GNU Basic Calculator (bc) Functions

My working collection of functions for GNU bc. The functions are split into two files: functions.bc containing simple “pure” functions, and routines.bc containing functions that I’ve found useful in practice as a math instructor.

functions.bc

Here is a list of the functions this file defines:

sgn abs heavyside int ln log logb pow rad2deg deg2rad 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 newton integrate prime

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

• Newton's Method (newton(x)) which approximates a zero of a smooth function f near x. Note that f and its derivative ff must be globally defined.
• Numerical Integration (integrate(a,b,n)) which numerically computes the value of a definite integral of a function f between a and b using Boole's Rule (boole_) with n subdivisions. Note that f must be globally defined.

routines.bc

Here is a list of the functions this file defines:

• Pythagorean Triple Generator (pythagtriple(m,n)) which prints the Pythagorean triple generated by two parameters m and n.
• Pythagorean Quadruple Generator (pythagquadruple(m,n,p,q)) which prints the primitive Pythagorean quadruple generated by four parameters m, n, p, and q.
• Newton's Method (newtoniter(x,n)) which iteratively approximates a zero of a smooth function f near x a total of n times, printing each successive approximation. Note that f and its derivative ff must be globally defined.
• Quadratic Polynomial Solver (quadratic(a,b,c)) which prints the roots and vertex of a degree-two polynomial given its coefficients as input.
• Rational Approximation (rational(x)) which displays subsequently better rational approximations to x — the convergents of its continued fraction presentation — until finding the first one equal to x up to scale.
• Different Base Expression (bases(n)) which displays a number n in bases 2, 3, …, 36.
• Prime Integer Factorization (factor(n)) which displays the prime integer factors of n.
• Rectangular/Polar Conversion (rect2pol(x,y)) and (pol2rect(r,θ)) which convert two-dimensional rectangular coordinates to polar coordinates and vice-versa respectively.
• Rectilinear/Cylindrical Conversion (rect2cyl(x,y,z)) and (cyl2rect(r,θ,z)) which convert three-dimensional rectilinear coordinates to cylindrical coordinates and vice-versa respectively.
• Rectilinear/Spherical Conversion (rect2sphere(x,y,z)) and (sphere2rect(ρ,θ,φ)) which convert three-dimensional rectilinear coordinates to spherical coordinates and vice-versa respectively.
• Collazt (Hailstone) Sequence (collatz(n)) which prints the hailstone sequence obtained by iteratively applying the transformation prescribed by the Collatz Conjecture.
• Sum of Consecutive Powers (sumofpowers(n,p)) which prints every way that n can be expressed as a sum of consecutive pth itegral powers.

Conventions

• 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.
• In the file routines.bc if a function changes/defines a variable globally, the function 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 is named f. If there must be two functions, the other will be named g. If a function wants to know the derivative of f, it'll expect that derivative to be named ff (e.g. newton).

Aspirations

• Add cubic and quartic functions that prints the details of a cubic and quartic polynomial.
• Address issues with the numerical integration integrate function: There must be a way to remove the need for the parameter n, Also, I'm concerned this function is inaccurate for large values of |b – a|.
• Add a function that finds constructable algebraic approximations to real numbers. (see this).
• Add a numerical derivative function (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
• "math and bitcoin tools in gnu bc and bash"