 High Performance Scientific Computing   AMath 483/583 Class Notes   Spring Quarter, 2013

# Homework 2 ¶

Warning

Assignment modified in Part 9.

Due Thursday, April 24, 2014, by 11:00pm PDT. See below for how to submit.

The goals of this homework are to:

• Get some more experience with Python and numpy.
• Get some experience with Fortran code and Makefiles.

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

You should be pleased to find that you are now starting to follow many of these best practices, but there are many good tips in the paper that have not been covered in lectures.

There will be a quiz on this paper to complete as part of the homework assignment. This quiz can be found on the Canvas page Homework 2 quiz.

2. Survey assignment

Please take this survey (worth 5 points) to give us some feedback on how the class is going so far.

3. Programming assignments

You should create a new subdirectory homework2 (of the same private repository you have used for submitting previous homeworks). Develop your code for the problems below in this directory and feel free to commit as often as you like, it will help you recover from blunders.

Note that to grade this homework, we will try using the functions you write with different input values than the ones used in the examples below, so you might want to test your functions for other reasonable input values.

4. Recall that computer hardware can only do basic arithmetic, so if you want to evaluate some special function such as square root or a trigonometric function, some algorithm must be implemented by someone to compute an approximation for arbitrary input values. To approximate $$\sqrt{y}$$, we saw that one approach is to use Newton’s method to approximate a zero of the function $$x^2 - y = 0$$, for example. (See also Special functions for more about this.)

Another approach is to use a Taylor series approximation, see e.g. the Wikipedia Taylor series page for a review.

Recall that the Taylor series for the exponential function (expanded about $$x_0 = 0$$) is given by

$$\exp(x) = \sum_{j=0}^\infty \frac{x^j}{j!} = 1 + x + \frac 1 2 x ^2 + \frac 1 6 x^3 + \cdots$$

with the convention that $$x^0 = 0! = 1$$.

Create a Python file taylor.py in the homework2 directory of your repository and write a function exp1(x,n) in this file that returns an approximation to $$\exp(x)$$ based on the Taylor series of degree n. (Note that this degree n polynomial is obtained by truncating the infinite series after n+1 terms.)

If you have done this properly, then if you start the Python or Ipython shell in the same directory, you should be able to do, for example:

>>> import taylor
>>> taylor.exp1(1.,0)
1.0
>>> taylor.exp1(1.,1)
2.0
>>> taylor.exp1(1.,2)
2.5
>>> taylor.exp1(1.,20)
2.7182818284590455


Hint: Note that term of degree j in the series can be computed from the previous term by multiplying by x and dividing by j. If you use this trick you won’t need the factorial function math.factorial. See Fortran examples: Taylor series and $UWHPSC/codes/fortran/taylor.f90 for an example of this same idea used in a Fortran version. 5. Add some debugging statements to your function, with an optional argument debug with the default value False (so the examples above still give the same output) but so that setting debug=True causes output similar to this: >>> taylor.exp1(1.,5,debug=True) j = 1, term = 1.000000000000000e+00 partial_sum updated from 1.000000000000000e+00 to 2.000000000000000e+00 j = 2, term = 5.000000000000000e-01 partial_sum updated from 2.000000000000000e+00 to 2.500000000000000e+00 j = 3, term = 1.666666666666667e-01 partial_sum updated from 2.500000000000000e+00 to 2.666666666666667e+00 j = 4, term = 4.166666666666666e-02 partial_sum updated from 2.666666666666667e+00 to 2.708333333333333e+00 j = 5, term = 8.333333333333333e-03 partial_sum updated from 2.708333333333333e+00 to 2.716666666666666e+00 2.7166666666666663  You probably won’t need these statements for this function, but similar statements might be useful in the next part. 6. Create a Python function sin1(x,n) (in the same file taylor.py as the function exp) that approximates the sine function at a point x by evaluating the Taylor series approximation of degree n. Use the Taylor series expansion about $$x_0=0$$, also known as the Maclaurin series: $$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$ Note that the degree 5 and 6 approximations only have three nonzero terms, the degree 7 and 8 approximations have four nonzero terms, etc. You should get results like: >>> taylor.sin1(pi/2, 2) 1.5707963267948966 >>> taylor.sin1(pi/2, 3) 0.9248322292886504 >>> taylor.sin1(pi/2, 4) 0.9248322292886504 >>> taylor.sin1(pi/2, 5) 1.0045248555348174  Add a debug option as in exp1. Hint: You might find it convenient to have a variable term that is updated as for the exponential function but then is multiplied by s before adding in to the partial sum, where s takes the appropriate value $$+1,~-1,$$ or 0 depending on j. 7. See what happens if you call your function exp1 or sin1 with negative values of n, or with non-integer real numbers. Add some input-checking to each function so that a non-negative integer value of n is required. If an invalid value is detected, print an error message and return the special value numpy.nan (“not a number”, similar to the Matlab NaN). For example: >>> taylor.exp1(1., -3) *** Invalid input -- n must be non-negative integer nan  8. The code$UWHPSC/codes/fortran/taylor.f90 contains a main program and subroutine for approximating exp(x) by a Taylor series. Split this code up into two separate files taylor_main.f90 and exptaylor.f90 and add a Makefile based on $UWHPSC/codes/fortran/multifile1/Makefile5 so that you can do: $ make exp_output.txt


to create a file exp_output.txt containing:

x =    1.0000000000000000
n =           20
exp_true  =    2.7182818284590451
exptaylor =    2.7182818284590455
error     =   4.44089209850062616E-016


Note: Also modify the main program so it prints the value of n as shown above.

9. Only 583 students need to do this part

(483 students are encouraged to do these parts too, but they will not count towards the score – the parts will be weighted differently for 583 students. Note that undergrads who registered for 583A will be treated as 483 students.)

Add another file sinetaylor.f90 that computes the approximation to the sine function, as you did in the Python version.

Warning

Assignment modified here:

Also create a new main program taylor_main2.f90 that calls this subroutine. Add these to the same Makefile so that

$make sine_output.txt  gives something sensible and make exp_output.txt still gives the previous results. 10. Only 583 students need to do this part The Wikipedia Taylor series page shows a nice plot of Taylor series approximations to the sine function for different orders. The gnuplot commands that created this plot can be found at http://en.wikipedia.org/wiki/File:Sintay_SVG.svg. Write a script plot_taylor.py to produce a similar plot that shows the sine function and approximations for n = 1,3,5,7 over the same range of x values. You don’t need to try to match the colors or add the grid lines. ## To submit¶ • At the end, you should have committed the following files to your repository: •$MYHPSC/homework2/taylor.py
• $MYHPSC/homework2/taylor_main.f90 •$MYHPSC/homework2/exptaylor.f90
• $MYHPSC/homework2/Makefile 583 students should also have the files •$MYHPSC/homework2/taylor_main2.f90

• $MYHPSC/homework2/sinetaylor.f90 •$MYHPSC/homework2/plot_taylor.py

You do not need to submit the png file of the figure this creates.

Make sure you push to bitbucket after committing.

• Submit the commit number that you want graded by following the link provided on the Canvas page for Homework 2. If you submit the wrong thing or make further changes to your work before the due date, you can simply resubmit new information at the same link.