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- MTH 373/573: Scientific Computing Homework 1

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Problem 1: Floating point hiccups

(50 points)

For computing the midpoint m of an interval [a, b], which of the following two formulas is

preferable in floating-point arithmetic?

(i) m = (a + b)/2.0

(ii) m = a + (b − a)/2.0

Why? When? (Hint: Devise examples for which the “midpoint” given by the formula lies

outside the interval [a, b].)

In addition, provide the following in your writeup.

(a) Specify the floating point system you are using.

(b) Specify values of a and b in floating-point representation (that is, if you choose a base-2

system, your numbers should be binary) within your floating point system.

(c) Write down all intermediate results for both cases in your chosen floating-point representation.

(d) Point out the steps where the problem occurs.

Problem 2: Bessel recurrence in floating point

(50 + 50 + 25 + 25 = 150 points)

Bessel functions Jn(·) obey the recurrence relation

Jn+1(z) = (2n/z)Jn(z) − Jn−1(z).

This means that, given J0(z) and J1(z) for a fixed z ∈ R \ {0}, Jn(z) for integer n ≥ 2 can

be computed – at least in theory.

You can evaluate Bessel functions using scipy.special.jv(n,z).

(a) Write a Python program that tests the accuracy to which the values returned from scipy

obey the Bessel recurrence relation. Print the left and right hand side values for each

n. Also print the relative (to left hand side) error between the left and right hand sides

for each n = 2, 3, . . . , 50. For your experiments, fix z = 20. (This should yield roughly

machine precision in each case.)

(b) Write a Python program that uses the values for J0(z) and J1(z) (as obtained from

scipy) and the recurrence relation to compute the values of J2(z), . . . , J50(z). Print

these recurrence computed values for each n, and also print the error relative to the

value returned by scipy as the true value. Again, for your experiments, fix z = 20.

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(c) You should find that the results from the recurrence relation rapidly start losing precision at around n = 30. Provide a reason for this loss of precision.

(d) Observe that the recurrence relation can be rearranged to compute Jn−1 from Jn+1 as

follows:

Jn−1(z) = (2n/z)Jn(z) − Jn+1(z).

Do you believe using this modified recurrence relation to compute J0, . . . , J48 from J49

and J50 will encounter loss of precision? Why or why not? (Run the experiment if you

are not sure!)

Problem 3: Floating point systems

(50 points)

Consider a floating point system with parameters (β, p, L, U) = (2, 4, −3, 3). Assume that

it is an unnormalized system. That is, a typical number is b0 · b1b2b3 × 2

e where b0 can be 0

or 1. (Of course the other bits b1, b2 and b3 are also 0 or 1.)

Give examples (one in each category) of nonzero numbers that have unique, two, three, and

four representations, respectively. Write the numbers in the forms b0 · b1b2b3 × 2

e

. Write

down all the representations in each category.

Problem 4: Basic linear algebra

(50 points)

Consider a 4×4 matrix B. We want to perform the following 6 operations on B in sequence:

1. Swap rows 1 and 4

2. Add 2 times column 3 to column 1

3. Swap columns 2 and 3

4. Add 5 times row 2 to row 4

5. Delete column 2

6. Delete row 3

(a) Express these operations as a sequence of matrix multiplications. Your final answer

should be written as a product of 7 matrices, one of which will be B. Do not carry out

the actual matrix product in this part.

(b) By doing the appropriate matrix multiplications express your answer as the matrix product ABC. That is, what A and C will yield an ABC which is identical to the matrix

obtained by the sequence of operations listed above?

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General Submission Guidelines

• This assignment should be answered individually.

• You will be penalized for copying or any plagiarism with an automatic zero.

• Start working on your solutions early and don’t wait until close to due date.

• The points for each problem is roughly indicative of effort needed for answering that

problem. For instance, 50 points can be taken to mean about 50 minutes of focussed

work. This can include reading and thinking if the problem is theoretical. For machine problems, this will include time spent in coding, debugging and producing output assuming some Python competency. Your mileage may vary!

• IIIT-Delhi academic policies on honesty and integrity apply to all HWs. This includes not copying from one another, from the internet, a book, or any other online

or offline source. A repeat offense will be reported to academic administration.

• You can ask questions about concepts, ideas and basics of coding on the Piazza discussion group for the class. You can also ask the instructor questions during office

hours. You should however provide your own solutions.

• Any technical question pertaining to the HW emailed to me will be ignored.

• If you discuss or read secondary sources (other than class notes), please list all your

discussion partners and/or secondary sources in your writeup. Failure to do so will

constitute violation of honor code.

• Each problem should have a clear and concise write-up.

• Please clearly show all steps involved in theoretical problems.

• The answer to all theoretical problems, and output, figures and analyses for computer

problems should be submitted in a PDF file or turned in handwritten.

• Handwritten submissions can be turned in during class, in person in my office or

under my office door before the submission deadline. I will be present in my office

during and for 5 minutes after every submission deadline to collect submissions.

• No late submission will be allowed unless I explicitly permit it.

• All Python files for computer problems should be submitted through Github Classroom.

• However, if your Python code generates an output figure or table, please provide all

such results in a single PDF file along with your code submission.

• You will need to write a separate Python code for each problem and sometimes for

each subproblem as well. You should name each such file as problem_n.py where

n is the problem number. For example, your files could be named problem_3a.py,

problem_3b.py and problem_4.py.

• You should import Python modules as follows:

from __future__ import division

import numpy as np

import scipy as sp

import matplotlib.pyplot as plt

import numpy.linalg as npla

import scipy.linalg as spla

Every code you write will have one or more of these import statements.

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