GNU Basic Calculator (bc) Functions

My working collection of functions for GNU bc. The functions are split into two files: functions.bc containing pure functions that only return a value, and routines.bc containing functions that 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 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 derivative newton integral prime

Alongside the ubiquitous mathematical functions in this list, this file contains implementations of the following:

• Numerical Differentiation derivative(x)
  Returns the derivative of the (global) function f at x using a finite central distance with eight points.
• Newton's Method newton(x)
  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.

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 angle x, presumed as a decimal number of degrees, in DMS (degrees° minutes′ seconds″) notation.
• 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, which are successively better rational approximations to the number 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)) and (pol2rect(r,θ))
  Convert two-dimensional rectangular coordinates to polar coordinates and vice-versa respectively.
• Rectilinear/Cylindrical Conversion (rect2cyl(x,y,z)) and (cyl2rect(r,θ,z))
  Convert three-dimensional rectangular coordinates to cylindrical coordinates and vice-versa respectively.
• Rectilinear/Spherical Conversion (rect2sphere(x,y,z)) and (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.
• 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.
• 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.
• Function names ending in _ are helper functions not intended to be called directly.
• Some functions that return the nth number in a sequence (e.g. fibonacci, prime) create an array of the same name as the function containing all previous terms in the sequence used to compute the nth term.
• There are certain things bc is not designed for — linear algebra, statistics, and complex arithmetic among others — 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).
• Browse the NIST Digital Library of Mathematical Functions for thoughts.

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