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jverzani
2024-05-22 07:55:20 -04:00
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quarto/derivatives/more_zeros.qmd
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@@ -111,7 +111,7 @@ f0,f1 = f1, sin(x1)
x1,f1
```
Like Newton's method, the secant method coverges quickly for this problem (though its rate is less than the quadratic rate of Newton's method).
Like Newton's method, the secant method converges quickly for this problem (though its rate is less than the quadratic rate of Newton's method).
This method is included in `Roots` as `Secant()` (or `Order1()`):
@@ -210,7 +210,7 @@ Though the above can be simplified quite a bit when computed by hand, here we si
An inverse quadratic step is utilized by Brent's method, as possible, to yield a rapidly convergent bracketing algorithm implemented as a default zero finder in many software languages. `Julia`'s `Roots` package implements the method in `Roots.Brent()`. An inverse cubic interpolation is utilized by [Alefeld, Potra, and Shi](https://dl.acm.org/doi/10.1145/210089.210111) which gives an asymptotically even more rapidly convergent algorithm than Brent's (implemented in `Roots.AlefeldPotraShi()` and also `Roots.A42()`). This is used as a finishing step in many cases by the default hybrid `Order0()` method of `find_zero`.
In a bracketing algorithm, the next step should reduce the size of the bracket, so the next iterate should be inside the current bracket. However, quadratic convergence does not guarantee this to happen. As such, sometimes a subsitute method must be chosen.
In a bracketing algorithm, the next step should reduce the size of the bracket, so the next iterate should be inside the current bracket. However, quadratic convergence does not guarantee this to happen. As such, sometimes a substitute method must be chosen.
[Chandrapatla's](https://www.google.com/books/edition/Computational_Physics/cC-8BAAAQBAJ?hl=en&gbpv=1&pg=PA95&printsec=frontcover) method, is a bracketing method utilizing an inverse quadratic step as the centerpiece. The key insight is the test to choose between this inverse quadratic step and a bisection step. This is done in the following based on values of $\xi$ and $\Phi$ defined within: