Homework 6

Due: Tues 5/1

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1. This problem concerns using numerical methods to calculate the integral

   \[ I = \int_1^2\ln(x)\,\mathrm{d}x. \]

   Whose exact value is, $I = 2\ln(2) -1$. MATLAB should be used to answer the following questions.

   1. Using the composite midpoint rule, and 4 subintervals, find an approximate value for the integral. What is the error?
   2. Using the composite trapezoidal rule, and 4 subintervals, find an approximate value for the integral. What is the error?
   3. Using the composite Simpson’s rule, and 4 subintervals, find an approximate value for the integral. What is the error?

2. Find the area of the region enclosed by the curve $x^4 + 2y^4 = 1$. Make sure to state which integration method you used, what integral it was used to evaluate, and how you selected your error tolerance

3. This problem considers what is known as Lobatto quadrature, or Gauss-Lobatto quadrature. It differs from Gaussian quadrature in that it assumes that the integration rule includes both endpoints, and possibly other points within the interval.

   1. The assumed form using two points is

      \[ \int_a^b f(x)\mathrm{d}x \approx A_1 f(a) + A_2f(b) \]

      Find the values of $A_1$, $A_2$ that maximizes the degree of the polynomials for which this is rule is exact.

   2. The assumed form using three points is

      \[ \int_a^b f(x)\mathrm{d}x \approx A_1 f(a) + A_2f(z) + A_3f(b) \]

      Find the values of $A_1$, $A_2$, $A_3$, and $z$ that maximize the degree of the polynomials for which this is rule is exact.

4. This problem involves finding Gaussian quadrature rules with weights. (Hint: Use orthogonal polynomials).

   1. Find a formula of the form

      \[ \int_0^1 xf(x)\mathrm{d}x \approx A_0 f(x_0) + A_1 f(x_1) \]

      that is exact for all polynomials of degree $\leq 3$.

   2. Find a formula of the form \[ \int_0^1 x^2 f(x) \mathrm{d}x \approx A_0 f(x_0) + A_1f(x_1) \]

      that is exact for all polynomials of degree $3$.

5. In this exercise Romberg rules are derived. Assume that $T(h)$ is an integration rule that uses $n$ subintervals and $h$ is the corresponding width of each subinterval

   1. It is known that the error term for the composite Simpson’s rule involves even powers of $h$. In particular,

      \[ \int_a^b f(x)\mathrm{d}x = T_S(h) + \alpha h^4 + \beta h^6 + \gamma h^8 + \ldots. \]

      Show that the integration rule

      \[ T_R(h) = \frac{1}{15}[16 T_S(h/2) - T_S(h)] \]

      has an error that is $\mathcal{O}(h^6)$.

   2. Suppose

      \[ \int_a^b f(x)\mathrm{d}x = T(h) + \alpha h^2 + \beta h^3 + \gamma h^4 + \ldots \]

      Show that the integration rule

      \[ T_R(h) = \frac{1}{21}[32 T(h/4) - 12T(h/2) + T(h)] \]

      has an error that is $\mathcal{O}(h^4)$.

   3. Suppose

      \[ \int_a^b f(x)\mathrm{d}x = T(h) + \alpha h^2 + \beta h^3 + \gamma h^4 + \ldots. \]

      Show that the integration rule

      \[ T_R(h) = \frac{1}{12}[27 T(h/3) - 16 T(h/2) + T(h)] \]

      has an error that is $\mathcal{O}(h^4)$.