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numerical analysis

Questions and Answers of Numerical Analysis

Show that for each Chebyshev polynomial Tn(x), we have
Determine all degree 2 Pade approximations for f (x) = e2x. Compare the results at xi = 0.2i, for i = 1, 2, 3, 4, 5, with the actual values f (xi).
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?
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
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.
To accurately approximate f (x) = ex for inclusion in a mathematical library, we first restrict the domain of f . Given a real number x, divide by ln√10 to obtain the relation x = M · ln√10 +
To accurately approximate sin x and cos x for inclusion in a mathematical library, we first restrict their domains. Given a real number x, divide by π to obtain the relation |x| = Mπ + s, where M
Determine all degree 3 Padé approximations for f (x) = x ln(x+1). Compare the results at xi = 0.2i, for i = 1, 2, 3, 4, 5, with the actual values f (xi).
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.
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.
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
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.
Table 8.10 lists results of the Padé approximation of degree 5 with n = 3 and m = 2, the fifthMaclaurin polynomial, and the exact values of f (x) = eˆ’x when xi = 0.2i, for i = 1, 2,
Express the following rational functions in continued-fraction form:a.b. c. d.
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?
Repeat Exercise 9 using m = 8. Compare the values of the approximating polynomials with the values of f at the points ξj = −π + 0.2jπ, for 0 ≤ j ≤ 10. Which approximation is better?
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.
a. Determine the discrete least squares trigonometric polynomial S4(x), using m = 16, for f (x) = x2 sin x on the interval [0, 1].b. Computec. Compare the integral in part (b) to
Show that for any continuous odd function f defined on the interval [ˆ’a, a], we have
Show that for any continuous even function f defined on the interval [ˆ’a, a], we have
Show that the functions (0(x) = 1/2, (1(x) = cos x. . . (n(x) = cos nx, (n+1(x) = sin x, . . . , (2n−1(x) = sin(n − 1)x are orthogonal on [−π, π] with respect to w(x) ≡ 1.
In Example 1 the Fourier series was determined for f (x) = |x|. Use this series and the assumption that it represents f at zero to find the value of the convergent infinite series
Show that the form of the constants ak for k = 0. . . n in Theorem 8.13 is correct as stated.
Find the general continuous least squares trigonometric polynomial Sn(x) for f (x)= ex on [−π, π].
Find the general continuous least squares trigonometric polynomial Sn(x) for
Find the general continuous least squares trigonometric polynomial Sn(x) in for
Determine the discrete least squares trigonometric polynomial Sn(x) on the interval [−π, π] for the following functions, using the given values of m and n: a. f (x) = cos 2x, m = 4, n = 2 b. f
Determine the trigonometric interpolating polynomial S2(x) of degree 2 on [ˆ’Ï€, Ï€] for the following functions, and graph f (x) ˆ’ S2(x):a. f (x) =
In the discussion preceding Algorithm 8.3, an example for m = 4 was explained. Define vectors c, d, e, f, and y as c = (c0, . . . , c7)t , d = (d0, . . . , d7)t , e = (e0, . . . , e7)t , f = (f0, . .
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.
Use the Fast Fourier Transform Algorithm to compute the trigonometric interpolating polynomial of degree 4 on [−π, π] for the following functions. a. f (x) = π(x − π) b. f (x) = |x| c. f (x)
a. Determine the trigonometric interpolating polynomial S4(x) of degree 4 for f (x) = x2 sin x on the interval [0, 1].b. Computec. Compare the integral in part (b) to
Use the approximations obtained in Exercise 3 to approximate the following integrals, and compare your results to the actual values.a.b. c. d.
Use the Fast Fourier Transform Algorithm to determine the trigonometric interpolating polynomial of degree 16 for f (x) = x2 cos x on [−π, π].
Use the Fast Fourier Transform Algorithm to determine the trigonometric interpolating polynomial of degree 64 for f (x) = x2 cos x on [−π, π].
Use a trigonometric identity to show that
Show that c0. . . c2mˆ’1 in Algorithm 8.3 are given byWhere Î¶ = eÏ€i/m.
Find the eigenvalues and associated eigenvectors of the following 3 Ã— 3 matrices. Is there a set of linearly independent eigenvectors?a.b. c. d.
Show that if A is a matrix and λ1, λ2, ..., λk are distinct eigenvalues with associated eigenvectors x1, x2, ..., xk, then {x1, x2, . . . , xk} is a linearly independent set.
Let {v1, . . . , vn} be a set of orthonormal nonzero vectors in Rn and x ˆˆ Rn. Determine the values of ck, for k = 1, 2, . . . , n, if
Consider the follow sets of vectors. (i) Show that the set is linearly independent; (ii) use the Gram- Schmidt process to find a set of orthogonal vectors; (iii) determine a set of orthonormal
Consider the follow sets of vectors. (i) Show that the set is linearly independent; (ii) use the Gram-Schmidt process to find a set of orthogonal vectors; (iii) determine a set of orthonormal vectors
Use the Geršgorin Circle Theorem to show that a strictly diagonally dominant matrix must be nonsingular.
Prove that the set of vectors {v1, v2 , . . . , vk} described in the Gram-Schmidt Theorem is orthogonal.
A persymmetric matrix is a matrix that is symmetric about both diagonals; that is, an N × N matrix A = (aij) is persymmetric if aij = aji = aN+1−i,N+1−j, for all i = 1, 2, . . . , N and j = 1,
Find the eigenvalues and associated eigenvectors of the following 3 Ã— 3 matrices. Is there a set of linearly independent eigenvectors?a.b.c.d.
Use the GerÅ¡gorin Circle Theorem to determine bounds for the eigenvalues, and the spectral radius of the following matrices.a.b.c.d.
Use the GerÅ¡gorin Circle Theorem to determine bounds for the eigenvalues, and the spectral radius of the following matrices.a.b.c.d.
For the matrices in Exercise 1 that have 3 linearly independent eigenvectors form the factorization A = PDP−1.
For the matrices in Exercise 2 that have 3 linearly independent eigenvectors form the factorization A = PDPˆ’1.a.b. c. d.
Show that the three eigenvectors in Example 3 are linearly independent.Example 3
Show that a set {v1, . . . , vk} of k nonzero orthogonal vectors is linearly independent.
Show that the following pairs of matrices are not similar.a.b. c. d.
Show that the following matrices are singular but are diagonalizable.a.b.
In Exercise 31 of Section 6.6, a symmetric matrixwas used to describe the average wing lengths of fruit flies that were offspring resulting from the mating of three mutants of the flies. The entry
Show that if A is similar to B and B is similar to C, then A is similar to C.
Show that if A is similar to B, then a. det(A) = det(B). b. The characteristic polynomial of A is the same as the characteristic polynomial of B. c. A is nonsingular if and only if B is
Show that the matrix given in Example 3 of Section 9.1,is similar to the diagonal matrices
Prove Theorem 9.10. Theorem 9.10 Suppose that Q is an orthogonal n × n matrix. Then (i) Q is invertible with Q−1 = Qt; (ii) For any x and y in Rn, (Qx)t Qy = xty; In addition, the converse of part
Show that there is no diagonal matrix similar to the matrix given in Example 4 of Section 9.1,
Prove that if Q is nonsingular matrix with Qt = Q−1, then Q is orthogonal.
Prove Theorem 9.13. Theorem 9.13 An n × n matrix A is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. In this case, D = S−1 AS, where the columns of S
Show that the following pairs of matrices are not similar.a.b.c.d.
Define A = PDPˆ’1 for the following matrices D and P. Determine A3.a.b. c. d.
Determine A4 for the matrices in Exercise 3.In exercisea.b. c. d.
For each of the following matrices determine if it diagonalizable and, if so, find P and D with A = PDPˆ’1.a.b. c. d.
For each of the following matrices determine if it diagonalizable and, if so, find P and D with A = PDPˆ’1.a.b. c. d.
(i) Determine if the following matrices are positive definite, and if so, (ii) construct an orthogonal matrix Q for which Qt AQ = D, where D is a diagonal matrix.a.b. c. d.
(i) Determine if the following matrices are positive definite, and if so, (ii) construct an orthogonal matrix Q for which QtAQ = D, where D is a diagonal matrix.a.b. c. d.
Show that each of the following matrices is nonsingular but not diagonalizable.a.b. c. d.
Find the first three iterations obtained by the Power method applied to the following matrices.a.Use x(0) = (1,ˆ’1, 2)t. b. Use x(0) = (ˆ’1, 0, 1)t. c. Use x(0) = (ˆ’1,
Use the Inverse Power method to approximate the most dominant eigenvalue of the matrices in Exercise 2. Iterate until a tolerance of 10ˆ’4 is achieved or until the number of iterations
Use the Symmetric Power method to approximate the most dominant eigenvalue of the matrices in Exercise 5. Iterate until a tolerance of 10ˆ’4 is achieved or until the number of iterations
Use the Symmetric Power method to approximate the most dominant eigenvalue of the matrices in Exercise 6. Iterate until a tolerance of 10ˆ’4 is achieved or until the number of iterations
UseWielandt deflation and the results of Exercise 7 to approximate the second most dominant eigenvalue of the matrices in Exercise 1. Iterate until a tolerance of 10ˆ’4 is achieved or until
UseWielandt deflation and the results of Exercise 8 to approximate the second most dominant eigenvalue of the matrices in Exercise 2. Iterate until a tolerance of 10ˆ’4 is achieved or until
Repeat Exercise 7 using Aitken's Î”2 technique and the Power method for the most dominant eigenvalue.Repeat exercise 7Use the Power method to approximate the most dominant eigenvalue of
Repeat Exercise 8 using Aitken's Î”2 technique and the Power method for the most dominant eigenvalue.Use the Power method to approximate the most dominant eigenvalue of the matrices in
Assume that the largest eigenvalue Î»1 in magnitude and an associated eigenvector v(1) have been obtained for the n Ã— n symmetric matrix A. Show that the matrixhas the same
Suppose the n × n matrix A has eigenvalues λ1, . . . , λn ordered by |λ1| > |λ2| > |λ3| ≥ ··· ≥ |λn|, with linearly independent eigenvectors v(1), v(2), . . . , v(n). a. Show that if
Following along the line of Exercise 11 in Section 6.3 and Exercise 15 in Section 7.2, suppose that a species of beetle has a life span of 4 years, and that a female in the first year has a survival
Find the first three iterations obtained by the Power method applied to the following matrices.a.Use x(0) = (1, 2, 1)t. b. Use x(0) = (1, 1, 0, 1)t. c. Use x(0) = (1, 1, 0,ˆ’3)t. d. Use
Show that the ith row of B = A−λ1v(1)xt is zero, where λ1 is the largest value of A in absolute value, v(1) is the associated eigenvector of A for λ1, and x is the vector defined in Eq. (9.7).
The (m ˆ’ 1) Ã— (m ˆ’ 1) tridiagonal matrixis involved in the Backward Difference method to solve the heat equation. For the stability of the method we need Ï(Aˆ’1) a. Î± = 1 / 4b.
The eigenvalues of the matrix A in Exercise 21 areCompare the approximation in Exercise 21 to the actual value of Ï(Aˆ’1). Again, when is the method stable?In exercisea. Î± = 1 / 4b. Î± = 1
The (m ˆ’ 1) Ã— (m ˆ’ 1) matrices A and B given byAnd are involved in the Crank-Nicolson method to solve the heat equation (see Section 12.2).Withm = 11, approximate
A linear dynamical system can be represented by the equationsDx / dt = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t),where A is an n Ã— n variable matrix, B is an n Ã— r variable matrix, C is
Repeat Exercise 1 using the Inverse Power method.In exerciseFind the first three iterations obtained by the Power method applied to the following matrices.a.Use x(0) = (1,ˆ’1, 2)t. b. Use
Repeat Exercise 2 using the Inverse Power method.In exerciseFind the first three iterations obtained by the Power method applied to the following matrices.a.Use x(0) = (1, 2, 1)t. b. Use x(0) = (1,
Find the first three iterations obtained by the Symmetric Power method applied to the following matrices.a.Use x(0) = (1,ˆ’1, 2)t. b. Use x(0) = (ˆ’1, 0, 1)t. c. Use x(0) = (0, 1,
Find the first three iterations obtained by the Symmetric Power method applied to the following matrices.a.Use x(0) = (1,ˆ’1, 2)t. b. Use x(0) = (ˆ’1, 0, 1)t. c. Use x(0) = (1, 0,
Use the Power method to approximate the most dominant eigenvalue of the matrices in Exercise 1. Iterate until a tolerance of 10ˆ’4 is achieved or until the number of iterations exceeds
Use the Power method to approximate the most dominant eigenvalue of the matrices in Exercise 2. Iterate until a tolerance of 10ˆ’4 is achieved or until the number of iterations exceeds 25.In
Use the Inverse Power method to approximate the most dominant eigenvalue of the matrices in Exercise 1. Iterate until a tolerance of 10ˆ’4 is achieved or until the number of iterations
Use Householder's method to place the following matrices in tridiagonal form.a.b. c. d.
Use Householder's method to place the following matrices in tridiagonal form.a.b. c. d.
Modify Householder's Algorithm 9.5 to compute similar upper Hessenberg matrices for the following nonsymmetric matrices.a.b. c. d.
Apply two iterations of the QR method without shifting to the following matrices.a.b. c. d. e. f.
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· ·
Jacobi's method for a symmetric matrix A is described byA1 = A,A2 = P1A1Pt1and, in general,Ai+1 = PiAiPti.The matrix Ai+1 tends to a diagonal matrix, where Pi is a rotation matrix chosen to eliminate

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