrangement of the integers 0, 1, . . . , n. Show that f [xi0 , xi1 , . . ., xin] = f [x0, x1, . . ., xn]. [Consider the leading coefficient of the nth Lagrange polynomial on the data {x0, x1, . . . , xn} = {xi0 , xi1 , . . . , xin}.] |
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| m54931 | Let P be the rotation matrix with pii = pjj = cos θ and pij = −pji = sin θ, for j < i. Show that for any n × n matrix A: |
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| m54933 | Let P3(x) be the interpolating polynomial for the data (0, 0), (0.5, y), (1, 3), and (2, 2). Use Neville s method to find y if P3(1.5) = 0. |
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| m54934 | Let Pk denote a rotation matrix of the form given in (9.17).
a. Show that Pt2 Pt3 differs from an upper triangular matrix only in at most the (2, 1) and (3, 2) positions.
b. Assume that Pt2 Pt3· · · Ptk differs from an upper triangular matrix only in at most the (2, 1), (3, 2), . . . , (k, k−1) positions. Show that Pt2 Pt3 · · · Ptk Ptk +1 differs from an upper triangular matrix only in at most the (2, 1), (3, 2), . . . , (k, k − 1), (k + 1, k) positions.
c. Show that the matrix Pt2 Pt3 · · · Ptn is upper Hessenberg. |
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| m54935 | Let {pn} be the sequence defined by
Show that {pn} diverges even though limn→∞ (pn−pn−1) = 0. |
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| m54936 | Let Pn(x) be the nth Taylor polynomial for f (x) = ex expanded about x0 = 0.
a. For fixed x, show that pn = Pn(x) satisfies the hypotheses of Theorem 2.14.
b. Let x = 1, and use Aitken s (2 method to generate the sequence 0, . . . , 8.
c. Does Aitken s method accelerate convergence in this situation? |
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| m54938 | Let P(x) = anxn + an−1xn−1 + · · · + a1x + a0 be a polynomial, and let x0 be given. Construct an algorithm to evaluate P(x0) using nested multiplication. |
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| m54942 | Let S be a positive definite n × n matrix. For any x in Rn define ||x|| = (xtSx)1/2. Show that this defines a norm on Rn. [Use the Cholesky factorization of S to show that xtSy = ytSx ≤ (xtSx)1/2(ytSy)1/2.] |
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| m54943 | Let S be a real and nonsingular matrix, and let ||·|| be any norm on Rn. Define ||·|| by ||x|| = ||Sx||.Show that || · || is also a norm on Rn. |
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| m54946 | Let T(a, b) and T(a, a+b/2 ) + T( a+b/2 , b) be the single and double applications of the Trapezoidal rule to
Derive the relationship between
And |
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| m54947 | Let the continuation method with the Runge-Kutta method of order four be abbreviated CMRK4.
After completing Exercises 4, 5, 6, 7, 8, and 9, answer the following questions.
a. Is CMRK4 with N = 1 comparable to Newton s method? Support your answer with the results of earlier exercises.
b. Should CMRK4 with N = 1 be used as a means to obtain an initial approximation for Newton smethod? Support your answer with the results of earlier exercises.
c. Repeat part (a) for CMRK4 with N = 2.
d. Repeat part (b) for CMRK4 with N = 2. |
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| m54954 | Let {v1, . . . , vn} be a set of orthonormal nonzero vectors in Rn and x ∈ Rn. Determine the values of ck, for k = 1, 2, . . . , n, if |
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| m55031 | Let (x0, y0) = (0, 0) and (x1, y1) = (5, 2) be the endpoints of a curve. Use the given guide points to construct parametric cubic Hermite approximations (x(t), y(t)) to the curve, and graph the approximations.
a. (1, 1) and (6, 1)
b. (0.5, 0.5) and (5.5, 1.5)
c. (1, 1) and (6, 3)
d. (2, 2) and (7, 0) |
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| m55049 | Let z0 = x0, z1 = x0, z2 = x1, and z3 = x1. Form the following divided-difference table.
Show that the cubic Hermite polynomial H3(x) can also be written as f [z0] + f [z0, z1](x − x0) + f [z0, z1, z2](x − x0)2 + f [z0, z1, z2, z3](x − x0)2(x − x1). |
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| m55086 | Modify Algorithms 12.2 and 12.3 to include the parabolic partial differential equation
∂u / ∂t − ∂2u / ∂x2 = F(x), 0< x < l, 0 < t;
u(0, t) = u(l, t) = 0, 0 < t;
u(x, 0) = f (x), 0≤ x ≤ l. |
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| m55088 | Modify Householder s Algorithm 9.5 to compute similar upper Hessenberg matrices for the following nonsymmetric matrices.
a.
b.
c.
d. |
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