# coding: utf-8

# # CSC338. Homework 1
# 
# Due Date: Wednesday January 15, 9pm
# 
# Please see the guidelines at https://www.cs.toronto.edu/~lczhang/338/homework.html
# 
# ### What to Hand In
# 
# Please hand in 2 files:
# 
# - Python File containing all your code, named `hw1.py`.
# - PDF file named `hw1_written.pdf` containing your solutions to the written parts of the assignment. Your solution can be hand-written, but must be legible. Graders may deduct marks for illegible or poorly presented solutions.
# 
# If you are using Jupyter Notebook to complete the work, your notebook can be exported as a .py file (File -> Download As -> Python). Your code will be auto-graded using Python 3.6, so please make sure that your code runs. There will be a 20% penalty if you need a remark due to small issues that renders your code untestable.
# 
# **Make sure to remove or comment out all matplotlib or other expensive code before submitting your code!**
# 
# Submit the assignment on **MarkUs** by 9pm on the due date. See the syllabus for the course policy regarding late assignments. All assignments must be done individually.

# In[ ]:


import math
import numpy as np


# ## Question 1.
# 
# For parts (a) and (b),
# consider the problem of evaluating the function $g(x) = x^2 + x - 4$.
# Suppose there is a small error, $h$ in the value of $x$.
# 
# ### Part (a) -- 6pt
# 
# What is the absolute error and relative error in computing $g(x)$?
# Implement function and `g_abs_err(x, h)` and `g_rel_err(x, h)` to
# compute those quantities.

# In[ ]:


def g_abs_err(x, h):
    """Returns the absolute error of computing `g` at `x` if `x` is
    perturbed by a small value `h`.
    """
    return None

def g_rel_err(x, h):
    """Returns the relative error of computing `g` at `x` if `x` is
    perturbed by a small value `h`.
    """
    return None


# ### Part (b) -- 3pt
# 
# Estimate the condition number for the problem. Simplify this answer.
# For what values of $x$ is this problem well-conditioned?
# Include your solution in your PDF file.
# 
# ### Part (c) -- 2pt
# 
# For parts (c) and (d), consider the problem of finding a root of
# the function $g(x) = x^2 + x + c$ by using the Quadratic Formula and
# taking the positive squareroot.
# Suppose there is a small error, $h$ in the value of $c$.
# 
# For what values of $c$ is this problem well-posed?
# Include your solution in your PDF file.
# 
# ### Part (d) -- 6pt
# 
# What is the absolute error and relative error in computing $g(x)$?
# Implement function and `g_root_abs_err(c, h)` and `g_root_rel_err(c, h)` to
# compute those quantities.

# In[ ]:


def g_root_abs_err(c, h):
    """Returns the absolute error of finding the (most) positive root of `g` when
    `c` is perturbed by a small value `h`.
    """
    return None

def g_root_rel_err(c, h):
    """Returns the relative error of finding the (most) positive root of `g` when
    `c` is perturbed by a small value `h`.
    """
    return None


# ## Question 2.
# 
# Consider the function, which is also implemented below. You can run the `plot_f` python function
# to see what the graph of $f$ looks like.
# 
# $f(x) = \frac{x - sin(x)}{x^3}$

# In[ ]:


def f(x):
    return (x - math.sin(x)) / math.pow(x, 3)

def plot_f():
    import matplotlib.pyplot as plt
    xs = [x for x in np.arange(-3.0, 3.0, 0.05) if abs(x) > 0.05]
    ys = [f(x) for x in xs]
    plt.plot(xs, ys, 'bo')
    
# plot_f() # please comment this out before submitting, or your code might be untestable


# ### Part (a) -- 2pt
# 
# What is $f(0.00000001)$? Save the results in the variable `q2_est`.
# 
# Given that $f$ is continuous except at $x = 0$,
# what should $f(0.00000001)$ be? Save the results in the variable `q2_true`.

# In[ ]:


q2_est = None
q2_true = None


# ### Part (b) -- 2pt
# 
# Why does the Python statement compute such inaccurate values of $f(0.00000001)$?
# Include your answer in your PDF File.
# 
# ### Part (c) -- 5pt
# 
# Define a Python function `f2(x)` that uses a different algorithm to compute
# more accurate values of $f(x)$ for $0 < x \le \frac{\pi}{2}$.
# More specifically, the relative error should be no more than 1% for those values of $x$.

# In[ ]:


def f2(x):
    return None


# ## Question 3. -- 4pt
# 
# The sine function is given by the infinite series
# 
# $\sin(x) = x - \frac{x^3}{3!}  + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$ 
# 
# Compute the absolute forward and backwards error if we approximate the 
# sine function by the first two terms of the series for elements of the
# array `xs` below.
# 
# Save your solution in the array `q3_forward` and `q3_backward`, 
# so that `q3_forward[i]` and `q3_backward[i]` are the absolute forward
# and backward errors corresponding to the input `xs[i]`.
# 
# You may find the functions `math.sin` and `math.asin` helpful.
# You may assume that the true value of $\sin(x)$ can be computed using 
# the function `math.sin`.
# 
# Update (Jan 9th): If the backward error does not exist, please enter "DNE".

# In[ ]:


xs = [0.1, 0.5, 1.0, 3.0]

q3_forward  = [None, None, None, None]
q3_backward = [None, None, None, None]

# math.sin(0.2)
# math.asin(0.2) # arcsin