Archive for the ‘numerical analysis’ Category

Converting from binary to integer

26 October 2010 1 comment

The following is an updated and edited version of my posts to this sage-support thread.


You have a bitstring as output by

and you want to convert that bitstring to an integer. Or in general, you want to convert a bit vector to its integer representation.


Here are two ways, assuming that you want the bits in little-endian order, i.e. you read the bits from right to left in increasing order of powers of 2.

sage: version()
'Sage Version 4.5.3, Release Date: 2010-09-04'
sage: from import blum_blum_shub
sage: b = blum_blum_shub(length=6, lbound=10**4, ubound=10**5); b
sage: type(b)
<class 'sage.monoids.string_monoid_element.StringMonoidElement'>
sage: # read in little-endian order
sage: # conversion using Python's built-in int()
sage: int(str(b), base=2)
sage: # conversion using Sage's built-in Integer()
sage: Integer(str(b), base=2)

Now assume you read the bitstring as output by blum_blum_shub in big-endian order, i.e. from left to right in increasing order of powers of 2. You simply convert the bitstring to a string, reverse that string, and apply any of the above two methods.

sage: # reversing a string
sage: str(b)
sage: str(b)[::-1]
sage: # read in big-endian order
sage: int(str(b)[::-1], base=2)
sage: Integer(str(b)[::-1], base=2)

Or you can do as follows:

sage: b = "100110"
sage: sum(Integer(i) * (2^Integer(e)) for e, i in enumerate(b))
sage: sum(Integer(i) * (2^Integer(e)) for e, i in enumerate(b[::-1]))

Another way is to use Horner’s method. Here’s a Sage function that computes the integer representation of a bit vector read using big-endian order. A usage example is also shown.

sage: def horner(A, x0):
...       # Evaluate the polynomial P(x) at x = x_0.
...       #
...       # INPUT
...       #
...       # - A -- list of coefficients of P where A[i] is the coefficient of
...       #   x_i.
...       # - x0 -- the value x_0 at which to evaluate P(x).
...       #
...       # OUTPUT
...       #
...       # An evaluation of P(x) using Horner's method.
...       i = len(A) - 1
...       b = A[i]
...       i -= 1
...       while i >= 0:
...           b = b*x0 + A[i]
...           i -= 1
...       return b
sage: A = [1, 0, 0, 1, 1, 0]
sage: horner(A, 2)

As an exercise, modify the function horner to output the integer representation of a bit vector that is read using little-endian order.

Lagrange interpolating polynomials

13 January 2010 1 comment

I’ve uploaded a draft of an expository paper presenting some basics of the Lagrange interpolating polynomials, with examples using Sage. The paper also covers Neville’s method for recursively generating Lagrange interpolating polynomials. The PDF version of the paper is available for your reading pleasure. The \LaTeX source tarball is also available under the GNU FDL in case you want to modify or extend the paper to suit your purposes.