Copyright (c) Edward Kmett 2010-2015 BSD3 ekmett@gmail.com experimental GHC only None Haskell2010

Contents

Description

Root finding using Halley's rational method (the second in the class of Householder methods). Assumes the function is three times continuously differentiable and converges cubically when progress can be made.

Synopsis

findZero :: (Fractional a, Eq a) => (Tower a -> Tower a) -> a -> [a] Source

The `findZero` function finds a zero of a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.

Examples:

````>>> ````take 10 \$ findZero (\x->x^2-4) 1
```[1.0,1.8571428571428572,1.9997967892704736,1.9999999999994755,2.0]
```
````>>> ````last \$ take 10 \$ findZero ((+1).(^2)) (1 :+ 1)
```0.0 :+ 1.0
```

inverse :: (Fractional a, Eq a) => (Tower a -> Tower a) -> a -> a -> [a] Source

The `inverse` function inverts a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.

Note: the `take 10 \$ inverse sqrt 1 (sqrt 10)` example that works for Newton's method fails with Halley's method because the preconditions do not hold!

fixedPoint :: (Fractional a, Eq a) => (Tower a -> Tower a) -> a -> [a] Source

The `fixedPoint` function find a fixedpoint of a scalar function using Halley's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)

If the stream becomes constant ("it converges"), no further elements are returned.

````>>> ````last \$ take 10 \$ fixedPoint cos 1
```0.7390851332151607
```

extremum :: (Fractional a, Eq a) => (On (Forward (Tower a)) -> On (Forward (Tower a))) -> a -> [a] Source

The `extremum` function finds an extremum of a scalar function using Halley's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.

````>>> ````take 10 \$ extremum cos 1
```[1.0,0.29616942658570555,4.59979519460002e-3,1.6220740159042513e-8,0.0]
```