GNU Basic Calculator (bc) Functions

My working collection of functions and stuff for GNU bc. The code is split into two files: functions.bc containing pure functions that only return a value, and routines.bc containing functions I’ve found useful in practice as an instructor that generally print information about some input. Files like these can be loaded automatically by bc at startup by setting this environment variable:

export BC_ENV_ARGS="-lq /PATH/TO/functions.bc /PATH/TO/routines.bc"

functions.bc

This file defines the following functions:

sgn abs heavyside max int frac trunc intmod isnearlyintegral ln log logb pow rad2deg deg2rad dms2dd cos sin tan sec csc cot arccos arcsin arctan atan2 arcsec arccsc arccot cosh sinh tanh sech csch coth arcosh arsinh artanh arsech arcsch arcoth factorial pick choose fibonacci contfrac2obase derivative newton integral gcd lcm prime minkowski

Most of these functions are either ubiquitous, or, if not, can be quickly understood from their signature and brief definition. Some of these functions however deserve an explanation:

• Numerical Differentiation derivative(x)
  Returns the derivative of the function f at x computed using a finite central distance with eight points.
• Newton's Method newton(x)
  If it terminates, returns the zero of the (global) function f that results from iteratively applying Newton's Method with initial parameter x.
• Numerical Integration integral(a,b)
  Returns the value of the definite integral of the (global) function f between a and b computed using Boole's rule applied recursively to progressively finer subintervals.
• Minkowski's Question-Mark Function minkowski(x)
  Returns a number constructed in binary from the continued fraction of x.

routines.bc

This file defines the following routines:

• Pythagorean Triple Generator pythagtriple(m,n)
  Print the Pythagorean triple generated by two parameters m and n, and return the hypotenuse.
• Pythagorean Quadruple Generator pythagquadruple(m,n,p,q)
  Print the Pythagorean quadruple generated by parameters m, n, p, and q, and return the hypotenuse.
• Degrees/Minutes/Seconds (DMS) dd2dms(x)
  Prints the decimal degree angle x in terms of DMS (degrees° minutes′ seconds″).
• Newton's Method newtoniter(x)
  Iteratively applies Newton's Method, printing each result, with initial guess x and (global) function f.
• Quadratic Polynomial Solver quadratic(a,b,c)
  Print the roots and vertex coordinates of ax²+bx+c.
• Simple Continued Fraction contfrac(x)
  Print the coefficients of the simple continued fraction representation of x as well as each convergent obtained by truncating at that coefficient, giving successively better rational approximations to x.
• Different Base Expression bases(n)
  Print n expressed in bases 2, 3, ..., 36.
• Prime Integer Factorization factor(n)
  Print the prime integer factorization of n.
• Rectangular/Polar Conversion rect2pol(x,y) / pol2rect(r,θ)
  Convert two-dimensional rectangular coordinates to polar coordinates and vice-versa respectively.
• Rectilinear/Cylindrical Conversion rect2cyl(x,y,z) / cyl2rect(r,θ,z)
  Convert three-dimensional rectangular coordinates to cylindrical coordinates and vice-versa respectively.
• Rectilinear/Spherical Conversion rect2sphere(x,y,z) / sphere2rect(ρ,θ,φ)
  Convert three-dimensional rectangular coordinates to spherical coordinates and vice-versa respectively, following the mathematician’s convention with the zenith angle θ in the range [0,2π) with θ = 0 along the positive x-axis, and the azimuth angle φ in the range [0,π] with φ = 0 along the positive z-axis.
• Collazt (Hailstone) Sequence collatz(n)
  Print the sequence of positive integers that results from iteratively applying the function prescribed by the Collatz Conjecture.
• Sum of Consecutive Powers sumofpowers(m)
  Print every way that m can be written as a sum of powers of consecutive positive integers.
• Digits → List of Digits intdigits(x) / fracdigits(x) / onlydigits(x, *d[])
  Create a list of the digits of x as a list. In particular, for any value of obase, intdigits will create a list of the digits of the integer part of a number, fracdigits will create a list of the digits of the fractional part of a number, and onlydigits will return true if the number contains only the digits specified in the list d[] as the indices with nonzero value.
• Zeckendorf Presentation zeckendorf(n)
  Prints the unique sum of non-adjacent Fibonacci numbers equal to n.

Conventions

• If a function in routines.bc defines or updates the value of a variable globally, it will print that variable assignment explicitly.
• Some functions in functions.bc that inherently entail a sequence of numbers (e.g. fibonacci, prime, etc) also create an array of the same name containing all computed terms in the sequence.
• There are functions not documented in this README with name ending in _. These are helper functions, usually for the sake of recursion, not intended to be called directly.
• Since bc doesn't accept functions as parameters to other functions, any functions that morally should be a parameter must be defined globally. Such a function will be named f.
• There are certain things bc is not designed for — linear algebra, statistics, complex arithmetic, etc — and should not be implemented in bc. These, if one so desires, should be implemented upstream within a fork of the bc program itself.

Aspirations

• Add cubic and quartic functions that prints the details of a cubic and quartic polynomial.
• Replace the discrete combinatorics functions (factorial, pick, choose, etc) with continuous (analytic?) analogous so I can remove the PRINT statements.
• Add a function that finds constructable algebraic approximations to real numbers. (see this).

Allusions

• The GNU Basic Calculator (bc): a Quick-Start Guide for Mathematicians
• GNU bc Manual
• Library of bc function on phodd.net
• Keith Matthews' number theory functions in bc
• NIST Digital Library of Mathematical Functions