.. _homework2:

==========================================
Homework 2
==========================================

Due Thursday, April 14, 2011, by 11:00pm PDT, by pushing to your bitbucket
repository.

The goal of this homework is to learn some basic Fortran 90, including
subroutines, functions, arrays, if statements, and loops.

Before tackling this homework, you should read some of the class notes and
links they point to.  In particular, the following sections are relevant:

 * :ref:`fortran`
 * :ref:`fortran_sub`
 * :ref:`fortran_taylor`

Make sure you update your clone of the class repository before doing this
assignment::

    $ cd $CLASSHG
    $ hg pull -u     # pulls and updates

You should also create a directory `$MYHG/homeworks/homework2`
to work in since this is where your modified versions of the files will
eventually need to be.

Assignment:

 #. Read the sections of the notes to become familiar with Fortran.

    The Fortran program `$CLASSHG/codes/fortran/taylor_converge.f90`
    contains a subroutine to compute :math:`\exp(x)` by Taylor series and an
    example of how it is used.  As you will see in lecture, this
    Taylor series works well for :math:`x\geq 0` but not so well for
    large negative :math:`x`.  Instead it is better to use the Taylor
    series to approximate :math:`y=\exp(-x)` and then use :math:`1/y`
    as the approximation.

    Modify `taylor_converge.f90` to create `taylor2.f90` that uses
    this technique to compute :math:`\exp(x)` when :math:`x` is
    negative.  It should produce more accurate values for :math:`x <
    0`.

 #. Modify the code further to create `taylor3.f90` so that the
    subroutine takes an array of `x` values as input and approximates
    the exponential function for each.  The subroutine should have the
    form::

        subroutine exptaylor(x,npts,nmax,y,nterms)

    where `npts` is the length of the arrays `x`, `y`, and `nterms`.

    Modify the main program so it calls this subroutine only once to
    compute all the values previously printed out, and print these out
    in the same format as before.  The output of running this code
    should look the same as that of Part 1.

 #. Consider the piecewise-defined function

    :math:`f_1(x) = \begin{cases} \cos(x), \quad & x< 0\\ \sin(x), \quad &
    0\leq x\leq 1\\ \sqrt{x}, \quad & 2\leq x<3\\ x-3, \quad &
    \text{otherwise} \end{cases}`

    The Fortran 90 file `$CLASSHG/homeworks/homework2/pwfcns.f90`
    contains a main program to print out the values of the function at
    a sequence of points, but is missing the function definition.
    Fill this in and make sure it works.

    Note: Be careful to check the function values where the function
    is discontinuous -- for example, :math:`f_1(0)` should be 0,
    not 1.  You can use the Python plotting script
    `$CLASSHG/homeworks/homework2/pwfcns_check.py` to compare your
    answers against the true solution; see the comments at the
    beginning of the script for more information.

 #. Make a directory $MYHG/homeworks/homework2 that contains your
    solutions, i.e the modified files:

     * taylor2.f90
     * taylor3.f90
     * pwfcns.f90

When you are done, don't forget to use the `hg add`, `hg commit`, and `hg
push` commands to push these to your bitbucket repository for grading.