CTK Insights

• Pages

19 Apr

The Secant Method

A straight line is uniquely determined by any two points. Let's there be given two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on a straight line (assumed neither vertical nor horizontal). We can find a generic third point $(x,y)$ by noticing similar triangles:

The resulting proportions have a simple geometric meaning: any two points on a straight line define the same slope. Therefore,

$\frac{y-y_1}{x-x_1} = \frac{y_{2}-y_1}{x_{2}-x_1}$.

This can be rewritten as

$y=y_1 + \frac{y_{2}-y_1}{x_{2}-x_1}(x-x_1)$,

or else, now expressing $x$ in terms of $y$, as

$x=x_1 + \frac{x_{2}-x_1}{y_{2}-y_1}(y-y_1)$.

If the line is not horizontal, it meets the $x$-axis in a point, say, $(t,0)$. In analytic geometry, $t$ is known as the $x$-intercept of the straight line. It can be determined from

$t=x_1 + \frac{x_{2}-x_1}{y_{2}-y_1}(0-y_1)=\frac{x_{1}y_{2}-x_{2}y_{1}}{y_{2}-y_{1}}$.

This gives a solution to a linear equation that describes a straight line through two given points.

The formula bears comparison with the one used in the Regula Falsi. The two are practically the same: given two unsuccessful attempts to guess a solution to a problem, the solution can be found as the $x$-intercept of a straight line determined by the two guesses. The Regula Falsi works under the assumption that the solution of a problem depends linearly on the data. This was certainly so for the sample word problem which was translated into a set of three linear equations in three variables.

The Regula False could be used even where the dependency of the result on the data is not linear, in which case it will only produce an approximate solution:

Assume we want to solve an equation $f(x)=0$. Find two points $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$, $x_{1}\lt (x_{2}$, on the graph of function $y = f(x)$, and see where the secant - the straight line through the two points - crosses the $x$-axis. There is a good chance that that value would serve a better guess for the solution of $f(x)=0$. There is something that can be claimed with certainty.

Assume $f$ is a well-behaved function and the two guesses $x_{1}$ and $x_{2}$ are so made as to have $f(x_{1})$ and $f(x_{2})$ of different signs. By the Bolzano theorem, equation $f(x)=0$ has a solution on the interval $(x_{1},x_{2})$ and so does the equation of the secant. If the latter is $t$, the former is located in one of the smaller intervals: $(x_{1},t)$ or $(t,x_{2})$. Which one? (Of course it could happen that we got an exact match and $f(t)=0$; but this is unlikely. Suppose that this did not happen.) The Bolzano theorem shows a way to choose. One of $f(x_{1})$ or $f(x_{2})$ has a sign different from that of $f(t)$. If, for example, $f(x_{1}f(t)\lt 0$ then we may be certain that a root of $f(x)=0$ lies in the interval $(x_{1},t)$.

To summarize,

1. We started with an equation $f(x)=0$ that we could not solve exactly.
2. We made two guesses $x_{1}$ and $x_{2}$ such that $f(x_{1})f(x_{2})\lt 0$.
3. The Bolzano theorem provided the assurance that in the interval $(x_{1},x_{2})$ there is a solution to $f(x)=0$.
4. Using Regula Falsi we were able to shorten the interval that contained a solution.

The Secant Method takes it from here: starting with the new interval use Regula Falsi to find even shorter interval and continue iteratively until the interval that contains a solution is short enough to insure acceptable accuracy.

8 − = zero