HOMEWORK ASSIGNMENTS:
| Week | Lecture Material | Relevant Sections | Homework |
|---|---|---|---|
| 2 | Gershgorin disks | NLA 24 | 24.2(a,b,c), 24.4(a) |
| 2 | The Bauer-Fike theorem | NLA 26 | 26.3 (We will do Exercise 26.1 in class on Wednesday 1/27) |
| 2 | The QR Algorithm | NLA 28 | 28.2, 28.3 |
| 3 | Singular value computation | NLA 31 | 31.1 |
| 3 | Iterative methods | NLA 33 | 33.1, 33.2 |
| 4 | GMRES | NLA 35 | 35.4, 35.6a (also describe the algorithm) |
| 4 | Lanczos iteration | NLA 36 | 36.1 |
| 5-6 | Orthogonal polynomials | INA 3.6 | Ch. 3: 14, 15, |
| 6 | Chebyshev polynomials (these problems have been/will be discussed in class) | INA 3.6 |
C.1: Show that T_k(x)=cos(k arccos(x)), for -1 < x < 1, is a polynomial
of degree k for all nonnegative integers k. T_k is known as the Chebyshev
polynomial of the first kind of degree k. Hint: Use the substitution x=cos(t)
and apply trigonometric identities to show that the T_k(x) satisfy a three-term
recurrence relations (like orthogonal polynomials). C.2: Show that the polynomials T_k(x) are orthogonal with respect to the inner product (f,g)=integral_{-1}^1 f(x)g(x)/(1-x^2)^{1/2} dx. Hint: Use the same substitution as above. C.3: At how many points does T_k(x)=cos(k arccos(x)) achieve its maximum magnitude on the interval [-1,1]? C.4: Use the above property to show that there is no polynomial p(x) of degree at most k with the same leading coefficient as T_k(x), such that max_{-1 < x < 1} |p(x)| < max_{-1 < x < 1}|T_k(x)|. Hint: Assume the contrary.
|
| 8 | Orthogonal polynomials | INA 3.6 | 3:18 (a)
|
| 8 | Interpolation | INA 2.1 | C.5: Consider the approximation of a smooth function on the interval [-1,1] by an interpolating polynomial, determined by m interpolation points. Explain why it may be a good idea to choose the zeros of the Chebyshev polynomial T_m(x) as interpolation points. Hint: Use the remainder formula for polynomial interpolation. |
| 8 | Interpolation | INA 2.1 | Ch. 2: 1(b), 2(a), 3, 4
|
| 9 | Interpolation | INA 2.1 | S.1: Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in MATLAB; see "help spline". Use m=10 and m=20. Compute splines that interpolate at equidistant nodes and at Chebyshev nodes. Provide tables of the errors and plots of the function f and the interpolating polynomials and splines. |
| 9 | Quadrature | INA 3.1 | Ch. 3: 1, 3, 4, 20(a): Derive a remainder formula for the midpoint rule analogous to the one of Problem 3 for the trapezoidal rule. Hint: use the Taylor expansion of the function at the midpoint with remainder term. 20(b): Using this result, derive a remainder formula for the composite midpoint rule analogous to the formula for the composite trapezoidal rule at the top of page 150. |
| 10 | Nonlinear equations | INA 5.1-5.2 | Ch. 5: 2, 5, 10, 11, and 19: The equation x=3exp(-x) has a root close to 1.05. Which one of the following iterative methods can be expected to converge the fastest when x_0=1.05? i) x_{k+1}=3 exp(-x_k), ii) x_{k+1}=(2x_k+3exp(-x_k))/3, iii) x_{k+1}=1.05x_k+3exp(-x_k), iv) x_{k+1}=(x_k+3exp(-x_k))/2, v) x_{k+1}=0.6x_k+1.2exp(-x_k). Explain. 20: The iterative method x_{k+1}=(x_k^3+3Nx_k)/(3x_k^2+N) with x_0=N/2 is considered to converge particularly quickly to sqrt(N). Determine the order of the method. Hint for 10: Try Newton's method. |