Multiline Functions¶

CalcpadCE lets a function definition span multiple lines using _ continuations and the if / $while constructs, so recursive, conditional and iterative routines fit into the same f(x; y; …) = … syntax as plain one-liners.

The four worksheets in this group show that pattern at work. Cubic Roots Function packages the closed-form solution of $x^3 + a x^2 + b x + c = 0$ as a single multiline call returning all three roots; Quadratic Roots Function does the same for $a x^2 + b x + c = 0$ with discriminant-based branching. Fibonacci Numbers defines $fib(n)$ recursively to illustrate self-referential calls, and Greatest Common Divisor - Euclid Algorithm builds an iterative $gcd(a, b)$ around a $while block.

Together the four pages serve as templates for any user-defined function that needs more than a single expression.

Cubic Roots Function¶

Multiline cubic-roots function returning all three roots of $x^3 + a x^2 + b x + c = 0$ via the Cardano formula, with discriminant-based real/complex branching.

Code:

"Cubic Equation Roots Function
'For solving equations of type:'f(x) = x^3 + a*x^2 + b*x + c'= 0

#rad
cubicRoots_1(a; Q; R) = $block{
  A = -sign(R)*(abs(R) + sqr(R^2 - Q^3))^(1/3);
  x_1 = if(A ≡ 0; -a/3; A + Q/A - a/3);
  x = hp([x_1; x_1; x_1]);
}
cubicRoots_3(a; Q; R) = $block{
  θ = acos(R/sqr(Q^3));
  x_1 = -2*sqr(Q)*cos(θ/3) - a/3;
  x_2 = -2*sqr(Q)*cos((θ - 2*π)/3) - a/3;
  x_3 = -2*sqr(Q)*cos((θ + 2*π)/3) - a/3;
  x = hp([x_1; x_2; x_3]);
}
cubicRoots(a; b; c) = $block{
  Q = (a^2 - 3*b)/9;
  R = (2*a^3 - 9*a*b + 27*c)/54;
  x = if(R^2 < Q^3; cubicRoots_3(a; Q; R); cubicRoots_1(a; Q; R));
}
x = cubicRoots(-6; 11; -6)
#hide
a = -6', 'b = 11', 'c = -6
d = max(abs(x.3 - x.1)/4; 0.5)
x_min = min(x.1; x.3) - d','x_max = max(x.1; x.3) + d
PlotWidth = 200', 'PlotHeight = 150
#show
'Plot
$Plot{f(ξ) & x.1|0 & x.2|0 & x.3|0 @ ξ = x_min : x_max}

Fibonacci Numbers¶

Recursive multiline definition of the Fibonacci function $fib(n)$, illustrating self-referential calls and conditional base cases.

Code:

'Function:
fib(n) = if(n ≡ 1; 1;
  $block{
    a = 0;
    b = 1;
    $repeat{
       c = a + b;
       a ← b;
       b ← c;
     @ k = 1 : n - 1} _
  } _
)
'Results:
#for i = 1 : 9
    fib(i)
#loop

Rendered Output:

Function:

fib ( n ) = {if n ≡ 1: 1
else: 🞀a = 0
b = 1
$Repeat 🞀for k = 1...n − 1
c = a + b
a ← b
b ← c

Results:

fib ( i ) = fib ( 1 ) = 1

fib ( i ) = fib ( 2 ) = 1

fib ( i ) = fib ( 3 ) = 2

fib ( i ) = fib ( 4 ) = 3

fib ( i ) = fib ( 5 ) = 5

fib ( i ) = fib ( 6 ) = 8

fib ( i ) = fib ( 7 ) = 13

fib ( i ) = fib ( 8 ) = 21

fib ( i ) = fib ( 9 ) = 34

Greatest Common Divisor - Euclid Algorithm¶

Iterative greatest common divisor built around a $while$ block, implementing the classical Euclid algorithm.

Code:

'Custom function:
mygcd(a; b) = $while{
  b > 0;
  t = b;
  b = a⦼b;
  a = t}
'Example:
mygcd(1071; 462)
'Internal gcd function:
gcd(1071; 462)

Rendered Output:

Custom function:

mygcd ( a; b ) = 🞀while b > 0
t = b
b = a mod b
a = t

Example:

mygcd ( 1071; 462 ) = 21

Internal gcd function:

gcd ( 1071; 462 ) = 21

Quadratic Roots Function¶

Multiline quadratic-roots function returning both roots of $a x^2 + b x + c = 0$ via the discriminant, with real/complex branching.

Code:

"Quadratic Equation Roots function
'Multiline code block example:
quadRoots(a; b; c) = _
$block{
    D = b^2 - 4*a*c;
    x_1 = (-b - sqrt(D))/(2*a);
    x_2 = (-b + sqrt(D))/(2*a);
    [x_1; x_2];
}
quadRoots(2; 3; -5)
'Inline code block example:
quadRoots(a; b; c) = $inline{D = b^2 - 4*a*c; x_1 = (-b - sqrt(D))/(2*a); x_2 = (-b + sqrt(D))/(2*a); [x_1; x_2]}
quadRoots(2; 3; -5)

Rendered Output:

Quadratic Equation Roots function

Multiline code block example:

quadRoots ( a; b; c ) = 🞀D = b² − 4 · a · c
x₁ = -b − √ D2 · a
x₂ = -b + √ D2 · a
[x₁; x₂]

quadRoots ( 2; 3; -5 ) = [-2.5 1]

Inline code block example:

quadRoots ( a; b; c ) = 🞀D = b² − 4 · a · c; x₁ = -b − √ D2 · a; x₂ = -b + √ D2 · a; [x₁; x₂]

quadRoots ( 2; 3; -5 ) = [-2.5 1]

Spotted an error? Edit these examples.