`Newton's method <https://en.wikipedia.org/wiki/Newton%27s_method>`__
=====================================================================

Let's use the Newton-Raphson method for finding the root of an equation
to write a function that can compute the square root of a number.

Cf. `"Why Functional Programming Matters" by John
Hughes <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>`__

.. code:: ipython2

    from notebook_preamble import J, V, define

A Generator for Approximations
------------------------------

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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This is the equation for computing the next approximate value of the
square root:

:math:`a_{i+1} = \frac{(a_i+\frac{n}{a_i})}{2}`

::

    a n over / + 2 /
    a n a    / + 2 /
    a n/a      + 2 /
    a+n/a        2 /
    (a+n/a)/2

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:

::

    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

.. code:: ipython2

    define('gsra == 1 swap [over / + 2 /] cons [dup] swoncat make_generator')

.. code:: ipython2

    J('23 gsra')


.. parsed-literal::

    [1 [dup 23 over / + 2 /] codireco]


Let's drive the generator a few time (with the ``x`` combinator) and
square the approximation to see how well it works...

.. code:: ipython2

    J('23 gsra 6 [x popd] times first sqr')


.. parsed-literal::

    23.0000000001585


Finding Consecutive Approximations within a Tolerance
-----------------------------------------------------

From `"Why Functional Programming Matters" by John
Hughes <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>`__:

    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 note that by “list” he means a lazily-evaluated list.)

Using the *output* ``[a G]`` of the above generator for square root
approximations, and further assuming that the first term a has been
generated already and epsilon ε is handy on the stack...

::

       a [b G] ε within
    ---------------------- a b - abs ε <=
          b


       a [b G] ε within
    ---------------------- a b - abs ε >
       b [c G] ε within

Predicate
~~~~~~~~~

::

    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)

.. code:: ipython2

    define('_within_P == [first - abs] dip <=')

Base-Case
~~~~~~~~~

::

    a [b G] ε roll< popop first
      [b G] ε a     popop first
      [b G]               first
       b

.. code:: ipython2

    define('_within_B == roll< popop first')

Recur
~~~~~

::

    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.)

Pretty straightforward:

::

    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

    b [c G] ε within

.. code:: ipython2

    define('_within_R == [popd x] dip')

Setting up
~~~~~~~~~~

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

    define('within == x 0.000000001 [_within_P] [_within_B] [_within_R] primrec')
    define('sqrt == gsra within')

Try it out...

.. code:: ipython2

    J('36 sqrt')


.. parsed-literal::

    6.0


.. code:: ipython2

    J('23 sqrt')


.. parsed-literal::

    4.795831523312719


Check it.

.. code:: ipython2

    4.795831523312719**2




.. parsed-literal::

    22.999999999999996



.. code:: ipython2

    from math import sqrt
    
    sqrt(23)




.. parsed-literal::

    4.795831523312719


