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## Introduction Part II and Solving Equations## 1 Numerical Solution to Quadratic EquationsRecall from last lecture that we wanted to find a numerical solution to a quadratic equation like
One obvious attempt at a solution would use the familiar quadratic formula:
But, while it is trivial for a computer to perform additions, subtractions,
multiplications, and ## 2 Finding Square Roots and Solving Quadratic Equations
As we discussed last time, there is a simple scheme for approximating square
roots to any given
using an iterative method that allows us to control the precision of the
solution. The idea is that
An interesting property of this algorithm is that the precision of the output
(x
Now that we have a scheme for solving a restricted kind of quadratic
equation, can we use the
We simply need to add another term to the denominator of the formula:
We can use this new formula iteratively to arrive at numerical solutions of
the quadratic equation So, it should now be clear that algorithms that are good for finding
numerical solutions are ## 3 Calculating Definite IntegralsAnother good example of the difference between numerical computation and
analytic solutions is
Figure 1: The Definite Integral of f(t) over [a, b] What is the best way to calculate the definite integral? Our experience in
math classes suggests
But this brings up two important questions: 2. How difficult is it to evaluate the anti-derivative function at a and b? The answer to question 1 may seem simple - we all learned to calculate
anti-derivatives in
So is there a method for calculating the definite integral that uses only
information we already
Figure 2: Approximating the Definite Integral Using Subintervals We have reduced the problem to estimating the definite integral over smaller
subintervals. Here
In the rectangle rule, we calculate the area of the rectangle with width h
and height determined
In the midpoint rule, we calculate the area of the rectangle with width h and
height determined
Figure 3: The Rectangle and Midpoint Rules These two methods are depicted in Figure 3. Using these methods, we can
calculate the definite
In general, we evaluate numeric algorithms using 3 major criteria: Cost - The amount of time (usually calculated as the number of operations)
the computation will Speed - How quickly the algorithm approaches our desired precision. If we
define the error as the Robustness - How the correctness of the algorithm is affected by different
types of functions and different
Applying these criteria to the midpoint and rectangle rules:
Rectangle Rule -For the rectangle rule, the error approaches 0 in direct
proportion to the
Through this evaluation, we have shown that, in general, the midpoint rule is
a far better choice ## 4 Loss of SignificanceOne final way that numerical computation is different than the way you have
done mathematics to π = 3.1415926535 If we subtract π from b, we will get the answer 0.0000000036, which has only
two significant Example 4.1 (Calculating Derivatives). Given the function f(t) = sin t, what is f′(2)? We all know from calculus that the derivative of sin t is cos t, so one could
calculate cos(2) in Instead, recall that the value of the derivative of a function f at a fixed
point t
Just as with the definite integral, we could approximate the derivative at t Preventing loss of significance in our calculations will be an important part of this course.
Figure 4: Tangent and Secant Lines on f(t) = sin t ## 5 Solving EquationsNow that we have some idea of what algorithms for computation are, we will
discuss algorithms
Figure 5: Graphical Solution for x^3 = sin x Consider the following equation, with solution depicted in Figure 5.
There is no analytic solution to this equation. Instead, we will present an
iterative method for
As in our solution to finding square roots, we would like to find a function
g, such that if we input Clearly, one property of g is that g(r) = r, where r is the real solution to
Equation 13. A |