The Secant Method is a numerical algorithm used to find the roots of a function. It is similar to the Newton-Raphson method, but instead of using the derivative of the function, it uses an approximation of the derivative based on two points. The algorithm starts with two initial points and then iteratively refines them to converge to the root of the function.

Here’s how the Secant Method works:

- Choose two initial points,
`x0`

, and`x1`

, such that`x0`

is not too close to`x1`

. - Calculate the function values at
`x0`

and`x1`

, i.e.,`f(x0)`

and`f(x1)`

. - Calculate the slope of the secant line passing through
`(x0, f(x0))`

and`(x1, f(x1))`

. - Calculate the x-intercept of the secant line.
- Set
`x0`

to`x1`

and`x1`

to the x-intercept of the secant line. - Repeat steps 2-5 until the difference between
`x1`

and`x0`

is less than a predetermined tolerance.

Here’s the Secant Method algorithm in mathematical notation:

Given a function `f(x)`

, initial points `x0`

and `x1`

, and a tolerance `tol,`

repeat the following steps until `|x1 - x0| < tol`

:

- Calculate the slope of the secant line passing through
`(x0, f(x0))`

and`(x1, f(x1)): m = (f(x1) - f(x0)) / (x1 - x0)`

- Calculate the x-intercept of the secant line:
`x2 = x1 - f(x1) / m`

- Set
`x0 = x1`

and`x1 = x2`

The value of `x1`

will be the root of the function.

Note that the Secant Method may not converge for some functions or initial points. It is also possible to encounter issues such as division by zero or numerical instability, so care must be taken when implementing the algorithm.

## Matlab code for secant method

Here’s how you can implement the Secant Method in MATLAB:

Step 1: Define the function You need to define the function for which you want to find the root. For example, let’s say you want to find the root of the function` f(x) = x^2 - 4`

. You can define the function in MATLAB as follows:

```
function y = f(x)
y = x^2 - 4;
end
```

Step 2: Define the initial points You need to define two initial points, `x0 `

and `x1`

, from which the algorithm will start. Make sure that `x0`

and `x1`

are not too close to each other. For example, let’s say you choose `x0 = 1`

and `x1 = 3`

.

```
x0 = 1;
x1 = 3;
```

Step 3: Define the tolerance You need to define a tolerance value that will determine the stopping criteria for the algorithm. For example, let’s say you choose a tolerance of `0.0001`

.

```
tol = 0.0001;
```

Step 4: Implement the Secant Method You can implement the Secant Method in MATLAB using a while loop. The loop will iterate until the difference between the two last approximations is smaller than the tolerance value.

```
while abs(x1 - x0) > tol
x2 = x1 - f(x1)*(x1 - x0)/(f(x1) - f(x0));
x0 = x1;
x1 = x2;
end
```

The value `x2`

will be the root of the function.

Step 5: Display the result Finally, you can display the result using the `disp()`

function.

```
disp(['The root is: ', num2str(x2)])
```

Here’s the complete code:

```
function y = f(x)
y = x^2 - 4;
end
x0 = 1;
x1 = 3;
tol = 0.0001;
while abs(x1 - x0) > tol
x2 = x1 - f(x1)*(x1 - x0)/(f(x1) - f(x0));
x0 = x1;
x1 = x2;
end
disp(['The root is: ', num2str(x2)])
```