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docs/Newton-Raphson.rst
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`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
Cf. `"Why Functional Programming Matters" by John
Hughes <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>`__
A Generator for Approximations
------------------------------
:math:`a_{i+1} = \frac{(a_i+\frac{n}{a_i})}{2}`
Let's define a function that computes the above equation:
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:
::
n a Q
---------------
(a+n/a)/2
a F
---------
a'
n a tuck / + 2 /
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
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:
::
Q == [tuck / + 2 /] unary
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('Q == [tuck / + 2 /] unary')
And a function to compute the error:
::
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.
::
err == [sqr - abs] nullary
define('codireco == cons dip rest cons')
define('make_generator == [codireco] ccons')
define('ccons == cons cons')
.. code:: ipython2
define('err == [sqr - abs] nullary')
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.)
::
n a ε square-root
-----------------
√n
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
Let's define the recursive function from here. Start with ``ifte``; the
predicate and the base case behavior are obvious:
::
n a' e ε [<] [popop popd] [J] ifte
Base-case
::
n a' e ε popop popd
n a' popd
a'
The recursive branch is pretty easy. Discard the error and recur.
::
w/ K == [<] [popop popd] [J] ifte
n a' e ε J
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
This fragment alone is pretty useful.
define('gsra == 1 swap [over / + 2 /] cons [dup] swoncat make_generator')
.. code:: ipython2
define('K == [<] [popop popd] [popd [Q err] dip] primrec')
.. code:: ipython2
J('25 10 0.001 dup K')
J('23 gsra')
.. parsed-literal::
5.000000232305737
[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('25 10 0.000001 dup K')
J('23 gsra 6 [x popd] times first sqr')
.. parsed-literal::
5.000000000000005
23.0000000001585
So now all we need is a way to generate an initial approximation and an
epsilon value:
Finding Consecutive Approximations within a Tolerance
-----------------------------------------------------
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.
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 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...
::
square-root == dup 3 / 0.000001 dup K
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('square-root == dup 3 / 0.000001 dup K')
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
J('36 square-root')
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')
.. code:: ipython2
J('23 sqrt')
.. parsed-literal::
6.000000000000007
4.795831523312719
.. code:: ipython2
J('4895048365636 square-root')
.. parsed-literal::
2212475.6192184356
.. code:: ipython2
2212475.6192184356 * 2212475.6192184356
4.795831523312719**2
.. parsed-literal::
4895048365636.0
22.999999999999996