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docs/sphinx_docs/notebooks/Newton-Raphson.rst
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*********************************************************************
`Newton's method <https://en.wikipedia.org/wiki/Newton%27s_method>`__
=====================================================================
*********************************************************************
Newton-Raphson for finding the root of an equation.
@@ -11,193 +11,194 @@ Newton-Raphson for finding the root of an equation.
Cf. `"Why Functional Programming Matters" by John
Hughes <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>`__
Finding the Square-Root of a Number
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Let's define a function that computes this equation:
A Generator for Approximations
==============================
In :doc:`Generator Programs` we derive a function (called ``make_generator`` in the dictionary) that accepts an initial value and a quoted program and returns a new quoted program that, when driven by the ``x`` combinator (:py:func:`joy.library.x`), acts like a lazy stream.
To make a generator that generates successive approximations let's start by assuming an initial approximation and then derive the function that computes the next approximation::
a F
---------
a'
A Function to Compute the Next Approximation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Looking at the equation again:
:math:`a_{i+1} = \frac{(a_i+\frac{n}{a_i})}{2}`
::
n a Q
---------------
(a+n/a)/2
n a tuck / + 2 /
a n over / + 2 /
a n a / + 2 /
a n/a + 2 /
a+n/a 2 /
(a+n/a)/2
We want it to leave n but replace a, so we execute it with ``unary``:
The function we want has the argument ``n`` in it::
F == n over / + 2 /
Make it into a Generator
^^^^^^^^^^^^^^^^^^^^^^^^
Our generator would be created by::
a [dup F] make_generator
With ``n`` as part of the function ``F``, but ``n`` is the input to the ``sqrt`` function we're writing. If we let 1 be the initial approximation::
1 n 1 / + 2 /
1 n/1 + 2 /
1 n + 2 /
n+1 2 /
(n+1)/2
The generator can be written as::
1 swap [over / + 2 /] cons [dup] swoncat make_generator
Example::
23 1 swap [over / + 2 /] cons [dup] swoncat make_generator
1 23 [over / + 2 /] cons [dup] swoncat make_generator
1 [23 over / + 2 /] [dup] swoncat make_generator
1 [dup 23 over / + 2 /] make_generator
.
.
.
[1 swap [dup 23 over / + 2 /] direco]
A Generator of Square Root Approximations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
::
Q == [tuck / + 2 /] unary
gsra == 1 swap [over / + 2 /] cons [dup] swoncat make_generator
.. code:: ipython2
define('Q == [tuck / + 2 /] unary')
Finding Consecutive Approximations ``within`` a Tolerance
=========================================================
Compute the Error
^^^^^^^^^^^^^^^^^
The remainder of a square root finder is a function *within*, which takes a tolerance and a list of approximations and looks down the list for two successive approximations that differ by no more than the given tolerance.
And a function to compute the error:
From `"Why Functional Programming Matters" by John
Hughes <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>`__
(And note that by "list" he means a lazily-evaluated list.)
Using the *output* ``[a G]`` of the above :doc:`generator <Generator Programs>` for square root approximations, and further assuming that the first term ``a`` has been generated already and epsilon ``ε`` is handy on the stack...
::
n a sqr - abs
|n-a**2|
This should be ``nullary`` so as to leave both n and a on the stack
below the error.
a [b G] ε within
---------------------- a b - abs ε <=
b
::
err == [sqr - abs] nullary
a [b G] ε within
---------------------- a b - abs ε >
.
[b G] x ε ...
b [c G] ε ...
.
----------------------
b [c G] ε within
.. code:: ipython2
define('err == [sqr - abs] nullary')
``square-root``
^^^^^^^^^^^^^^^
Now we can define a recursive program that expects a number ``n``, an
initial estimate ``a``, and an epsilon value ``ε``, and that leaves on
the stack the square root of ``n`` to within the precision of the
epsilon value. (Later on we'll refine it to generate the initial
estimate and hard-code an epsilon value.)
Predicate
^^^^^^^^^^^^^
::
n a ε square-root
-----------------
√n
a [b G] ε [first - abs] dip <=
a [b G] first - abs ε <=
a b - abs ε <=
a-b abs ε <=
abs(a-b) ε <=
(abs(a-b)<=ε)
If we apply the two functions ``Q`` and ``err`` defined above we get the
next approximation and the error on the stack below the epsilon.
::
n a ε [Q err] dip
n a Q err ε
n a' err ε
n a' e ε
P == [first - abs] dip <=
Let's define a recursive function ``K`` from here.
Base-Case
^^^^^^^^^^^^^
::
n a' e ε K
K == [P] [E] [R0] [R1] genrec
Base-case
~~~~~~~~~
The predicate and the base case are obvious:
a [b G] ε roll< popop first
[b G] ε a popop first
[b G] first
b
::
K == [<] [popop popd] [R0] [R1] genrec
B == roll< popop first
::
n a' e ε popop popd
n a' popd
a'
Recur
~~~~~~~~~~
The recursive branch is pretty easy. Discard the error and recur.
^^^^^^^^^^^^^
::
K == [<] [popop popd] [R0] [R1] genrec
K == [<] [popop popd] [R0 [K] R1] ifte
a [b G] ε R0 [within] R1
1. Discard ``a``.
2. Use ``x`` combinator to generate next term from ``G``.
3. Run ``within`` with ``i`` (it is a ``primrec`` function.)
::
n a' e ε R0 [K] R1
n a' e ε popd [Q err] dip [K] i
n a' ε [Q err] dip [K] i
n a' Q err ε [K] i
n a'' e ε K
a [b G] ε R0 [within] R1
a [b G] ε [popd x] dip [within] i
a [b G] popd x ε [within] i
[b G] x ε [within] i
b [c G] ε [within] i
b [c G] ε within
This fragment alone is pretty useful. (``R1`` is ``i`` so this is a ``primrec`` "primitive recursive" function.)
.. code:: ipython2
define('K == [<] [popop popd] [popd [Q err] dip] primrec')
.. code:: ipython2
J('25 10 0.001 dup K')
.. parsed-literal::
5.000000232305737
.. code:: ipython2
J('25 10 0.000001 dup K')
.. parsed-literal::
5.000000000000005
Initial Approximation and Epsilon
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So now all we need is a way to generate an initial approximation and an
epsilon value:
b [c G] ε within
::
square-root == dup 3 / 0.000001 dup K
.. code:: ipython2
define('square-root == dup 3 / 0.000001 dup K')
Examples
~~~~~~~~~~
.. code:: ipython2
J('36 square-root')
R0 == [popd x] dip
.. parsed-literal::
Setting up
^^^^^^^^^^
6.000000000000007
The recursive function we have defined so far needs a slight preamble: ``x`` to prime the generator and the epsilon value to use::
[a G] x ε ...
a [b G] ε ...
.. code:: ipython2
``within``
^^^^^^^^^^
J('4895048365636 square-root')
Giving us the following definitions::
_within_P == [first - abs] dip <=
_within_B == roll< popop first
_within_R == [popd x] dip
within == x ε [_within_P] [_within_B] [_within_R] primrec
.. parsed-literal::
Finding Square Roots
====================
2212475.6192184356
.. code:: ipython2
2212475.6192184356 * 2212475.6192184356
.. parsed-literal::
4895048365636.0
::
sqrt == gsra within