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Ridder's Method: f(x) = 0 Brief Description of the method. Given the initial bracketing interval [$x_0$, $x_2$], containing one solution, and such that $f(x_0)$ and $f(x_2)$ have opposite signs, compute the midpoint $x_1 = (x_0 + x_2)/2$. Then define a new function $h(x) = f(x)e^{ax}$. Find the parameter $a$ that ensures $h(x_1) = \frac{1}{2}(h(x_0) + h(x_2))$. Then define $x_3$ as the $x$-intercept of the line passing through ($x_0$, $h(x_0)$) and ($x_2$, $h(x_2)$). Use $x_3$ as one of the endpoints of the next interval bracketing the solution. The other endpoint is $x_1$ if $f(x_1)f(x_3) < 0$. Otherwise, choose either $x_0$ or $x_2$ based on the requirement that the sign of $f(x)$ at the chosen point must be opposite to the sign of $f(x_3)$. Continue until the desired accuracy is reached. 1) Implement Ridders' method on a computer together with the standard bisection method for finding all solutions of $e^x - x^2 = 0$. 2) what is the order of convergence for ridder's method ?

          Ridder's Method: f(x) = 0
Brief Description of the method. Given the initial bracketing interval [$x_0$, $x_2$], containing one solution, and such that $f(x_0)$ and $f(x_2)$ have opposite signs, compute the midpoint $x_1 = (x_0 + x_2)/2$. Then define a new function $h(x) = f(x)e^{ax}$. Find the parameter $a$ that ensures $h(x_1) = \frac{1}{2}(h(x_0) + h(x_2))$. Then define $x_3$ as the $x$-intercept of the line passing through ($x_0$, $h(x_0)$) and ($x_2$, $h(x_2)$). Use $x_3$ as one of the endpoints of the next interval bracketing the solution. The other endpoint is $x_1$ if $f(x_1)f(x_3) < 0$. Otherwise, choose either $x_0$ or $x_2$ based on the requirement that the sign of $f(x)$ at the chosen point must be opposite to the sign of $f(x_3)$. Continue until the desired accuracy is reached.
1)
Implement Ridders' method on a computer together with the standard bisection method for finding all solutions of
$e^x - x^2 = 0$.
2) what is the order of convergence for ridder's method ?

Ridder's Method: f(x) = 0
Brief Description of the method. Given the initial bracketing interval [x0, x2], containing one solution, and such that f(x0) and f(x2) have opposite signs, compute the midpoint x1 = (x0 + x2)/2. Then define a new function h(x) = f(x)e^ax. Find the parameter a that ensures h(x1) = (1)/(2)(h(x0) + h(x2)). Then define x3 as the x-intercept of the line passing through (x0, h(x0)) and (x2, h(x2)). Use x3 as one of the endpoints of the next interval bracketing the solution. The other endpoint is x1 if f(x1)f(x3) < 0. Otherwise, choose either x0 or x2 based on the requirement that the sign of f(x) at the chosen point must be opposite to the sign of f(x3). Continue until the desired accuracy is reached.
1)
Implement Ridders' method on a computer together with the standard bisection method for finding all solutions of
e^x - x^2 = 0.
2) what is the order of convergence for ridder's method ?

Added by Domingo R.

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