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Questions and Answers of Numerical Analysis

Prove that if || · || is a vector norm on Rn, then ||A|| = max||x||=1 ||Ax|| is a matrix norm.
The following excerpt from the Mathematics Magazine [Sz] gives an alternative way to prove the Cauchy-Buniakowsky-Schwarz Inequality.a. Show that when x ‰ 0 and y ‰ 0, we haveb. Use
Show that the Cauchy-Buniakowsky-Schwarz Inequality can be strengthened to
a. Verify that the function ||·||1, defined on Rn byis a norm on Rn.b. Find ||x||1 for the vectors given in Exercise 1.c. Prove that for all x ˆˆ Rn, ||x||1 ‰¥ ||x||2.
Prove that the following sequences are convergent, and find their limits. a. x(k) = (1/k, e1−k ,−2/k2)t b. x(k) = (e−k cos k, k sin(1/k), 3 + k)t c. x(k) = (ke−k2 , (cos k)/k, √(k2 + k)
The following linear systems Ax = b have x as the actual solution and x as an approximate solution. Compute ||x − x||∞ and ||Ax − b||∞a. 1/2 x1 + 1/3 x2 = 1/63,1/3 x1 + 1/4 x2 = 1/168,x = (
Show by example that || · ||∞, defined by ||A||∞ = max1≤i, j≤n |ai j|, does not define a matrix norm.
Show that || · ||①, defined by is a matrix norm. Find || · ||① for the matrices in Exercise 4.
a. The Frobenius norm (which is not a natural norm) is defined for an n Ã— n matrix A byShow that || · ||F is a matrix norm.b. Find || · ||F for the matrices in Exercise 4.c. For any matrix A,
Compute the eigenvalues and associated eigenvectors of the following matrices.a.b. c. d. e. f.
LetAndShow that A1 is not convergent, but A2 is convergent.
An n × n matrix A is called nilpotent if an integer m exists with Am = On. Show that if λ is an eigenvalue of a nilpotent matrix, then λ = 0.
Show that the characteristic polynomial p(λ) = det(A − λI) for the n × n matrix A is an nth-degree polynomial. [Expand det(A−λI) along the first row, and use mathematical induction on n.]
a. Show that if A is an n × n matrix, then where λi , . . . , λn are the eigenvalues of A.b. Show that A is singular if and only if λ = 0 is an eigenvalue of A.
Let λ be an eigenvalue of the n × n matrix A and x = 0 be an associated eigenvector. a. Show that λ is also an eigenvalue of At. b. Show that for any integer k ≥ 1, λk is an eigenvalue of Ak
Show that if A is symmetric, then ||A||2 = ρ(A).
Find matrices A and B for which ρ(A+B) > ρ(A)+ρ(B). (This shows that ρ(A) cannot be a matrix norm.)
Show that if || · || is any natural norm, then (||A−1||)−1 ≤ |λ| ≤ ||A|| for any eigenvalue λ of the nonsingular matrix A.
Compute the eigenvalues and associated eigenvectors of the following matricesa.b. c. d. e. f.
Find the complex eigenvalues and associated eigenvectors for the following matrices.a.b.
Find the complex eigenvalues and associated eigenvectors for the following matrices.a.b.
Find the first two iterations of the Jacobi method for the following linear systems, using x(0) = 0: a. 3x1 − x2 + x3 = 1, 3x1 + 6x2 + 2x3 = 0, 3x1 + 3x2 + 7x3 = 4. b. 10x1 − x2 = 9, −x1 + 10x2
The linear system x1 + 2x2 − 2x3 = 7, x1 + x2 + x3 = 2, 2x1 + 2x2 + x3 = 5, has the solution (1, 2,−1)t .a. Show that ρ(Tj) = 0.b. Use the Jacobi method with x(0) = 0 to approximate the solution
The linear system x1 − x3 = 0.2, −1/2 x1 + x2 – 1/4 x3 = −1.425, x1 – 1/2 x2 + x3 = 2. Has the solution (0.9,−0.8, 0.7)t .a. Is the coefficient matrixStrictly diagonally dominant?b.
Repeat Exercise 11 using the Jacobi method.In Exercise 11a. Is the coefficient matrixStrictly diagonally dominant?b. Compute the spectral radius of the Gauss-Seidel matrix Tg.c. Use the Gauss-Seidel
a. Prove that ||x(k) − x|| ≤ ||T||k ||x(0) − x|| and ||x(k) − x|| ≤ ||T||k /1 − ||T|| ||x(1) − x(0) ||, Where T is an n × n matrix with ||T|| < 1 and x(k) = Tx(k−1) + c, k = 1, 2,
Show that if A is strictly diagonally dominant, then ||Tj ||∞ < 1.
Use (a) the Jacobi and (b) the Gauss-Seidel methods to solve the linear system Ax = b to within 10ˆ’5 in the lˆž norm, where the entries of A areAnd those of b are bi =
Suppose that an object can be at any one of n+1 equally spaced points x0, x1. . . xn. When an object is at location xi, it is equally likely to move to either xi−1 or xi+1 and cannot directly move
Suppose that A is a positive definite.a. Show that we can write A = D − L − Lt, where D is diagonal with dii > 0 for each 1 ≤ i ≤ n and L is lower triangular. Further, show that D − L is
Find the first two iterations of the Jacobi method for the following linear systems, using x(0) = 0: a. 4x1 + x2 − x3 = 5, −x1 + 3x2 + x3 = −4, 2x1 + 2x2 + 5x3 = 1. b. −2x1+ x2 + 1/2 x3 =
Repeat Exercise 1 using the Gauss-Seidel method. 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.
Repeat Exercise 2 using the Gauss-Seidel method. 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 =
Use the Jacobi method to solve the linear systems in Exercise 1, with TOL = 10−3 in the l∞ norm. In Exercise 1 a. 3x1 − x2 + x3 = 1, 3x1 + 6x2 + 2x3 = 0, 3x1 + 3x2 + 7x3 = 4. b. 10x1 − x2 =
Use the Jacobi method to solve the linear systems in Exercise 2, with TOL = 10−3 in the l∞ norm. In Exercise 2 a. 4x1 + x2 − x3 = 5, −x1 + 3x2 + x3 = −4, 2x1 + 2x2 + 5x3 = 1. b. −2x1+ x2
Use the Gauss-Seidel method to solve the linear systems in Exercise 1, with TOL = 10−3 in the l∞ norm. In Exercise 1 a. 3x1 − x2 + x3 = 1, 3x1 + 6x2 + 2x3 = 0, 3x1 + 3x2 + 7x3 = 4. b. 10x1 −
Use the Gauss-Seidel method to solve the linear systems in Exercise 2, with TOL = 10−3 in the l∞ norm. In Exercise 2 a. 4x1 + x2 − x3 = 5, −x1 + 3x2 + x3 = −4, 2x1 + 2x2 + 5x3 = 1. b.
The linear system 2x1 − x2 + x3 = −1, 2x1 + 2x2 + 2x3 = 4, −x1 − x2 + 2x3 = −5 has the solution (1, 2,−1)t .a. Show that ρ(Tj) = √5/2 > 1.b. Show that the Jacobi method with x(0) =
Find the first two iterations of the SOR method with ω = 1.1 for the following linear systems, using x(0) = 0: a. 3x1 − x2 + x3 = 1, 3x1 + 6x2 + 2x3 = 0, 3x1 + 3x2 + 7x3 = 4. b. 10x1 − x2 =
The forces on the bridge truss described in the opening to this chapter satisfy the equations in the following tableThis linear system can be placed in the matrix forma. Explain why the system of
Use the SOR method to solve the linear system Ax = b to within 10ˆ’5 in the lˆž norm, where the entries of A areAnd those of b are bi = Ï€, for each i = 1, 2, . . . , 80.
In Exercise 17 of Section 7.3 a techniquewas outlined to prove that the Gauss-Seidel method converges when A is a positive definite matrix. Extend this method of proof to show that in this case there
Find the first two iterations of the SOR method with ω = 1.1 for the following linear systems, using x(0) = 0: a. 4x1 + x2 − x3 = 5, −x1 + 3x2 + x3 = −4, 2x1 + 2x2 + 5x3 = 1. b. −2x1+ x2 +
Repeat Exercise 1 using ω = 1.3. 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 =
Repeat Exercise 2 using ω = 1.3. 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
Use the SOR method with ω = 1.2 to solve the linear systems in Exercise 1 with a tolerance TOL = 10−3 in the l∞ norm. In Exercise 1 a. 3x1 − x2 + x3 = 1, 3x1 + 6x2 + 2x3 = 0, 3x1 + 3x2 + 7x3 =
Use the SOR method with ω = 1.2 to solve the linear systems in Exercise 2 with a tolerance TOL = 10−3 in the l∞ norm. In Exercise 2 a. 4x1 + x2 − x3 = 5, −x1 + 3x2 + x3 = −4, 2x1 + 2x2 +
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 +
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 +
Prove Kahan's Theorem 7.24. [Hint: If Î»1, . . . , Î»n are eigenvalues of TÏ‰,Since det Dˆ’1 = det(D ˆ’ Ï‰L) ˆ’1 and the determinant of a product of matrices is the product of the
Using Exercise 9, estimate the condition numbers for the following matrices:a.b.
The n Ã— n Hilbert matrix H(n) defined byIs an ill-conditioned matrix that arises in solving the normal equations for the coefficients of the least-squares polynomial a. Show thatAnd compute
Use four-digit rounding arithmetic to compute the inverse H−1 of the 3 × 3 Hilbert matrix H, and then compute = (H−1)−1. Determine ||H − ||∞.
The following linear systems Ax = b have x as the actual solution and x as an approximate solution. Using the results of Exercise 1, compute a. 1/2 x1 + 1/3x2 = 1/63, 1/3 x1 + 1/4 x2 = 1/168, x =
The following linear systems Ax = b have x as the actual solution and as an approximate solution. Using the results of Exercise 2, compute a. 0.03x1 + 58.9x2 = 59.2, 5.31x1 − 6.10x2 = 47.0, x =
(i) Use Gaussian elimination and three-digit rounding arithmetic to approximate the solutions to the following linear systems. (ii) Then use one iteration of iterative refinement to improve the
Repeat Exercise 5 using four-digit rounding arithmetic. In Exercise 5 a. 0.03x1 + 58.9x2 = 59.2, 5.31x1 − 6.10x2 = 47.0
Show that if B is singular, then
The linear system x1 + 1/2 x2 = 5/21, 1/2 x1 + 1/3 x2 = 11/84
Solve the linear system in Exercise 16(b) of Exercise Set 7.3 using the conjugate gradient method with C−1 = I.
LetAnd Form the 16 Ã— 16 matrix A in partitioned form, Let b = (1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6)t. a. Solve Ax = b using the conjugate gradient method with tolerance
Use the transpose properties given in Theorem 6.14 on page 390 to prove Theorem 7.30.
a. Show that an A-orthogonal set of nonzero vectors associated with a positive definite matrix is linearly independent. b. Show that if {v(1), v(2), . . . , v(n)} is a set of A-orthogonal nonzero
Prove Theorem 7.33 using mathematical induction as follows: a. Show that (r(1), v(1)) = 0. b. Assume that (r(k), v(j)) = 0, for . each k ≤ l and j = 1, 2, . . . , k, and show that this implies that
In Example 3 the eigenvalues were found for the matrix A and the conditioned matrix AH. Use these to determine the condition numbers of A and AH in the l2 norm, and compare your results to those
The linear system 0.1x1 + 0.2x2 = 0.3, 0.2x1 + 113x2 = 113.2
The linear system x1 + 1/2 x2 + 1/3 x3 = 5/6, 1/2 x1 + 1/3 x2 + 1/4 x3 = 5/12, 1/3 x1 + 1/4 x2 + 1/5 x3 = 17/60
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
Perform only two steps of the conjugate gradient method with C = C−1 = I on each of the following linear systems. Compare the results in parts (b) and (c) to the results obtained in parts (b) and
Repeat Exercise 5 using C−1 = D−1/2.In Exercise 5a. 3x1 − x2 + x3 = 1,−x1 + 6x2 + 2x3 = 0,x1 + 2x2 + 7x3 = 4.b. 10x1 − x2 = 9,−x1 + 10x2 − 2x3 = 7,− 2x2 + 10x3 = 6.c. 10x1 + 5x2 =
Repeat Exercise 5 with TOL = 10−3 in the l∞ norm. Compare the results in parts (b) and (c) to those obtained in Exercises 5 and 7 of Section 7.3 and Exercise 5 of Section 7.4.In Exercise 5a. 3x1
Repeat Exercise 7 using C−1 = D−1/2.In Exercise 7a. 3x1 − x2 + x3 = 1,−x1 + 6x2 + 2x3 = 0,x1 + 2x2 + 7x3 = 4.b. 10x1 − x2 = 9,−x1 + 10x2 − 2x3 = 7,− 2x2 + 10x3 = 6.c. 10x1 + 5x2 =
Approximate solutions to the following linear systems Ax = b to within 10−5 in the l∞ norm. (i) and b = (1.902207, 1.051143, 1.175689, 3.480083, 0.819600,−0.264419, − 0.412789, 1.175689,
In a paper dealing with the efficiency of energy utilization of the larvae of the modest sphinx moth (Pachysphinx modesta), L. Schroeder [Schr1] used the following data to determine a relation
Show that the normal equations (8.3) resulting from discrete least squares approximation yield a symmetric and nonsingular matrix and hence have a unique solution. [Let A = (aij), where and x1, x2,
Find the least squares polynomials of degrees 1, 2, and 3 for the data in the following table. Compute the error E in each case. Graph the data and the polynomials.
Find the least squares polynomials of degrees 1, 2, and 3 for the data in the following table. Compute the error E in each case. Graph the data and the polynomials.
Given the data:a. Construct the least squares polynomial of degree 1, and compute the error. b. Construct the least squares polynomial of degree 2, and compute the error. c. Construct the least
Repeat Exercise 5 for the following data. In Exercise 5 a. Construct the least squares polynomial of degree 1, and compute the error. b. Construct the least squares polynomial of degree 2, and
Find the linear least squares polynomial approximation to f (x) on the indicated interval if 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,
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
Use the Laguerre polynomials calculated in Exercise 11 to compute the least squares polynomials of degree one, two, and three on the interval (0,∞) with respect to the weight function w(x) = e−x
Suppose {(0, (1. . . (n} is any linearly independent set in ((n. Show that for any element Q ˆˆ(n, there exist unique constants c0, c1, . . . , cn, such that
Show that if { (, (1, . . . , (n} is an orthogonal set of functions on [a, b] with respect to the weight function w, then {(0, (1, . . . , (n} is a linearly independent set.
Show that the normal equations (8.6) have a unique solution. [Show that the only solution for the function f (x) ≡ 0 is aj = 0, j = 0, 1. . . n. Multiply Eq. (8.6) by aj , and sum over all j.
Find the linear least squares polynomial approximation on the interval [−1, 1] for the following functions. a. f (x) = x2 − 2x + 3 b. f (x) = x3 c. f (x) = 1/(x + 2) d. f (x) = ex e. f (x) = 1/2
Find the least squares polynomial approximation of degree two to the functions and intervals in Exercise 1. In Exercise 1 a. f (x) = x2 + 3x + 2, [0, 1]; b. f (x) = x3, [0, 2]; c. f (x) = 1/x, [1,
Find the least squares polynomial approximation of degree 2 on the interval [−1, 1] for the functions in Exercise 3. In Exercise 3 a. f (x) = x2 + 3x + 2, [0, 1]; b. f (x) = x3, [0, 2]; c. f (x) =
Compute the error E for the approximations in Exercise 3. 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 +
Compute the error E for the approximations in Exercise 4.In Exercise 4a. 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
Use the Gram-Schmidt process to construct φ0(x), φ1(x), φ2(x), and φ3(x) for the following intervals. a. [0, 1] b. [0, 2] c. [1, 3]
Repeat Exercise 1 using the results of Exercise 7. In Exercise 1 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
Obtain the least squares approximation polynomial of degree 3 for the functions in Exercise 1 using the results of Exercise 7. In Exercise 1 a. f (x) = x2 + 3x + 2, [0, 1]; b. f (x) = x3, [0, 2]; c.
Use the zeros of 3 to construct an interpolating polynomial of degree 2 for the following functions on the interval [−1, 1]. a. f (x) = ex b. f (x) = sin x c. f (x) = ln(x + 2) d. f (x) = x4
Show that for each n, the Chebyshev polynomial Tn(x) has n distinct zeros in (−1, 1).
Show that for each n, the derivative of the Chebyshev polynomial Tn(x) has n − 1 distinct zeros in (−1, 1).
Use the zeros of T4 to construct an interpolating polynomial of degree 3 for the functions in Exercise 1. In Exercise 1 a. f (x) = ex b. f (x) = sin x c. f (x) = ln(x + 2) d. f (x) = x4
Use the zeros of 3 and transformations of the given interval to construct an interpolating polynomial of degree 2 for the following functions. a. f (x) = 1/x , [1, 3] b. f (x) = e−x , [0, 2] c. f
Find the sixth Maclaurin polynomial for xex , and use Chebyshev economization to obtain a lesser degree polynomial approximation while keeping the error less than 0.01 on [−1, 1].
Show that for any positive integers i and j with i > j, we have Ti(x)Tj(x) = 1/2[Ti+j(x) + Ti−j(x)].

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