# Tutorial 5 - 2010

 Due date(s): 25 October 2010 (PDF) Tutorial questions Other instructions Hand-in at class.

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.. rubric:: Tutorial objectives: solution of a single, nonlinear equation.

.. note:: All questions below will consider the following problem.

The heat of reaction for a certain reaction is given by :math:\Delta H_{r}^{0}(T)= -24097 -0.26 T+1.69\times 10^{-3}T^2 + {\displaystyle{\frac{1.5\times10^5}{T}}}\; cal/mol. Compute the temperature at which :math:\Delta H_{r}^{0}(T)= -23505 cal/mol.

Question 1 [1]

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Show the first 4 iterations of the bisection method to solve for :math:T, justifying your choice for the initial bracket.

For iteration 2, 3, and 4, please also report these two relative errors:

.. math::

\varepsilon_\text{rel,x}^{(k)} = \displaystyle \left|\frac{x_m^{(k)} - x_m^{(k-1)}}{x_m^{(k)}} \right| \qquad\qquad \varepsilon_\text{rel,f}^{(k)} = \displaystyle \left|\frac{f(x_m^{(k)}) - f(x_m^{(k-1)})}{f(x_m^{(k)})} \right|

Question 2 [1]

###### ===
1. . Derive a :math:g(x) = x function to use in the fixed-point algorithm.
2. . Show the first 3 iterations of using the fixed-point algorithm, starting with an initial guess of :math:T = 380 K.
3. . Will the fixed-point method converge for this problem, using your :math:g(x)?

Question 3 [1]

###### ==
1. . Write the Newton-Raphson iteration formula that you would use to solve this nonlinear equation.
2. . Apply 3 iterations of this formula, also starting from :math:T = 380 K, and calculate the error tolerances.

Question 4 [1]

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Comment on the 3 approaches used so far. Are your calculations what you would expect from each method?

Bonus question [1]

###### =======

A naive code for the bisection algorithm would evaluate the function :math:f(x) at the three points, :math:[x_\ell, x_m, x_u] in every iteration. Fewer function evaluations can be obtained though.

Write a function, in either MATLAB or Python, that implements the bisection method, that evaluates :math:f(x) as few times as possible. You should report the following 8 outputs in each iteration: :math:[x_\ell^{(k)},\, x_m^{(k)},\, x_u^{(k)},\, f(x_\ell^{(k)}),\, f(x_m^{(k)}),\, f(x_u^{(k)}),\, \varepsilon_\text{rel,x}^{(k)},\, \varepsilon_\text{rel,f}^{(k)}].

Use this code to find the solution to the above problem, within a tolerance of :math:\sqrt{\text{eps}} based on :math:\varepsilon_\text{rel,x}^{(k)}.

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